SScuola Internazionale Superiore di Studi Avanzati - Trieste
SISSA - Via Bonomea 265 - 34136 TRIESTE - ITALY
Doctoral Thesis
FQHE and tt ∗ Geometry
Candidate : Riccardo Bergamin
Supervisor : Sergio Cecotti
Opponents : Michele Del ZottoCumrun Vafa
Academic Year 2018 – 2019 a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r bstract Cumrun Vafa proposed in [17] a new unifying model for the principal series ofFQHE which predicts non-Abelian statistics of the quasi-holes. The many-bodyHamiltonian supporting these topological phases of matter is invariant under foursupersymmetries. In the thesis we study the geometrical properties of this Landau-Ginzburg theory. The emerging picture is in agreement with the predictions in [17].The 4-SQM Vafa Hamiltonian is shown to capture the topological order of FQHE andthe tt ∗ monodromy representation of the braid group factors through a Temperley-Lieb/Hecke algebra with q = ± exp( πi/ν ). In particular, the quasi-holes have thesame non-Abelian braiding properties of the degenerate field φ , in Virasoro minimalmodels. Part of the thesis is dedicated to minor results about the geometricalproperties of the Vafa model for the case of a single electron. In particular, westudy a special class of models which reveal a beautiful connection between thephysics of quantum Hall effect and the geometry of modular curves. Despite it isnot relevant for phenomenological purposes, this class of theories has remarkableproperties which enlarge further the rich mathematical structure of FQHE.The thesis is based on the following papers:R. Bergamin and S. Cecotti, “ FQHE and tt ∗ geometry ”, arXiv:1910.05022 [hep-th],2019.R. Bergamin, “ tt ∗ Geometry of Modular Curves”, arXiv:1803.00489 [hep-th],published in JHEP 1908 (2019) 007. cknowledgements
Voglio ringraziare Mamma, Pap´a e Stefano per tutto quel che hanno fatto per mein questi anni di studi. Un supporto non solo morale, ma anche alimentare. Unamenzione speciale va infatti alle torte salate e dolci che mi ritrovavo in valigia ognidomenica sera di ritorno a Trieste. Di questo anche i miei coinquilini ringrazianosentitamente.Ringrazio di cuore tutti i coinquilini con cui ho abitato la Geppa in questi quattroanni intensissimi: Sofy, Bruno, Ane, Ale, Mizu, StronzEle, Milly, Segret, Leo, Fra,Carly e Skodella. Li ringrazio per il loro grande affetto e supporto morale, nonch´eper tutti i bei momenti trascorsi assieme come una vera famiglia.Ringrazio di cuore anche Giovanni, Sara, Luca, Matteo, Giulia, Alessandro, Andreae Jenny per la loro amicizia incondizionata. Perch´e anche se trascorro molto tempolontano da casa, al mio ritorno trovo sempre delle persone su cui contare.Ringrazio anche Fabio e Cristina per la loro grande ospitalit´a. Quella di Fabio ´e lanaturale evoluzione dell’ osmiza: se magna, se beve e se ride, ma al posto del vinelloscrauso trovi Whisky americano di grande qualit´a.Ci tengo molto a ringraziare i miei colleghi ed amici in SISSA per il tempo trascorsoassieme e le interessanti discussioni scentifiche. In particolare ringrazio Giulio Ruzzae Matteo Caorsi, che sono sicuramente i ragazzi che ho importunato di pi´u con lemie domande.Voglio ringraziare il prof. Sergio Cecotti per avermi fatto conoscere l’argomento delfractional quantum Hall effect ed in generale delle fasi topologiche della materia,che trovo uno dei pi´u interessanti argomenti della fisica teorica di oggi. Lo ringrazioin particolare di avermi reso partecipe di nuove ed interessanti scoperte al riguardo.Ringrazio anche tutti i professori che mi hanno dato tempo e disponibilit´a, oltre alpersonale della caffetteria e la mensa, sempre allegro, gentile e disponibile.Ringrazio infine i ragazzi della palestra Audace per avermi fatto conoscere il cross-fit e per la loro grande simpatia, disponibilit´a ed entusiasmo durante gli intensiallenamenti. Come recita il famoso detto: “Mens sana in corpore sano”.2 ontents
Contents 31 Introduction and Overview 52 Topological Order, FQHE and Anyons 10 tt ∗ Geometry 36 N = 2 SQM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 N = 4 SQM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 tt ∗ Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.1 tt ∗ Geometry in SQM . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 The Integral Formulation of tt ∗ Geometry . . . . . . . . . . . 563.2.3 Computing the tt ∗ Monodromy Representation . . . . . . . . . 60 tt ∗ Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.1 Abelian Covers . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.2 tt ∗ Equations with θ -Vacua . . . . . . . . . . . . . . . . . . . 785.3 The Problem of Statistics . . . . . . . . . . . . . . . . . . . . . . . . 825.4 tt ∗ Geometry of the ν = 1 Phase . . . . . . . . . . . . . . . . . . . . 853.5 The Vacuum Space of the Interacting Theory . . . . . . . . . . . . . 905.5.1 The Heine-Stieltjes Problem . . . . . . . . . . . . . . . . . . . 905.5.2 Fermionic Truncation . . . . . . . . . . . . . . . . . . . . . . . 92 tt ∗ Geometry 94 tt ∗ Monodromy and the Universal Pure Braid Representation . . . . . 946.2 Complete and Very Complete tt ∗ Geometries . . . . . . . . . . . . . . 966.3 Ising Model and SQM . . . . . . . . . . . . . . . . . . . . . . . . . . 986.4 Hecke Algebra Representations of the tt ∗ Connection . . . . . . . . . 103 tt ∗ Geometry of the Vafa model 109 λ ( θ ) and the allowed filling fractions . . . . . . . . . . . 1177.5 Comparison with the Homological Approach . . . . . . . . . . . . . . 120 A The Long-Range Limit of the Vafa Hamiltonian 123
A.1 One Electron in Uniform Magnetic Field . . . . . . . . . . . . . . . . 123A.2 The Single Field Model with the Quasi-Holes . . . . . . . . . . . . . . 128
B Weak Limit of ∂W C.1 T n,g Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134C.2 Matrix Models and 2 d/ d Correspondence . . . . . . . . . . . . . . . 137 D tt ∗ Geometry of the One Electron Model with Two Quasi-Holes 144E tt ∗ Geometry of Modular Curves 151
E.1 Quantum Hall Effect and Modular Curves . . . . . . . . . . . . . . . 151E.2 Classification of the Models . . . . . . . . . . . . . . . . . . . . . . . 154E.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154E.2.2 Modular Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 156E.2.3 The Role of Number Theory . . . . . . . . . . . . . . . . . . . 160E.3 Geometry of the Models . . . . . . . . . . . . . . . . . . . . . . . . . 161E.3.1 The Model on the Target Manifold . . . . . . . . . . . . . . . 161E.3.2 Abelian Universal Cover . . . . . . . . . . . . . . . . . . . . . 162E.4 tt ∗ Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166E.4.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 166E.4.2 Peculiarities of the N=2 Level . . . . . . . . . . . . . . . . . . 173E.5 Modular Properties of the Models . . . . . . . . . . . . . . . . . . . . 176E.5.1 Modular Transformations of ∂ z W ( N,l ) ( z ; τ ) . . . . . . . . . . . 1764.5.2 Modular Transformations of the Physical Mass . . . . . . . . . 177E.5.3 Modular Transformations of the Ground State Metric . . . . . 186E.6 Physics of the Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . 189E.6.1 Classification of the Cusps . . . . . . . . . . . . . . . . . . . . 189E.6.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 196E.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 F Modular Transformations of log E u ,u ( τ ) The theory of topological phases of matter is one of the most interesting subjectin theoretical physics. The remarkable discovery of these quantum states showedthe limit of the Landau-Ginzburg theory of phase transitions. The topological uni-versality classes cannot be classified with the concepts of symmetry and symmetrybreaking: we can have either states in the same topological class but with differentsymmetries, or states in inequivalent topological classes but with the same symme-tries. These quantum phases of matter encode a new kind of order, the topologicalorder, which is not associated to any local order parameter, but rather to globalnon-local observables [13]. From these considerations one can understand the im-portance of the discovery of quantum Hall effect. A system of interacting electronsmoving on a 2 d surface in a strong magnetic field at low temperature exhibits verysurprising properties like the quantization of the Hall conductivity σ xy = e π (cid:125) ν. Initially ν was found to be integer. The quantization of a physical quantity is notnew in quantum mechanics, but in this context it acquires a new meaning. Therobustness of the conductivity under deformations of the magnetic field within acertain range reveals the topological nature of the Hall states. Subsequently, it wasfound that ν can also assume very specific rational values. The most prominentfractions experimentally are ν = 1 / ν = 2 /
5, but many others have beenobserved. This quantum number has the interpretation of filling fraction of thelowest Landau levels. It is for a rational ν that the most interesting phenomena ofthe quantum Hall effect happen. It turns out that the charged excitations of thesesystems, the quasi-holes, carry a fraction of the charge of the electron, and, moreremarkably, they behave like particles of anyonic statistics. From a theoretical point5f view what provides the connection between quantum mechanics and geometry isthe Berry’s connection. The definition of this mathematical object arises from theSchroedinger equation when we study the adiabatic evolution of quantum systems.The Berry’s holonomy induced by taking a quasi-hole around another one containsthe informations about the statistics of these quantum objects. In chapter 2 webriefly review the theory of quantum Hall states and the relation between topologi-cal order and anyonic particles.Despite the discovery of FQHE [2] dates back more than thirty years ago, sayingif the quasi-holes are abelian or non-abelian particles still represents a challengingproblem for both theorists and experimentalists. From the theoretical point of view,many models have been developed to explain the observed filling fractions. Amongthese, the Laughlin’s proposal [3] and the idea of hierarchy states of Haldane andHalperin [4, 5], as well as Jain’s composite fermion theory [6], predict abelian any-onic statistics for the principal series of FQHE. Also the possibility of non-abelianstatistics has been explored by several models for other filling fractions [7, 8, 9, 10].More recently, C.Vafa proposed in [17] a unifying model of FQHE which leads tonew predictions for the statistics of the quasi-holes. He claims that the effectivetheory of FQH systems with ν = n n ± N = (2 ,
0) theory of A type on a punctured Riemannsurface leads to a correspondence between the Nekrason partition functions [42] ofcertain SU (2) gauge theories and the correlators of the Liouville CFT. In this setup the punctured Riemann surface is identified with the target manifold of FQHE,where the punctures correspond to insertions of quasi-holes, and the Liouville con-formal blocks are the wave functions of FQHE. In Appendix C we provide a shortreview about the Gaiotto theory and the relation between topological string ampli-tudes and Liouville chiral blocks which inspires the Vafa proposal. This constructionmotivates in addition a microscopic description of FQHE states in terms of a N = 4supersymmetric Hamiltonian. Choosing the plane as Riemann surface, we have aLandau-Ginzburg model with superpotential W ( z ) = N (cid:88) i =1 (cid:32) n (cid:88) a =1 log( z i − x a ) − M (cid:88) k =1 log( z i − ζ k ) (cid:33) + 1 ν (cid:88) i V → Y n (1.8)provides a monodromy representation of the braid group of the quasi-holes B n = π ( Y n ). In the chapters 3,5,6 we discuss all the necessary tools to compute the tt ∗ monodromy representation of the Vafa model. The chapter 3 contains a review ofsupersymmetric quantum mechanics and basics of tt ∗ geometry that the reader canfind in literature. The chapters 5,6 cover more advanced topics in tt ∗ geometrywhich are relevant for our problem. In chapter 5 we discuss the concept of statisticsin tt ∗ geometry. In particular, since the electrons are fermionic particles, we study infull generality the fermionic sector of a LG model of N identical particles. We alsodiscuss the formulation of tt ∗ geometry in models with a non simply connected targetmanifold and multivalued superpotential. The Vafa superpotential 1.2 belongs tothis family of theories. In order to treat properly these models one has to introducethe concept of covering spaces and extend the vacuum space with the so called “ θ -sectors”. The common denominator of the chapter is the tt ∗ functoriality, whichturns out to be a very powerful tool to generate isomorphisms between quantumtheories. In chapter 6 we study a special class of tt ∗ geometries which we call“ very complete”. These models are quite peculiar. In the language of 2 d N = (2 , SL (2 , C ) Knizhnik-Zamolodchikov connection [75, 77] on C n withglobal coordinates given by the critical values of the superpotential w i . The UVlimit consists in rescaling the critical coordinates w i → βw i and taking β → 0. Inthis regime the Lax connection and the Berry connection coincide and, if the model8s symmetric under permutations of the w i , the UV Berry connection provides aunitary representation of the full braid group B n .In chapter 7 we study the tt ∗ geometry of the Vafa Hamiltonian. We show that themodel is very complete and symmetric. The corresponding UV Berry connectiontakes the form of a Kohno connection D = d + λ (cid:88) i 1) Virasoro model. The ratio of the twoeigenvalues is q = exp[2 πi ( h , − h , )]exp[2 πi ( h , − h , )] = exp(2 πi/ν ) . (1.15)As predicted in [17], the quasi-holes have the same non-Abelian braiding propertiesof the degenerate fields φ , in Virasoro minimal models. On the other hand, theless natural solution q = − e iπ/ν gives other two series of filling fractions. These arerespectively ν = mm + 2 , m = k + 2 ∈ N ≥ , ν = m m − , m = k + 2 ≥ , (1.16)where the first series contains the values of ν corresponding to the Moore-Read [8]and Read-Rezayi models [9].The reader can find in appendix other minor results about the geometrical propertiesof the Vafa model for the case of a single electron. In particular, in E we study aspecial class of models which reveal a beautiful connection between the physics ofquantum Hall effect and the geometry of modular curves. The analysis is based on[64]. Despite it is not relevant for phenomenological purposes, this class of theorieshas remarkable properties which enlarge further the rich mathematical structureof FQHE. Among the main results, the theorems about the cusps counting andclassification are recovered in a physical language. From our investigation of thisfamily of models, the algebraic properties of the modular curves emerge in an elegantmanner. The fractional quantum Hall effect opened a new chapter in condensed matterphysics. For a long time, before the discovery of the quantum Hall states, Lan-10au’s symmetry breaking theory defined the fundamental paradigm of many-bodyphysics. We know that at sufficiently high temperature matter is in form of gas. Inthis regime the particles are weakly interacting and the motion of a single constituentis not influenced by the other ones. When temperature decreases the particles be-come more and more correlated and start to develop a regular pattern. In thisregime we observe an emergent collective behaviour revealing an internal structure,which we also call order. The concept of order allows to classify different statesof matter and is intimately related to the concept of phase transition. One saysthat two many-body states has the same order if we can change one states intothe other with a smooth deformation of the Hamiltonian. If this is not possible weencounter a phase transition. The definition of order organizes the states of matterin equivalence classes which are said universality classes in the Wilsonian language.According to Landau’s theory there is a deep relation between order and symmetry.Different orders are associated with different symmetries, which are described bylocal parameters, and a phase transition involves the breaking of some symmetry.This picture is able to explain a large class of experimentally observed states ofmatter and predict the existence of gaplessness excitations in the materials due tosymmetry breaking. The existence of the quantum Hall states reveals the limits ofLandau’s paradigm and shows the existence of a new type of order, usually calledtopological order [13]. These states of matter are created by confining electrons on a2 d interface between two different semiconductors under strong magnetic fields andlow temperature. In this regime the electrons are strongly correlated and behavecollectively like a quantum liquid. The topological phases are not classified by sym-metries and cannot be described by local order parameters. Indeed, all the quantumHall states have the same symmetries and a phase transition between them does notinvolve any symmetry breaking. One can define a topological universality class as afamily of Hamiltonians with gapped spectrum which can be smoothly deformed toeach other without the emergence of gappless excitations. The topological order canbe characterized by global, non-local observables, which are robust under any localperturbations that can break symmetries. These are essentially the degeneracy ofthe ground state, which depends only on the topology of the target space, and thenon-local behaviour of the quasi-hole and quasi-particle excitations. We recall some basic facts about the phenomenology of quantum Hall effect. Moredetails can be found in [13, 12] and references there. The Hall effect was originallydiscovered by Edwin Hall in 1879. The physical set-up is very simple to construct:one has to take a bunch of electrons, restrict them to move on a two dimensional11lane and turn on an electric field (cid:126)E and magnetic field (cid:126)B respectively parallel andorthogonal to the plane. The classical Hall effect is simply the consequence of themotion of charged particles in a magnetic field. In a static regime the electric forceacting on the charge carriers is balanced by the Lorentz force of the magnetic field.In natural units we have the condition q e (cid:126)E = q e (cid:126)v × (cid:126)B, (2.1)where q e = − e is the charge of the electron and (cid:126)v is the velocity of the particles.Denoting with n the density of the electrons, a current (cid:126)j = n(cid:126)v is made to flow inthe normal direction to the electric field. Moreover, the norms E, j are related by E = j Bnq e c = j hq e ν (2.2)where h, c are respectively the Planck constant and the speed of light, while ν = nhcq e B = nB/ Φ = number of particlesnumber of flux quanta (2.3)is the filling fraction. In the above equality we introduced Φ = hcq e as the quantumunit of magnetic flux. Recalling the Ohm’s law (cid:126)E = ρ(cid:126)j (2.4)where ρ = (cid:18) ρ xx ρ xy − ρ xy ρ yy (cid:19) (2.5)is the resistivity tensor, from 2.2 we obtain the relations ρ xx = 0 , ρ xy = hq e ν . (2.6)According to the classical theory, the Hall resistance ρ xy is proportional to themagnetic field at fixed electron density. Indeed, experimentally one finds that ρ xy ∝ B at weak fields. However, in the early 1980s it was found that for strong magneticfields ( ∼ 10 T) and very low temperatures ( ∼ ν is quantizedin this regime and labels the different plateaus. These new states of matter behavelike an incompressible fluid of uniform density. Initially ν was found to be aninteger. This phenomenon is called integer quantum Hall effect (IQHE) and wasdiscovered by von Klitzing in 1980 [1]. In the experiment the electrons are confinedon an interface between two different semiconductors which play the role of 2 d plane. The quantization of a physical quantity is rather common at the microscopiclevel, but the case of the filling fraction is different from the usual quantizationphenomena in quantum mechanics. The profile of the resistivity shows that thequantum Hall states are robust under deformations of the magnetic field within acertain range. Moreover, the plateau spectrum turns out to be indipendent fromthe local, microscopic details of the material. These features reveal the topologicalnature of these phases of matter.Figure 1: Plateau structureSubsequently, it was found that ν can also take very specific rational values. Thisphenomenon is called fractional quantum Hall effect (FQHE) and was discoveredby Tsui and Stormer in 1982 [2]. The most prominent fractions experimentally are ν = 1 / ν = 2 / 5, but many other fractions have been seen. The majorityof them has odd denominator and can be recasted in the so called principal series ν = n n ± . It is in these states that the most remarkable things happen. The chargedexcitations of the Hall fluid, the quasi-holes, carry a fraction of the charge of theelectron, as if the electron split itself into several pieces. It is not just the chargeof the electron that fractionalises: this happens to the statistics of the electronas well. It is known that quantum particles in two spatial dimensions can havestatistics which does not correspond to the classical bosonic or fermionic one. Afterthe braiding of two identical particles the wave function of the many-body systempicks up a phase e iθ , where the parameter θ determines the statistics. The statisticsof the particles in two dimensions are 13 θ = 0 boson0 < θ < π anyon θ = π fermion (2.7)In more complicated examples even this description breaks down: the resulting ob-jects are called non-Abelian anyons and provide physical realization of non-localentanglement in quantum mechanical systems. In this case the Hilbert space of thesystem is degenerate and a braiding operation induces a unitary transformation ona generic state. While the fractional charge of quasi-holes has been measured exper-imentally, a direct detection of their statistics is more challenging. It is confirmedthat the quasi-holes of the fractional Hall states do not posses classical statistics,but if they are Abelian or non-Abelian anyons is still an open question. The quantum Hall effect is based at the microscopic level on the dynamics of chargedparticles in a magnetic field. The Hamiltonian of a quantum particle in a constant,uniform magnetic field has discrete spectrum. The structure of the energy levels,known as Landau levels, allows to describe the plateaus at integer values of the fillingfraction [13]. In particular we are interested in the lowest Landau level, namely theground state. We ignore for the moment the Coulomb interaction and assume theelectrons to be spin polarized. An electron of mass m in a uniform magnetic field isdescribed by the Hamiltonian H = − m ( ∂ i − iq e A i ) , (cid:125) = c = 1 , (2.8)where we choose the symmetric gauge( A x , A y ) = B − y, x ) , B = ∂ x A y − ∂ y A x . (2.9)To find the energy levels is convenient to introduce the complex coordinate z = x + iy and the holomorphic and anti-holomorphic derivatives ∂ z = 12 ( ∂ x − i∂ y ) , ∂ ¯ z = 12 ( ∂ x + i∂ y ) . (2.10)One can rewrite the Hamiltonian in complex coordinates as14 = − m ( D z D ¯ z + D ¯ z D z ) (2.11)where D z , D ¯ z are the covariant derivatives D z = ∂ z − iq e A z , A z = 12 ( A x − iA y ) = B i ¯ z,D ¯ z = ∂ ¯ z − iq e A ¯ z , A ¯ z = 12 ( A x + iA y ) = − B i z. (2.12)One can exploit the commutation relation [ D z , D ¯ z ] = mω B , where ωB = q e B/m isthe cyclotron frequency, to simplify the expression of H as H = − m D z D ¯ z + 12 ω B . (2.13)The constant E = ω B is the energy of the first Landau level. A wave functionΨ( z, ¯ z ) in this subspace satisfies the condition D ¯ z Ψ( z, ¯ z ) = 0 . (2.14)Assuming q e B > l B defined by l B = 1 q e B (2.15)which represents the characteristic length scale governing the magnetic phenomenain the quantum regime. The Hamiltonian above is conjugated by the relation H = e −| z | / l B ˜ He + | z | / l B (2.16)to the operator ˜ H = − m ˜ D ¯ z ˜ D z + E (2.17)where 15 D z = e + | z | / l B D z e −| z | / l B = ∂ z − l B ¯ z, ˜ D ¯ z = e + | z | / l B D ¯ z e −| z | / l B = ∂ ¯ z . (2.18)We note that the non unitary transformation e + | z | / l B makes the anti-holomorphicpart of the gauge connection vanishing. Rewriting a generic state in the Hilber spaceas Ψ( z, ¯ z ) = ˜Ψ( z, ¯ z ) e −| z | / l B , (2.19)where ˜Ψ( z, ¯ z ) is a smooth function, the condition to be in the lowest Landau levelbecomes ˜ D ¯ z ˜Ψ( z, ¯ z ) = ∂ ¯ z ˜Ψ( z, ¯ z ) = 0 , (2.20)namely ˜Ψ( z, ¯ z ) must be an holomorphic function on the complex plane. A naturalbasis of holomorphic functions is given by the monomials z k with k (cid:62) 0. Hence, abasis of vacua is Ψ k = z k e −| z | / l B . (2.21)These wave functions have a circular shape and k -th state is peaked at the radious r k = √ kl B . Noting that the ring of the k -th vacuum encloses m flux quantaΦ = 2 πl B , we learn that there is one state for every flux quantum. Hence, thenumber of states in the lowest Landau level is equal to the number of flux quantaΦ / Φ , where Φ is the magnetic flux of B . One can obtain the wave functionscorresponding to the n -th excited Landau levels as followsΨ n ; k = D nz Ψ k , (2.22)where the energy of the n -th state is E n = ( + n ) ω B . It is clear that all the Landaulevels have the same degeneracy.In the case of N free electrons in a uniform magnetic field one has to fill the Landaulevels with Fermi statistics. It is not difficult to guess that for a quantum Hall stateat an integer filling fraction ν = n ∈ N the first n Landau levels are completely filled.The finite gap ∆ = ω B for the excitations explains why at low temperature we ob-serve these plateau, while the Pauli exclusion principle explains the incompressibilityof the Hall fluid. The wave function for the ν = 1 state is16( z , ..., z N ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ...z z ...z z .... . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − (cid:80) i | z i | / l B = (cid:89) i 1. For suchvalues of the filling fraction we naturally conclude that the lowest Landau level ispartially filled by the electrons, whose number must be N = ν N , where N = Φ / Φ is the capacity of the lowest Landau level. It is immediate to see that the freeelectrons description cannot work in the fractional case. Indeed, if we fill a fractionof the lowest Landau level with Fermi statistics, in the thermodynamical limit weget a hugely degenerate ground state. The number of states is (cid:18) N ν N (cid:19) , which isapproximately (cid:0) ν (cid:1) ν N (cid:0) − ν (cid:1) (1 − ν ) N ∼ for N ∼ . Hence, it is clear that thefractional quantum Hall states are strongly correlated system in which the Coulombinteraction plays a fundamental role. This partially removes the degeneracy of thelowest Landau level in such a way that we can observe plateaus at fractional valulesof ν . Solving the Schoredinger equation and find the exact wave function for thesesystems is hopeless. The first approach to describe these plateaus is due to Laughlin[3]. He proposed a wave function for the ν = m states, with m an odd integer. Hisguess is based on some physical considerations. First of all, any wave function inthe lowest Landau level must have the formΨ( z , ..., z N ) = f ( z , ..., z N ) e − (cid:80) i | z i | / l B , (2.24)where f ( z , ..., z N ) must be holomorphic and anti-symmetric under the exchange ofany two particle positions because of the Fermi statistics. Moreover, this wave func-tion should describe a fluid of uniform density on a disk of radious R = √ mN l B ,17ompatibly with the fact that the number of states in the full Landau level is N = mN . The Laughlin proposal isΨ( z , ..., z N ) = (cid:89) i 1. One definesthe vacuum bundle V → M where the fiber V λ at a point λ j is the vacuum space of H ( x i ; λ j ). Let us a set a local frame | a ; λ (cid:105) , a = 1 , ..., N. (2.27)The definition of the Berry’s connection arises when we study the adiabatic evolutionof the system. The adiabatic theorem states that, if we prepare the quantum systemin a certain vacuum | a ; λ (cid:105) for certain λ j and vary slowly the parameters comparedto the energy gap of the excitations, at every point λ (cid:48) j of the path in the space ofcouplings the perturbed state | ψ a ( λ (cid:48) ) (cid:105) is in the ground state V λ (cid:48) . More precisely,the adiabatic ansatz for the evolved state is | ψ a ( λ (cid:48) ) (cid:105) = U ab ( λ (cid:48) ) | b ; λ (cid:48) (cid:105) (2.28)where U ∈ U ( N ) since time evolution in quantum mechanics is unitary. We seefrom this formula that a vacuum state is parallely transported along a certain inpath in the space of couplings. Plugging the adiabatic ansatz in the Schoredingerequation provides the definition of a unitary connection on the vacuum bundle whichgenerates the time evolution. This is the Berry’s connection( A i ) ba = − i (cid:104) a ; λ | ∂ λ i | b ; λ (cid:105) . (2.29)The unitary matrix introduced in 2.28 contains also the dynamical phase e − i (cid:82) dλE o ( λ ) which we always have in the time evolution of a quantum state. One can ignore thiscontribution or simply set E o ( λ ) = 0 by adding a constant to the Hamiltonian. It isinteresting in general to evaluate the transformation of a state after a closed loop inthe space space of couplings. The associated unitary matrix is the Berry’s holonomy U = P exp (cid:18) − i (cid:73) A i dλ i (cid:19) , (2.30)where P stands for path ordering. In the case of degenerate ground state the Berry’sconnection is non-Abelian and the associated holonomy group is a subgroup of U ( N )with N > 1. If the system has a unique vacuum state | λ (cid:105) , the Berry’s connectionis Abelian and the holonomy 2.30 reduces to the Berry’s phase e iγ = exp (cid:18) − i (cid:73) A i ( λ ) dλ i (cid:19) , (2.31)19ith A i ( λ ) = − i (cid:104) λ | ∂ λ i | λ (cid:105) . Topological phases are beleaved to support anyonic particles. These have not tobe thought as fundamental particles existing in nature. Indeed, they are artificiallyrealized as charge excitations of quantum liquids. The fusion and braiding rules ofanyons are among the geometrical observables which characterize the topologicalorder. Moreover, as we will discuss in the next section, the existence of anyonicparticles is deeply related to the topological degeneracy of the ground state. Giventheir importance in the phenomenology of topological phases, we spend some wordsto recall some basic properties about a theory of anyons. We know that in d = 3spatial dimensions we can have only fermionic and bosonic statistics. This is relatedto the topology of the configuration space of N identical particles in three dimen-sions. Denoting with r i , i = 1 , ..., N the positions of the particles in R , the space is (cid:0) R N \ { r i = r j } (cid:1) / S N , where S N is the permutation group of N objects. The funda-mental group of this space is Z . This means that all the possible ways to exchangetwo particles are topologically equivalent, while a double exchange is topologicallytrivial. In quantum mechanics the exchange of particles acts as unitary operator onthe wave function ψ ( r , r , ..., r N ). Hence, the statistics are in correspondence withunitary irreducible representations of Z . These are one dimensional and the uniquepossibilities are given by ψ ( r , r , ..., r N ) = ± ψ ( r , r , ..., r N ) (2.32)where ± N particles on the plane sitting along a line and order them. The imagein space-time of the particle worldlines under an exchange of their order is calleda braid. One distinguishes braids by their topological class, which means that twobraids are considered the same if we can smoothly change one into the other withoutcrossing of the worldlines. Such braidings form an infinite group called braid group B N . This coincide with the fundamental group of the confirguration space of N identical particles on the plane. This is (cid:0) C N \ { z i = z j } (cid:1) / S N , where z i are thecoordinates of the particles in C . The braid group of N objects is generated by N − R , ..., R N − , where R i exchanges the i -th and ( i + 1)-th particlesin an anti-clockwise direction. These satisfy the defining relations R i R j = R j R i | i − j | > R i R i +1 R i = R i +1 R i R i +1 , i = 1 , ..., N − . (2.34)Different statistics in two dimensions are associated with unitary irreducible repre-sentations of the braid group. Particles that form a one dimensional representationof B N are called Abelian anyons. In this case the braid matrices act on the wavefunction as phases R i = e iπα i . (2.35)The Yang-Baxter equations for these operators require that e iπα i = e iπα i +1 , implyingthat all identical particles have the same phase. If α = 0 , < α < a, b, c.... which denote the different type of anyons in our theory. These objects satisfy amultiplicative algebra, usually called fusion algebra, which reads a (cid:63) b = (cid:88) c N cab c (2.36)where N cab are non-negative integers telling how many different ways there are toget the anyon c when we bring together two anyons a and b . The fusion algebrais commutative and associative, implying that the order in which we fuse anyons isirrelevant. Each anyonic species identifies an irreducible representations of the braidgroup. Hence, the Hilbert of two anyons a, b decomposes in irreducible representa-tions of dimension N cab corresponding to the fusion channels c appearing in a (cid:63) b .The anyons are said non-Abelian if N cab ≥ c . The vacuum of the theorycorresponds to the identity element of the algebra 1 and satisfies a (cid:63) a (2.37)for any a . This relation implies that the Hilbert space of a single anyon is alwaysone dimensional. Hence, we see that a single anyonic particle does not have anyinternal degree of freedom. The information contained in the Hilbert space H ab oftwo non-Abelian anyons a, b is a property of the pair and cannot be associated to asingle particle. We learn that anyonic particles are mutually non local objects and21he corresponding Hilbert space is said topological since the stored information isrelated to global properties of the system, namely how the anyons braid. The Berry’s connection allows to study the topological properties of the quantumHall states and in particular the physics of the charged exctitations. One can intro-duce M quasi-holes in the Hall fluid at positions ζ a , a = 1 , ..., M . These are createdexperimentally by inserting thin solenoids in the fluid and have a three dimensionalnature of magnetic fluxes. However, from the 2 d perspective they behave like pointlike particles. A Laughlin state ν = 1 /m with M quasi-holes is described by thewave functionΨ( z , ..., z N ; ζ , ..., ζ M ) = (cid:89) i,a ( z i − ζ a ) (cid:89) i 2, where J is the electron charge density and J i , i = 1 , ν = 1 /m hascharge density J = − e/ πl B = e νB/ π and the charge current induced by anexternal electrostatic field E i is J i = σ xy (cid:15) ij E j , where σ xy = νe π is the Hall conduc-tance. Hence, the vector field J µ has the following response to a variation of theelectromagnetic field δJ µ = σ xy (cid:15) µνλ ∂ ν δA λ . (2.45)We want to write an action giving the above equation as equation of motion. Theelectron density can be parametrized in term of a U (1) gauge field a µ as J µ = e π ∂ ν a λ (cid:15) µνλ . (2.46)This current satisfies automatically the conservation law ∂ µ J µ = 0. The gauge field a µ is the emergent topological degree of freedom of the FQH liquid. The Lagrangian24escribing the dynamics of this field and reproducing the above equation of motionis L ( a µ , A ν ) = − e m π a µ ∂ ν a λ (cid:15) µνλ + e π A µ ∂ ν a λ (cid:15) µνλ . (2.47)The first piece of the Lagrangian is the well known Abelian Chern-Simons term. Asone could expect, the effective theory describing the geometrical properties of FQHstates is a topological field theory in 2 + 1 dimensions. The identification of theinverse filling fraction ν = m with the Chern-Simons coupling, which is quantizedby the requirement of gauge invariance at quantum level, leads to a correspondencebetween Chern-Simons levels and FQH states.To provide a complete description of the FQH liquids one has to include in theLagrangian the quasi-hole and quasi-particle excitations. This requires to introduceanother current j µ which couples to a µ . The Lagrangian gains a new term∆ L = a µ j µ . (2.48)The gauge invariance of the action is preserved if the current is conserved: ∂ µ j µ = 0.Turning off the electromagnetic field A µ , the equation of motion is e π f µν = 1 m (cid:15) µνρ j ρ . (2.49)Let us place a static quasi-hole in the origin. The current is given by j = eδ ( x ) , j = j = 0. The equation of motion becomes12 π f = 1 em δ ( x ) . (2.50)We see that the effect of the Chern-Simons term is to attach a magnetic fluxΦ = 2 πem (2.51)to a particle of charge q = e . Consequently, If we take a quasi-hole around anotherone we get an Aharonov-Bohm phase e iq Φ = e πi/m , which we interpret as thestatistical phase generated by a double exchange of the particles. This result agreeswith the fractional statistics that we find for the the Laughlin state ν = 1 /m . Usingthe definition of electromagnetic current density in 2.46 we also get the equality25 = e π f = em δ ( x ) , (2.52)from which we correctly the recover the fractionalization of the quasi-hole charge e ∗ = em .Recent reviews about the Chern-Simons approach to FQHE can be found in [12, 16]. So far we discussed only FQH liquids at the fractions ν = m . As we anticipated insection 2.4 one can describe the other plateaus with the idea of hierarchical statesproposed by Haldane and Halperin. One starts with a Laughlin state at ν = 1 /m and vary the magnetic field to make the system change the plateau. Given theirnature of magnetic fluxes, we can increase or decrease the magnetic field by injectingrespecticely quasi-particles or quasi-holes in a ν = 1 /m state such that they formthemselves a Laughlin condensate. The effective Lagrangian of the model is modifiedas follows [12, 13]. Let us set e = 1 for simplicity. One writes the quasi-holes currentas j µ = 12 π (cid:15) µνρ ∂ ν ˜ a ρ , (2.53)where ˜ a ρ is another emergent Chern-Simons field describing the quasi-hole conden-sate. The current couples to a µ and the new Lagrangian reads L ( a µ , ˜ a ν ) = 12 π A µ ∂ ν a λ (cid:15) µνλ − m π a µ ∂ ν a λ (cid:15) µνλ + 12 π a µ ∂ ν ˜ a λ (cid:15) µνλ − ˜ m π ˜ a µ ∂ ν ˜ a λ , (2.54)where ˜ m is an integer. To compute the Hall conductivity one has to solve theequations of motion for ˜ a µ and then for a µ . One finds that this theory describes aHall state with filling fraction ν = 1 m − m . (2.55)We can compute the charge and statistics of quasi-holes in this new state. Thereare two type of excitations for this fluid, the ones which couple to a µ and the oneswhich couple to ˜ a µ . For the first type we find26 f − ˜ f = 2 πδ ( x ) , ˜ m ˜ f − f = 0 , (2.56)from which we get f = 2 πm − m δ ( x ) . (2.57)For the second type we have mf − ˜ f = 0 ˜ m ˜ f − f = 2 πδ ( x ) (2.58)which gives f = 2 πm ˜ m − . (2.59)If we set for instance m = 3 and ˜ m = 2 we find the state ν = 2 / 5, which is amongthe most prominent among the observed plateau. The resulting charges of the quasi-holes are e ∗ = 2 / e ∗ = 1 / 5, which have been measured experimentally. One cannow repeat this construction: the quasi-particles of the new state form a condensatedescribed another gauge field, for which we introduce a new Chern-Simons term andcurrent of type 2.53 which couples to ˜ a µ . Iterating this procedure we obtain Abelianquantum Hall states with filling fraction ν = 1 m − m − m − .... . (2.60)Using the idea of hierarchy one can write down the most general Abelian quantumHall state [13, 12]. We consider N emergent gauge fields a iµ , with i = 1 , ..., N . Theeffective theory for these fields is L ( a iµ ) = 14 π K ij (cid:15) µνρ a iµ ∂ ν a jρ + 12 π t i (cid:15) µνρ A µ ∂ ν a iν (2.61)where the K matrix specifies the Chern-Simons coupling and the t vector tells whichlinear combination of currents play the role of electron current. So far we onlyconsidered the so called single-layer FQH states, where the electrons are confineda single 2 d interface. This setting is described by the choice t = (1 , , , ..., K and t . The Hall conductance can be computed byintegrating out the gauge fields and is given by (cid:0) K − (cid:1) ij t i t j , (2.62)while the charge of the quasi-holes which couple to a iµ is( e (cid:63) ) i = (cid:0) K − (cid:1) ij t j . (2.63)The statistics between quasi-holes which couple to a i and a j is α ij = (cid:0) K − (cid:1) ij . (2.64)Moreover, one can show that the degeneracy of the ground state on a surface ofgenus-g is | det K | g . When can recover the single-layer model given in 2.54 with thechoice K = (cid:18) m − − m (cid:19) t = (1 , . (2.65)It is possible to write down a wave function for the hierarchy states, also in themulti-layer case, which extends the Laughlin proposal to more general filling frac-tions. The interested reader can refer to [13] for the details of the construction.An alternative way to describe the hierachy states is given by the composite fermiontheory proposed by Jain [6]. The basic proposal is that FQH states for electronscan be thought as integer Hall states for new weakly interacting degrees of freedom.These are the composite fermions, namely electrons bounded to magnetic flux vor-tices. One can find a discussion about this alternative model in [12]. Despite theapproach is very different from the hierarchy theory, the composite fermion approachreproduces the same filling fractions. Among these we get in particular the principalseries ν = n n ± , (2.66)also known as Jain’s series. As the hierarchy model, the composite fermion theorypredicts Abelian fractional statistics for the quasi-holes.28he possibility of non-Abelian statistics has been explored for certain FQH statesat higher Landau levels [7, 8, 9, 10]. Among these the most prominent is observedat ν = 5 / 2, which consists of fully filled lowest Landau level for both spin up andspin down electrons, followed by a spin polarized Landau level at half filling. This isalso known as Moore-Read state and is believed to support non-Abelian anyons ofIsing type [12]. One can construct effective low energy theories also for non-Abelianquantum Hall states. These naturally involve emergent non-Abelian Chern-Simonsfields associated with certain non-Abelian gauge groups. It has been noticed in [8] a remarkable connection between wave functions of FQHstates and correlators of certain 2 d conformal field theories. Let us consider a ν =1 /m Laughlin state with N electrons at positions z i , i = 1 , ..., N and M quasi-holesat positions ζ a , a = 1 , .., M . We consider the c = 1 theory describing a free scalarfield φ ( z ) and associate to electrons and quasi-holes respectively the chiral vertexoperators V ( z i ) = exp ( iφ ( z i ) / √ ν ) and W ( ζ a ) = exp ( i √ νφ ( ζ a )). The holomorphicpart of the Laughlin wave function is captured up to a prefactor depending on the ζ a ) by the correlator (cid:104) (cid:89) i,a V ( z i ) W ( ζ a ) (cid:105) = (cid:89) i,a ( z i − ζ a ) (cid:89) i 1. The immaginary partof the sum − Φ( a ) (cid:80) Λ( a ) ∩ D ( R ) log( z − ζ m,n ( a )) oscillates very rapidly for a (cid:28) a → 0. Instead, the real partcoincides with the 2-dimensional electrostatic potential generated by a discrete setof charges, which tends to a continuous distribution for a (cid:28) 1. So, we obtain thelimit − Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) log( z − ζ m,n ( a )) a → , R →∞ −−−−−−−→ − (cid:88) i | z | / l B , (2.71)which reproduces the non-holomorphic part of the FQHE wave functions.The relation between FQHE and 2 d CFT is not accidental. The c = 1 chiraltheory describes the edge excitations of the Abelian Chern-Simons theory [13, 12].Contrary to the bulk topological theory, which is gapped by definition, the edgetheory is a conformal chiral liquid with gapless excitations. This bulk to boundarycorrespondence is one of the main property of topological phases and apply also tonon-Abelian quantum Hall states. The physics of the edge modes is a reflectionof the topological order in the bulk and provides the most powerful way tool toclassify topological universality classes. It turns out that the edge excitations havethe same spectrum of the quasi-particle and quasi-hole excitations of the fluid. Therole of the different kinds of anyons is now played by the different representations ofthe conformal algebra that appear in a given conformal field theory. Each of theserepresentations is labelled by a primary operator O i . In classifying topological order30e consider rational conformal field theories, since they have a finite number ofprimary operators. The CFT algebra encodes also the concept of fusion. Given twooperators O i , O j , the operator product expansion (OPE) between them can containother representations labelled by some O k . The OPEs are naturally interpreted asthe fusion rules satisfied by the anyonic particles. One can write the OPE as O i (cid:63) O j = (cid:88) k N ijk O k (2.72)where N kij are integers.The CFT description of FQHE provides also the definition of braiding matrices. Ingeneral a CFT contains both chiral and anti-chiral modes. A correlation function ofprimary operators can be written as (cid:104) (cid:89) i O ( z i , ¯ z i ) (cid:105) = (cid:88) p |F p ( z i ) | , (2.73)where F p ( z i ) are multi-branched holomorphic functions of the z i depending on theset of operators inserted in the correlator. These are also known as conformal blocksand provide a basis of quantum Hall wave functions. As we exchange the positions z i of the quasi-particles the conformal blocks will be analytically continued ontodifferent branches. The result can be written as some linear combination of theoriginal functions, from which one derives the braiding properties of the anyons. According to the previous discussion, to study the topological properties of FQHEis enough to identify the correct CFT. This encodes all the necessary physical infor-mations to characterize the universality classes of FQH states, such as fusion andbraiding rules of the quasi-particles excitations. Moreover, it provides a systematicway to construct the quantum Hall wave functions without any guesswork. Fol-lowing this approach, it has been recently proposed by Vafa a new unifying modelfor the principal series of FQHE systems with ν = n n ± [17]. The identification ofthe CFT describing the Hall states comes from consistency conditions between theabove description and the CFT paradigms. The starting point is again the Laughlinstate ν = 1 /m , which according to the previous models is described by the c = 1CFT. The wave function interpretation of the correlators implies that we have tointegrate them over the electron positions z i . However, in the context of a CFT thisis allowed only if the vertex operator exp( iφ ( z i ) / √ ν ) has dimension 1. This can beachieved by modifying the c = 1 theory with the addition of a background term31 (cid:48) R φ to the scalar action, where R is the Ricci curvature and Q (cid:48) is the backgroundcharge. This is given by Q (cid:48) √ Q = 1 b + b = i (cid:18) √ ν − √ ν (cid:19) , (2.74)where b = − i √ ν . (2.75)Hence, it turns out that FQHE systems are described by a Liouville theory withaction [20, 21] S = (cid:90) d z (cid:20) π ∂φ ¯ ∂φ + iQ (cid:48) Rφ (cid:21) . (2.76)The central charge of the theory reads c = 1 + 6 Q = 1 − ν − ν . (2.77)We relax the restriction ν = m and consider a generic rational filling fraction ν = nm .We obtain c = 1 − n − m ) nm (2.78)which is the central charge of the 2 d CFT minimal model (2 n, m ) [23]. Here m needs to be odd in order to have 2 n and m relatively prime. Moreover, since theedge modes in the FQHE are supposed to have correlations which fall off with thedistance (see [11]) we should restrict to unitary CFT’s [11]. This puts the furtherconstraint m = 2 n ± 1, from which we recover the Jain’s series ν = n n ± . Theoperator algebra of the (2 n, m ) minimal models is realized by the degenerate fields[22]. Φ r,s = exp (cid:20) i ( r − φ √ ν + i ( s − √ νφ (cid:21) (2.79)for 1 ≤ r < n, ≤ s < m , which satisfy the relations [24]32 r ,s × Φ r ,s = k = r − r − l = r − r − (cid:88) k =1+ | r − r | ,k + r + r +1=0 mod 2 l =1+ | r − r | ,l + r + r +1=0 mod 2 Φ k,l (2.80)In particular, the Φ ,s are identified with the quasi-holes operators, where Φ , is theminimal excitation, and generate insertions of ( z i − ζ ) s − factors in the wave function.This leads to the main difference between this model and the previous ones, wherethe quasi-holes of the states ν = n n ± are predicted to be Abelian anyons. On thecontrary, in this theory the quasi-holes should have the same fusion rules and non-Abelian braiding properties of the (2 n, n ± 1) minimal models. The embeddingof minimal models in Liouville theory allows also to identify the bulk topologicaltheory, which turns out to be a Chern-Simons theory based on an SL (2 , C ) gaugegroup. Among the several proposals of Vafa in [17], there is a connection between FQHEand supersymmetric N = 2 gauge theories in four dimensions, which arise fromcompactification of six dimensional (2 , 0) theories on a punctured Riemann surfaceΣ, the so called Gaiotto curve [18]. These are labelled by semisimple Lie algebrasin the ADE classification. Picking an ADE group, the rank r of the correspondingDynkin graph is identified with the number of layers of FQHE. We associate to eachnode in the graph an integer N a , a = 1 , ..., r which denotes the number of electronsin the a -th layer. The worldvolume of FQHE is identified with Σ × R , where R isthe time direction and the Riemann surface Σ is the target space of FQHE. For eachnode we assign also a meromorphic (1 , 0) form W a ( z ) (cid:48) dz on Σ. The meromorphicfunctions W a ( z ) (cid:48) encode the interaction between an electron of coordinate z in the a -th layer and the quasi-holes in the quantum Hall fluid. We focus on the caseof target space P , where W a ( z ) (cid:48) are just rational functions on the complex plane.Finally, we introduce for each node of the graph the set of chiral superfields z a,k a , k a = 1 , ..., N a , which play the role of electron coordinates for the different layers ofFQHE. Hence, with the ingredients defined above, a Dynkin diagram of ADE typeidentifies a 4-SQM Landau-Ginzburg theory with superpotential33 ( z a,k a ) = r (cid:88) a =1 N a (cid:88) k a =1 W a ( z a,k a ) + β r (cid:88) a =1 (cid:88) ≤ k a 0) six dimensional theory. In the single layercase we have a unique rational function W (cid:48) ( z ) containing the interaction betweenelectrons and quasi-holes. It is natural to describe the quasi-holes with simplepoles. Indeed, the function W (cid:48) ( z ) has the physical interpretation of electrostaticfield generated by the quasi-holes and the two dimensional Coulomb interactionbetween point-like particles decays with the inverse of the distance. This is also themost general case, since higher degree poles can be obtained by confluent limit ofsimple poles. Hence, denoting with ζ (cid:96) , (cid:96) = 1 , ..., M the positions of the punctureson P , the holomorphic superpotential corresponding to this setting is W ( z i ; ζ (cid:96) ) = N (cid:88) i =1 W ( z i ; ζ (cid:96) ) + β (cid:88) ≤ i Supersymmetry and tt ∗ Geometry Before exploring the connections between FQHE and supersymmetry, we need torecall some basic knoledge about supersymmetric quantum mechanics and relatedtools to study the geometry of vacua. We begin with the case of two supercharges. N = 2 SQM N = 2 supersymmetric quantum mechanics is essentially the Witten’s reformulationof Morse theory [37, 40]. The theory has a graded Hilbert space H = H + ⊕H − , where H ± are spaces of bosonic and fermionic states respectively. These are eigenspacesof the Fermi parity operator I = ( − F which counts the fermion number F of astate modulo two. This operator is Hermitian and satisfies I = 1. By definitionof N = 2 supersymmetry, the theory has an Hermitian operator Q (and its adjoint Q † ) which maps bosonic states to fermionic states and viceversa. The susy charges Q , Q † and the Fermi parity satisfy the algebraic relations {Q , I } = (cid:8) Q † , I (cid:9) = 0 Q = Q † = (cid:8) Q , Q † (cid:9) = 0 (3.1)Since they are symmetry operators of the system, the susy charges commute withthe Hamiltonian H . In particular, the Hamiltonian is the “Laplacian” H = 12 (cid:0) QQ † + Q † Q (cid:1) . (3.2)The operators Q , Q † , F, H generate the supersymmetry algebra of the N = 2 theory.Since it is a nilpotent operator, one can define the cohomology of Q in the Hilbertspace. The cohomology classes of Q are in correspondence with the ground statesof the theory. The vacuum wave functions | Ψ (cid:105) are eigenstates of the Hamiltonianwith zero energy and coincide with the harmonic representatives of the cohomologyclasses H | Ψ (cid:105) = 0 ⇔ Q | Ψ (cid:105) = Q † | Ψ (cid:105) = 0 . (3.3)36e consider a smooth compact manifold X of dimension dim R X = n , with a localcoordinate system x i , i = 1 , ..., n . To each coordinate x i we associate a couple ofcreation-annihilation operator ψ i , ψ i which satisfy the Clifford algebra (cid:8) ψ i , ψ j (cid:9) = { ψ i , ψ j } = 0 (cid:8) ψ i , ψ j (cid:9) = δ ij . (3.4)The Hilbert space is a representation of this algebra and a general wave functionreads Φ = (cid:88) k φ ( x ) i ,...,i k ψ i ...ψ i k | (cid:105) , (3.5)where | (cid:105) is the Clifford vacuum and the functions φ ( x ) i ,...,i k are totally antisym-metric in the indices. The commutation relation between creation and annihilationoperators can be read as the pairing between forms and vectors dx i ( ∂ j ) = δ ij . Theidentification ψ i → dx i allows to rewrite the above wave function as a differentialform on M . Hence, the space of differential forms Ω ∗ ( X ) plays the role of Hilbertspace, where the Hermitian scalar products between states coincides with the Hodgeinner product of forms (cid:104) Φ | Ψ (cid:105) = (cid:90) X Ψ ∧ ∗ Φ ∗ . (3.6)In this representation the susy charges act as differential operators on the space offorms, while the degree k of a wave form has the interpretation of Fermi number.Moreover, the Hermitian conjugation of operators is defined with respect to theinner product 3.6 and has the usual expression of Hodge theory.The easiest realization of N = 2 supersymmetric quantum mechanics is a sigmamodel on a Riemannian manifold. The supersymmetric charges Q , Q † are given bythe exterior derivative d and the adjoint d † = ( − k ∗ − d ∗ . The correspondingHamiltonian is the usual Laplacian on a Riemannian manifold H = 12 ∆ = 12 ( d † d + dd † ) . (3.7)The vacuum of this theory is isomorphic to the de Rham cohomology H ∗ ( X ; C ) andthe Witten index coincides the Euler character of X W = Tr H ( − F = n (cid:88) k =0 ( − k b k = χ ( X ) , (3.8)where the Betti numbers b k count the number of vacua with degree k .One can deform the sigma model by introducing a smooth real superpotential V on X . The susy algebra generators of this theory are the generalized differentialoperators d V = d + dV ∧ , d † V = ( − k ∗ − ( d + dV ∧ ) ∗ 12 ∆ V = 12 (cid:0) ∆ + g ij ∂ i V ∂ j V + ∂ i ∂ j V [ ∗ dx i , dx j ] (cid:1) , (3.9)where g ij is a Riemannian metric on the target manifold. The susy charge d V isconjugated to the exterior derivative by the relation d V = e − V de V . It follows thatthe cohomology classes [ ˜Ψ] of d V are in one to one correspondence with de Rhamclasses [Ψ] through the map ˜Ψ = e − V Ψ . (3.10)The isomorphism H vacuum (cid:39) H ∗ ( X ; C ) implies that on a compact manifold thedimension of the ground state depends only on the topology of the manifold andnot on the choice of the superpotential.One can study the ground states of the theory also with a perturbative approach.We introduce the planck constant (cid:125) and rescale the superpotential as V → (cid:125) V .In the semiclassical limit (cid:125) (cid:28) (cid:125) g ij ∂ i V ∂ j V becomes verylarge except around the critical points of V . For small (cid:125) the eigenfunctions of theHamiltonian are concentrated around the classical vacua and can be computed withan asymptotic expansion in terms of local data. This is the essence of Morse theory:we reconstruct the cohomology from the local data of a smooth function around itscritical points. We consider the case in which V is a Morse function. This impliesthat the critical points of V are isolated and non degenerate, namely det ∂ i ∂ j V (cid:54) = 0.Since the set of Morse functions is open and dense in the space of functions, theseproperties are very general. One can associate to each perturbative vacuum a Morseindex I M which counts the number of negative eigenvalues of the Hessian of V at thecritical point. The Morse index has the physical interpretation of fermion number.Since the Witten index is invariant under smooth deformation of the theory, one canperform its computation also in the classical limit. Denoting with c k the number ofclassical vacua with Morse index k , one has38 W = n (cid:88) k =0 ( − k c k = n (cid:88) k =0 ( − k b k . (3.11)The classical computation of vacua is exact at every order in perturbation theory, butis in general not correct at the non-perturbative level. Indeed, instanton correctionscan lift some of the classical vacua. Hence, the true quantum vacua are genericallyless than the classical ones and we have c k ≥ b k . Since the Witten index must bepreserved, bosonic and fermionic vacua can be lifted by instantons only in pairs.One can extend N = 2 SQM to a non compact Riemannian manifold. In the noncompact case the Hilbert space is the space of differential forms whose coefficentsare L functions on X . In absence of a superpotential, the vacua are harmonicrepresentatives in cohomology classes of L forms. These states are exact in theabsolute cohomology of X and one can write for a vacuum Ψ the relationΨ = d Φ . (3.12)However, Φ is not square integrable on the target space and so the wave form Ψdefines a non trivial cohomology class in the Hilbert space.As before, we can introduce a real superpotential V on the target space whichgenerates a potential term in the Hamiltonian. Differently from the compact case,now the structure of vacua depends on the choice of the superpotential. Morsetheory admits an extension for non compact manifolds and one can show that thespace of vacua is isomorphic to the relative de Rham cohomology H ∗ ( X, X V ; C ),where X V = { x ∈ X (cid:12)(cid:12) V < − Λ; Λ → ∞} (3.13)is the asymptotic region of X where the superpotential tends to −∞ . In particular,the relative de Rham classes in H ∗ ( X, X V ; C ) are mapped to d V -cohomology classesin the space of L forms by the map 3.10. We can consider as an example thepolynomial superpotential on the real line V = ax n + lower order terms , x ∈ R . (3.14)The structure of the ground state depends on the sign of the coefficient a in fron ofthe highest power of the polynomial. If a is positive the superpotential is boundedfrom below and so we have X V = ∅ . The vacuum space is isomorphic to the deRham cohomology of the real line 39 ∗ ( R ) = (cid:40) H ( R ) = C H ( R ) = 0 . (3.15)One can easily check that the eigenfunction of the Hamiltonian is the zero form ψ = ce − V , (3.16)where c is some normalization constant. If a is negative the superpotential isbounded from above and tends to −∞ as x → ±∞ . In this case the vacuum isisomorphic to the cohomology with compact support on R . This is Poincar´e dualto the de Rham cohomology and reads H ∗ c ( R ) = (cid:40) H ( R ) = 0 H ( R ) = C . (3.17)The corresponding vacuum is the fermionic state ψ = ce V dx. (3.18) N = 4 SQM Supersymmetric quantum mechanics with four supercharges has the same structureof Hodge theory on a Kahler manifold X [32, 37]. For any neighborhood of themanifold one can choose a set of holomorphic coordinates z i , i = 1 , ..., n = dim C X and a local Kahler potential K . This defines also the corresponding Kahler metric g ij = ∂ i ¯ ∂ j K and Kahler form ω K = ig i ¯ j dz i d ¯ z j . The fermionic partners of the bosonicdegrees of freedom z i , ¯ z i are four creation and annihilation operators ψ i , ψ i , ¯ ψ i , ¯ ψ i which satisfy the Clifford relations (cid:8) ψ i , ψ j (cid:9) = { ψ i , ψ j } = 0 (cid:8) ¯ ψ i , ¯ ψ j (cid:9) = (cid:8) ¯ ψ i , ¯ ψ j (cid:9) = 0 (cid:8) ψ i , ψ j (cid:9) = δ ij (cid:8) ¯ ψ i , ¯ ψ j (cid:9) = δ ij . (3.19)Similarly to the case with two supercharges, one can identify the creation and an-nihilation operators with forms and vectors on the target space. The map ψ i → dz i , ¯ ψ i → d ¯ z i realizes the isomorphism between the space of L smooth forms on X ,endowed with the inner product 3.6, and the Hilbert space H of a N = 4 theory.40ompared to the Riemannian case, the richer structure of Kahler geometry leadsto a larger supersymmetry algebra. The Hilbert space can be decomposed in irre-ducible representations of the Lefschetz SU (2) symmetry group, whose algebra isgenerated by the Lefschetz operators L = ω K ∧ , Λ = ( − k ∗ − L ∗ (3.20)and the shifted Fermi number ˜ F = k − n , where k = p + q is the degree of a formwhich decomposes in holomorphic and anti-holomorphic degree p, q . These operatorssatisfy the algebraic relations [ ˜ F , L ] = 2 L, [ ˜ F , Λ] = − L, [ L, Λ] = ˜ F . (3.21)The Lefschetz group has the physical interpretation of R -symmetry group of thesupersymmetry algebra.The easiest realization of N = 4 SQM is a sigma model with target space X . Inthis theory the supercharges are given by the Dolbeault operators on the Kahlermanifold Q = ∂, Q = ¯ ∂, Q † = ¯ δ, Q † = δ. (3.22)The susy charges are nilpotent and anticommuting operators ∂ = ¯ ∂ = 0 ∂ ¯ ∂ + ∂ ¯ ∂ = 0 δ = ¯ δ = 0 ¯ δδ + δ ¯ δ = 0 (3.23)and with the R-symmetry generators satisfy the Kahler identities (cid:2) ¯ ∂, L (cid:3) = [ ∂, L ] = 0 (cid:2) ¯ ∂ † , Λ (cid:3) = (cid:2) ∂ † , Λ (cid:3) = 0 (cid:2) ¯ ∂ † , L (cid:3) = i∂ (cid:2) ∂ † , L (cid:3) = − i ¯ ∂H = ¯ ∂ ¯ ∂ † + ¯ ∂ † ¯ ∂ = ∂∂ † + ∂ † ∂ = 12 ( dd † + d † d ) , (3.24)41here H is the Hamiltonian. The four susy charges, the R-symmetry generatorsand the Hamiltonian provide a complete basis of generators for the supersymmetryalgebra.If the Kahler manifold is non compact we can add to the theory an holomorphicsuperpotential W . A N = 4 theory with a superpotential is called Landau-Ginzburg(LG) theory. In presence of W the supercharges become the generalized Dolbeaultoperators ∂ W = ∂ + ¯ dW ∧ , ¯ ∂ W = ¯ ∂ + dW ∧ , ¯ δ W = δ † W , δ W = ¯ ∂ † W . (3.25)One can check that these operators satisfy the same algebraic relations of ∂, ¯ ∂ . Inparticular, the Hamiltonian is the generalized Laplacian H = ∂ W ∂ † W + ∂ † W ∂ W = ¯ ∂ W ¯ ∂ † W + ¯ ∂ † W ¯ ∂ W = 12 ( d V d † V + d † V d V ) , (3.26)where d V = ∂ W + ¯ ∂ W is the generalized exterior derivative of V = W + W .We are interested in the space of supersymmetric vacua of a N = 4 Landau-Ginzburgtheory on a non compact space. By the Kahler identities 3.24, we can study thecohomology in the Hilbert space of one of the two Dolbeault derivatives. For con-venience, one typically considers the complex of Q = ¯ ∂ W . Similarly to Hodgetheory, the vacuum wave forms coincide with the harmonic representatives of the¯ ∂ W -cohomology classes. There exists a subclass of manifolds that are very com-mon in the applications in which this problem simplifies considerably, namely Steinmanifolds. There are several characterizations of this type of spaces, in particularcapturing the property of having “many” holomorphic functions. Denoting with O ( X ) the ring of holomorphic functions on X , the following two conditions aresatisfied: • X is holomorphically convex, namely for every compact subset C ⊂ X , theso-called holomorphically convex hull¯ C = (cid:8) z ∈ X (cid:12)(cid:12) | f ( z ) | ≤ sup w ∈ K | f ( w ) | ∀ f ∈ O ( X ) (cid:9) (3.27)is also a compact subset of X . • X is holomorphically separable, i.e. ∀ x, y ∈ X with x (cid:54) = y there is an holo-morphic function f such that f ( x ) (cid:54) = f ( y ).42xamples of Stein manifolds are C n , as well as its holomorphic domains, and all thenon compact Riemann surfaces. Among the properties of a Stein space, we havethat X is holomorphically spreadable, i.e. for every point x ∈ X there is a set of n = dim C X globally defined holomorphic functions which form a local coordinatesystem when restricted to a neighborhood of x . Another relevant property is thatthe Dolbeault cohomologies H p,q ¯ ∂ ( X ) have a simple structure, i.e. H p,q ¯ ∂ ( X ) = (cid:40) , q ≥ p ( X ) , q = 0 (3.28)where Ω p ( X ) is the space of holomorphic p -forms on X . The cohomologies H p,q∂ ( X )have a similar structure and can be obtained by complex conjugation. This resultimplies that a Stein space admits a globally defined Kahler potential K , since thecorresponding Kahler form is exact in the Dolbeault cohomology and can be writtenglobally as ω = i∂ ¯ ∂K . Moreover, a Stein space admits geodesically complete Kahlermetrics.In order to describe the ground states of a Landau-Ginzburg theory it is relevant todefine the BRST cohomology of Q in the space of operators. The supercharge actson an operator φ as (cid:2) Q, φ (cid:3) , (3.29)where the brackets denote respectively a commutator or anticommutator if the op-erator has even or odd fermion number. An operator is closed if the commutatorwith the susy charge is vanishing, while it is exact if there exist an operator ϕ suchthat φ = (cid:2) Q, ϕ (cid:3) . The BRST cohomology of Q forms a ring under multiplication ofoperators, also known as chiral ring [32, 37] R = Q − closed opers Q − exact opers . (3.30)In particular, R is a unital associative algebra defined over C . Let us consider anoperator which multiplies a wave form by a smooth function χ . This operator isclosed if (cid:2) Q, χ (cid:3) = ¯ ∂χ = 0 , (3.31)namely if χ is holomorphic. Moreover, the operator is exact if there exists anholomorphic section υ of the tangent bundle on X such that43 ι υ , Q (cid:9) = υ i ∂ i W, (3.32)where ι υ is the contraction of differential forms by the vector field υ . It turns outthat the chiral ring is isomorphic to the commutative quotient algebra R (cid:39) O ( X ) /J W , (3.33)where J W = (cid:104) ∂ W, ..., ∂ n W (cid:105) is the Jacobian ideal generated by the partial derivativesof W . It is know from singularity theory that if the critical point of W are isolated,the quotient algebra is finite dimensional and localizes around the critical points.Denoting with p i , i = 1 , ..., N the set of critical points, we have the isomorphism ofcomplex algebras R (cid:39) N (cid:89) i =1 R i , R i = O p i /J W (cid:39) C µ i (3.34)where O p i are the germs of holomorphic functions at p i . The dimension of the localring µ i = dim C R i is called Milnor number of the critical point p i and is equal tothe degeneracy of p i . The dimension of the algebra is computed by the sum of theMilnor numbers dim C R = (cid:80) i µ i . If W is non degenerate at the critical points, thechiral ring is a semisimple algebra and can be written as product of simple factors R (cid:39) (cid:81) Ni =1 C i (cid:39) C N . In particular, we have R i (cid:39) C i and µ i = 1 for each nondegenerate p i . One can introduce a basis of orthogonal idempotents e i , i = 1 , ..., N in which the chiral ring is completely diagonal. This is defined by the condition e i ( p j ) = δ ij (3.35)and provides a canonical isomorphism between R and C N as complex algebra. Thisbasis can always be constructed if X is a Stein manifold and the set of non degeneratecritical points has no accumulation points. An holomorphic operator φ is identifiedby this map with its set of critical values φ → φ ( p ) ...φ ( p N ) . (3.36)44ne can introduce a non degenerate symmetric pairing between chiral operators (cid:104) φ i φ j (cid:105) = η ij (3.37)which is defined by the Grothendieck formula [28] (cid:104) φ (cid:105) = 1(2 πi ) n (cid:73) φ dz ...dz n ∂ W...∂ n W , (3.38)where the integration contour encircles the critical points of W . Endowed with theabove pairing, the chiral ring is a symmetric Frobenius algebra.The structure of the vacuum space V of N = 4 Landau-Ginzburg theory is describedby the following theorem [32, 33]. Under the condition that X is a Stein manifold,with dim C X = n and the critical points of W are isolated • The supersymmetric vacua Ψ i are invariant n -forms under the Lefschetz sym-metry group, namely L Ψ i = ΛΨ i = 0 (3.39) • The vacuum wave forms are “essentially independent of the Kahler metric”.Denoting with Ψ a wave function for a Kahler potential K , given anotherpotential K the corresponding wave function isΨ = Ψ + [ n/ (cid:88) k =0 a k L k Λ k Ψ (3.40)for some coefficients a k . In particular Ψ n, = Ψ n, and Ψ ,n = Ψ ,n . • The Hamiltonian is compatible with complex conjugation: if Ψ i is a basis ofvacuum wave functions, also the Hermitian conjugatesΨ ∗ i = M j ¯ i Ψ j (3.41)form a complete basis, where the complex conjugation is defined by the realstructure M of the ground state. • The space of vacua is isomorphic to the chiral ring as vector space45 (cid:39) R . (3.42)Denoting with φ i a basis of R , the vacuum wave function Ψ i representing the φ i -class reads Ψ i = φ i dz ∧ .... ∧ dz n + Q − exact term . (3.43)We make a few comments about this result. The vacuum wave functions can be seenas representatives of the ¯ ∂ W -cohomology in the Hilbert space. One can show thatif the ¯ ∂ -cohomology has the structure 3.28, which is guaranteed by the fact that X is a Stein manifold, the ¯ ∂ W -cohomology is concentrated in the middle cohomologygroup H n ¯ ∂ W , which is isomorphic to the vector space H n ¯ ∂ W (cid:39) ˜ R = Ω n dW ∧ Ω n − , (3.44)namely the space of holomorphic n -forms modulo those that can be written as dW ∧ α for some n − α . The map φ ( z ) → φ ( z ) dz ....dz n makes explicitthe isomorphism between ˜ R and the chiral ring˜ R (cid:39) R (3.45)as modules over R . Moreover, H n ¯ ∂ W has the same local structure of the chiral ring.The fact that the map dW ∧ : Ω n − ( M ) → Ω n ( M ) fails to be surjective only atthe critical points p i of W , which are assumed to be isolated, implies that thecohomology is localized around the classical vacua. Indeed, one can prove that˜ R (cid:39) (cid:77) i ˜ R i , ˜ R i = Ω ni ( M ) dW ∧ Ω n − i ( M ) , (3.46)where Ω ni ( M ) are the germs of holomorphic n -forms at p i . Moreover, each space ˜ R i is isomorphic to the local ring R i = O p i I W (3.47)as module over R i . It is proved in [32] that each ¯ ∂ W -class has representativeswith compact support, hence in the Hilbert space, and viceversa a class in the46ilbert space determines a class in the space of smooth forms Λ ∗ ( M ). Therefore,the cohomology H ¯ ∂ W ( M ) is equivalent to the cohomology in the Hilbert space andwe have the isomorphism of vector spaces H ¯ ∂ W ( M ) (cid:39) H ¯ ∂ W ( H ) (cid:39) R . (3.48)The vacuum space has vanishing Lefschetz angular momentum ˜ F and forms a trivialrepresentation of the R-symmetry group. In particular, the Witten index I W =Tr H ( − ˜ F + n counts the number of vacua up to the sign and is given by the sum ofthe Milnor numbers I W = ( − n (cid:88) i µ i . (3.49)If W is a Morse function, all the critical points are non degenerate and the dimensionof the ground state is equal to the number of classical vacua. Indeed, since they haveequal fermion number, the classical vacua cannot be lifted by instantons correctionsand, differently from the case with two supercharges, the perturbative computationis always exact.By the Frobenius structure of the chiral ring, the space of vacua is naturally endowedwith a symmetric pairing η ij = 1(2 πi ) n (cid:90) Ψ i ∧ ∗ Ψ j = (cid:73) φ i φ j dz ...dz n ∂ W...∂ n W = (cid:104) φ i φ j (cid:105) , (3.50)where φ i , φ j are representative in the chiral ring of the vacua Ψ i , Ψ j . Viceversa, thescalar product of the Hilbert space allows to define an Hermitian pairing betweenchiral and anti-chiral operators, also known as tt ∗ metric [28] g j ¯ i = (cid:104) φ i | φ j (cid:105) = (cid:104) Ψ i | Ψ j (cid:105) = (cid:90) Ψ j ∧ ∗ Ψ ∗ j . (3.51)Using the identities 3.41, 3.50 one finds that the ground state metric g , the realstructure M and the symmetric pairing η are related by g j ¯ i = M k ¯ i (cid:90) Ψ j ∧ ∗ Ψ k = η jk M k ¯ i . (3.52)Moreover, since the real structure satisfies M M ∗ = 1, where M ∗ denotes the complex(not Hermitian) conjugate matrix, one can write an identity between η and g − g ( η − g ) ∗ = 1 (3.53)also known as reality constraint.A Landau-Ginzburg theory with four supercharges is in particular N = 2 super-symmetric. In the N = 4 supersymmetry algebra one can define a family of N = 2subalgebras parametrized by an angle θ as d V θ = d + ( e iθ ∂W + e − iθ ∂W ) ∧ , δ h θ = d † h θ , (3.54)where the real superpotential V θ is the harmonic function V θ = 2 Re ( e iθ W ) . (3.55)On a Kahler space the complex structure is compatible with the Riemannian struc-ture and one can see the vacuum space also as L -cohomology of d V θ .The cohomology classes which label the vacua of the supersymmetric system aredefined over the complex numbers. However, it is possible to endow the vacua alsowith an integral structure. By Morse cobordism, we have an isomorphism betweenthe vacuum space and the relative de Rham cohomology [28] V = H ∗ ( X, X Re( e iθ W ) ; C ) , (3.56)where the set X Re( e iθ W ) ⊂ X is defined according to 3.13. The vacuum wave formsΨ i are conjugated to some closed n -forms ω i in the relative de Rham classes byΨ i = e − e iθ W ) ω i , i = 1 , ..., N. (3.57)The classes in the dual homology group H n ( X, X Re( e iθ W ) ; C ) are represented by noncompact cycles with boundary in the region X Re( e iθ W ) . This space has a naturalintegral structure given by the homology with integer coefficients V ∨ (cid:39) H ∗ ( X, X Re( e iθ W ) ; Z ) ⊗ Z C . (3.58)In case of Morse superpotential a canonical integral basis is given by the Lefschetzthimbles B a ( θ ) , a = 1 , ..., N describing the gradient flow of the superpotential e iθ W [29, 37]. These are special Lagrangian middle-dimensional submanifolds of X whicharise from the critical point of W . The images of these cycles on the W -plane48re straight lines stretched in the e iθ direction starting at the critical values of thesuperpotential. The relative homology is also called the space of branes, becausein 2 d the corresponding objects have the physical interpretation of half-BPS branes[29]. In particular, the angle θ specifies which linear combinations of the original4 supercharges leave the brane invariant. One can use the above formula and thestandard pairing between cycles and forms to introduce the so called tt ∗ braneamplitudes (cid:104) B a ( θ ) | Ψ i (cid:105) = Ψ a,i ( θ ) = (cid:90) B a ( θ ) e e iθ W ) Ψ i , (3.59)which define a non-degenerate N × N matrix whose components are not univaluedas function of the couplings and the phase e iθ . tt ∗ Geometry tt ∗ Geometry in SQM Once we solve the Schroedinger equation for the zero energy level, it is interestingto study the evolution of the ground states as we vary the parameters of the theory.We consider a family of Landau-Ginzburg models W ( z i , t a ) parametrized by a setof holomorphic couplings t a . These are local holomorphic coordinates on a complexmanifold P . We assume for this family of theories that the target manifold X isa Stein space and the critical points of W are isolated. One defines a complexvector bundle H → P , also called Hilbert bundle, where the fiber is the Hilbertspace H . This is a trivial bundle and can be written globally as H × P → P . Thevacuum bundle V → P is the subbundle of the Hilbert bundle whose fiber is thespace of vacua H vac . Differently from the Hilbert bundle, the bundle of vacua is nontrivial, since the ground state of the Hamiltonian vary non trivially as we changethe couplings of the superpotential. The canonical trivial connection of the Hilbertbundle induces by projection P : H → H vac a non trivial connection on the vacuumbundle, which is by definition the Berry connection. The equations which prescribethe curvature of the Berry connection are known as tt ∗ equations [28, 32]. We recallthat a generic wave form Ψ in the Hilbert space has a unique orthogonal Hodgedecomposition Ψ = Ψ + ¯ ∂ W α + ¯ ∂ † W β (3.60)where Ψ is W -harmonic. The operator P projects the state along the vacuum49pace, namely P : Ψ → Ψ . (3.61)The vacua Ψ k , k = 1 , ..., N , satisfy the equations( ¯ ∂ + dW ∧ )Ψ k = 0 , ( ¯ ∂ + dW ∧ ) † Ψ k = 0 . (3.62)We take the derivative of these equations with respect to some coupling t a . Denoting W a = ∂ t a W , we find ∂ t a (cid:2) ( ¯ ∂ + dW ∧ )Ψ k (cid:3) = ¯ ∂ W ( ∂ t a Ψ k ) + d ( ∂ a W ) ∧ Ψ k = ¯ ∂ W ( ∂ t a Ψ k ) + ∂ W ( W a Ψ k ) = 0 . (3.63)and ∂ t a (cid:2) ( ¯ ∂ + dW ∧ ) † Ψ k (cid:3) = ¯ ∂ † W ( ∂ t a Ψ k ) = 0 . (3.64)We introduce the forms λ a,k which satisfy¯ ∂ W λ a,k = W a Ψ k − P ( W a Ψ k ) , ¯ ∂ † W λ a,k = 0 . (3.65)In particular, the first equation is consistent with the fact that W a Ψ k is ¯ ∂ W -closed.One can rewrite the equations 3.63 and 3.64 as¯ ∂ W ( ∂ t a Ψ k − ∂ W λ a,k ) = 0 , ¯ ∂ † W ( ∂ t a Ψ k − ∂ W λ a,k ) = 0 . (3.66)So, we see that ∂ t a Ψ k − ∂ W λ a,k is in the ground state and can be written as linearcombination of Ψ k . The coefficients of the linear combination defines the Berryconnection ∂ t a Ψ k − ∂ W λ a,k = − ( A a ) lk Ψ l . (3.67)50ne can introduce the corresponding covariant derivative D a and rewrite this rela-tion as D a Ψ k = ∂ W λ a,k . (3.68)We can repeat the same steps also for the other vacuum equations( ∂ + dW ∧ )Ψ k = 0 , ( ∂ + dW ∧ ) † Ψ k = 0 . (3.69)We find the definition of the antiholomorphic part of the Berry connection D a Ψ k = ¯ ∂ W ˜ λ a,k (3.70),where ˜ λ a,k are wave forms satisfying ∂ W ˜ λ a,k = W a Ψ k − P ( W a Ψ k ) , ∂ † W ˜ λ a,k = 0 . (3.71)The bundle of vacua has a natural structure of holomorphic vector bundle. A basisof holomorphic sections is given by the operators in the chiral ring R , which areholomorphic representatives of the vacua in the ¯ ∂ W -cohomology. The holomorphicoperators provide also a basis for the tangent space of the couplings manifold. In-deed, the deformations of the superpotential W a are elements of the chiral ring.Denoting with φ k a basis for R , the chiral ring algebra is encoded in the relation φ i φ j = C kij φ k , (3.72)where the numbers C kij are the structure constants of the ring. These matricesdescribe the action of the chiral operators on the vacua. Projecting on the groundstate the action of W a on a wave function Ψ k one finds P ( W a Ψ k ) = W a φ k dz ∧ ... ∧ dz n + ¯ ∂ W ( ... ) = C lak φ l dz ∧ ... ∧ dz n + ¯ ∂ W ( ... ) . (3.73)51he curvature of the Berry connection can be computed in terms of the chiral ringcoefficients ( C a ) lk with Hodge theoretical arguments. One gets the tt ∗ equations [28][ D a , D b ] = − [ C a , C b ][ D a , D b ] = [ C a , C b ] = 0[ D a , D b ] = [ C a , C b ] = 0 (3.74)In addition, the chiral ring coefficients satisfy D a C b = D a C b = 0 D a C b = D b C a D a C b = D b C a . (3.75)The structure of the tt ∗ equations depends uniquely on the holomorphic data ofthe chiral ring, while there is no dependence on the Kahler metric of the targetspace. In general, once the topology of the target manifold is fixed, the geometryof the vacuum bundle is completely determined by the superpotential. From theabove equations one can see that the Berry connection preserves the holomorphicstructure of the bundle and is compatible with the metric. In particular, only the(1 , 1) component of the curvature is non vanishing. Hence, the Berry connection isthe unique Chern connection of the bundle of vacua. In an holomorphic trivializationthe coefficients of the connections read A jak = g k ¯ l ( ∂ a g − ) ¯ lj , A j ¯ ak = 0 , (3.76)where g k ¯ l = (cid:104) φ l | φ k (cid:105) is the ground state metric. One can express C a in this basis interms of C a and g as ( C a ) kl = g k ¯ j ( C † ) ¯ j ¯ r g ¯ rl = ( gC † g − ) kl . (3.77)Plugging the expression 3.76 in the tt ∗ equations one obtains a set of differentialequations for g ¯ ∂ i ( g∂ j g − ) = (cid:2) C j , C i (cid:3) . (3.78)Solving this equation with the appropriate boundary conditions and the realitycontraint 3.53 allows to determine the geometry of the vacuum bundle.We specialize the analysis to the class of Landau-Ginzburg models in which thecritical points of the superpotential p i , i = 1 , ..., N are non degenerate. In this52ase, as we said in the previous section, the chiral ring is semisimple and factorizesas product of simple factors R (cid:39) (cid:81) Ni =1 C i . Moreover, one can introduce a basisof orthogonal idempotents e i which provide a canonical isomorphism between R and C N . This is also called ‘point basis ’, since the operators satisfy the property e i ( p j ) = δ ij . In this basis the chiral ring algebra diagonalizes completely e i e j = δ ij e j , (3.79)and a generic operator Φ = (cid:80) Ni =1 Φ( p i ) e i can be identified with its set of criticalvalues Φ( p i ). It is convenient to rescale the generators e i in order to normalize thesymmetric pairing η ij of the chiral ring to the identity. For a Morse superpotentialthe Grothendieck residue 3.38 becomes η ij = (cid:104) φ i φ j (cid:105) = (cid:88) p i φ i φ j (det ∂ k ∂ l W ) − . (3.80)In the basis of orthogonal idempotent we have η ij = (det ∂ k ∂ l W ( p i )) − δ ij , (cid:104) e i (cid:105) = (det ∂ k ∂ l W ( p i )) − . (3.81)The symmetric pairing can be normalized to the identity with the rescaling e i → (cid:112) det ∂ k ∂ l W ( p i ) e i . The so redefined basis is called canonical basis and is uniqueup to the sign. Since the Berry connection is compatible also with the symmetricpairing, in the canonical basis we have A ti = − A i , (3.82)namely the Berry connection takes value in the algebra of the orthogonal group.The LG theories with a Morse superpotential posses a natural system of coor-dinates in the coupling constant space. The critical map w : P → C N givenby t a → ( w ( t a ) , ..., w N ( t a )) is a local immersion and the set of critical values w i ( t a ) = W ( p i ; t a ) form a local coordinate system on the Frobenius manifold of allcouplings of the theory which contains the physical coupling space P as a subman-ifold. These are usually called canonical coordinates [30, 48, 49]. In the canonicalbasis, the chiral ring coefficients associated to the variations of w i have the simplestructure ( C i ) kl = δ il δ kl . (3.83)53n these coordinates the tt ∗ equations become universal and inequivalent physicalsystems are distinguished by different boundary conditions.It is known that the vacuum bundle of a N = 4 Landau-Ginzburg theory and itstwo dimensional counterpart have the same tt ∗ geometry [28]. Even if the physi-cal interpretation is different, the formulas are exactly the same and one can usethe language of 2 d N = (2 , 2) quantum field theory to study the tt ∗ geometry ofthe model. The usual strategy to solve the tt ∗ equations requires to rescale thesuperpotential W → βW and consider variations of the overall coupling β . In twodimensions this is equivalent to study the RG flow of the theory [28]. Even thoughthe superpotential is protected by non-renormalization theorems, the F-term picksup a factor due to the rescaling of the superspace coordinates. From z → βz and θ → β − / θ we get (cid:90) d θd z W −→ β (cid:90) d θd z W. (3.84)Variations in the overall factor β generate a flow which has UV limit for β → β → ∞ .We denote with ˙ P ∈ P the dense open domain in the space of couplings in which R is semisimple and w i (cid:54) = w j for i (cid:54) = j . In other words, ˙ P is the domain in which W is strictly Morse. A quantity of great interest is the Berry’s connection in thedirection of the RG flow [30, 38] Q = ι ε A = (cid:88) i w i A i , (3.85)where ε = (cid:80) i w i ∂ w i = β∂ β is the Euler vector in ˙ P . The equations D i C j = D j C i in the canonical coordinates become[ A i , C j ] = [ A j , C i ] . (3.86)From this equality and the definition of Q we find (cid:2) A i , w j C j (cid:3) = (cid:2) A j w j , C i (cid:3) = [ Q, C i ] . (3.87)The k, l component of the above expression is( A i ) kl w l − w k ( A i ) kl = Q ki − Q li , (3.88)54hich allows to write the Berry’s connection in terms of Q only as [48]( g∂g − ) kl = Q kl d ( w k − w l ) w k − w l , (3.89)where we used the fact that Q is antisymmetric in the canonical basis. Hence, it isenough to know Q in order to specify completely the solution of the tt ∗ equations.In 2 d (2 , 2) models Q ij is a pseudo-index which plays two roles [38]. First, it capturesthe half-BPS solitons with boundary conditions given by the i, j -th vacuum repsec-tively at x → ±∞ . In the IR limit the tt ∗ solution can be written in the form ofsoliton expansion, with the boundary condition represented by the soliton spectrum.The IR region in canonical coordinates is identified by large masses m ij = 2 | w i − w j | of the kinks interpolating the i -th and j -th vacua. In the canonical basis the IRexpansion ( β → ∞ ) of the metric and Q ij are [30] g i ¯ j (cid:39) δ ij − iπ µ ij K ( m ij β ) Q ij (cid:39) − i π µ ij m ij βK ( m ij β ) (3.90)where K , K are the modified Bessel function and µ ij is the soliton matrix whichcounts with sign the number of soliton species in the i, j sector. The existence ofregular solutions requires µ ij to be real, although they are integral in the physicalcase. Moreover, the CPT invariance imposes µ ji = − µ ij .The second role of Q ij is that it captures the U (1) R charges of the Ramond vacuaat the UV fixed point. As we said, the UV fixed point is reached when β → 0. Ifthere are no Landau poles along the RG flow and the UV theory exists, the modelcan flow to a SCFT or an asymptotically free theory. In this limit the operators inthe chiral ring are the chiral primary fields of the conformal theory [28, 30]. It isknown that a generic Landau-Ginzburg theory at the critical point gains the U (1) V R-symmetry, which is broken off-criticality by the superpotential. The associatedcharge is the scaling dimension of the conformal fields. One can choose a basis ofchiral primaries for the chiral ring and order the R-charges q i , i = 1 , ..., N = dim R in a non-decreasing sequence0 = q ≤ q ≤ ... ≤ q N = c/ q max , (3.91)where c is the central charge of the CFT. In particular, there is always an operatorwith 0 charge in the spectrum which is given by the identity operator. The Berryconnection in the direction of the RG flow has the same eigenvalues of the generator55f the U (1) R symmetry Q of the (2 , 2) SCFT [28, 30, 38]. In this sense, the Berry’sconnection associated to the RG flow can be seen as an off-criticality definition ofthe U (1) R generator. The eigenvalues of Q at the critical point are the Ramondcharges of the vacua [28] q Ri = q i − ˆ c/ , (3.92)which are symmetrically distributed between − ˆ c/ c/ tt ∗ Geometry We previously introduced D-branes to provide the vacua with an integral structure.One can define a dual bundle whose fiber is the lattice of relative homology cycleswhich pair with the vacuum wave functions. This bundle is parametrized by a phase ζ ∈ P , | ζ | = 1 which defines which half of supersymmetry is preserved by the D-branes. The pairing between D-branes and vacua in the Hilbert space allows todefine a basis of sections for the vacuum bundleΨ i,a = (cid:104) φ i | B a ( ζ ) (cid:105) . (3.93)We remind that, because of the integral structure of D-branes, these amplitudes arelocally constant in the couplings w i and the spectral parameter ζ . We can define afamily of flat connections parametrized by ζ [30] ∇ ζi = ∂ i + ( g∂ i g − ) − ζC i , ¯ ∇ ζi = ¯ ∂ i − ζ − C i , (3.94)anso known as tt ∗ Lax connection. The tt ∗ equations for the Berry connection canbe rephrased as the flatness conditions for the above connection, i.e.( ∇ ζ ) = ( ¯ ∇ ζ ) = ∇ ζ ¯ ∇ ζ + ¯ ∇ ζ ∇ ζ = 0 . (3.95)The D-brane amplitudes are horizontal sections of the Lax connection ∇ ζi Ψ a ( ζ ) = ¯ ∇ ζi Ψ a ( ζ ) = 0 . (3.96)56ifferently from the Berry connection, the Lax connection is not compatible withthe ground state metric and provides a non unitary representation of π ( P ). LetΨ( w i , ζ ) be a N × N matrix whose columns are linearly independent solutions ofthe above linear equation. Taking the analytic continuation of the solutions along anon trivial loop γ in the space of couplings we obtain the monodromy ρ ζ ( γ ) definedby Ψ( w i , ζ ) → Ψ( w i , ζ ) ρ ζ ( γ ) . (3.97)Since the Lax connection is flat, the matrix ρ ζ ( γ ) depends only on the homotopyclass of γ . The tt ∗ monodromy representation ρ ζ in GL ( N, C ) may be conjugatedsuch that it lays in the arithmetic subgroup SL ( N, Z ). Since the branes are repre-sentatives of integral homology classes, for each ζ ∈ P they define a local systemon P canonically equipped with a flat connection, the Gauss-Manin one. Dually,the brane amplitudes define a P - family of flat connections on the vacuum bun-dle which is naturally identified with the P -family of tt ∗ Lax connections. Hence,modulo conjugation we have ρ ζ : π ( P ) → SL ( N, Z ) . (3.98)Since the entries of the matrix ρ ζ ( γ ) and its inverse are integers, they are locallyindipendent from the couplings and the spectral parameter ζ . This implies that the tt ∗ monodromy representations is indipendent from the point t ∈ P and ζ .The tt ∗ geometry is a set of isomonodromic equations for the Lax linear system andadmits a reformulation as Riemann-Hilbert problem [30, 49]. It is known that thesolution is captured by the monodromy of the flat sections around the movable sin-gularities. These are poles for the equations and have the 2 d physical interpretationof UV fixed points in the parameter space. A condition we need to incorporate inthis setup is the indipendence of the ground state metric from an overall rotationsof w i . Indeed, an overall phase can always be absorbed in the fermionic measure ofthe superspace. We can consider the dependence of Ψ a on the RG scale β ∈ R ≥ and a phase e iθ by redefining the canonical coordinates as w i → βe iθ w i . (3.99)After the identification ζ = e iθ , the Lax equations become57 ∂ ζ Ψ = (cid:0) βζC + Q − βζ − C (cid:1) Ψ ,β∂ β Ψ = (cid:0) βζC + Q + βζ − C (cid:1) Ψ . (3.100)The compatibility of these equations automatically implies the indipendence of the tt ∗ solution from the angle θ . Indeed, Q, C, C are consistently indipendent from ζ .The identification of the overall phase with the spectral parameter allows to extendits domain to whole complex plane. Thus, the equation above has two singularpoints in ζ = 0 , ∞ and Ψ( ζ, w i ) undergoes monodromy as w i → e πi w i . In this casethere are two Stokes sector which can be chosen to be the upper and lower half ζ -plane. The solutions on these two half planes Ψ + , Ψ − overlap on the real line andare related by Ψ − ( y ) = Ψ + ( y ) S Ψ − ( − y ) = Ψ + ( − y ) S t (3.101)where y > 0. The stokes matrix S is given by the phase-ordered product of theStokes jumps M ij = δ ij − A ij (3.102)which are generated when ζ crosses the BPS central charge Z ij = 2( w i − w j ) ofa primitive soliton connecting the j -th to the i -th vacuum. The matrix A ij has aunique non vanishing coefficient in the ij entry. In the basis of Lefschetz thimblesthis is equal to the BPS multiplicity A ij = µ ij of the ij sector. When ζ spans thewhole upper half plane in the anticlockwise sense we find the definition of the Stokesmatrix S = (cid:89) < arg Z ij <π M ij , (3.103)where we are assuming that there are no solitons with central charges aligned withthe Stokes axes. This can always be achieved with a rotation of the axes.One can solve the Riemann-Hilbert problem 3.101 imposing the correct boundarycondition at the infinity of the ζ -plane 58im ζ →∞ Ψ( x ) exp (cid:2) β (cid:0) xC + x − C † (cid:1)(cid:3) = 1 . (3.104)Using this boundary condition and the known identity1 x − y ∓ i(cid:15) = P x − y ± iπδ ( x − y ) (3.105)one rewrites the Riemann-Hilbert problem as the integral equationΦ( x ) ij = δ ij + 12 πi (cid:88) k (cid:90) ∞ dyy − ζ + i(cid:15) Φ( y ) ik A kj e − β ( yδ kj + y − ¯ δ kj + 12 πi (cid:88) k (cid:90) −∞ dyy − ζ + i(cid:15) Φ( y ) ik A t kj e − β ( yδ kj + y − ¯ δ kj , (3.106)where δ kj = w k − w j . The Riemann-Hilbert problem has a unique solution given bythe piecewise constant function Ψ = (Ψ + , Ψ), whereΨ + ( x ) = Φ( x ) exp (cid:2) − β (cid:0) ζC + ζ − C † (cid:1)(cid:3) . (3.107)The solution Ψ = (Ψ + , Ψ) satisfy [30, 49]Ψ( x )Ψ t ( − x ) = 1Ψ(1 / ¯ x ) = g − Ψ( x ) , (3.108)where the second relation means that in the canonical basis the complex conjugationacts on the vacuum wave functions as the ground state metric. Indeed, in this basiswe have η = 1 and so the ground state metric and the real structure coincide. Fromthe above relations and the boundary condition 3.104 one can extract the groundstate metric g i ¯ j = lim ζ → Φ( x ) i ¯ j . (3.109)The object which specifies the tt ∗ solution is the monodromy of the D-brane statesaround the singular point ζ = 0, also known as the quantum monodromy [30]59 = S ( S t ) − . (3.110)According to the theory of isomonodromic deformations, the monodromy group ofthe Lax connection is invariant up to conjugacy under deformations of the parame-ters and can be computed in the limit that we prefer. The eigenvalues of H are thephases e πiq Rj which encode the Ramond charges of the chiral primary fields at theUV conformal point. Given the dependence of the Stokes matrix on the BPS mul-tiplicities, the quantum monodromy puts in relation the spectrum of solitons andchiral primaries, providing a map between IR and UV fixed points. The quantummonodromy is an element of SL ( N, Z ) and may have non trivial Jordan blocks. Itis known that a non trivial Jordan structure is related to logarithmic violations ofscaling and reveals that the theory is asymptotocially free in the UV [30]. tt ∗ Monodromy Representation Our main problem is to compute the SL ( N, Z ) monodromy representation of π ( P )given by the tt ∗ Lax connection. We are mostly interested in the case in whichthe chiral ring of the theory is semi-simple. As we mentioned before, since tt ∗ isan isomonodromic problem, we have the freedom to deform continuously the modelin the coupling constant space. The monodromy groups we find will be related byan overall conjugation. In particular, the Jordan block structure and the eigenval-ues of the monodromy matrices are invariant under finite continuous deformations.However, the useful limits in which the computations really simplify are typicallysingular limits in the coupling constant space. The monodromy representation thatwe find in these limits may be related to the original by a singular conjugation ma-trix. The eigenvalues of the matrices will not be modified, but the Jordan blocksmay decompose in smaller ones. Indeed, having a certain spacetrum is a closedcondition in the matrix space, while having Jordan blocks of dimension > π ( P )is a complicated non-Abelian group the Jordan blocks are severely restricted by thegroup relations and so it is plausible that they can be recovered from the knowledgeof the eigenvalues. The limits that are relevant for our purposes are the so-calledasymmetric limit and the UV limit (in the 2 d language). In the first approach werescale the critical values w i → βw i (3.111)60o that the tt ∗ flat connection becomes ∇ ζ = D + βζ C, ∇ ζ = D + ζβC, (3.112)while the Berry curvature changes as F = − β (cid:2) C, C (cid:3) . (3.113)The asymmetric limit consists in taking the unphysical limit β → ζ → ∞ with β/ζ fixed. The Berry curvature vanishes in the limit, so the metric connection A is pure gauge. The tt ∗ linear problem 3.100 reduces formally to (cid:18) ∂ i + A i + βζ C i (cid:19) Ψ = ¯ ∂ i Ψ = 0 . (3.114)A basis of solutions to this equation is given by the integralsΨ ai = (cid:90) Γ a φ i ( z ) e βζ W ( z ; w ) dz ∧ ... ∧ dz n , (3.115)where the cycles Γ a are an integral basis of branes and φ i ( z ) is a basis of the chiralring. The homology classes of the branes with given ζ are locally constant in couplingconstant space, but jump at loci where (in the 2d language) there are BPS solitonswhich preserve the same two supercharges as the branes [30]. The jump in homologyat such a locus are given by the Picard-Lefshetz transformations of the forms 3.102.In the canonical basis of branes, namely the Lefshetz thimbles, the Stokes jumpscan be written in terms of the soliton multiplicities computed in the IR limit of the2 d theory. Taking into account all the jumps in homology one encounters along thepath (controlled by the 2d BPS spectrum), one gets the monodromy matrix whichis automatically integral of determinant 1. The full monodromy representation isgiven by the combinatorics of the PL transformations. This approach is convenientfrom the practical point of view and the monodromy matrices computed is thislimit are manifestly integral. On the other side, the fact that we consider a limitwhich do not correspond to any unitary quantum system tends to make the physicssomewhat obscure. For our present purposes the UV approach seems more natural.As discussed before, this just requires to send β → 0, while the spectral parameter ζ remain fixed and can be taken in the unitary locus | ζ | = 1. One can see that inthis limit the Lax connection reduces to the Berry one61 ζ + ∇ ζ β → −−→ D + D. (3.116)Hence, the Berry connection becomes flat in this limit. Since the monodromy ofthe flat connection is independent of β , the flat UV Berry connection should havethe same monodromy modulo the subtlety with the size of the Jordan blocks. Thisconcept can be clarified by comparing the quantum monodromy computed in the UVand IR limit of the theory. The holonomy on the ζ -plane of the UV Berry connection,which corresponds to the Lax connection in the same limit, is conjugated to theunitary matrix e πi Q . On the other hand, as we discussed previously, the quantummonodromy is computed in the IR limit in terms of the Stokes matrices as H = SS − t . The two matrices have the same unitary spectrum, but for asymptotically freetheories the IR monodromy has non trivial Jordan blocks. It is clear in general thatthe monodromy representation computed in the UV limit is unitary, since the Berryconnection is metric, but differently from the homological approach the integralstructure of the matrices is not manifest. In order to give an explicit description ofthe holonomy representation of the UV Berry connection we need additional detailsabout the tt ∗ geometry that we want to study. In this section we start to explore the connections between supersymmetry and thephysics of quantum Hall effect. It turns out that the structure of the lowest Landaulevel naturally admits a N = 4 supersymmetric description. One can describethe Landau levels of a single electron on a Riemann surface Σ of arbitrary genusby rephrasing the construction in section 2.3 in a more geometrical language. Weconsider a complex line bundle L → Σ with first Chern class c ( L ) = Φ / π , whereΦ > ψ : Σ → L ofthe line bundle. Every complex line bundle over a one dimensional complex manifoldis holomorphic and so one can always find a local holomorphic trivialization. Theline bundle is endowed with an hermitian metric h which allows to define an innerproduct on the Hilbert space 62 ψ | ψ (cid:105) = (cid:90) Σ dzd ¯ zh ψ ∗ ψ . (4.1)The lowest Landau level is defined as the kernel of the Hamiltonian operator˜ H = − D z D ¯ z + E (4.2)where D is the connection with respect to the U (1) structure group of the bundle.More precisely, D is the Chern connection associated to h . In an holomorphictrivialization one has D z = − h∂ z h − , D ¯ z = ∂ ¯ z . (4.3)The magnetic field is encoded in the curvature of the connection, which is a closedand real (1 , 1) form F h = ¯ ∂∂ log h, (4.4)where ∂, ¯ ∂ are the Dolbeault operators of the complex manifold. Choosing naturalunits, in the case of the complex plane one has h = e − B | z | / and F h = B dzd ¯ z . TheRiemann surface Σ endowed with the curvature of the line bundle, which is positivefor B > 0, is naturally a Kahler manifold. The fact that the configuration space of anelectron in a constant magnetic field is a Kahler manifold, in particular a Stein spacewith a globally defined Kahler potential K = log h , provides a first important hintthat quantum Hall effect may be related to N = 4 supersymmetry. Isomorphismclasses of line bundles over a compact Riemann surface are in correspondence withthe divisors of the surface modulo linear equivalence. These form a free abelian groupon the points of the surface. We write a divisor D as a finite linear combination ofpoints D = (cid:88) i n i p i , (4.5)where p i ∈ Σ and n i are integers. We demand the divisor to be effective, namelywith n i > 0. Since the electrons are particles with spin, in this discussion we havealso to endow the line bundle with a spin structure. This is associated with a divisor S on the Riemann surface such that 2 S is in the class of canonical divisors. Thedivisor identifying the twisted line bundle is the sum D + S and is unique up to63inear equivalence. The vacuum wave functions satisfy D ¯ z φ = 0 and so we have thedefinition H LLL = Γ(Σ , L ( D + S )) , (4.6)where H LLL is the lowest Landau level and Γ(Σ , L ( D + S )) is the space of holomorphicsections of the twisted line bundle L ( D + S ). The Riemann-Roch theorem for acompact genus- g Riemann surface states that the dimension (cid:96) ( D + S ) of the vectorspace Γ(Σ , L ( D + S )) is (cid:96) ( D + S ) = deg( D + S ) − g + 1 (4.7)where deg( D + S ) is the degree of the divisor. This result holds as long as D + S has degree at least 2 g − 1. For definiteness we put the system in a finite box withperiodic boundary conditions. This is equivalent to choose an elliptic curve E astarget manifold for the electron coordinate. It is convenient to pick an even spinstructure associated to a divisor S = p − q , where p , q are distinct points on theelliptic curve which satisfy 2 p = 2 q . The choice of p does not affect the discussionand we can translate it as we prefer. In the case of a genus-1 surface the aboverelation becomes (cid:96) ( D + S ) = deg( D + S ) = (cid:88) i n i . (4.8)Since S has vanishing degree, the dimension of the lowest Landau level is entirelydetermined by D . This is consistent with the fact that the spin of the electrons inFQH systems is a frozen degree of freedom and is irrelevant for the physics of theground state. From this equality we see that the divisor D parametrizes topologicallythe magnetic flux of the system. In particular, the degree of the divisor is equal to thetotal magnetic flux Φ / π which defines the degeneracy of the lowest Landau level.The divisor D + S has a defining meromorphic section ψ with zeros of order n i at p i ,a simple zero at p and a single pole at q . The map ψ → ψ/ψ provides a canonicalidentification between holomorphic sections in Γ( E, L ( D + S )) and meromorphicfunctions on the elliptic curve with poles at most given by D ∞ = D + p andvanishing in q .Taking the limit of infinite volume we recover a system of electrons moving on thecomplex plane. In this case the spin structure is associated to a divisor S = − q ,where q is a generic point on the plane. The contribution of deg S = − ( D − q ) = deg D = (cid:88) i n i . (4.9)The states in the lowest Landau level are represented by elements ψ ∈ Γ( P , L ( D − q )), which are canonically associated by the map ψ → ψ/ψ to meromorphic func-tions on P with poles at most given by D and vanishing at q .In order to get the Hilbert space of the N electrons system we simply need to takethe antisymmetric tensor product of the single particle space H Φ = N (cid:94) Γ(Σ , L ( D + S )) , (4.10)where the dimension is given by Fermi statistics (cid:18) Φ / πN (cid:19) .We can show that the problem of studying the Landau levels of a charged particle inmagnetic field is equivalent to find the vacua of a N = 4 supersymmetric Hamilto-nian. In particular, the holomorphic structure of Landau levels require to have foursupercharges. As we recalled in the previous section, a theory in supersymmetricquantum mechanics with four supercharges is specified by the choice of a Kahlerpotential K and an holomorphic superpotential W . The Kahler potential prescribesan hermitian metric on the target space. Once the topology of the Riemann surfaceis fixed, the choice of the Kahler metric does not affect the structure of the vacua,which depends only on the superpotential. Hence, we can choose K = log h asglobally defined Kahler potential on the Riemann surface. The number of vacua isgiven by the Witten index, which is equal to the number of zeros counted with mul-tiplicitly of the 1-form dW . In order to compare this description with the previousone we choose as target space the manifold K = E \ supp F , where E is an ellipticcurve and F is an effective divisor on E . As we discussed above, a divisor identifiesup to linear equivalence a line bundle with its set of holomorphic sections. However,for a linear combination of points on the Riemann surface one can associate also a N = 4 supersymmetric system. Given an effective divisor D = (cid:80) i n i p i , we assign aclosed meromorphic 1-form dW on E with zeros in p i of order n i . We choose W (cid:48) ( z )such that its polar divisor is given by F and in making a precise dictionary betweenthe two models we can set p ∈ supp F . The ground states of a N = 4 theory arein correspondence with cohomology classes of the susy charge Q = ¯ ∂ + dW ∧ in thespace of differential forms, which are labelled by holomorphic operators of the chiralring R . The meromorphic functions ψ/ψ defines canonically a basis of wave formsfor the supersymmetric vacuum space V through the map ψψ → ψψ W (cid:48) dz + Q ( .... ) . (4.11)65espite we can write ψψ W (cid:48) dz = Q ( ψψ ), the 1-forms above define non trivial repre-sentatives in the Q cohomology, since the meromorphic functions ψψ are singular atthe zeros of the superpotential and so are not elements of the chiral ring.When the electrons move on the complex plane the corresponding N = 4 Landau-Ginzburg model is defined by a one-form dW which is a rational differential on theplane with a pole of order ≥ ∞ and zero-divisor D = (cid:80) i n i z i , namely dW ( z ) = (cid:81) i ( z − z i ) n i P ( z ) , deg P ( z ) ≤ (cid:88) i n i = Φ2 π . (4.12)The prescription about the behaviour of dW at ∞ implies that the scalar poten-tial | W (cid:48) | in the supersymmetric hamiltonian is non vanishing at infinity for all thecomplete Kahler metrics on the complex plane. This condition ensures the normaliz-ability of the vacuum wave functions and the existence of an energy gap between theground state and the first excited level of the Hamiltonian. Moreover, it guaranteesthe absence of run-away vacua in the 2 d N = (2 , 2) version of the theory. In thecorrespondence with the Landau description we choose the reference point of thespin structure such that q (cid:54)∈ Supp D ∪{∞} . As in the case of periodic boundary con-ditions, the meromorphic functions ψψ W (cid:48) define a basis of chiral operators labellingthe supersymmetric vacua of the model. In conclusion, we have the isomorphismsof vector spaces U Φ : H LLL ∼ −→ V , U Φ : H Φ ∼ −→ V N = N (cid:94) V (4.13)which confirms the first prediction in the list 2.12. The correspondence between the lowest Landau level and the vacuum space of theLG model does not guarantee that the two systems have the same geometry of thevacuum bundle. One can conclude that the vacuum sector of a supersymmetric sys-tem and the lowest Landau level are two equivalent description of the same physicalsystem if the above isomorphism is also an isometry between Hilbert spaces. If thenorm of the Hilbert space is preserved by the above map, the Berry connection in-duced on the vacuum bundle and consequently the topological order of the groundstate are the same in the two systems.We focus on the case of the complex plane as target manifold and consider a systemwith a single electron. The generalization to the multi-particle case is straightfor-66ard. In the Landau description the probability of finding an electron at a position z is given by P ( z ) LLL = | f ( z ) | e − B | z | , (4.14)where f ( z ) is an holomorphic function and B > 0. On a macroscopic volume theprobability measure satisfieslog P ( z ) LLL = − B | z | + subleading as | z | → ∞ . (4.15)An exact identification between the hermitian structures of the Hilbert spaces is atoo strong requirement. What we can ask is an equivalence in measure, i.e. measure-ments of long range observables on small but macroscopic domains U ⊂ C give thesame answers on the two sides. This is enough for our purposes, since the long rangeobservables are the ones that characterize the quantum topological order. Accordingto the previous discussion, the supersymmetric representation of the Landau levelsrequires to introduce a meromorphic 1-form dW on the plane, whose primitive playsthe role of superpotential. We consider a differential form dW ( z ) with Φ / π zeroesand polar divisor of the type F = F f + 2 ∞ . This has the form dW ( z ) = µ + Φ / π (cid:88) i =1 a i z − ζ i dz, F f = (cid:88) i ζ i (4.16)where µ, a i ∈ C × and ζ i are all distinct. Since we can always redefine the overallphase of the superpotential, we assume a i ∈ R without loss of generality. In order toreproduce a macroscopically uniform magnetic field we should take the residues a i all equal and consider a uniform distribution of the flux sources ζ i in C . The typicalseparation of the punctures should be much smaller than the size of the macroscopicdomain U ⊂ C on which we want to measure observables. In the present contextbeing macroscopic means12 π (magnetic flux through U) = (cid:90) U B π = { ζ i ∈ U } (cid:29) , (4.17)namely the domain U contains a large number of fluxes. The supersymmetric wavefunctions have the form ψ ( z ) SUSY = Φ( z ) dz + ˜Φ( z ) dz (4.18)67nd the corresponding probability distribution is P ( z ) SUSY = | Φ( z ) | + | ˜Φ( z ) | . (4.19)We can choose a real basis of wave functions such that ˜Φ( z ) = Φ( z ). Then, theSchroedinger equation for the zero energy levels is [32] (cid:32) − ∂ ∂z∂ ¯ z + (cid:12)(cid:12)(cid:12)(cid:12) dWdz (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) Φ W (cid:48) = 0 . (4.20)An asymptotic behaviour of the solution in the macroscopic limit which is consistentwith the Schroedinger equation isΦ( z ) = e ± W ( z )+ subleading as | z |→∞ , (4.21)where the subleading terms are smooth and bounded functions in the domain U which ensure that the wave function is single-valued and normalizable on the wholecomplex plane. We notice that the function2Re W = µz + µz + (cid:88) i a i log | z − ζ i | (4.22)is the electrostatic potential of a system of point-like charges of size a i at positions ζ i superimposed to a constant background electric field µ . When averaged on amacroscopic region U , it looks like the potential for a continuous charge distributionwith density σ ( z ) such that (cid:90) U d z σ ( z ) = (cid:88) ζ i ∈ U a i (4.23)for any U ⊂ C . The conclusion is that for any macroscopic domain U ⊂ C we have magnetic flux through U = i (cid:90) U ¯ ∂∂ log P ( z ) SUSY ≈ ± i (cid:90) U ¯ ∂∂ (2Re W ) = ∓ π (cid:88) ζ i ∈ U a i , (4.24) Recall that the volume form of R is dx ∧ dy = i dz ∧ d ¯ z . d electrostatics. It isclear that the background electric potential does not contribute to the magneticflux of the system. We see that log P ( z ) SUSY matches the behaviour of log P ( z ) LLL in 4.15 when averaged on any macroscopic domain U ⊂ C iff we set respectively a i = − a i = +1 for all ζ i . The two choices are related by a change of orientation.We fix the conventions so that the external magnetic field is modelled in the susyside by 4.16 with a i = − i . It is very natural to introduce defects on the supersymmetric side. One just needto flip the sign of the residues a i for a bunch of punctures. Now there is a smallmismatch between the number of vacua and the effective magnetic field measuredby the fall-off of the wave function at infinity: we have two extra vacua per defect.The extra vacua are localized near the position of the corresponding defect in theplane and may be interpreted as internal states of the defect. We identify thesedefects with the quasi-holes of FQHE.We have shown in the previous sections that the low energy physics of a system ofcharged electrons in a uniform magnetic field is described at large B by a quan-tum system with Hilbert space H Φ . Moreover, this system admits two equivalentdescriptions which are related by the isomorphism U Φ in 4.13. In particular, theoriginal FQHE Hamiltonian H FQHE = U − ˆ HU Φ is mapped to some Hamiltonian ˆ H which can be seen as a deformation of a 4-SQM model. The free part of the Hamil-tonian si supersymmetric and the corresponding superpotential is given by the sumof N copies of the single particle superpotential describing the interaction betweena single electron and the punctures. The interacting part of the Hamiltonian canbe splitted in two groups: the interactions which preserve supersymmetry and theones which are susy-breaking. Including the susy-preserving interaction we get adifferential d W of the form d W = N (cid:88) i (cid:32) dW ( z i ) + h (cid:88) a =1 dz i z − x a (cid:33) + N (cid:88) i =1 U i ( z , ..., z N ) dz i . (4.25)The term dW ( z ) models the interaction between the electrons and the macroscopicmagnetic field. The Vafa proposal for this term is dW Vafa ( z ) = − Φ / π − h (cid:88) k =1 dzz − ζ k , (4.26)69here ζ k form a regular lattice. Working on the plane is convenient to add a constantto dW Vafa ( z ) in order to regularize the double pole at infinity. This may be seen asintegration constant for the electrostatic Poisson equation satisfied by 2Re W anddoes not affect the magnetic flux of the system. Hence, one obtains dW ( z ) = dW Vafa ( z ) + µdz (4.27)where µ (cid:54) = 0. The meromorphic 1-form U i dz i describes the Coulomb interactionbetween an electron and the other ones. As a function of the position z i of the i-electron at fixed z j (cid:54) = i , this form can have poles only when z i = z j . Generically U i dz i has only simple poles and the residues must be entire bounded functions on theplane, namely they are constants. Since W must be symmetric under permutationof the electron coordinates, the most general interacting part of the superpotentialreads d W int = 2 β (cid:88) ≤ i 1, at x a with residue +1 and at z j , j (cid:54) = i withresidue 2 β . Since the sum of the residues of an elliptic function must vanish, wehave the condition0 = − (cid:18) Φ2 π − h (cid:19) + h + 2 β ( N − ≈ (2 βν − 1) Φ2 π , (4.29)where in the last equality we have used N (cid:29) N = ν Φ / π .Hence, we obtain the quantization condition2 β = 1 /ν ∈ Q > , (4.30)which is the value given in [17]. So, the Vafa model on E N is described by thedifferential 70 W = N (cid:88) i =1 (cid:32)(cid:88) a U ( z i , x a ) − (cid:88) k U ( z i , ζ k ) + 1 ν (cid:88) j (cid:54) = i U ( z i , z j ) (cid:33) , (4.31)where U ( z, w ) = ℘ (cid:48) ( w/ dz℘ ( z − w/ − ℘ ( w/ . (4.32) The isomorphism U Φ maps the original Landau Hamiltonian to the new Hamiltonianˆ H = H W + H su . br , (4.33)which provides an equivalent description of FQHE systems. The piece H W is the su-persymmetric Hamiltonian associated to the superpotential W discussed previously,while H su . br is the susy-breaking part. For large magnetic fields H W is of order O ( B ), while H su . br is of order O (1) and represents a small perturbation. However,this does not mean and we can neglect it when we study the topological order ofFQHE. Indeed, the supersymmetric part alone cannot be in the same universalityclass of FQHE states. As we are going to discuss in the next section, the vacuum sec-tor with Fermi statistics V Fer of the Vafa Hamiltonian has dimension d = (cid:18) Φ / πN (cid:19) .The vacuum bundle of the theory is an holomorphic vector bundle of rank d whichis endowed with the tt ∗ flat connection ∇ which extends holomorphically the Berryconnection D . In particular, the topological order of H W is captured by the mon-odromy representations of ∇ . According to the Laughlin argument in the originaldescription of FQHE, the degeneracy of the ground state is lifted by the Coulombinteractions which select a single vacuum. In a similar way, the susy-breaking partof the Hamiltonian should select a unique ground state | vac (cid:105) in the Hilbert space H Φ which encodes the topological quantum order of FQHE. The sub-bundle over thecoupling constant space with fiber spanned by | vac (cid:105) is endowed with two canonicalsub-bundle connections ∇ vac , D vac which are induced from ∇ and D respectively.In general the sub-bundle curvature is different from the curvatural of the originalone and the monodromy of ∇ vac is a priori neither well defined nor simply relatedto the one of ∇ . Hence, a priori there is no relation between the topological ordercaptured by H W and the FQHE one. In order to have such relation the followingtwo conditions must be satisfied: 71 The monodromy representation of ∇ must be reducible with an invariant sub-bundle of rank 1. The fiber of this eigenbundle defines a unique preferredvacuum for the N = 4 Hamiltonian. • The physical vacuum | vac (cid:105) is mapped by the isomorphism U Φ to the preferredvacuum of the susy Hamiltonian.The first question is purely related to the supersymmetric model. It is suggested in[17] that such a preferred vacuum exists and should correspond to the identity oper-ator. While this sounds as a natural guess, it is in general not true in tt ∗ geometrythat the identity operator spans an invariant subspace of the flat connection. Thisis an extremely non trivial fact that we have to check by studying the monodromyrepresentation of ∇ . The validity of the second point is based on the fact that thepreferred vacuum, if it exists, should be the most symmetric one under permutationsof the quasi-holes. Then one may argue euristically that [17], indipendently fromthe details of the interactions between electrons , as long as the susy-breaking part H su . br respects the permutation symmetry of electrons and quasi-holes, the uniquevacuum | vac (cid:105) will also be the maximally symmetric one. This state should corre-spond to the preferred vacuum of the susy Hamiltonian.The conclusion is that, under our mild assumptions, the quantum order of the FQHEis captured by the 4-susy SQM model proposed in [17]. In the present discussionwe are actually arguing more than this. Since our considerations do not depend onthe details of the interactions between the electrons, we claim that the supersym-metric model represents the correct universality class of any multi-particle systemin a strong uniform magnetic field. tt ∗ Functoriality Supersymmetric quantum mechanics is functorial with respect to branched cover-ings [28]. Let us consider a Landau-Ginzburg theory with target space X andsuperpotential W ( x ; λ ) depending on a set of couplings λ ∈ P . We denote with f : X → X an holomorphic but not globally invertible map between a covering A fundamental requirement the interactions between electrons should satisfy is to vanish inthe limit | z i − z j | → ∞ . This ensures that the dynamics of the system at large | z | is governed bythe magnetic field B , so that on any macroscopic domain the probability of finding the electrons atpositions z i satisfies log P ( z i ) ∼ − B (cid:80) i | z i | . The interaction term may diverge when two particlesbecome closer. This is not an issue, since the hyperplanes z i = z j are not part of the configurationspace of N identical particles and the wave functions of the Hilbert space are vanishing there. X and X . We assume that the map is indipendent from the point of thecoupling constant space. One can use f to pull-back the LG model on X . Denotingwith x i , y i , i = 1 , ..., n a set of coordinates on X , X respectively, the superpoten-tial of the covering model can be obtained by making the substitution x i = f i ( y j ),namely f ∗ W ( x i ) = W ( f i ( y j )) = W f ( y j ) . (5.1)For the susy charges we have Q = f ∗ Q = ∂ + dW f , Q = f ∗ Q = ¯ ∂ + dW f , (5.2)while the Lefschetz operators can be defined by pulling-back the Kahler form on thecover space. The algebraic relations satisfied by the generators of the supersymmetryalgebra are preserved by the pull-back operation. The pull-back of the vacuum waveforms of the original theory are not in general ground states for the covering model.Given a basis of vacuum wave functions Ψ k , k = 1 , ..., N , the pulled-back forms f ∗ Ψ k are cohomologous to the true vacua on X . Indeed, the conditions solved bythe vacua which are compatible with the pull-back are¯ ∂ f ∗ W f ∗ Ψ k = f ∗ ( ¯ ∂ W Ψ k ) = 0 ∂ f ∗ W f ∗ Ψ k = f ∗ ( ∂ W Ψ k ) = 0 f ∗ ω ∧ f ∗ ψ k = f ∗ ( ω ∧ Ψ k ) = 0 , (5.3)where f ∗ ω is the pulled-back Kahler form. The vacuum equations involving thecharges Q † , Q † depend D-term and in general are not preserved by the pull-back.The covering theory has generically more vacua than the original LG model. Thenumber of vacua in the new theory N is related to N by N = deg f · N + J f , (5.4)where deg f is the degree of the cover and J f is the Jacobian of f . From the pulled-back expression of the vacuum wave functions f ∗ Ψ k = f ∗ (cid:0) φ k ( z i ) dz ∧ .... ∧ dz n + ¯ ∂ W ( .... ) (cid:1) = φ k ( f i ( y j )) J f dy ∧ .... ∧ dy n + ¯ ∂ W f ( .... ) (5.5)73ne can read how the chiral ring roperators of the vacua transform by pull-back f (cid:93) ( φ k ) = f ∗ ( φ k ) J f . (5.6)The linear map f (cid:93) : R → R is an isometry for the topological metric (cid:104) f (cid:93) ( φ k ) , f (cid:93) ( φ j ) (cid:105) X = (cid:104) φ k , φ j (cid:105) X . (5.7)and is compatible with the R -module structure f (cid:93) ( φ k · φ j ) = f ∗ ( φ k ) · f (cid:93) ( φ j ) ∈ R . (5.8)The chiral ring R decomposes as direct sum R = f (cid:93) ( R ) ⊕ f (cid:93) ( R ) ⊥ , (5.9)where ( · ) ⊥ denotes the orthogonal complement with respect to the tt ∗ metric. The tt ∗ functoriality is the statement that f (cid:93) : R → R is an isometry also for the tt ∗ metric. In order to show this fact one has to check that the two tt ∗ metrics solve thesame equations and satisfy the same boundary conditions. Since the classes in R ofthe operators ∂ λ W belong to the subspace f (cid:93) ( R ), the chiral ring coefficients C λ arefunctorial by 5.8. Since also the topological metric η ij is functorial by the 5.7, weconclude that the tt ∗ equations are preserved by f (cid:93) . The boundary conditions whichselect the correct solution to the tt ∗ equations are encoded in the 2d BPS solitonmultiplicities. The BPS solitons are the connected preimages of straight lines in the W -plane ending at critical points [30, 37]. Since the map W f : X → C : factorizesthrough W : X → C , so do the counterimages of straight lines, and therefore thecounting of solitons agrees in the two theories. tt ∗ functoriality preserves also the integral structure of D-branes. If B ⊂ X is aD-brane for the LG model on X , then f ( B ) is a D-brane for the model on X .In particular, B is a special Lagrangian submanifold of X and this property isinvariant under pull-back and push-forward operations by smooth functions. Thefunctorial property of SQM regards also the integral pairing between branes andvacua (cid:104) f ( B ) | Ψ (cid:105) X = (cid:104) B | f ∗ Ψ (cid:105) X . (5.10)74ne gives in general the following definition: a tt ∗ -duality between two 4-susy the-ories is a Frobenius algebra isomorphism between their chiral rings R → R whichis an isometry for the tt ∗ metric and so for the brane amplitudes. tt ∗ functorial-ity produces several interesting tt ∗ -dual pairs and for an appropriate choice of therespective D-terms implies the equaivalence of the full quantum theories. The functorial property of SQM is particularly important when we study Landau-Ginzburg models in which the target manifold X is not simply connected and thesuperpotential differential dW is a closed meromorphic 1-form. Despite it is notpossible to define a primitive W on X , the model is still well defined, since theHamiltonian and the susy charges depend only on the derivatives of the superpoten-tial. However, in these systems one can consider deformations of the theory whichare not described by operators in the chiral ring [31]. For instance, the chiral op-erator associated to the RG flow deformation is precisely the superpotential, whichis not an holomorphic function on X . In order to write and solve the tt ∗ equationswith respect to these variations we need to pull-back the model on the universalcover K of the target manifold. This can be defined in an abstract way as the spaceof curves K = { p : [0 , −→ X, p (0) = p ∗ ∈ X } / ∼ , ∼ : p ∼ q = (cid:40) p (1) = q (1) ,p · q − = 0 in the fundamental group π ( X, Z ) . (5.11)We can pull-back dW on this space and give a formal definition of superpotential: W ( p ) = (cid:90) p dW. (5.12)The fundamental group π ( X ) plays the role of Galois group of the cover and is asymmetry group of the model on K . The action of a loop generator (cid:96) of π ( X ) on W is 75 ∗ W ( p ) = (cid:90) (cid:96) · p dW = W ( p ) + (cid:90) (cid:96) dW. (5.13)Compatibily with the definition of symmetry, the superpotential is left invariantup to a constant factor. The Galois group of the universal cover is genericallynon abelian and may not have unitary representations. In the context of quantummechanics, since the symmetry transformations must be unitary, one considers tipi-cally the abelian universal cover of the target space. We note that the action ofthe fundamental group is abelianized at the level of superpotential and the modelnaturally descends on the abelian universal cover. This space is the minimal simplyconnected cover of the target space on which W is single-valued. The Galois groupof the abelian cover is obtained by dividing the fundamental group π ( X ) by thecommutator subgroup [ π ( X ) , π ( X )]. This subgroup is normal in π ( X ) and so thequotient π ( X ) Ab = π ( X ) / [ π ( X ) , π ( X )] is still a Galois group. More precisely, itis the first homology group H ( X ; Z ) of the target space. By replacing the funda-mental group with the homology group in 5.11, one gets the definition of abelianuniversal cover A of X . The first homology of the target space is an abelian groupwith a freely generated part and a torsion part. It is clear by the above formulathat the torsion subgroup has a trivial action on the superpotential. Hence, one canconsider only the torsion-free part H ( X ; Z ) / tor (cid:39) Z b , where b is the first Bettinumber.The vacuum space of the LG model on A decomposes in a direct sum of unitaryirreducible representations of the homology group: V A = (cid:77) χ ∈ Hom( H ( X ; Z ) ,U (1)) V χ , dim V χ = d, (5.14)where d = dim V X is the dimension of the vacuum space of the theory on X . Identi-fying H ( X ; Z ) with Z b , the characters labelling the unitary representations V χ canbe written as χ (cid:126)θ : (cid:126)n → e i(cid:126)n · (cid:126)θ (5.15)and we can call (cid:126)θ -vacua the states in the eigenspace V χ = V (cid:126)θ . Since it is metric,the orthogonal decomposition of the vacuum space in θ -sectors is preserved by theparallel transport with the Berry connection. However, it is not generically leftinvariant by the tt ∗ Lax connection. The subrepresentation of the tt ∗ monodromyare associated with subgroups of the Galois group. Let H ∈ π ( X ) Ab a subgroup andlet A H = A /H . We have an Abelian cover A H → X with Galois group π ( X ) Ab /H .76his cover is not simply connected and the fundamental group π ( A H ) is the kernelof the surjective homomorphism β : π ( X ) Ab → π ( X ) Ab /H, (5.16)which is precisely the normal subgroup ker β = H . One can consistently formulatethe 4-SQM model on the target space A H . The vacuum space of the LG theory on A H is identified with the H -invariant subspace of the theory on the universal cover V H = (cid:77) χ : χ | H =trivial V χ . (5.17)The covering map A → A /H allows to pull-back on A also the branes and the tt ∗ brane amplitudes of the A /H -theory. Hence, we know by tt ∗ functoriality that V H must be preserved by the tt ∗ monodromy. We conclude that for each subgroup H of π ( X ) Ab we have a monodromy subrepresentation Mon H of the Lax connection.Moreover, to a sequence of subgroups ... ⊂ H k ⊂ H k − ⊂ .... ⊂ H ⊂ H = π ( X ) Ab , (5.18)there corresponds an inverse sequence of tt ∗ monodromy representationsMon H ⊂ Mon H ⊂ .... ⊂ Mon H k − ⊂ Mon H k ⊂ .... (5.19)where Mon H is the monodromy representation for the original model defined on X .We can choose a subgroup H of finite index in π ( X ) Ab in such a way that π ( X ) Ab /H is a finite Abelian torsion group. In this case the theory on A H has finite Wittenindex d H = (cid:2) π ( X ) Ab : H (cid:3) · d. (5.20)A periodic character χ of π ( X ) Ab is identified by theta angles (cid:126)θ ∈ (2 π Q ) b . (5.21)Denoting with J χ the finite cyclic group generated by χ and H χ = ker χ ∈ π ( X ) Ab the corresponding finite-index normal subgroup, we have77 ( X ) Ab /H χ (cid:39) J χ . (5.22)In this case we may reduce from the infinite universal cover to a finite cover withGalois group J χ . From the physical viewpoint, torsion characters χ have the specialproperty that they allow a consisten truncation of the chiral ring R A to a finite-dimensional ring R χ . In this way the (cid:126)θ -vacua become normalizable, which is abasic requirement in quantum mechanics, while they are never normalizable fornon-torsion χ . tt ∗ Equations with θ -Vacua On the universal cover we have more vacua than on the target space, but also a largersymmetry group to classify them. We denote with z i , i = 1 , ..., d the zeroes of dW on X , which we assume to be non degenerate. Choosing a representative p i of z i on theabelian cover, all the critica points of W can be obtained by composing p i with thegenerators (cid:96) , ..., (cid:96) b of the torsion-free part of the homology group H ( X, Z ) / torsion.Hence, the points z i label equivalence classes of vacua which are isomorphic to theGalois group of the cover.A basis for the chiral ring of the theory on A can be constructed as follows. Since X is a Stein space we can find a basis of holomorphic one forms ρ k , k = 1 , ..., b ∈ Ω( X )dual to (cid:96) k whose classes generates H ( X, Z ) / tor. Since the homology has an integerstructure, we can choose the one forms ρ k such that (cid:90) (cid:96) k ρ k = c k ∈ Z . (5.23)Without loss of generality we can normalize the constants c k to 1. On A we canfind holomorphic functions h k such that dh k = ρ k . Let { φ a } ∈ R X be holomorphicfunctions on X forming a basis for the chiral ring of the model, with φ = 1 X andrelations φ a φ b = C cab φ c . Then, let (cid:36) : A → X be the projection map from theabelian universal cover to the target space. The holomorphic functions on A Φ a ( (cid:126)θ ) = (cid:36) ∗ φ a · e i(cid:126)θ · (cid:126)h , (cid:126)θ ∈ [0 , π ) b , (5.24)form a basis for R A which is diagonal in the characters of the Galois group. Indeed,the action of a generator of the homology (cid:96) k on the above state is (cid:96) k ( (cid:36) ∗ φ a · e i(cid:126)θ · (cid:126)h ) = (cid:36) ∗ φ a · e i(cid:126)θ · ( (cid:126)h + (cid:82) (cid:96)k d(cid:126)h ) = e iθ k (cid:36) ∗ φ a · e i(cid:126)θ · (cid:126)h (5.25)78he product table of R A isΦ a ( (cid:126)θ ) · Φ a ( (cid:126)ϕ ) = C cab Φ c ( (cid:126)θ + (cid:126)ϕ ) . (5.26)This equation implies that the Ramond charges of the chiral operators are piece-wiselinear functions of the angles. The same applies to the UV Berry connection A ( (cid:126)θ ) UV ,which is function of the U (1) R charges. The discontinuous jumps of A ( (cid:126)θ ) UV corre-spond to gauge transformations. On the contrary, the characters of the monodromyrepresentation are continuous. For generic (cid:126)θ the eigenvalues of the monodromy ma-trices are distinct, and hence no Jordan blocks are present; at characters where wehave jumps typically non-trivial Jordan blocks appear.Let H a subgroup of π ( X ) Ab and J the subgroup of characters which are trivial on H . Then, the chiral ring R H of the model on A H is spanned by the chiral operators { Φ a ( (cid:126)θ ) } (cid:126)θ ∈ J . (5.27)Using the fact that (cid:96) k are symmetries of the model, one finds that the ground statemetric diagonalizes with respect to the angles: (cid:104) Φ j ( (cid:126)θ ) | Φ k ( (cid:126)θ (cid:48) ) (cid:105) = δ ( (cid:126)θ − (cid:126)θ (cid:48) ) g k, ¯ j ( (cid:126)θ ) , g k, ¯ j ( (cid:126)θ ) = (cid:88) (cid:126)r ∈ Z b e i(cid:126)θ · (cid:126)r g k, ¯ j . (5.28)In presence of theta sectors, the complex conjugate operator g ∗ ( (cid:126)θ ) must be intendedas the Fourier series of the complex conjugated coefficients, namely g ∗ ( (cid:126)θ ) = [ g ( − (cid:126)θ )] ∗ . (5.29)Hence, using the fact that g ( (cid:126)θ ) must be hermitian, the tt ∗ reality constraint in thecanonical basis becomes g ( − (cid:126)θ ) t = g ( (cid:126)θ ) − . (5.30)We can use the action of the homology generators 5.13 to compute the critical valuesof the superpotential. These reads W (( (cid:96) n .... (cid:96) n b b ) · p i ) = ( (cid:96) n .... (cid:96) n b b ) ∗ W ( p i ) = W ( p i ) + b (cid:88) i =1 n i ω i , (5.31)79here ω i = (cid:82) (cid:96) i dW are constants in the chiral fields. We choose ω i as the first b local coordinates on the coupling constant space P and denote with t a ∈ P the remaining couplings such that ∂ t a W are well defined holomorphic functionsrepresenting elements of R X . The tt ∗ metric can be thought as function of thevariables ( ω i , θ i ) ∈ ( C × S ) b (5.32)at fixed t a . One can consistently define the action of the chiral operator C ω i asso-ciated to the coupling ω i on the theta vacua. This can be see as a U ( d ) covariantderivative in the θ i -direction C ω i = D ϑ i = ∂∂θ i + M ω i , C ω i = − D ϑ i = ∂∂θ i − g∂ θ i g − − gM † ω i g − , (5.33)where M ω i is a d × d matrix. At fixed t a , the component of the tt ∗ flat connectiontake the form D ω i + 1 ζ D ϑ i = D ( ζ )1 ,i , D ω i − ζD ϑ i = D ( ζ )2 ,i (5.34)We can see the ϑ i as complex coordinates with real part θ i and introduce the newcomplex coordinates ( η ζi , ξ ζi ) , i = 1 , ..., b η ζi = ω i − ζϑ i , ξ ζi = ω i + 1 ζ ϑ i , (5.35)which defines a P family of complex structures parametrized by the twistor variable ζ and a flat hyperKhaler geometry with holomorphic symplectic structures dξ ζi ∧ dη ζi .The tt ∗ Lax connection annihilates with spectral parameter ζ all the holomorphiccoordinates ( η ζi , ξ ζi ) D ( ζ ) α,i η ζj = D ( ζ ) α,i ξ ζj = 0 , α = 1 , ζ , it is the (0 , 1) part of a connection A on the hyperkahler space ( R × T ) b . The Lax equations D ( ζ ) α,i Ψ( ζ ) = 0 , α = 1 , ζ ) are holomorphic in complex structure ζ andindependent of Im ϑ i [31]. The tt ∗ equations then say that the curvature of theconnection D ( ζ ) on the flat hyperKahler manifold is of type (1 , 1) in all complexstructures, i.e. Ψ( ζ ) is a section of a hyperholomorphic vector bundle [31]. The hy-perholomorphic condition, supplemented by the condition on translation invariancein Im( ϑ i ), is equivalent to the higher dimensional generalization of the Bogomolnjimonopole equations on ( R × S ) b .The tt ∗ geometry decomposes into an Abelian U (1) monopole and a non-Abelian SU ( d ) monopole. Restricted to the Abelian part, the tt ∗ equations become linear.Writing L ( (cid:126)θ ) = − log(det g ( (cid:126)θ )), they read (cid:18) ∂ ∂ω i ∂ω j + ∂ ∂θ i ∂θ j (cid:19) L ( (cid:126)θ ) = 0 ∂ ∂t a ∂ω j L ( (cid:126)θ ) = ∂ ∂t a ∂t b L ( (cid:126)θ ) = 0 . (5.38)The Abelian part of the Berry connection is A Ab = ∂L ( (cid:126)θ ) = ∂ ω i L ( (cid:126)θ ) dω i + ∂ t a L ( (cid:126)θ ) dt a . (5.39)The tt ∗ relation [ A ω i , C t a ] = [ A t a , C ω i ], together with 5.33, implies ∂ θ i A t a = [ A t a , M ω i ] − [ A ω i , C t a ] . (5.40)Taking the trace gives ∂ θ i A Ab t a = 0. Since A Ab is odd in (cid:126)θ by 5.30, we conclude thatthe t a -components of the U (1) connection vanish.For the sake of comparison with the literature on representation of braid groupsand the Knizhnik-Zamolodchikov equation [71, 72] we state the above result in adifferent way. We write q i = e iθ i for i = 1 , ..., b . The Frobenious algebra R A is amodule over the ring C [ q ± ] of Laurent polynomials in q , ..., q b . The isomorphismwith the space of branes R A (cid:39) B A ( ζ ) = H ∗ ( A , A Re( ζW ) ; Z ) ⊗ Z C (5.41)allows us to restrict the scalars to Z . Thus, we have that B ( ζ ) A (cid:39) R A (cid:39) V A is afree Z [ q ± ]-module of rank d . Moreover, the tt ∗ Lax connection provides the grouphomomorphism 81 : π ( P ) → GL ( d, Z [ q ± ]) . (5.42) The concept of statistics is crucial in understanding the physics of FQHE. We wantnow to discuss this problem in the context of N = 4 SQM. We consider for simplicitlythe case in which the target space is C N . We suppose to have a superpotential W ( z , ..., z N ) which is symmetric under arbitrary permutations of the coordinates z i ↔ z j . In other words, the particles of our theory are indistinguishable. Sinceit is a symmetry of the model, the Hilbert space and in particular its subspace ofvacua V must decompose in irreducible representations of the permutation group of N object S N V = (cid:77) η ∈ irrep( S N ) V η . (5.43)The relevant representation for physical applications are the trivial representation V s , which corresponds to symmetric wave functions, and the sign representation V a , which corresponds to antisymmetric wave functions. The symmetric group actsalso on the chiral ring R , whose elements labels the zero energy solutions of theSchroedinger equation. Similarly to the Hilbert space, the chiral ring decomposeslinearly as R = (cid:77) η ∈ irrep( S N ) R η . (5.44)The space of symmetric chiral operators R s is a ring, while all the other components,as the space of antisymmetric chiral operators R a , are R s -modules. One can see thatthe map between chiral operators φ ∈ R and vacuum wave functions φ → φ dz ∧ .... ∧ dz N + Q ( .... ) (5.45)provides the isomorphism of R s -modules R η (cid:39) V a · η . (5.46)82his follows from the antisymmetric property of the fermionic operator dz ∧ .... ∧ dz N in the vacuum wave form. Counter intuition, we claim that Fermi (Bose)statistics corresponds to have symmetric (antisymmetric) wave functions. To justifyour definition let us count the number of ground states in an important special case.We consider the class of superpotentials which can be written as sum of single fieldsuperpotentials W ( z , ..., z N ) = N (cid:88) i =1 W ( z i ) , (5.47)where each superpotential W ( z i ) is defined on a target space C . We denote with R the one-particle chiral ring and with d = dim R the dimension of the single particleground state. The chiral ring of the N -particle model is given by the tensor product R = N (cid:79) R . (5.48)The symmetric and antisymmetric subspaces of R are R s = N (cid:75) R , R a = N (cid:94) R , (5.49)which have dimensions respectivelydim R s = (cid:18) N + d − N (cid:19) , dim R a = (cid:18) dN (cid:19) , (5.50)which correspond to Bose and Fermi statistics. The above relations remain true alsoif we add to the superpotential an arbitrary supersymmetric interactions which donot change the behaviour at infinity in field space, since the dimension of the chiralring is the Witten index I W = d . Using the isomorphism R (cid:39) V we have V B = N (cid:75) V , V F = N (cid:94) V (5.51)and the tt ∗ metric, connection and brane amplitudes are induced by the singleparticle ones.Coming back to the general case, one can see that any antisymmetric operator φ ∈ R a can be written as 83 = ˆ φ (cid:89) i We want to study the fermionic sector of the N particles LG theory with superpo-tential W ( z , ..., z N ) = N (cid:88) i =1 (cid:32) µz i + N (cid:88) (cid:96) =1 α (cid:96) log( z i + ζ (cid:96) ) (cid:33) (5.59)where we let α (cid:96) , µ to be generic complex couplings and take ζ (cid:96) all distinct. Thecoupling constant manifold of the theory at fixed α (cid:96) , µ is given by the space of N distinct ordered points on C C N = (cid:8) ( ζ , ..., ζ N ) ∈ C N | ζ i (cid:54) = ζ j , i (cid:54) = j (cid:9) . (5.60)If we demand the residues α (cid:96) to be equal, the manifold of couplings is naturallyprojected on Y N = C N /S N , namely the space of N identical particles on C . Inparituclar, it is clear from the discussion in section 4 that if we set the couplings α (cid:96) = − ν = 1 phase of quantum Hall effect.Writing the superpotential as W ( z , ..., z N ) = N (cid:88) i =1 µz i + N (cid:88) (cid:96) =1 α (cid:96) log (cid:32) N (cid:89) i =1 ( z i + ζ (cid:96) ) (cid:33) (5.61)we note that the argument of the logarithm can be expanded as N (cid:89) i =1 ( z i + ζ (cid:96) ) = N (cid:88) k =0 e k ζ N − k(cid:96) = P ( ζ (cid:96) ) . (5.62)85he polynomials P ( ζ (cid:96) ) are linear in the dynamical fields and depend holomorphicallyon the couplings. Hence, one can do a linear redefinition of variables e (cid:96) → P ( ζ (cid:96) ) = P (cid:96) . (5.63)Rewriting the interaction µ (cid:80) i z i as linear combination of P (cid:96) , the superpotential asfunction of the new variables reads W ( P ( ζ ) , ..., P ( ζ N )) = N (cid:88) (cid:96) =1 ( µ (cid:96) P (cid:96) + α (cid:96) log P (cid:96) ) . (5.64)We note that in the dynamical fields u (cid:96) = P (cid:96) the superpotential of the theory canbe written as sum of N copies of the single field superpotential W ( u ) = µu + α log u. (5.65)From this example we see how much powerful tt ∗ functoriality can be. In the startingtheory 5.59 we have a large number of punctures, equal to the dimension of thelowest Landau level, and an equal number of electrons which are correlated by theFermi statistics. On the other hand, in the tt ∗ dual we have just a unit of magneticflux concentrated in a single puncture and the particles are not correlated by thestatistics. The tt ∗ functoriality ensures that we can study the Berry connection andits holonomy representations in the tt ∗ dual of the model. Since the Hilbert spaceof the dual model is simply the tensor producs of single particle Hilbert spaces, itis enough to study the tt ∗ geometry of the LG theory with superpotential 5.65.The equation for the vacua ∂ P k W = 0 is given by a set of N indipendent equations µ (cid:96) + α (cid:96) P (cid:96) = 0 , (5.66)which has unique solution P (cid:96) = − α (cid:96) µ (cid:96) . (5.67)As expected, the dimension of the Fermi chiral ring is dim R F = 1.We want to study the tt ∗ geometry of the vacua for this theory. The essentialcouplings are the positions of the punctures ζ (cid:96) which are coordinates on the space C N . Since we have a unique vacuum, the Berry connection must be Abelian. Thechiral operators which describe the deformation of the theory with respect to ζ (cid:96) are86 ζ (cid:96) W = N (cid:88) i =1 α (cid:96) z i + ζ (cid:96) . (5.68)The chiral ring coefficients C (cid:96) which represent the action of these operators on thevacua are 1 × C (cid:96) , C m ] are vanishing andthe tt ∗ equations read ¯ ∂ (cid:96) ( g∂ m g − ) = 0 , (5.69)where the ground state metric g is just a real function of the couplings. This set ofequations is solved by g = | f ( ζ , ..., ζ N ) | , (5.70)where f ( ζ (cid:96) ) is an holomorphic function of the couplings ζ (cid:96) . We also have to imposethe reality constraint η − g ( η − g ) ∗ = 1 . (5.71)In the canonical basis the Frobenious pairing is the identity and we have gg ∗ = 1.Since g is a real function we conclude that g = 1 . (5.72)So, as expected for the ν = 1 phase, the Berry connection on C N is vanishing andits monodromy representation in the UV limit is trivial.We note that the target manifold is not simply connected and the theory admits nontrivial theta sectors. The solution that we found above corresponds to the sector ofthe Hilbert space with trivial character. To study the dependence of tt ∗ geometryon the theta angles we have to consider deformations of the theory which are not inthe chiral ring. The corresponding couplings are the residues α (cid:96) of d W at the poles ζ (cid:96) . The associated operators ∂ α (cid:96) W = N (cid:88) i =1 log( z i + ζ (cid:96) ) (5.73)87re not univalued functions on the target space so are not elements of R . By thefunctoriality of tt ∗ geometry, the ground state metric of the full theory factorizes astensor product of metrics of the single field models with superpotential 5.65. Thissuperpotential describes in 2 d a free chiral superfield with a twisted mass. Thecorresponding tt ∗ geometry has already been studied in [31]. The model is definedon C \ { } and the Galois group of the universal cover is generated by the loop whichencircles the origin. The universal cover A is Abelian and is simply the complexplane. An explicit map between the covering space and the target manifold is givenby the exponential u = e Y . Pulling-back the model on the complex plane one hasthe superpotential W ( Y ) = e Y + αY, (5.74)where the action of the Galois group is given by the shift T : Y → Y + 2 πi . Wecan easily construct the unique theta-vacua of this theory. We denote with | (cid:105) the vacuum state corresponding to some idempotent element e of R A . Then, thevacuum space is spanned by | x (cid:105) = (cid:88) n ∈ Z e πinx T n | (cid:105) , (5.75)where x is the theta-angle normalized in the interval [0 , T with eigenvalue the character e − πix . Acting with thegenerator of the Galois group T on the classical vacuum corresponding to | (cid:105) weobtain all the vacua of the covering model. We can set to 0 the critical valuecorresponding to | (cid:105) by adding a constant to the superpotential. Then, the wholeset of critical values is simply W n = 2 πiαn. (5.76)We want to derive the tt ∗ equation in the parameter α and study the critical limitof the solution. The chiral ring operator C α acts on the theta-vacuum as differentialoperator in the angle C α | x (cid:105) = (cid:88) n ∈ Z e πinx πinT n | (cid:105) = ∂∂x | x (cid:105) . (5.77)We define the ground state metric on the vacuum bundle88 ( t, x ) = (cid:104) x | x (cid:105) = e L ( t,x ) , (5.78)where L ( t, x ) is a real function of the angles and the RG scale t = | α | . The tt ∗ metric can be expanded in Fourier series as g ( x ) = (cid:88) r ∈ Z e πirx (cid:104) r | (cid:105) . (5.79)In order to solve the tt ∗ equation we have to impose the reality constraint of themetric. In presence of theta sectors the coplex conjugation of the metric must beintended as g ∗ ( x ) = [ g ( − x )] ∗ . (5.80)Hence, using the fact that g is real, the tt ∗ reality constraint in the canonical basisbecomes g ( − x ) = g − ( x ) , (5.81)which implies L ( − x ) = − L ( x ) . (5.82)Thus, the tt ∗ equation for ground state metric reads (cid:18) ∂ α ∂ ¯ α + ∂ ∂x (cid:19) L ( t, x ) = 0 . (5.83)which is a U (1) monopole equation on R × S as expected from the general discussionin 5.2.2. The solution can be expanded in terms of Bessel functions as L ( t, x ) = ∞ (cid:88) m =1 a m sin(2 πmx ) K (4 πmt ) (5.84)where the coefficients a m are determined by the boundary conditions. In the limit t → x (cid:54) = 0 we must have the asymptotics89 ( t, x ) t → ∼ − q ( x ) − ˆ c/ 2) log t (5.85)where q ( x ) is the SCFT U (1) charge of the chiral primary e xY at the UV fixed point.Since e Y has charge 1, one has q ( x ) = x ∈ [0 , 1] and ˆ c = q max = 1. Therefore, fromthe limit K ( x ) x → ∼ − log( x/ 2) (5.86)we get the relation 12 ∞ (cid:88) m =1 a m sin(2 πmx ) = (cid:18) λ π − (cid:19) . (5.87)From this equality we learn that the coefficients a m are those of the Fourier expansionof the first periodic Bernoulli polynomial, i.e. a m = − π m . (5.88)Consistently with the discussion in 5.2.2, also for non vanishing theta angles thecomponents of the U (1) Berry connection in the ζ (cid:96) -directions are vanishing. In thecase of trivial character we recover the constant solution g = 1. We consider a LG model with N chiral fields and superpotential differential d W = 2 β (cid:88) ≤ i 0: in a classical vacuum configuration for W the z i areclose to the vacua of the single particle model and several of them may take differentvalures in the vicinity of the same one-field vacuum. Since these values differ by92rders O ( β ), in the β → N , ...., N d ) which denote how many particles we put in thevacua of the single field model z , ..., z d . A natural guess is that, in order to get thecorrect FQHE phenomenology, one should consider only the subspace V Fer ⊂ V , dim V Fer = (cid:18) dN (cid:19) (5.97)which contains the vacua that survive in the β → ∈ V ⊥ Fer escape at the infinite end of M N where two or more z i coincide.The fermionic truncation from V to V Fer is geometrically consistent if and only ifit is preserved under parallel transport by the tt ∗ flat connection, i.e. if V Fer is asubrepresentation of the monodromy representation. Since the flat tt ∗ connection isthe Gauss-Manin connection of the local system on P provided by the BPS branes(for a fixed ζ ∈ P ), this is equivalent to the condition that the model has (cid:18) dN (cid:19) preferred branes which remain regular as β → V Fer .The fermionic truncation has already been studied by Gaiotto and Witten in astrictly related context [65]. They show that preferred branes with the requiredmonodromy properties do exist. We review their argument in our notation. Weassume that the rational one-form dW has a double pole at infinity of strength µ and d simple poles in general positions. Then we can write B ( z ) = µA ( z ) + lower degree . (5.98)Hence, the Heine-Stieltjes equation becomes2 βA ( z ) P ( z ) (cid:48)(cid:48) + ( µA ( z ) + ... ) P (cid:48) ( z ) = µ ˜ f ( z ) P ( z ) , (5.99)where f ( z ) = µ ˜ f ( z ). The monodromy representation is independent of µ as long asit is non-zero. Taking β ∼ O (1) and µ finite but very large (the reasonable regimefor FQHE), the above equation up to a O (1 /µ ) correction becomes A ( z ) P (cid:48) ( z ) = ˜ f ( z ) P ( z ) (5.100)which implies that up to O (1 /µ ) corrections the zeros of P ( z ) coincide with thezeros of A ( z ), namely the positions of the punctures. At large µ these approximatealso the zeros of B ( z ), i.e. the vacua of the single field model. The fermionictruncation amounts to require that their multiplicities are at most one, namely93he polynomials P ( z ) and P (cid:48) ( z ) are coprime. In this regime, the product of N one-particle Lefshetz thimbles starting at distinct zeros of B ( z ) is approximatively abrane for the full interacting model. While the actual brane differs from the productof one-particle ones by some O (1 /µ ) correction, they are equivalent in homology andthis is sufficient to study the tt ∗ monodromy. Since it is dual to the space of branes ofthe N -particles Fermi model, the fermionic sector V Fer ∈ V defines by constructiona sub-representation of the tt ∗ monodromy. So, differently from the case of non-interacting electrons, the fermionic sector is selected by the tt ∗ solution and not bythe statistics. tt ∗ Geometry To compute the monodromy representation of the Vafa model in the UV approachwe need a more in-depth understanding of tt ∗ geometry. It turns out that for aspecial class of theories the UV Berry connection is a Kohno connection [72, 75].This appears in the theory of the braid group representation [71, 74] as a solutionof the Knizhnik-Zamolodchikov equations [73]. In this section we go through thedetails of this beautiful relation. tt ∗ Monodromy and the Universal Pure Braid Repre-sentation Following the strategy discussed in 3.2.3 we rescale the critical values w i → βw i .We consider w i ∈ ˙ P , where ˙ P is an open domain in the space of couplings such thatthe chiral ring is semi-simple. We note that if w i ∈ ˙ P also βw i ∈ ˙ P for all β > P of the semi-simple domain. Aswe approach the UV fixed point of the RG the tt ∗ equations imply that ¯ ∂Q → Q is Hermitian, we have also ∂Q → 0, so that lim β → Q is a constant matrix.Naively, to get the UV Berry connection we just replace this constant matrix in thethe basic formula 3.89. However, this is not the correct way to define the β → R is believed to be regular(even as a Frobenius algebra) in the UV limit but, since the limit ring is no longersemi-simple, its generators are related to the canonical ones by a singular change ofbasis. A trivialization which is better behaved as β → e i . Starting from 3.89 and performing the diagonalgauge transformation, we get 94 kl = h k Q kl h − l d ( w k − w l ) w k − w l − δ kl d log h l , (6.1)where h l = (cid:112) (cid:104) e l (cid:105) . By the residue formula 3.80, the norm (cid:104) e l (cid:105) should be a mero-morphic function of the critical coordinates with poles at w l = w i for l (cid:54) = i . Indeed,the superpotential becomes degenerate when two critical points coincide and, for astrictly Morse superpotential, the approach of two critical values imply the same forthe corresponding critical points. In the β → A kl becomeslocally a meromorphic one form with simple poles at w l = w j for l (cid:54) = j and invariantunder w j → w j + c and w j → λw j . In addition, its contraction with the Euler vector ξ = (cid:80) k w k ∂ w k has no poles. Hence, in the UV limit the Berry connection D = D + ¯ ∂ can be written locally as D = d + (cid:88) ≤ i We recall the definition of configuration space C d of d ordered distinct points in theplane C d = (cid:8) ( w , ..., w d ) ∈ C d | w i (cid:54) = w j for i (cid:54) = j (cid:9) . (6.6)The cohomology ring H ∗ ( C d , Z ) is generated by the (cid:18) d (cid:19) ω ij = ω ji = 12 πi d ( w i − w j ) w i − w j (6.7)which satisfy the relations [76] ω ij ∧ ω jk + ω jk ∧ ω ki + ω ki ∧ ω ij = 0 . (6.8)The fundamental group P d = π ( C d ) is called the pure braid group of d strings. Theconfiguration space of d unordered points is the quotient space Y d = C d /S d (6.9)and its fundamental group B d = π ( Y d ) is the Artin braid group of d strings. It isan extension of the symmetric group S d by the pure braid group1 → P d ι −→ B d β −→ S d → . (6.10) B d has a presentation with d − σ i and the relations σ i σ i +1 σ i = σ i +1 σ i σ i +1 = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i for | i − j | ≥ . (6.11)We define the critical value map w : ˙ P → Y d , t → { w , ..., w d } . (6.12)For semi-simple chiral rings the critical values provides a set of local coordinates forthe space of couplings, implying that the above map is an holomorphic immersion.96e say that the tt ∗ geometry is complete if, in addition, w is also a submersion.This is equivalent to say that w is a local isomorphism and so a covering map from ˙ P to Y d . The perturbations of the UV fixed point are generated by the chiral primaryoperators φ ∈ R . Not all these deformations can be included in the superpotential,since for some of them the couplings can be UV relevant and the theory can developLandau poles. Hence, being complete means that all the chiral primary operators areIR relevant or marginal non-dangerous. In this case the dimension of the manifoldof physical couplings is precisely d .We also say that tt ∗ geometry is very complete if the canonical projection p : C d → Y d factors through the critical value map w . This means that the manifold C d plays the role of cover space for ˙ P and denoting with s : C d → ˙ P the covermap we have p = w ◦ s . If tt ∗ geometry is very complete we can pull-back thevacuum bundle V → ˙ P to a bundle over C d and consider the tt ∗ geometry on theconfiguration space C d . In a very complete tt ∗ geometry, pulled-back to C d , thelocal expression 6.2 becomes global, since in this case the w i are global coordinatesand the partials ∂ w i W define a global trivialization of the bundle R → C d . In thevery complete case the entries of the matrices B ij are holomorphic functions on C d ,homogenous of degree zero and invariant under overall translation, which satisfy6.3. We conclude that the matrices B ij should be constant. In the general case the B ij are only locally constant and can have jumps under certain perturbations of thecritical values w k . Being very complete means essentially that the theory has nowall-crossing phenomena (see [30]). The B ij are further constrained by the flatnesscondition D = 0, which leads to the theory of Kohno connections [75, 77]. If aconnection D = d + A of the form 6.2 satisfies D = 0, then the following relationshold [ B ij , B ik + B jk ] = [ B ij + B ik , B jk ] = 0 , for i < j < k, [ B ij , B kl ] = 0 , for distinct i , j , k , l . (6.13)The above equations are called the infinitesimal pure braid relations. A connectionof the form 6.2 where the constant d × d matrices B ij satisfy the above relations iscalled a rank- d Kohno connection. A connection of this type defines a representationof the pure braid group P d of d strings σ : P d → GL ( d, C ) (6.14)through the parallel transport on the configuration space C d σ : γ ∈ C d → P exp( − (cid:90) γ A ) ∈ GL ( d, C ) . (6.15)97he family of representations σ parametrized by the matrices B ij satisfying the in-finitesimal pure braid relations is called the universal monodromy [77]. We concludethat for a very complete tt ∗ geometry the UV Berry connection is the universal mon-odromy representation of the pure braid group P d of d strings specialized to Kohnomatrices B ij computed in terms of the U (1) R spectrum of the UV chiral ring. Avery important class of very complete tt ∗ geometries are the symmetric ones. In thiscase the Kohno matrices satisfy( B π ( i ) π ( j ) ) π ( k ) π ( l ) = ( B ij ) kl , for all π ∈ S d (6.16)In this case the connection D descends to a flat connection on a suitable bundle V → Y d , providing a representation of the full Braid group [72, 75, 77]. This happenswhen the critical map w is an isomorphism. It is known that N = 4 SQM is closely related to the Ising model in 2 d [48]. The isingmodel consists in a free massive Majorana field Ψ( w ) which satisfies the EuclideanDirac equation ( /∂ − m )Ψ( w ) = 0 , (6.17)where m is the mass of the field and /∂ = (cid:18) ∂ ¯ ∂ (cid:19) , Ψ( w ) = (cid:18) Ψ + ( w )Ψ − ( w ) (cid:19) . (6.18)The space of solutions to the Dirac equation is endowed with an inner product,which coincide with the hermitian norm of the first quantized Hilbert space (cid:107) Ψ (cid:107) = (cid:90) (cid:0) | Ψ+ | + | Ψ − | (cid:1) d w, (6.19)where Ψ + , Ψ − must be L functions on the plane. It is possible to rephrase the Diracequation as cohomological problem. The Majorana field Ψ( w ) is not univalued onthe W -plane, but has complicated branching properties because of the insertion oftopological defect operators O k ( w k ) at points w k . The OPE98( w ) ± O k ( w k ) (6.20)is singular as w → w k . One considers two possible defect insertions at w k , which wedenote by σ k , µ k [48]. The OPEs of these operators with the Majorana field areΨ + ( u ) σ k ( w ) ∼ i u − w ) − / µ k ( w ) , Ψ − ( u ) σ k ( w ) ∼ − i u − ¯ w ) − / µ k ( w )Ψ + ( u ) µ k ( w ) ∼ 12 ( u − w ) − / σ k ( w ) , Ψ − ( u ) µ k ( w ) ∼ 12 (¯ u − ¯ w ) − / σ k ( w ) (6.21)up to O ( | u − w | / ) contributions. Although the defect operators µ k , σ k have thesame OPE singularities with the fermion field Ψ( w ) as the Ising order/disorderoperators, they are not in general mere Ising order/disorder operators since globallythey have different topological properties. The fermion field Ψ( w ) is univalued on asuitable connected cover of the plane punctured at the insertion points. By Riemannexistence theorem [78], this can be extended to a cover of the Riemann sphere W : Σ → P (6.22)branched at { w , ..., w d , ∞} . In case of a cover of finite degree m , f is specified byits Hurwitz data at the ( d + 1) branching points [78]. The Hurwitz data consist ofan element π k ∈ S m for each finite branching point w k , while π ∞ = ( π π ....π d ) − .The monodromy group of the cover is the subgroup of the permutation group S m generated by the π k ’s. When m is infinite the monodromy group is an infinite groupand the geometry is a bit more involved. Hence, the topological defect operator O k ( w k ) inserted at the k -th branching point is specified by the choice between σ -type and µ -type and the monodromy element π k ∈ S m .One can replace the Majorana field Ψ( w ) with the 1-form on the complex plane ψ ( w ) = Ψ + dw + Ψ − dw. (6.23)The map Ψ → ψ is consistent with the structure of the Hilbert space of the Diracequation. Indeed, the hermitian product of the wave functions is identified with theinner product of forms (cid:107) Ψ (cid:107) = (cid:90) ψ ∧ ∗ ψ ∗ . (6.24)It is natural to introduce the operators D , D which act on forms as99 ψ = ¯ ∂ψ + dw ∧ ψ, D ψ = ∂ψ + dw ∧ ψ. (6.25)These operators are nilpotent D = D = DD + DD = 0 , (6.26)and satisfy the Kahler-Hodge identites 3.24 with the Lefschetz operators L, Λ on theplane. In particular, we have D † = i [Λ , D ] . (6.27)It is easy to see that the Dirac equation for m = 1 can be written as the cohomo-logical relations D ψ = D ψ = 0 , (6.28)and using the above Kahler identity we also have D † ψ = 0 . (6.29)Hence, by standard Hodge theory, the normalizable solutions to the Dirac equationare the harmonic representatives of D -cohomology classes in the space of L formson the complex plane. One considers the following system of solutions [48] ψ i ( z ; ζ ) = (cid:104) Ψ + ( W ( z )) µ ( w ) ...σ j ( w j ) ...µ d ( w d ) (cid:105) W (cid:48) ( z ) dzζτ ( w j ) + (cid:104) Ψ − ( W ( z )) µ ( w ) ...σ j ( w j ) ...µ d ( w d ) (cid:105) W (cid:48) (¯ z ) ζd ¯ zτ ( w j ) (6.30)where z is a local coordinate on the cover Σ, ζ ∈ P is an arbitrary parameter suchthat | ζ | = 1 and the normalization constant is the Sato-Miwa-Jimbo τ -function τ = (cid:104) µ ( w ) ...µ ( w n ) (cid:105) . (6.31)The wave functions ψ i ( w, ζ ) are singular at the branching points w i . These singu-larities can be computed using the OPEs 6.21 and encode the cohomology class of100he j -th vacuum. We can use the hermitian scalar product 6.24 to define a groundstate metric on the Hilbert space spanned by the above Ising one-forms. Studyingvariations of the metric in the branching points w k , one finds that the bundle withfiber spanned by the local system of solutions 6.30 has the same tt ∗ geometry of thevacuum bundle of a 4-SQM model with semi-simple d -dimensional chiral ring [48].In this formal correspondence the insertion points w k are identified with the criticalcoordinates on the coupling constant space of the SQM model. Hence, 4-SQM andthe off-critical Ising model provide two equivalent description of the same geometry.The relation between the Ising model on the W -plane and 4-SQM is especially sim-ple when the superpotential depends on a single chiral field z . Let us consider a N = 4 Landau-Ginzburg model on a Stein space Σ and a Morse superpotential W .Comparing the expression of D , D with the susy charges Q = ∂ + dW ∧ , Q = ∂ + dW ∧ , (6.32)we see that the holomorphic map W : Σ → C (6.33)provides the relations Q = W ∗ D , Q = W ∗ D . (6.34)Namely, one can map the Schroedinger equation for the vacua of the LG model onΣ to the Euclidean Dirac equation on the W plane. In this correspondence, the one-form ψ i ( z, ζ ) is identified with the Schroedinger wave-function of the i -th vacuumcorresponding to the idempotent e i ∈ R w .We see that the formulation in critical coordinates is universal. The informationswhich distinguish different models and make precise the dictionary between SQMand Ising model are encoded in the Hurwitz data. The wave functions are multi-valued on the W -plane with branching points just given by w k . The Hurwitz datashould be chosen such that ψ i ( z, ζ ) are univalued in the coordinate z of the targetmanifold Σ. This space can be seen as a cover of the W -plane with cover map W : Σ → C branched at the critical points w k . By the Riemann existence theorem,the Hurwitz data specify uniquely up to isomorphism the cover ( W, Σ) of the punc-tured plane.A similar formula for the vacuum wave functions in the N -fields case can be con-structed as follows. We consider the inverse image of a point w on the W -plane.This has the homotopy type of a bouquet of N − S α ( w ) , α = 1 , ..., d for the middle dimensional homology of the fiber. Avacuum wave function Ψ is a N -form on the target space X . Pulling-back Ψ on the w -level curve of the superpotential and integrating it over the cycles S α ( w ), we geta set of d one-forms on the W -plane ψ α ( w ) = (cid:90) S α ( w ) Ψ . (6.35)If we transport the homology cycles along a closed loop along in the W -plane(punctured at the critical values) we come back with a different (integral) basisof ( N − S (cid:48) α ( w ) = M αβ S β ( w ). The integral matrix M αβ is described by thePicard-Lefshetz theory [30]. Thus, the 6.35 is best interpreted as a single but mul-tivalued wave-function ψ ( w ) on the W -plane branched at the critical points, whosemonodromy representation is determined by the Picard-Lefshetz formula. Let Σ bethe minimal branched cover of the W -plane such that ψ α ( w ) is single valued. Themap W : X → C factorizes through Σ. Pulling-back the wave forms on Σ by thecovering map w : Σ → C , one can replace the original LG model with N chiral fieldsand Morse superpotential W with an LG model with target space Σ and superpo-tential w given by the factorization of W thorugh Σ. Hence, we can use the formulafor the single field case and write ψ i ( w ; ζ ) = (cid:104) Ψ + ( w ) µ ( w ) ...σ i ( w i ) ...µ d ( w d ) (cid:105) dwζτ ( w j ) + (cid:104) Ψ − ( w ) µ ( w ) ...σ i ( w i ) ...µ d ( w d ) (cid:105) ζd ¯ wτ ( w j ) . (6.36)The branes B k ( ζ ) of the model on the W -plane are straight lines starting at thecritical points w k and strethced towards the Re w/ζ = −∞ direction. Denotingwith Γ k ( ζ ) the support of B k on Σ, we can define the brane amplitudes (cid:104) Γ k ( ζ ) | Ψ i (cid:105) = (cid:90) Γ k ( ζ ) e βw/ζ + βζ ¯ w ψ i ( w ; ζ ) . (6.37)In the physical 2 d (2,2) LG model, the UV limit consists in sending to zero the overallcoupling β . But, as one can see comparing the Dirac equation with the vacuumequation 6.28, β is also the mass m of the Majorana fermion in the context of theQFT on the W -plane. Hence, the physical UV limit of the 2 d LG model coincideswith the UV limit of the fermion theory on the W -plane. As β → tt ∗ quantities: in the UV they become some complicate combination of conformalblocks. Then the differential equations they satisfy, the tt ∗ equations, should berelated in a simple way to the PDEs for the conformal blocks. From this viewpointthe fact that in the UV limit the Berry connection, coinciding with the tt ∗ Lax one inthe β → tt ∗ monodromy with the braidrepresentation of conformal blocks, we need to use the precise disctionary betweenthe two. From 6.30 we see that the wave functions, being normalized, are to be seenas ratios of n -point functions in the W -plane CFT (cid:104) Ψ ± ( w ) µ ( w ) ...σ i ( w i ) ...µ d ( w d ) (cid:105)(cid:104) µ ( w ) ...µ n ( w n ) (cid:105) (6.38)rather than correlators. Hence, the actual braid representation on the CFT operatorsis the tt ∗ one twisted by the one defined by the τ -function. tt ∗ Connection We want to determine the matrices B ij for a very complete tt ∗ geometry. B ij is theresidue of the pole of the UV Berry connection as w i → w j . In order to compute B ij it is enough to consider this limit from the point of view of the 2 d model.Without loss of generality, we may deforme the D -terms so that the only lightdegrees of freedom are the BPS solitons interpolating between vacua i and j of mass2 | w i − w j | . We may integrate out all other degrees of freedom and we end up withan effective IR description with just these two susy vacua. The theory that weget may be not UV complete, but this is not an issue, since it is just an effectivedescription valid up to some non-zero energy scale. From the viewpoint of SQM,the 2 d BPS solitons are BPS instantons. The effect of these BPS instantons is tosplit the two vacua not in energy as it happens in non-susy QM, but in the charge q of the U (1) R symmetry which emerges in the w i − w j → Z symmetry which interchanges the classical vacua. The U (1) R symmetry is broken off-criticality to Z and the eigenstates are the symmetricand antisymmetric combination of the classical vacua [28, 30]. The correspondingcharges must be opposite by PCT symmetry. We renumber the critical values w k such that w i , w j are w , w . With a convenient choice of the relative phases of thetwo states, the upper-left 2 × U (1) R generator Q takes the form Q upper − left = − λ (cid:18) (cid:19) = − λσ , (6.39)103or some constant λ . We have to remember that in the trivialization given by theidempotent basis the UV Berry connection gains the extra piece δ kl d log h l whichgenerates an additional term in the above formula proportional to the identity. Tobe general we consider a shift by a constant µ − λ (cid:18) (cid:19) → − λ (cid:18) (cid:19) + µ (cid:18) (cid:19) (6.40)From the Ising model point of view, at each of the two critical points w , wemay insert either a σ -like defect or a µ -like one and different choices correspond todifferent vacua of the original LG theory. The matrix σ has the effect of flippingthe two-vacua system, which corresponds to the exchange σ ↔ µ in the correlators.It is therefore convenient to introduce a two-component notationΣ k,α ( w ) = (cid:18) σ k ( w ) µ k ( w ) (cid:19) . (6.41)The UV conformal blocks for the effective theory with two vacua have the form (cid:104) Ψ ± ( w )Σ ,α ( w )Σ ,α ( w ) (cid:105) ∈ V ⊗ V (6.42)where V a (cid:39) C , a = 1 , 2, are two copies of the fundamental representation spaceof sl (2 , C ). We note that the amplitudes representing the vacua span only a twodimensional subspace of V ⊗ V of dimension 2. The action of B can be extendedon the full four dimensional space of conformal blocks as B = − λ (cid:0) σ ⊗ σ − + σ − ⊗ σ (cid:1) − µ σ ⊗ σ , (6.43)where σ (cid:96)a is the Pauli matrix acting on the a -th copy V a of C . We conclude thatthe UV Berry connection D of a very complete tt ∗ geometry with d vacua has thegeneral form D = d − (cid:88) i 2. So, we have V = V − d/ . (6.48)The constants λ ij , µ ij are restricted by further conditions: • D is flat acting on V• Since D coincide with the Lax connection in the UV limit, the monodromyrepresentation must be arithmetic. • If the very complete tt ∗ geometry is symmetric, the constant λ ij , µ ij must beindipendent from i, j , i.e. λ ij = λ, µ ij = µ .A well known solution to the Knizhnik-Zamolodchikov equations is [75, 77] λ ij = λ , µ ij = 0 , (6.49)105hich gives D = d + λ (cid:88) i In this last section we want to study the tt ∗ geometry of the Vafa model for FQHE.We consider N electrons moving on the plane C . We denote with z i , i = 1 , ..., N and x a , a = 1 , ..., n the positions of electrons and quasi-holes respectively, and with ζ α , α = 1 , ..., M the support of the polar divisor D ∞ parametrizing the magneticflux. We take the d = M + n points ζ α , x a all distinct. Hence, the target space ofthe SQM Vafa model of FQHE is X d,N = (cid:110) ( z , ..., z N ) ∈ ( C \ { x a , ζ α } ) N (cid:12)(cid:12) z i (cid:54) = z j , for i (cid:54) = j (cid:111) (cid:14) S N . (7.1)In the experimental set-up N is very large, while N/d = ν and n are fixed. Despitethis, we shall keep N arbitrary as our arguments apply to any N . The superpoten-tial of the LG model is W = β (cid:88) i 0. Since the monodromy is independent of β (as long as it is not zero), andits limit as β → β = 0 whilekeeping track of the non-trivial topology of the target space through the character θ . In the language of [41] this is called θ -limit. According to the discussion in 5.2.2,switching a non zero θ is equivalent to pull-back the operators φ a of the fermionicchiral ring R F on A H and define the basisΦ a ( θ ) = (cid:36) ∗ φ a e iθh , (7.14)where (cid:36) : A θ → X and the chiral field e ih is proportional todiscr( P ) θ/ π = (cid:89) i 3. In particular, the minimal b -torsion character, a = 1yields the FQHE principal series ν = b b ± , b ∈ N . (7.18)Although this series is the most natural LG quantum systems of the form 7.2, itis by no means the only possibility in the present framework. We are going todiscuss the allowed values of the filling fraction in 7.4. The above analysis recallsin some sense the idea of Jain’s composite fermion theory [6]. If we consider thecharacter e πi (1+ θ/π ) as the Aharonov-Bohm phase generated by taking an electronaround another one, we can interpret the degrees of freedom of the Vafa model inthe θ -limit as free fermions bounded to θπ = 1 /ν − The fermionic truncation allows us to effectively put the coupling β to zero andconsider the LG Fermi model of N chiral fields with superpotential W ( e i ) = N (cid:88) i =1 W ( z i ) (7.19)where the one-field model is W ( z ) = µz + n (cid:88) a =1 log( z − x a ) − M (cid:88) α =1 log( z − ζ α ) . (7.20)Following [65] we can prove that the tt ∗ geometry of the single particle model is verycomplete. Since the monodromy representation is indipendent from µ , we can take µ (cid:29) 1. The equations for the classical vacua µ = (cid:88) α = 1 z − ζ α − (cid:88) a z − x a , (7.21)have d = n + M solutions of the form z = x a + O (1 /µ ) or z = ζ α + O (1 /µ ). Up toan irrelevant rescaling by a factor µ − , the set of critical values is113 w , ..., w n + M } = { x , ..., x n , ζ , ..., ζ M } + O (cid:18) µ log µ (cid:19) . (7.22)The cover of the coupling constant space P = C n + M /S n × S M is the configurationspace of n + M distinct points C n + M , which can be seen also as the cover of thespace Y n + M = C n + M /S n + M where the critical map w : P → Y n + M take values. Bythe above equality we see that the canonical projection p : C n + M → Y n + M factorizesthorugh the critical map. This shows that the tt ∗ geometry of the one-field modelis very complete and so the UV Berry connection is a flat sl (2) Kohno connection D = d + λ (cid:88) i 1) Virasoro model. The ratioof the two eigenvalues is q = exp[2 πi ( h , − h , )]exp[2 πi ( h , − h , )] = exp(2 πi/ν ) , (7.49)which is in agreement with the tt ∗ result. The model admits also another list offilling fractions with denominatior divisible by 4, namely ν = b b ± , b odd (7.50)which correspond to the second solution of 7.47.We may consider also the less natural class of solutions in which C is an eveninteger. In this case we have q ( θ ) = − e iπ/ν = e ± πi/ ( k +2) (7.51)Demanding 1 ≤ /ν ≤ k + 2 = 1 + θπ mod 2 = 1 ν mod 2 (7.52)which implies ν = mm + 2 , m = k + 2 ∈ N ≥ . (7.53)This series contains the values of ν corresponding to the Moore-Read [8] and Read-Rezayi models [9]. The second possibility is3 − k + 2 = 1 ν mod 2 (7.54)which gives ν = m m − , m = k + 2 ≥ . (7.55) We can compare the tt ∗ monodromy representation computed in the UV limit withthe one obtained in the asymmetric limit. In general the two are different unless theUV limit is regular. However, for a very complete tt ∗ geometry, where π ( P ) = B d ,the asymmetric limit monodromy yields the so-called homology representation ofthe braid group, which is essentially equivalent to the monodromy of the Knizhnik-Zamolodchikov connection. Following again [65], we can write an explicit integralrepresentation of solutions to the sl (2) Knizhnik-Zamolodchikov equationΨ KZ = f ( ζ , ..., ζ d ) (cid:90) Γ (cid:89) i,α ( z i − ζ α ) b e ˜ µ (cid:80) i z i (cid:89) i A.1 One Electron in Uniform Magnetic Field We discussed in section 4 the equivalence between the Landau description of anelectron in a uniform magnetic field and a certain class of 4-SQM Landau-Ginzburgtheories. In the case of a particle moving on the complex plane, the Vafa superpo-tential is given by W = (cid:88) i Φ i log( z − ζ i ) , (A.1)where Φ i ∈ C × and ζ i are all distinct. In order to reproduce a large macroscopicallyuniform magnetic field we have to take the residues Φ i all equal and consider auniform distribution of the flux sources ζ i in C . We may proceed by analogy with theCFT description of FQHE and consider a lattice of fundamental units of magneticflux Φ( a ) = a B π at positions ζ m,n ( a ) = am + ian , m, n ∈ Z , where a is a realparameter. The lattice Λ( a ) = a Z + ia Z is contained in a disk D ( R ) of radius R (cid:29) a which defines the size of the sample. As we already discussed in section2.10, one can recover the uniform constant B field by taking the continuous limit ofthe lattice and then sending R to infinity. The immaginary part of the logarithmicsuperpotential oscillates very rapidly for a (cid:28) a → 0. Instead, the real part of W coincides with the 2-dimensional electrostatic potential of a discrete set of charges and tends in the samelimit to the potential of a continuous charge distribution. Thus, we remain with areal superpotentialΦ( a ) (cid:88) Λ( a ) ∩ D ( R ) log( z − ζ m,n ( a )) a → , R →∞ −−−−−−−→ B | z | / . (A.2)This convergence is natural from the point of view of electrostatics, but it is non-trivial in the context of SQM. The model is based on N = 4 supersymmetry whichrequires the superpotential to be holomorphic. So, the fact that only the real part ofthe superpotential survives in the limit seems to be inconsistent with supersymmetry.Moreover, the very different analytical properties of the final limit compared to thefinite series suggest that this convergence cannot be intended in the strong topology.In the present section we want to clarify the interpretation of this limit from the123oint of view of supersymmetry. The function W ( z ; a, R ) = Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) log( z − ζ m,n ( a )) (A.3)is multivalued and cannot be really considered as the superpotential of the theory.Conversely, the derivative ∂ z W ( z ; a, R ) = Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) z − ζ m,n ( a ) (A.4)is a well defined meromorphic function on the plane. The dimension of the degen-erate vacuum space is given by the number of classical vacua, i.e. the solutions tothe equation ∂ z W = 0. These are counted by the Witten index of the theory whichgrows with the number of lattice points I W = Tr( − F = − ( a ) ∩ D ( R )) − , (A.5)where the mignus sign keeps into account that the vacua have fermion number F = 1. In the limit a → , R → ∞ the Witten index goes like | I W | (cid:39) πR /a ,which is an extremely large number of vacua. The meaning of the a → a . As physically expected, it turns out that for this specific classof measures the outcomes are correctly reproduced by an effective model with asingle vacuum.Let us begin with understanding the physics of the system in the continuous limit.It is clear that the wave functions of the theory must vanish at the points of thelattice Λ( a ) and, in the limit of a → 0, they oscillate rapidly on a length scalecomparable with the lattice parameter a . If we consider measures of observables ona scale much larger than a , we are not able to detect anymore the fluctuation of thewave functions and see the discrete structure of the set of charges. One can checkthat long-range measurements do not see the many vacua also by the correlators oftopological obervables in the two dimensional N = (2 , 2) version of the theory. Ina generic N = (2 , 2) Landau-Ginzburg model with fundamental chiral fields X i andsuperpotential W , the topological sector is described by operators Φ k in the chiralring R = C [ X i ] ∂ j W , which are in one to one correspondence with the vacua of the model.We recall that, if the critical points p α of W are non degenerate, the chiral ring issemisimple and we have R (cid:39) C . The canonical isomorphism betweenthese C -algebras is provided by a basis of minimal orthogonal idempotents definedby 124 β ( z α ) = δ αβ . (A.6)In this basis each operator of the chiral ring is represented by its set of values at thecritical points.Coming back to our theory, we see that for a → α on the target space must vanish on the critical points of W except z α and so become rapidly oscillating for a (cid:28) 1. Hence, it is clear that these op-erators are irrelevant for long-range measurements and one should consider a basisof generators for the chiral ring which are smooth in this limit. A suited basis isgiven by the monomials z k , k ∈ N , which are defined indipendently from the vacuastructure. Let us consider the correlators of these observables in the limit a → k = z k , Φ j = 1 in the Grothendieck formula 3.38 for the topological twopoints function we get (cid:104) z k (cid:105) = 1(2 πi ) (cid:90) Γ z k dz Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) z − ζ m,n ( a ) . (A.7)In this computation one has to choose a contour Γ which encircles all the criticalpoints of W . It is clear that these will be located in the region of the disk containgthe holes and so one must take a contour encircling D ( R ). In the limit of a → R , the discrete set of charges approaches a continuous distribution on D ( R ).If the coordinate z is outside the disk, the derivative of the superpotential tends toΦ( a ) (cid:88) Λ( a ) ∩ D ( R ) z − ζ m,n ( a ) ∼ BR z + O ( a ) , (A.8)in which we recognize the 2-dimensional electrostatic field generated by a radiallysymmetric charge distribution. Hence, we find (cid:104) z k (cid:105) ∝ (cid:90) Γ dzz k +1 + O ( a ) a → −−→ . (A.9)The triviality of the topological correlators reflect the fact that measurements at ascale z much larger than a do not detect the many vacua of the theory.It is already expected from A.2 which should be the effective model which capturesthe relevant physics of the FQHE. The fact that in the limit we obtain a real su-perpotential implies a passage from the initial N = 4 to a N = 2 theory in which125he holomorphicity is not required. Now we are able to give a precise physical andmathematical sense to the transition between the two formalisms.The considerations above justify the introduction of an effective Hilbert space, dif-ferent from the previous one, on which we can make long-range measurements ofobservables without seeing the lattice structure. This is the Schwartz space S ( C )of rapidly decreasing functions with bounded derivatives. The concept is that ex-pectation values of observables in this Hilbert space coincide with the expectationvalues in the original Hilbert space in the long-wave regime. It is clear now thatthe limit of the superpotential and all the other operators must be intended in thecontext of the weak topology induced on the space of operators by the Hermitianscalar product of S ( C ). Hence, the correct limit we have to study regards the matrixelement lim R →∞ lim a → Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) (cid:68) φ (cid:12)(cid:12)(cid:12) z − ζ m,n ( a ) (cid:12)(cid:12)(cid:12) ψ (cid:69) =lim R →∞ lim a → (cid:90) C d z Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) φ ∗ ( z ) ψ ( z ) z − ζ m,n ( a ) . (A.10)with φ, ψ ∈ S ( C ). The details of the computation are given in B. As one can expectwe find lim R →∞ lim a → (cid:104) φ | ∂ z W ( z ; a, R ) | ψ (cid:105) = (cid:68) φ (cid:12)(cid:12)(cid:12) B z (cid:12)(cid:12)(cid:12) ψ (cid:69) (A.11)which gives the same limit discussed in the previous section, (actually the derivative) ∂ z W ( z ; a, R ) → B z. (A.12)Hovewer, the mathematical structure that we have introduced here gives to theabove formula a precise mathematical meaning. It is evident that this convergencemust be intended only in the weak sense of quantum measure. A convergence innorm L p is excluded for evident reasons of integrability, as well as the uniform one,given the complitely opposite behavour between the initial function A.4 and thefinal limit. Indeed, the former is a meromorphic function with simple poles at thelattice points and vanishing at infinity, while the latter is a pure antiholomorphicobject with no singularities at finite points. This is consistent with the previous126onsiderations: long-distance experiments cannot detect the lattice structure.Contrary to ∂ z W , we cannot define the weak limit of | ∂W | and ∂ W because thecorresponding matrix elements are divergent. Indeed, the weak convergence doesnot preserve in general products and derivative of operators (otherwise it would bea strong convergence). This fact, together with the lost of holomorphicity of thesuperpotential, implies that part of the supersymmetry algebra is not preserved bythe weak limit (as expected), included the Hamiltonian.Adopting the formalism of section 3.1.2 we can write the susy charges as the gener-alized Dolbeault operators¯ ∂ W = ¯ ∂ + ∂W ∧ , δ W = ¯ ∂ † W ,∂ W = ∂ + ∂W ∧ ¯ δ W = ∂ † W . (A.13)The limit A.12 is not compatible with the structure of the susy charges because itexchanges holomorphic with antiholomorphic derivatives of W . Instead, there is noambiguity if we consider real combinations of the generators. Only the real part ofthe superpotential has a limit and so only the N = 2 subalgebra generated by d h = ¯ ∂ W + ∂ W = d + dh ∧ , δ h = δ W + ¯ δ W = d † h , (A.14)where h = Re W , is compatible with the weak limit. This corresponds to the θ = 0algebra in the family of N = 2 subalgebras. Learning from A.12 that dh → B z + ¯ z ) (A.15)we conclude that the initial N = 4 theory is equivalent for long-range measurementsto a N = 2 one with real superpotential h ( z, ¯ z ) = B | z | . (A.16)Depending on the sign of B we obtain the superpotential of a two dimensionalharmonic oscillator or repulsor. Both the models are studied in [37]. In this sectionwe assume B to be negative. One can count and classify the vacua of this theoryby exploiting the similarity relation between d h and the exterior derivative. The d h -cohomology classes in the Hilbert space are conjugated by e − h to relative deRham classes of smooth forms which decrease rapidly at infinity. These classes are127n the cohomology with compact support H ∗ c ( C ). Since only H ( C ) is non trivial,the theory has a unique ground state represented by the two formΦ( z, ¯ z ) = e − | B | | z | dzd ¯ z, (A.17)in which we recognize the typical B field factor which appears in the wave functionsof the Landau levels.We know that the Witten index of a supersymmetric theory is robust under contin-uous deformations and, the fact that we started from a model with a huge numberof vacua and we found in the end a theory with just one vacuum, may lead to somedoubts. However, this paradox does not arise because the N = 2 model is not thelimit of the initial one in a strict sense. Indeed, the new Hamiltonian is not directlyrelated to the previous one and acts on a different Hilbert space. The meaning ofthe weak limit is that for long distance measurements we can replace the original N = 4 theory with an effective N = 2 one with superpotential A.16. A.2 The Single Field Model with the Quasi-Holes Now that we have consistently recovered in this mathematical framework the stan-dard non-holomorphic description of the magnetic field, we can include the quasi-holes in the system. Let us consider a generic set S of quasi-holes at position p ( s ).Correspondingly, the holomorphic superpotential A.1 gains the new piece W → W + (cid:88) s ∈ S e ( s ) log( z − p ( s )) , (A.18)where we assume e ( s ) ∈ R . The target manifold of the theory is now C \ S and,taking the weak limit as before, we find a N = 2 model with superpotential h S ( z, ¯ z ) = B | z | + (cid:88) s ∈ S e ( s ) log | z − p ( s ) | . (A.19)The logarithmic terms in the superpotential do not spoil the polynomial divergenceat infinity, but now the relative cohomology classes representing the vacua dependalso on the behaviour of h S ( z, ¯ z ) near the quasi-holes positions p ( s ). If we demandthe e ( s ) to be all positive, the superpotential is bounded from above also in the regionof the quasi-holes and tends to −∞ as z → p ( s ). The map e − h S provides a corre-spondence between states in the Hilbert space and smooth forms with a sufficiently128apid decay on the region B − h S of C where h S → −∞ . Compared to the previ-ous case, now the forms have to vanish also approaching the quasi-holes positions.Hence, it follows that the L representatives of the d h S -classes in the Hilbert spaceare conjugated to closed forms in the relative cohomology H ∗ ( C , B − h S ) = H ∗ c ( C \ S ).One can easily see that the vacuum space contains again a two formΦ S ( z, ¯ z ) = e − | B | | z | (cid:89) s ∈ S | z − p ( s ) | e ( s ) dzd ¯ z, (A.20)which is a deformation of A.17 vanishing at the positions of the quasi-holes. In thecase of FQHE one should set all the charges to 1. In presence of the quasi-holesthe vacuum space gains new fermionic states in addition to the bosonic one above.These correspond to cohomology classes which are dual to the 1-homology cycleswith boundary on B − h S . By Poincar´e duality with the de Rham cohomology we findthat there are as many homology classes as the number of quasi-holes.One can count the fermionic states of the vacuum space also with the followingargument. We note that h S contains an harmonic part which can be written asreal part of an homolomorphic superpotential W . The definition of this functionis ambigous because we have the freedom to shift the electron coordinate. So,to be more general one should perform the translation z → z + ∆. In this waythe holomorphic superpotential depends also on this extra parameter and has theexpression W ( z ) = ∆ B z + (cid:88) s ∈ S e ( s ) log( z − p (cid:48) ( s )) , (A.21)where we setted p (cid:48) ( s ) = p ( s ) − ∆. This function defines a N = 4 theory in whichthe vacuum wave forms are in correspondence with the solutions to the equation ∂W = 0. For m quasi-holes and generic parameters the superpotential has m distinct critical points and so the vacuum space has dimension m . Moreover, asdiscussed in section 3.1.2, the vacua of a N = 4 theory with Morse superpotentialon C n are in correspondence with the relative cohomology classes of H n ( C n , B − ReW ),where B − ReW is the region of C n where Re W → −∞ . In the present case, denotingwith Φ α , α = 1 , ..., m a basis of d ReW -closed representatives in the Hilbert space, onecan use the 3.57 and write Φ α = e − ReW ω α , (A.22)where ω α are d -closed forms in H ( C , B − ReW ). In the same way the relative de Rhamclasses are trivial in the absolute cohomology, also the states Φ α become exact in129he d ReW -cohomology with smooth coefficients and we can find 0-forms χ α such thatΦ α = d ReW χ α . (A.23)However, since χ α are not L forms, Φ α represent non trivial classes in the Hilbertspace cohomology. The forms ω α are dual to homology cycles in H ( C , B − ReW ). Werecall that for a Morse superpotential one can define a canonical basis of cycles D α which begin from the critical points of W and have boundary on B − ReW (see3.1.2). The real part of W preserves the behaviour of h S near the quasi-holes andpartially also at infinity. The boundary set for the non compact cycles contains smallneighborhoods around p ( s ) and a connected region at infinity where Re( − ∆ z ) →−∞ . Hence, one can easily see that the relative classes [ D α ] provide a complete basisalso for H ( C , B − h S ), implying that the two homology groups are isomorphic. Wenote further that the susy generators of the full theory are related by the similaritytransformations d h S = e − B | z | / d ReW e B | z | / e B | z | / δ ReW e − B | z | / (A.24)to the charges of the N = 2 subalgebra with θ = 0 of the W -theory. By theseconjugation relations we can map representatives Φ α of d ReW -classes to d h S -closedforms Ψ α = e − B | z | / Φ α . (A.25)The multiplication by e − B | z | / is not an isometry between Hilbert spaces and couldspoil in general the integrability of the wave functions. However, one can alwaysfind in each d ReW -cohomology class a representative Φ α with compact support, orwith sufficiently rapid decay on C \ S , such that the corresponding Ψ α is an L -wavefunction of the Hilbert space. It is clear that these states remain closed but not ex-act in the d h S -cohomology because the multiplication by e − B | z | / cannot make theprimitive χ α normalizable. Therefore, cohomology classes of d ReW defines cohomol-ogy classes of d h S and both are dual to the Lefschetz thimbles D α . In this sense thefermionic subspace of the vacuum space represent the N = 4 sector of the theory.We note that ∆ = 0 represents a critical limit in which the superpotential has adifferent behaviour at infinity. In this case we find a degenerate model in which oneof the critical point of W is sent to infinity and decouples from the other ones.We said that the great advantage of the Vafa model of FQHE compared to the pre-vious ones is supersymmetry. In particular, this shoud provide more tools to studythe monodromy of the ground states induced by the quasi-holes braiding. However,130n order to make this problem resonable to solve, we need the tools of tt ∗ geometryensured by N = 4 supersymmetry that we lost in the weak limit. A strategy onecan follow at this point is to consider some region of the parameter space in whichwe have an enhancement of supersymmetry from N = 2 to N = 4. In such regimethe bosonic state should decouple from the system in such a way that one could seeonly the fermionic vacua. We note that, since it is the unique 2-form in the vacuumspace, the wavefunction A.20 must be an eigenstate of the Berry’s holonomy. More-over, since this form is real for generic holes positions, the corresponding Berry’sphase is trivial. The fact that the non trivial part of the holonomy regards only thefermionic sector of the vacuum space provides a further justification to consider thislimit.It is already clear from the discussion above that the effective superpotential we arelooking for is given by the harmonic part of h S ( z, ¯ z ). This IR description corre-sponds to the region of the parameter space where the quasi-holes are placed in aneighborhood of a point ∆ ∈ C \ { } of radious r (cid:28) (cid:96) B = 1 / (cid:112) | B | . Denoting with ζ ( s ) = ( p ( s ) − ∆) /r the quasi-holes positions in r units and redefining the electroncoordinate z → rz + ∆, one has to expand h S (cid:0) rz + ∆ , r ¯ z + ∆ (cid:1) for r (cid:28) (cid:96) B andfinite ζ ( s ). At the leading order one obtains (up to constant terms) h S ( z, ¯ z ) ∼ − z ¯∆ r (cid:96) B − ¯ z ∆ r (cid:96) B + (cid:88) s ∈ S log | z − ζ ( s ) | , (A.26)where we setted the quasi-holes charges to 1. The holomorphic superpotential thatwe get is the A.21 with rescaled coordinates in 1 /r unit W ( z ) = µz + (cid:88) s ∈ S log( z − ζ ( s )) , (A.27)where we introduced the notation µ = − ∆ r (cid:96) B . This theory allows to capture exclu-sively the physics of the fermionic vacuum space, restoring the initial amount ofsupersymmetry. As we already mentioned, the case of µ = 0 represents a criticalpoint in the parameter space where also one of the fermionic vacua decouples fromthe system. In the language of the (2 , 2) version of the model, this turns out tobe an IR fixed point of the RG flow, while by deforming µ away from the originthe number of vacua computed by the Witten index is consistently stable. So, weshould keep µ (cid:54) = 0 if we want to study the physics of the full fermionic sector of thetheory. From the point of view of FQHE, the linear interaction in the effective su-perpotential above describes the electric field induced by the magnetic field. Takingthe limit µ → N = 4 theory which is131efinitely easier to study with respect to the initial one. The virtue of the simplifica-tions we have considered is to make the problem resonable to approach analitically,remaining at the same time in a realistic setting for the physics of FQHE. B Weak Limit of ∂W Let us take φ, ψ in the Schwartz space S ( C ) of smooth functions with boundedderivatives. We want to compute the limit A.10lim R →∞ lim a → Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) (cid:68) φ (cid:12)(cid:12)(cid:12) z − ζ m,n ( a ) (cid:12)(cid:12)(cid:12) ψ (cid:69) =lim R →∞ lim a → (cid:90) C d z Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) φ ∗ ( z ) ψ ( z ) z − ζ m,n ( a ) , (B.1)where Λ( a ) = a Z + ia Z , D ( R ) = { z ∈ C , | z | ≤ R } and Φ( a ) = a B π . Let us first take a → R . Performing the change of variable z → z + ζ m,n ( a ), we getlim a → (cid:90) C d zz Φ( a ) (cid:88) Λ( a ) ∩ D ( R ) φ ∗ ( z + ζ m,n ( a )) ψ ( z + ζ m,n ( a )) =lim a → (cid:68) z (cid:12)(cid:12)(cid:12) Φ( a ) φ ∗ ( z + ζ m,n ( a )) ψ ( z + ζ m,n ( a )) (cid:69) . (B.2)Given that z acts on the space of test functions as a linear continuous functional ,one can take the limit under the sign of integration. In the expression a (cid:88) Λ( a ) ∩ D ( R ) φ ∗ ( z + ζ m,n ( a )) ψ ( z + ζ m,n ( a )) (B.3)we recognize the Riemann sum of φ ∗ ( z + ζ ) ψ ( z + ζ ) with partition of D ( R ) givenby squares of area a . The smoothness and boundedness of the Schwartz functionsallow to write It is straighforward to see that 1 /z is a tempered distribution of function type. a → a (cid:88) Λ( a ) ∩ D ( R ) φ ∗ ( z + ζ m,n ( a )) ψ ( z + ζ m,n ( a )) = (cid:90) D ( R ) d ζ φ ∗ ( z + ζ ) ψ ( z + ζ ) . (B.4)Then, after another shift of z we remain withlim R →∞ (cid:90) C d z φ ∗ ( z ) ψ ( z ) B π (cid:90) D ( R ) d ζ z − ζ . (B.5)Let us focus on the integral in ζ, ¯ ζ . Since we are going to take the limit of infinitevolume, we expand the integral for | z | /R (cid:28) 1. Introducing the new variable w = ζ − z and denoting with R z ( θ ) = | ζ ∂D ( θ ) − z | = (cid:113) | z | + R − zζ ∂D ( θ ) − ¯ zζ ∂D ( θ ) , the distance between z and the point ζ ∂D ( θ ) ∈ ∂D ( R ) with phase θ ∈ [0 , π ], wefind (cid:90) D ( R ) d ζ z − ζ = − (cid:90) π dθR z ( θ ) e − iθ = − (cid:90) π dθ (cid:16) R − z R ζ ∂D ( θ ) − ¯ z R ζ ∂D ( θ ) + O (1 /R ) (cid:17) e − iθ . (B.6)Using ζ ∂D ( θ ) = Re iθ + O (1) and sending R to infinity one gets (cid:90) D ( R ) d ζ ζ − z = π ¯ z + O (1 /R ) . (B.7)Plugging this large R expansion in the integral B.5 one obtains a convergent seriesof integrals weighted by powers of 1 /R . In the limit of infinite R we keep only theleading term and write finallylim R →∞ lim a → (cid:104) φ | ∂ z W ( z ; a, R ) | ψ (cid:105) = (cid:68) φ (cid:12)(cid:12)(cid:12) B z (cid:12)(cid:12)(cid:12) ψ (cid:69) . (B.8)133 FQHE and Gauge Theories In this section we review some well known facts about class S theories and the 2 d/ d correspondence. We also recall the correspondence between matrix models andgauge theories. These are the basic ingredients which motivates the Vafa proposaland allow to formulate a precise dictionary between FQHE and supersymmetricgauge theories in 4 d . C.1 T n,g Gauge Theories We recall some known facts about class S theories, namely four dimensional N = 2gauge theories which arise from the compactification of six dimensional theories. Inparticular, we focus on the subclass which according to Vafa are related to FQHE.A theory in this class is denoted with T n +3 ,g [ A ] and is associated to a Riemannsurface C n +3 ,g of genus g and n + 3 equivalent punctures. These models are identified[18] with the twisted compactification of the A (2 , 0) six-dimensional SCFT on the g -Riemann surface in the presence of n defect operators. In the context of FQHEthe surface C n +3 ,g has also the interpretation of target space for the electrons, hencewe are interested in the case of g = 0. The first model of the series correspond tothe three-punctured sphere. This theory simply contains four free hypermultipletsand the gauge dynamics is absent. Much more interesting are the T n +3 , [ A ] modelswith n > N = 2 SU (2) gauge theories with massive deformations.The gauge group contains n SU (2) factors, each of them coupled to N f = 4 flavours.These theories are therefore conformal and admit a space of exactly marginal gaugecouplings. The parameter space of gauge couplings turns out to coincide with themoduli space M n +3 , of complex structures of the Riemann sphere with n + 3 punc-tures C n +3 , . The space of couplings has boundaries where one of the SU (2) gaugegroups become weakly coupled. To each puncture corresponds also a flavour sub-group SU (2). So, in total we have an SU (2) n +3 flavour group. Each punture isalso associated to a mass parameter m a , a = 1 , ..., n + 3 of the corresponding flavoursubgroup SU (2) a . These theories enjoy S -duality, which has the geometric inter-pretation of fundamental group π of M n, . The action of S-duality on the flavoursymmetry groups is a permutation action and coincides with the permutation ofthe n + 3 punctures on the sphere and the corresponding mass parameters. The S -duality can rearrange in different ways the matter content of the theory, but ineach possible S -frame we remain always with n SU (2) gauge groups coupled tofour hypermultiplet doublets and n + 3 SU (2) flavour groups. The various weaklycoupled S -dual frames of the theory coincide with the different ways a sphere with n + 3 points can degenerate completely to a set of n + 1 three-punctured spheresattached together at n nodes. Labelling with SU (2) i , i = 1 , ..n the gauge groups, in134 possible weakly coupled S -duality frame we have two hypermultiplet doublets inthe fundamental of SU (2) with masses m ± m , n − SU (2) i × SU (2) i +1 with masses m i +2 and other two hypermultipletsin the fundamental of SU (2) n with masses m n +2 ± m n +3 . The linear gauge quiverdescribing this frame is SU (2) SU (2) SU (2) SU (2)where the rectangular nodes on the two sides represent the two couples of fundamen-tal hypermultiplets, the circular nodes are the gauge groups and the lines betweenthe gauge nodes are the bifundamental hypermultiplets.A canonical example is the N = 2 SU (2) gauge theory with N f = 4 fundamen-tal flavours. The theory has an exact marginal coupling τ = θπ + πig , since thenumber of flavours is twice the number of colours. The flavour group SU (2) isenhanced to SO (8) and the four hypermultiplet doublets transform in the eight di-mensional vector representation. In this case the S -duality group acts by fractionallinear transformation of SL (2 , Z ) on τ and by triality on SO (8). Hence, the space ofmarginal couplings parametrized by τ is H /SL (2 , Z ), which is the complex structuremoduli space M , of a sphere with four equivalent punctures. If we quotient theupper half plane only by the subgroup Γ(2) we get the moduli space of a spherewith four marked punctures, i.e. the modular curve H / Γ(2) (cid:39) P \ { , , ∞} . Anatural parametrization of this space is given by the cross-ratio q of the positions ofthe four punctures. This can be seen as a coordinates on P \ , , ∞ and is relatedto τ by the Γ(2)-invariant modular lambda function λ ( τ ) = θ (0 ,τ ) θ (0 ,τ ) which realizesthe isomorphism H / Γ(2) (cid:39) P \ { , , ∞} . The action of Γ(2) does not permute thepunctures of the sphere and the moduli space has three cusps at τ = 0 , , ∞ corre-sponding to the three weakly coupled frames of the theory. The full S -duality grouppermutes the punctures among themselves and simultaneously the associated massparameters, mapping between each other also the three weakly coupled descriptionof the theory which are therefore physically equivalent.The compactification of A (2 , 0) theories on a puctured Riemann surface providesalso the construction of a canonical Seiberg-Witten curve encoding the physics ofthe Coulomb phase of the theory. This curve is a ramified double cover of C n +3 ,g defined by the equation y = φ ( z ) , (C.1)where ( y, z ) are local coordinates in the cotangent bundle of C n +3 ,g . The SW dif-135erential is the canonical one form dλ = ydz , while dλ = φ ( z ) dz is the associatedJerkin-Strebel quadratic differential with appropriate poles at the punctures. In thecase of the T n +3 , the quadratic differential on the n + 3-punctured sphere has doublepoles at the punctures ζ a with coefficients given by the square of the correspondingmass parameters.More general theories in the A class are associated to a quadratic differential withpoles of higher order [45]. These theories are obtained from superconformal theoriesby tuning some mass parameter m a to be very large, adjusting at the same time thecoupled marginal gauge coupling in the UV so that the running coupling in the IRremains finite. From a six-dimensional perspective, the limiting procedure bringstwo or more standard punctures together to produce a single puncture with a largerdegree of divergence. It is relevant for FQHE the case in which two punctures in the T n +3 , theory collide to generate an irregular puncture. This appears as a quarticpole in the quadratic differential. In the n = 0 case one obtains an Argyres-Douglassystem of D type, namely a theory of a free hypermutliplet doublet and SU (2) flavour group. For n > SU (2) n gauge theory in which n − SU (2)factors are coupled to N f = 4 flavour and one SU (2) gauge group is coupled tothree fundamentals. Hence, one of the couplings has negative beta function and aYang-Mills scale Λ YM is generated at one loop. This coupling vanishes in the UVlimit, while the other ones are marginal. So, the UV limit of the theory is decribedby a C n +2 , punctured sphere and corresponds to the superconformal T n +2 , theory.The space of couplings is identified with the complex structure moduli space of asphere with n + 1 equivalent punctures and a marked puncture. Each puncturecontributes to the flavour group with an SU (2) factor, giving in total a SU (2) n +2 flavour group. The S -duality group which permute the equivalent punctures in theconformal case now is broken to a subgroup which act only on the regular ones, act-ing simultaneously on the punctures and the corresponding mass parameters. Thisresidual group plays the role of fundamental group of the space of couplings. Thetheory admits different weakly coupled description which correspond to the possibleways in which the punctured sphere degenerates to n spheres with three regularpunctures, corresponding to T , models, and a sphere with a regular and an irregu-lar puncture which is identified with a D system. Labelling with SU (2) i , i = 1 , ..n the gauge groups, in a possible weakly coupled S -frame we have one hypermultipletdoublet in the fundamental of SU (2) with masses m , n − SU (2) i × SU (2) i +1 with masses m i and other two hypermultipletsin the fundamental of SU (2) n with masses m n ± m n +1 . The linear gauge quiverdescribing this frame is SU (2) SU (2) SU (2) SU (2)136ompared to the superconformal quiver C.1, now attached to the the first SU (2)node we have a rectangular node representing a single hypermultiplet doublet, im-plying that SU (2) is coupled only to three flavours. A simple example is given bythe N = 2 SU (2) gauge theory with N f = 3 fundamental flavours. Since N f < N c the theory is asymptotically free in the UV. The flavour group SU (2) is enhanced to SO (6) and the three hypermultiplet doublets transform in the sixth dimensional vec-tor representation. The fundamental group of the complex structure moduli spaceof a sphere with two equivalent and a marked puncture has a unique generator.This acts by permuting the two regular punctures and corresponds to the residual S -duality transformation that we have in all the asymptotically free SU (2) gaugetheories with N f < 4. In the UV limit the theory flows to the T , theory of fourfree hypermultiplets described by the three punctured sphere. C.2 Matrix Models and d/ d Correspondence One of the key ingredients to connect FQHE to the gauge theories introduced aboveis the relation between matrix models and four dimensional N = 2 gauge theories[26, 27, 41]. This connection arises from string theory as follows. One considers thetype IIB superstring on a background CY × R , where CY denotes a fixed Calabi-Yau threefold with a non compact holomorphic curve Y ⊂ CY , with N D Y × R . The theory living on R admits Ω-deformation [42] withparameters (cid:15) for a rotation transverse to the brane and (cid:15) along the brane. Thecomplex structure moduli of CY is controlled by a set of parameters t k which playthe role of couplings in the four dimensional theory. In this set up one considers atopological string on CY with N branes on Y . It has been found [27, 43, 44] thatthe Nekrasov partition function Z ( t k , (cid:15) , (cid:15) ) of the 4 d gauge theory coincides withthe open topological string partition function Z ( t k , g s ) with (cid:15) = − (cid:15) = g s . Thisadmits the representation of matrix model partition function. In the case of SU (2)gauge theories only a single N × N matrix Φ is required and we have Z ( t k , (cid:15) , (cid:15) ) = (cid:90) N × N d Φ exp (cid:18) − g s Tr W ( t k , Φ) (cid:19) = (cid:90) N (cid:89) i =1 dz i (cid:89) ≤ i 2) Landau-Ginzburgmodels with superpotential W ( t k ; z i ). The key point is that the open string am-plitudes can be identified with the flat sections of the tt ∗ Lax connection in theasymmetric limit [41]. This can be realized by compactifying the theory on a spatialcircle of radious R in such a way that the superpotential gets rescaled by W → R W .With a rotation of the fermionic measure of the superspace one can also introducean overall phase ζ multiplying the superpotential W → W /ζ, W → ζ W . Then onehas to consider an analytic continuation of the phase ζ away from the locus | ζ | = 1and take the limit R → , ζ → (cid:15) = ζ/R finite. This is equivalent to a nonunitary deformation of the theory in which we set W = 0 and rescale W → W/(cid:15) .Among the tt ∗ brane amplitudes, a set of distinguished elements is given by thepairing of the D-brane states | D α (cid:105) with the topological vacuum | (cid:105) correspondingto the identity operator. We denote such wave functions as ψ α = (cid:104) | D α (cid:105) . (C.24)In the asymmetric limit one has an explicit formula for the above overlap whichreads lim asym ψ α ( (cid:15) , (cid:15) ) = Z α ( (cid:15) , (cid:15) ) , (C.25)where one has to redefine the Nekrasov deformation (cid:15) with respect to the openstring convention by setting (cid:15) = ζ/ ˜ R . As one can see from the expression abovethe Nekrasov parameter (cid:15) plays the role of Van der Monde coupling and has massdimension 1. One can naturally extend the 2 d/ d correspondence also to SQM. Theone dimensional N = 4 version of the theory has the same structure of vacua andBPS spectrum, which implies that one can study the geometry of the vacuum bundleeither in one or two dimensions [28]. The superpotential in SQM is dimensionlessand arises from the compactification of the 2 d theory. The matching scale is givenby the radious R of the tt ∗ circle and we have W SQM = −W d /(cid:15) . According tothe AGT correspondence, the amplitudes of the Penner matrix models compute theconformal blocks of the Liouville CFT and hence the FQHE wave functions in theVafa’s model. To match the normalization of the Liouville correlators given in [17]143ne has to identify − m a /(cid:15) = 1 and 1 /ν = (cid:15) /(cid:15) . The AGT correspondence has beengeneralized in [45] also to class S theories with irregular punctures in the Gaiottosurface. It turns out that the Nekrasov partition functions of these models reproduceirregular conformal blocks of the Liouville CFT. The presence of higher degree polesin the quadratic differential corresponds to insertions in the Liouville correlators ofthe so called irregular vertex operators [46, 47]. As for the mass parameters, therelevant coupling µ d of the 2 d theory is related to its dimensionless counterpart µ d in SQM by µ d = − µ d /(cid:15) . D tt ∗ Geometry of the One Electron Model withTwo Quasi-Holes We consider a class of 2 d N = (2 , 2) Landau-Ginzburg theory described by thesuperpotential W ( z ) = µz + log z + log( z − ρ ) . (D.1)The 4-SQM version of this theory corresponds to the Vafa model for a single electronon the plane with two quasi-holes at distinct positions 0 , ρ ∈ C and a backgroundelectrostatic field µ ∈ C × . The above function is multi-valued and cannot be strictlyconsidered as the superpotential of the model. We want to define this theory on theAbelian universal cover of the target space and study the correspoding tt ∗ geometry.As discussed in section 3.2.2, the solution to the tt ∗ equations is captured by theStokes matrices which describe the jumps of the brane amplitudes Ψ( ζ ) on the ζ -plane. The monodromy data and the leading IR behaviour of the tt ∗ metric andconnection are determined by the BPS spectrum of the model. The inital step isto find the classical vacua of the theory on the universal cover. The solutions z ± tothe equation ∂ z W = 0 denote two equivalence classes of vacua which are isomorphicto the homology group H ( C \ (0 , ρ ); Z ). A natural basis for this group is givenby the two loop generators (cid:96) , (cid:96) ρ encircling the holes positions. The vacuum spaceof the theory on the Abelian cover A decomposes in a direct sum of irreduciblerepresentations of the homology group: V A (cid:39) L (cid:0) Hom (cid:0) Z , U (1) (cid:1)(cid:1) ⊗ C , (D.2)where H ( C \ (0 , ρ ); Z ) = Z . We denote the orthogonal idempotents of R A corre-sponding to the classical vacua on the universal cover with144 ± ; m, n (cid:105) = (cid:96) m (cid:96) nρ |± ; 0 , (cid:105) (D.3)where |± ; 0 , (cid:105) corresponds to some representative of z ± on the universal cover. Interms of these states one can construct eigenstates of the loop generators |± ; φ, ϕ (cid:105) = (cid:88) m,n e − i ( mφ + nϕ ) |± ; m, n (cid:105) (D.4)where the angles φ, ϕ ∈ [0 , π ] label representations of H ( C \ (0 , ρ ) , Z ). By thefact that the homology is an abelian symmetry of the model, the tt ∗ geometrydiagonalizes completely with respect to the angles φ, ϕ . In particular, introducingthe labels k, j = ± , the ground state metric in the point basis D.4 can be expandedin Fourier series as g k, ¯ j ( φ, ϕ ) = (cid:88) r,s e i ( rφ + sϕ ) g k, ¯ j ( r, s ) ,g k, ¯ j ( r, s ) = (cid:104) j ; r, s | k ; 0 , (cid:105) . (D.5)From the IR expansion (cid:104) j ; r, s | k ; 0 , (cid:105) ∼ δ k,j δ r, δ s, − iπ µ k, , j,r,s K (2 | w k, , − w j,r,s | ) (D.6)we see that the one-soliton multiplicities µ k, , j,r,s are weighted by the Fourier phasefactors e i ( rφ + sϕ ) . The number of solitons saturating the Bogomonlyi bound can beobtained by solving the BPS equation [30] ∂ σ z = α∂ z W (D.7)where σ is the spatial variable of the field z and α = (cid:52) W/ |(cid:52) W | identifies the BPSsector we are considering. At spatial infinity σ = ±∞ one has also to impose thecorresponding critical points as boundary condition. One can count the number ofsolutions to the BPS equations by plotting with a program the flow of the vectorfield V = (cid:18) Re z Im z (cid:19) = (cid:18) Re ¯ α∂W − Im ¯ α∂W (cid:19) . (D.8)145et us start with the case of α = ± i . The solitons in this sector projected on thetarget manifold are respectively anti-clockwise and clockwise loops based at z ± . Thespectrum that we find is the same of the models W − ( z ) = µz + 2 log z,W + ( z ) = log z + log( z − ρ ) , (D.9)which describe the dynamics of the two vacua at z ± respectively in the IR limits ρ → µ → 0. In the case of α = i one finds | µ − ,m,n ; − ,r,s | = δ m,r +1 δ n,s +1 , | µ + ,m,n ;+ ,r,s | = δ m,r +1 δ n,s + δ m,r δ n,s +1 , (D.10)where the sign of the soliton multiplicities will be fixed by the computation of theStokes matrices.Let us consider now the solitons connecting z − to z + . Given that the quantummonodromy is not affected by the choice of the parameters (up to conjugation) wecan count the soliton solutions for µ = ρ = 1. The soliton equation for these BPSstates must be solved with the central charges∆ W + − k = z + − z − + log (cid:12)(cid:12)(cid:12)(cid:12) z + ( z + − z − ( z − − (cid:12)(cid:12)(cid:12)(cid:12) + 2 πi ( k + 1 / , k ∈ Z (D.11)and boundary conditions x ( −∞ ) = z − , x (+ ∞ ) = z + , where z ± = (1 ± √ / 2. Letus fix a representative z − ;0 , of z − on the spectral cover and denote with |− ; 0 , (cid:105) the corresponding state in the Hilbert space. z − z + x xFigure 2: Definition of γ +0 , z ± on the spectral cover with homotopyclasses of curves which connect a fixed point z ∗ ∈ C \ { , ρ } to z ± . Let us choose z ∗ = z − . In this way the curves γ − r,s corresponding to the points |− ; r, s (cid:105) are simplythe closed cycles (cid:96) r · (cid:96) sρ based at z − . Then, we identify the curve γ +0 , in figure 2 whichconnects z − to z + with the state | +; 0 , (cid:105) . All the other curves γ + r,s going from z − to z + which label the representatives | +; r, s (cid:105) of z + are in the same homotopy classof γ +0 , · (cid:96) r · (cid:96) sρ . The solutions to the BPS equation which interpolate z − ;00 and z +; r,s belong to homotopy classes γ + r,s for certain couples of integers ( r, s ). The multiplicitymatrix that we obtain by studying the flow of D.8 for α = ∆ W + − k / | ∆ W + − k | is | µ − , , ,r,s | = δ r,s + δ r +1 ,s (D.12) z − z + x x γ +0 , γ +0 , γ + − , Figure 3: Some soliton solutionsand the central charge of the BPS particle in the (+ , r, s ; − , , 0) sector turns out tobe ∆ W + − r,s = z + − z − + log (cid:12)(cid:12)(cid:12)(cid:12) z + ( z + − z − ( z − − (cid:12)(cid:12)(cid:12)(cid:12) + 2 πi ( r + s + 1 / . (D.13)One can easily see that the braiding of the quasi-holes generates a spectral flow ofthe BPS spectrum. As we exchange the two singularities in the anticlockwise sensethe curves γ ± r,s transform respectively as γ − r,s → γ − s,r , γ + r,s → γ + s,r +1 , (D.14)which implies that the entire BPS spectrum can be generated by the fundamental147oliton γ +0 , by iterating the braiding of the quasi-holes. At the same time the states |± ; r, s (cid:105) transform as |− ; r, s (cid:105) → |− ; s, r (cid:105) , | +; r, s (cid:105) → | +; s, r + 1 (cid:105) , (D.15)which implies (cid:104) +; r, s |− ; 0 , (cid:105) → (cid:104) +; s, r + 1 |− ; 0 , (cid:105) . (D.16)On the other hand, from the IR expansion of the metric one gets also (cid:104) +; r, s |− ; 0 , (cid:105) ∼ − iπ µ − , , ,r,s K (2 | w k, , − w j,r,s | ) → − iπ µ − , , ,r,s K (2 | w k, , − w j,s,r +1 | ) . (D.17)The compatibility of these two expressions puts the condition on the soliton matrix µ − , , ,r,s = µ − , , ,s,r +1 (D.18)which implies that the sign of the BPS multiplicity in the ( − , , 0; + , r, s ) sector isthe same for each couple ( r, s ). This sign has not an intrinsic meaning, since it canalways be absorbed by changing the sign of one of the theta vacua in D.4 withoutaffecting the diagonal elements g k, ¯ k ( φ, ϕ ).We are able now to compute the Stokes operators of the system. In the basis of thetavacua |− ; φ, ϕ (cid:105) , | +; φ, ϕ (cid:105) these are 2 × φ, ϕ . We choose the real axis of the ζ -plane as Stokes axis. The Stokesmatrix can be written as ordered product of three matrices S = S S S (D.19)which we are going to describe. The first contribution S is generated by the phase-ordered product of the solitons with positive real and immaginary part. For µ = ρ = 1 we have Re W ( z − ) > Re W ( z + ). So, these BPS states have central charges − ∆ W + − r,s and connect the points z +; r,s to z − ;0 , with r < , s ≤ 0. Each of themgenerates a Stokes jump 148 r,s = (cid:18) µ − , , ,r,s e i ( rφ + sϕ ) (cid:19) (D.20)where the phase factor comes from the Fourier expansion of the metric and we takethe soliton multiplicities µ − , , ,r,s = − ( δ r,s + δ r +1 ,s ) with the mignus sign. Takingthe product of these factors for r < , s ≤ S = (cid:89) r< ,s ≤ M r,s = (cid:32) − (cid:80) r< ,s ≤ e i ( rφ + sϕ ) ( δ r,s + δ r +1 ,s )0 1 (cid:33) = (cid:32) e iϕ − e i ( φ + ϕ ) (cid:33) . (D.21)When ζ crosses the positive immaginary axis the Stokes matrix picks up the contri-bution S of the closed solitons bases at z ± . This matrix has a diagonal structure andthe entries S ± correspond to the Stokes jumps of the abelian models D.9. In presenceof aligned critical values there are subtleties in the computation of the Stokes factors.Let us suppose to have N vacua and a set of critical values w k , k = 1 , ...., m ≤ N on a straight line in the W-plane with one physical soliton (and the time-reversed)connecting only the consecutive ones. As discussed in [50], one finds in the leadingIR behavour of the ground state metric terms looking like there were solitons in the k, k + n sector with multiplicity ± /n . This could seem strange, since the number ofBPS particles should be an integer. However, what is really required to be integralis the Stokes matrix S ∈ SL ( N, Z ) and not the soliton matrix µ ij . In this case onehas to define a matrix F belonging to the Lie algebra sl ( N, Q ) that encodes alsothe BPS particles with rational multiplicities. In order to have an integral Stokesoperator we have to exponentiate the algebra element S = e F . (D.22)Coming back to our model, one has to deal with an infinite number of vacua onthe spectral cover with pure immaginary critical values. The correct Stokes jumpsas function of the angles can be extracted by the asymmetric limit of the tt ∗ braneamplitudes for the Landau-Ginzburg models W ± ( z ) in D.9. The computation in [41]provides the following functions 149 − ( φ, ϕ ) = log (cid:0) − e − i ( φ + ϕ ) (cid:1) F + ( φ, ϕ ) = log (cid:0) − e − iφ (cid:1) + log (cid:0) − e − iϕ (cid:1) − log (cid:0) − e − i ( φ + ϕ ) (cid:1) (D.23)from which one can obtain the Stokes jump S = (cid:18) e F − e F + (cid:19) . (D.24)The matrix S includes the contribution of the solitons with negative real partand positive immaginary part. These BPS particles connect the points z − ,r,s with r, s ≤ z + , , and have central charges ∆ W + −− r, − s . The corresponding multiplicitiesare µ +;0 , − ,r,s = − µ − ; r,s ;+ , , = − µ − ;0 , , − r, − s . Each of these particle generates acontribution to the Stokes matrix which reads M r,s = (cid:18) − µ − , , , − r, − s e i ( rφ + sϕ ) (cid:19) . (D.25)Multiplying the Stokes jumps of these solitons for r, s ≤ S = (cid:89) r,s ≤ M r,s = (cid:32) (cid:80) r,s ≤ e i ( rφ + sϕ ) ( δ − r, − s + δ − r +1 , − s ) 1 (cid:33) = (cid:32) e − iϕ − e − i ( φ + ϕ ) (cid:33) . (D.26)Putting all the pieces together we get S ( φ, ϕ ) = (cid:32) e F − ( φ,ϕ ) 1+ e iϕ − e i ( φ + ϕ ) e F − ( φ,ϕ )1+ e − iϕ − e − i ( φ + ϕ ) e F − ( φ,ϕ ) e F + ( φ,ϕ ) + ϕ )(1 − e i ( φ + ϕ ) )(1 − e − i ( φ + ϕ ) ) e F − ( φ,ϕ ) (cid:33) . (D.27)The Stokes matrix encoding the contribution of the BPS states with central chargesin the lower part of the ζ -plane corresponds to the operator S − t ( φ, ϕ ). This hasnot to be intended as the inverse transpose in the 2 × − t ( φ, ϕ ) = [ S ( − φ, − ϕ )] − t , (D.28)which requires also to change the sign of the angles.In the present case we obtain S − t ( φ, ϕ ) = (cid:32) e − F − ( − φ, − ϕ ) + ϕ ) e − F +( − φ, − ϕ ) (1 − e i ( φ + ϕ ) )(1 − e − i ( φ + ϕ ) ) − (1+ e iϕ ) e − F +( − φ, − ϕ ) − e i ( φ + ϕ ) − (1+ e − iϕ ) e − F +( − φ, − ϕ ) − e − i ( φ + ϕ ) e − F + ( − φ, − ϕ ) (cid:33) . (D.29)So, the quantum monodromy H = S − t S of this system is H ( φ, ϕ ) = (cid:32) e F − ( φ,ϕ ) − F − ( − φ, − ϕ ) 1+ e iϕ − e i ( φ + ϕ ) (cid:0) e F − ( φ,ϕ ) − F − ( − φ, − ϕ ) − e F + ( φ,ϕ ) − F + ( − φ, − ϕ ) (cid:1) e F + ( φ,ϕ ) − F + ( − φ, − ϕ ) (cid:33) = (cid:18) − e − i ( φ + ϕ ) − (1 + e iϕ ) e − i ( φ + ϕ ) − (cid:19) (D.30)where in the second equality we have used the expression for F ± in D.23. We see thatdespite the Stokes matrices are singular for some values of the angles, the quantummonodromy has smooth coefficients. This is equivalent to the statement that thephases of the quantum monodromy e πiq R ± ( φ,ϕ ) are smooth in the angles, while theRamond charges q R ± ( φ, ϕ ) are allowed to have integral jumps. The Ramond chargesof the theta-vacua reads q R − = − B (( φ + ϕ ) / π ) , q R + = B (( φ + ϕ ) / π ) − B ( φ/ π ) − B ( ϕ/ π ) , (D.31)where B ( x ) = x − / x ∈ [0 , E tt ∗ Geometry of Modular Curves E.1 Quantum Hall Effect and Modular Curves In this section we are going to study a special class of models which reveal a beautifulconnection between the physics of quantum Hall effect and the geometry of modular151urves [64]. Despite it is not relevant for phenomenological purposes, this classof theories has remarkable properties which enlarge further the rich mathematicalstructure of FQHE. We recall that the prototype of one-particle supersymmetricmodel which is relevant for the FQHE physics is given by the N = 4 Landau-Ginzburg theory with superpotential W ( z ) = (cid:88) ζ ∈ L e ( ζ ) log( z − ζ ) , (E.1)where L is a discrete set of C and e ( ζ ) are real numbers. The variable z is inter-preted as the electron coordinate and the term e ( ζ ) log( z − ζ ) is the two dimensionalcoulombic potential which describes the interaction between the electron and anexternal charge. In the standard setting of the FQH systems the source of electro-static interaction is taken to be L = Λ ∪ S , with Λ a lattice and S a set of positionsof quasi-holes. The effect of the lattice is to reproduce the constant macroscopicmagnetic field with e ( λ ) = 1 units of magnetic flux at a point λ ∈ Λ. At this levelthe expression of the superpotential is just symbolic, since the sum is taken over aninfinite set of points and the function is multi-valued. Therefore it requires a moreprecise definition according to the class of models that one is considering.For a generic choice of the parameters defining W ( z ), the classical vacua are iso-lated, and the elements in the chiral ring are identified with their set of values at thecritical points. However, finding the classical vacua and studying the tt ∗ geometry ofthese models is rather complicated, unless one arranges the set of quasi-holes in somespecial configuration to have an enhancement of symmetry. In this way we can con-struct degenerate models of FQHE which are not realistic for phenomenology, butat least analitically treatable. We want to consider a particular class of theories ofthis type which have an abelian subgroup of symmetry acting transitively on the setof vacua. This is the most convenient limit, since in this case the Berry’s connectioncan be completely diagonalized in a basis of eigenstates of such symmetry and the tt ∗ equations can be derived. It turns out that these models are parametrized moduloisomorphisms by the family of Riemann surfaces Y ( N ) = H / Γ ( N ), labelled by aninteger N ≥ 2, also known as modular curves for the congruence subgroup Γ ( N )of the modular group SL (2 , Z ) [55]. Each point of the curve of level N identifiesa theory where ∂ z W ( z ) is an ellitpic function with a Z N symmetry generated by atorsion point of the elliptic curve C / Λ. For each connected component of the spaceof models one can define its spectral cover as the complex manifold whose pointsidentify a model and a vacuum [31]. These are the modular curves Y ( N ) = H / Γ( N )for the principal congruence subgroup Γ( N ). More precisley, Y ( N ) is a cover of Y ( N ) of degree N , which is the number of vacua in the fundamental cell of thetorus. The tt ∗ equations simplify considerably on the spectral curve and can benormalized in the form of ˆ A N − Toda equations [28]. These appear in all the models152ith a Z N symmetry group which is transitive on the vacua. The modular curvesare manifolds with cusps, which represent physically the RG flow fixed points ofthe theory. These are in correspondence with the equivalence classes of rationalswith respect to the congruence subgroups. An outstandig fact is that for a given N all the ˆ A Q − models with Q | N are embedded in this class of theories as criticallimits, providing the regularity conditions for the solutions to the equations. Anexception is the case of N = 4, where only ˆ A models appear. The beauty of themodular curves is that they possess various geometrical structures. For instance, inthe modular curves Y ( N ) of level N = 3 , , Q ( ζ N + ζ − N ), with ζ N = e πi/N . At the level of superpo-tential, the action of the Galois group is reproduced by a third congruence subgroupwhich enters in this theory, i.e. Γ ( N ). An interesting implication is that the solu-tions of the N -Toda equations are related by the action of the Galois group, sincethe UV cusps described by the ˆ A N − models are all in the same orbit of Γ ( N ).Another interesting phenomenon appearing in this class of theories is that, despitethe covariance of the tt ∗ equations, neither the superpotential nor the ground statemetric, and therefore the Berry’s connection, are invariant under the action of Γ( N ).This apparent contraddiction finds a consistent explanation in the context of theabelian universal cover of the model.The rest of the chapter is organized as follows: In section E.2 we classify up toisomorphisms all the models of the type E.1 with an abelian subgroup of symmetryacting transitively on the vacua. The underlying structure of the modular curves andthe relation between geometry and number theory arise naturally in the derivation.In section E.3 we provide an explicit description of these models. The target mani-fold is not simply connected and one needs to pull-back the model on the universalcover in order to define the Hilbert space and write the tt ∗ equations. On this spacethe symmetry group contains also the generators of loops around the poles in thefundamental cell. The symmetry algebra is non-abelian on the universal cover andthe abelian physics of the punctured plane can be recovered at the level of quantumstates by considering trivial representations of the loop generators. We show thatthis can be done consistently with the tt ∗ equations in E.4. In section E.5 we studythe modular properties of these systems. First we consider the transformation of ∂ z W ( z ) under the congruence subgroups and then of the superpotential. In partic-ular we focus on the critical value of one the vacua which we use to write the tt ∗ equations in the Toda form. This can be defined as holomorphic function only onthe upper half plane and shifts by a costant under a transformation of Γ( N ). Weconnect this phenomenon to the geometry of the modular curves in the simple cases153f genus 0, i.e. with 2 ≤ N ≤ 5. The details of the computation of the constant fora generic N are instead given in F. The modular transformations have also the effectof changing the basis of the symmetry generators and act on the states by modify-ing the representation of the symmetry group. We study this action and how thecomponents of the ground state metric are transformed. In E.6 we provide a clas-sification of the cusps. In particular we study the behavour of the superpotentialsaround these points and distinguish between UV and IR critical regions. Finally,we discuss the boundary conditions of the solutions and how they are related by theGalois group. E.2 Classification of the Models E.2.1 Derivation Our first aim is to classify (up to isomorphisms) all the models in the class of FQHEtheories E.1 with an abelian subgroup of symmetry acting transitively on the vacua.Since the punctured plane is not a simply connected space, one cannot define forthese models a superpotential on the target manifold. So, we have to start theclassification from the derivative ∂ z W ( z ) = (cid:88) ζ ∈ Λ z − ζ + (cid:88) s ∈ S e ( s ) z − s , where Λ = 2 π Z ⊕ πτ Z , τ ∈ H ,e : S −→ C . We stress again that the expression above is just formal and represents a meromor-phic function with a simple pole at each point of L = Λ ∪ S . Moreover, in thisclassification we allow the charges of the quasi-holes to be complex. This is mathe-matically consistent, since the superpotential is a complex function.The action of the abelian group is transitive on the vacua and, in the case of a nontrivial kernel, can always be made faithful. The transitivity implies that the set ofvacua is a copy of the abelian group. In particular, given that the zeroes of ∂ z W ( z )cannot have accumulation points, it must be finitely generated. The abelian sub-groups of C satisfying this property are lattices. Since the group acts freely also on154he set of poles and the principal divisor of ∂ z W ( z ) has degree 0 , it is immediateto conclude that also L is a lattice, as well as (the faithful representation of) theabelian subgroup of symmetries of our model.If we want L to be at least a pseudosymmetry for ∂ z W ( z ), the function e ( s ) mustbe extended to a multiplicative periodic character: ∂ z W ( z + ζ ) = e ( ζ ) ∂ z W ( z ) , e ( ζ + λ ) = e ( ζ ) , for each ζ ∈ L , λ ∈ Λ. By definition of homomorphism, the kernel of e must be asubgroup of L . Up to a redefinition of the initial set of holes, this is represented bythe sublattice Λ, which is a symmetry for ∂ z W ( z ) in a strict sense.Up to this point, we have a model for each lattice L and a character e which isperiodic of a sublattice Λ ⊂ L . Given the periodicity of e , we can equivalently restrict the analysis to primitivecharacters, i.e. with trivial kernel: e : L/ Λ −→ C , where L/ Λ (cid:39) Z N ⊕ Z N for two positive integers N , N such that N | N . The factthat e is primitive implies the isomorphism Z N ⊕ Z N (cid:39) Z N N . But, according to thechinese remainder theorem, this can be true only if the two integers are coprime.The consistency between the two conditions on the integers N , N requires that N = 1 and N = N ≥ 2, with L/ Λ (cid:39) Z N , N ≥ . If we set N = 1 we obtain the trivial case in which there are no quasi-holes.In conclusion, the models are classified by couples ( E Λ , Q ), where • E Λ is the elliptic curve C / Λ, • Q ∈ E [ N ] = { P ∈ C / Λ | N P ∈ Λ } such that e ( Q ) = e πiN .The set E [ N ] is called the N-torsion subgroup of the additive torus group C / Λ.Once Λ and the level N are fixed, the choice of the torsion point specifies an em-bedding of the cyclic subgroup L/ Λ (cid:39) Z N in the elliptic curve. In particular, thetorsion point must be of order N , i.e. such that N Q ∈ Λ but nQ (cid:54)∈ Λ for 1 < n < N . This is true for a compact Riemann surface, as it turns out to be the target manifold. We are going to show that e is non trivial. .2.2 Modular Curves Since we are classifying models up to strict equivalence, we have to identify thosewhich are related by an isomorphism. It is known that there is a bijection betweenthe set of equivalence classes of elliptic curves endowed with a N -torsion point andthe space Y ( N ) = H / Γ ( N ) , where Γ ( N ) is a subgroup of SL (2 , Z ) defined by the congruence conditionΓ ( N ) = (cid:26) γ ∈ SL (2 , Z ) : γ = (cid:18) a bc d (cid:19) = (cid:18) ∗ (cid:19) mod N (cid:27) . A complete proof of this result can be found in [55]. It is worth to recall that amatrix γ ∈ SL (2 , Z ) acts on the upper half plane by the usual fractional lineartransformation τ (cid:48) = (cid:18) a bc d (cid:19) τ = aτ + bcτ + d , (E.2)and induces on the points of the elliptic curve the isogeny z + Λ τ −→ mz + Λ τ (cid:48) , forsome m ∈ C such that m Λ τ = Λ τ (cid:48) . These maps are the only bijections which preservethe group structure of the elliptic curve. With these definitions it is immediate toshow that the enhanced elliptic curve ( E Λ τ (cid:48) , Q ) is isogenous to ( E Λ τ , π/N + Λ τ ),where τ (cid:48) = γ ( τ ) for some γ ∈ SL (2 , Z ), and that transformations of Γ ( N ) arethe only ones which preserve the choice of the torsion point. So, we can define themoduli space for Γ ( N ) as S ( N ) = { ( E Λ τ , π/N + Λ τ ) , τ ∈ H } / ∼ where τ ∼ τ (cid:48) if and only if Γ ( N ) τ = Γ ( N ) τ (cid:48) , and state the bijection S ( N ) ∼ −→ Y ( N ) . The space Y ( N ) is topologically a complex manifold with cusps and can be com-pactified. One first has to extend the action of SL (2 , Z ) to the rational projectiveline P ( Q ) = Q ∪ {∞} . Given (cid:18) p qr t (cid:19) ∈ SL (2 , Z ), we have156 −→ H = H ∪ P ( Q ) , (cid:18) p qr t (cid:19) ac = pa + qcra + tc , (cid:18) p qr t (cid:19) ∞ = pr with a, c ∈ Z such that gcd( a, c ) = 1.The set of cusps is given by C Γ ( N ) = P ( Q ) / Γ ( N ) and is finite. By adding thesepoints to Y ( N ) one obtains Y ( N ) −→ X ( N ) = H / Γ ( N ) . The space X ( N ) is called modular curve for Γ ( N ) and can be shown to have thestructure of a compact Riemann surface. Since the two dimensional lattice becomesdegenerate when τ approaches the rational projective line, the cusps cannot bestrictly considered as members of this class of theories, but rather as critical limits .So, we learn that the whole space of models can be written as union of connectedcomponents labelled by the integer N : A = (cid:91) N ≥ X ( N ) , where X ( N ) parametrizes the subclass of theories of level N . It follows from thederivation that each point on A identifies a model up to isomorphisms.The next step is to classify the vacua for this family of theories. Let us consider themodular curve of level N . The derivative of the superpotential is an elliptic functionon the elliptic curve C / Λ τ which has a simple pole at each point kQ + Λ τ , k =0 , ..., N − P + kQ + Λ τ , k = 0 , ..., N − P ∈ C/ Λ τ such that ∂ z W ( P ) = 0. Given that the principal divisor is vanishing byAbel’s theorem, we getdiv( ∂ z W ( z )) = (cid:88) ≤ k Another congruence subgroup of SL (2 , Z ) which plays an important role in thisclassification isΓ ( N ) = (cid:26) γ ∈ SL (2 , Z ) : γ = (cid:18) a bc d (cid:19) = (cid:18) ∗ ∗ ∗ (cid:19) mod N (cid:27) . The three congruence subgroups that we have defined satisfy the chain of inclusionsΓ( N ) ⊂ Γ ( N ) ⊂ Γ ( N ) ⊂ SL (2 , Z ). Moreover, Γ ( N ) is a normal subgroup ofΓ ( N ). The action of Γ ( N ) on the space of models is more clear when we choose Q = πN . It is evident that γ ∈ Γ ( N ) does not leave the model invariant and mapsthe torsion point into an inequivalent one γ ∗ Q = 2 π ( cτ + d ) N ∼ πdN , for an integer d coprime with N . The effect on the model corresponds to modify the character bythe formula σ a ( e ( Q )) = e ( Q ) a = e πiaN , where a is the inverse of d in Z N . Therefore,it is manifest that Γ ( N ) reproduces the action of the Galois group of the cyclotomicextension [ Q ( ζ N ) : Q ]. This number field is obtained by adjoining a primitive N -throot of unity ζ N to the rational numbers. Such remarkable connection reveals thealgebraic nature of the modular curves. The above formula suggests to define acharacter e l ( Q ) = e πilN depending on an integer l coprime with N , which we callco-level of the modular curve. It is clear that, for a fixed lattice Λ τ , a differentchoice of the co-level corresponds to pick a different point on the space of models.With this definition, a transformation of Γ ( N ) can be seen as a permutation of theco-levels.We can be more precise about these statements by keeping into account that Γ ( N )contains the matrix − I . We know that a point on X ( N ) parametrizes an elliptic160urve and a specific embedding of Z N . However, a cyclic group has always twogenerators which are one the inverse of the other. It is clear that − I acts on themodel as a parity transformation, since it does not change the point on X ( N ) butinverts the sign of the torsion point. This means that the co-levels l and N − l areactually two descriptions of the same model. So, except for the trivial case of N = 2,since − I is in Γ ( N ) but not in Γ ( N ) the degree of the cover H / Γ ( N ) → H / Γ ( N )is [Γ ( N ) : Γ ( N )] / φ ( N ) / 2, where φ ( N ) is the Euler totient function whichcount the elements of { , ..., N − } coprime with N . As one can expect, this isalso the degree of the number field defining this class of theories, which is actuallythe real cyclotomic extension Q ( ζ N + ζ − N ). The cusps of Γ ( N ) fall in equivalenceclasses of the Galois group described by the set C Γ ( N ) = P ( Q ) / Γ ( N ). We willdiscuss the orbits of Γ ( N ) in the set of critical theories in section 5. E.3 Geometry of the Models E.3.1 The Model on the Target Manifold Now we want to translate our abstract classification into an explicit description ofthese models. A fundamental property of the elliptic functions is that they areuniquely specified (up to multiplicative constants) by the positions and orders oftheir zeroes and poles. Let us focus on the modular curve of level N and co-level l and choose ( P, Q ) = ( − πlτN + Λ τ , πN + Λ τ ) as torsion points. For convenience weinvert the lattices of poles and vacua with respect to the previous derivation. Thiscan be done with the translation z → z − πlτN . The derivative of the superpotentialfor this class of theories is ∂ z W ( N,l ) ( z ; τ ) = N − (cid:88) k =0 e πilkN (cid:20) ζ (cid:18) z − πN ( lτ + k ); τ (cid:19) + 2 η kN (cid:21) , (E.5)where z ∈ S = C \ (cid:8) πN ( lτ + k ) + Λ τ , k = 0 , ...., N − 1; Λ τ = 2 π Z ⊕ πτ Z (cid:9) .The Weierstrass zeta function ζ ( z ; τ ) is defined with the conventions η = ζ ( π ; τ ), η = ζ ( πτ ; τ ) [57]. This function is meromorphic on S with simple poles and simplezeroes respectively in π ( lτ + k ) N + Λ τ and πkN + Λ τ , k = 0 , ..., N − 1. By definitionof elliptic function, the sum of the residue inside the fundamental cell is vanishing.From a physical point of view this means that the flux of the magnetic field iscancelled by that of the quasi-holes charges.The algebra of the abelian symmetry group of this model is generated by threeoperators σ, A, B , defined by the actions161 : z −→ z + 2 π/N ; ∂ z W ( N,l ) ( z + 2 π/N ; τ ) = e πilN ∂ z W ( N,l ) ( z ; τ ) ,A : z −→ z + 2 π ; ∂ z W ( N,l ) ( z + 2 π ; τ ) = ∂ z W ( N,l ) ( z ; τ ) ,B : z −→ z + 2 πτ ; ∂ z W ( N,l ) ( z + 2 πτ ; τ ) = ∂ z W ( N,l ) ( z ; τ ) , (E.6)with the additional relation σ N = A . These transformations follows from thequasi-periodicity properties of the Weierstrass function. The double periodicityof ∂ z W ( N,l ) ( z ; τ ) allows to identity points wich differ by an element of Λ τ , so thatthe model is naturally projected on the torus K = S / Λ τ . With this identifica-tion we can work with just N critical points, denoted by the equivalence classes (cid:2) πkN (cid:3) , k = 0 , ..., N − 1. From the property ζ ( − z ; τ ) = − ζ ( z ; τ ), we find that thesuperpotentials of co-levels l and N − l are actually related by a parity transformation ∂ z W ( N,N − l ) ( − z ; τ ) = − ∂ z W ( N,l ) ( z ; τ ) (E.7)and therefore, as pointed out in the previous section, they describe the same modelwith inverse torsion points generating the Z N symmetry.We can also write for this class of theories a symbolic ‘primitive’ W ( N,l ) ( z ; τ ) = “ N − (cid:88) k =0 e πilkN log Θ (cid:20) − lN − kN (cid:21) (cid:16) z π ; τ (cid:17) ” , (E.8)where Θ (cid:20) αβ (cid:21) ( z ; τ ) is the theta function of character ( α, β ) ∈ R and quasi-periods1 , τ . This function is multi-valued and cannot be really considered as superpotentialof the model. Indeed, the target manifold is not simply connected and we cannotfind a true primitive on this space. E.3.2 Abelian Universal Cover In order to define a superpotential, as well as the Hilbert space of the theory, weneed to pull-back the model on the abelian universal cover of S . We remind thedefinition of universal cover as space of curves162 = { p : [0 , −→ S , p (0) = p ∗ ∈ S} / ∼ , ∼ : p ∼ q = (cid:40) p (1) = q (1) ,p · q − = 0 in the homology group H ( S , Z ) . (E.9)By the pull-back of ∂ z W ( N,l ) ( z ; τ ) on this space we can define the superpotential ofthe theory: W ( N,l ) ( p ; τ ) = (cid:90) p ∂ z W ( N,l ) ( z ; τ ) dz. (E.10)Contrary to E.8, this is a well defined function which assigns a single value to eachequivalence class of paths in H .In the covering model we have many more vacua than before, but at the same timea larger symmetry group to classify them. An evident subgroup is the deck groupof the cover, i.e. the homology. Let us fix a point z ∈ S . We choose as local basisof H ( S ; Z ) the one given by the curves γ a , γ b , obtained by applying respectivelythe lattice vectors a = 2 π, b = 2 πτ to the base point z , and the anticlockwise loops (cid:96) , ..., (cid:96) N − encircling the first N − γ a , γ b (Figure 4). The action on the superpotential of the corresponding operators is A ∗ W ( N,l ) ( p ; τ ) = (cid:90) γ a · p ∂ z W ( N,l ) ( z ; τ ) dz = W ( N,l ) ( p ; τ ) + (cid:90) γ a ∂ z W ( N,l ) ( z ; τ ) dz,B ∗ W ( N,l ) ( p ; τ ) = (cid:90) γ b · p ∂ z W ( N,l ) ( z ; τ ) dz = W ( N,l ) ( p ; τ ) + (cid:90) γ b ∂ z W ( N,l ) ( z ; τ ) dz,L ∗ k W ( N,l ) ( p ; τ ) = (cid:90) (cid:96) k · p ∂ z W ( N,l ) ( z ; τ ) dz = W ( N,l ) ( p ; τ ) + (cid:73) (cid:96) k ∂ z W ( N,l ) ( z ; τ ) dz. (E.11)Compatibly with the definition of symmetry, the superpotential shifts by a constantunder the action of the homology generators.In order to define the action of σ on H we need to specify a path γ σ connecting z to z + πN . Let us fix the point p ∗ in the definition E.9. We choose a path γ ∗ a γ b z x x x x x. . . . . . . . . . . . . . . . . . . . . (cid:96) (cid:96) (cid:96) k (cid:96) N − Figure 4: Homology generatorsconnecting p ∗ to p ∗ + πN . If the point is not orizontally aligned with the poles wecan take for instance a straight line. Now, given a curve p in H , we define γ σ suchthat γ σ · p = σ ( p ) · γ ∗ in H ( S ; Z ), where σ ( p ) denotes the curve p shifted by πN (Figure 5). Choosing the integration constant so that W ( N,l ) ( γ ∗ ; τ ) = 0, we get thetransformation of the superpotential σ ∗ W ( N,l ) ( p ; τ ) = (cid:90) γ σ · p ∂ z W ( N,l ) ( z ; τ ) dz = (cid:90) σ ( p ) · γ ∗ ∂ z W ( N,l ) ( z ; τ ) dz = e πilN W ( N,l ) ( p ; τ ) + W ( N,l ) ( γ ∗ ; τ ) = e πilN W ( N,l ) ( p ; τ ) . (E.12)Inequivalent definitions of γ σ differ by compositions with the loops (cid:96) k and gener-ate a different constant W ( N,l ) ( γ ∗ ; τ ) in the transformation. Indipendently from thechoice, by composing N times γ σ we obtain an curve in H ( S ; Z ) homologous to γ a ,consistently with the operatorial relation σ N = A . γ ∗ γ σ p ∗ σ ( p ) p σ ( p ∗ ) p (1) σ ( p )Figure 5: Definition of γ σ The generators σ, A, B, ( L k ) k =0 ,...,N − represent a complete basis for the symmetrygroup of the covering model. It is clear from Figure 6 that the algebra is non-abelianon the space of curves. Indeed, the generator σ do not commute with B and L k according to the algebraic relations 164 B = L Bσ, σL k = L k +1 σ, k = 0 , ..., N − , (E.13)where we have L N − = N − (cid:81) k =0 L − k . We see that the obstruction to abelianity on theuniversal cover is represented by the generators of loops. In the faithful representa-tion on the plane these act trivially and A, B, σ commute. This is not inconsistent,since the physics must be abelian on the target manifold but not necessarily on thecovering model. The abelianity which we have asked in the classification can berecoverd at the quantum level by projecting the Hilbert space on the trivial rep-resentation of L k . We will discuss this point in the next section. There are notproblems instead in the commutation between σ and A and clearly the subgroupgiven by the homology is abelian. B B (cid:48) B − B (cid:48) x xx x σ zσ σL k L k +1 L Figure 6: Algebra of curvesNow we can classify the vacua of the model defined on H . The vacua on the targetmanifold are represented by the points z k = πkN , k = 0 , ..., N − , up to periodicidentification of Λ τ . For these points there is an ambiguity in the definition of thecycle γ b , since there is a pole along the side of the fundamental cell. We chooseconventionally the homology path which bypasses the pole on the left. Now thatthe homology based on the critical points is well defined, we fix p ∗ away from the setof vacua and pick a curve p connecting p ∗ to z as representative of z in H . All theother representatives in the corresponding fiber can be obtained by composing p with the cycles of H ( S , Z ) based on z . Moreover, we can map fibers over differentpoints to each other with the action of σ . Therefore, the vacua on H are labelledby the curves 165 k ; m,n,n ,...,n N − = γ ma · γ nb · N − (cid:89) r =0 (cid:96) n r r · γ kσ · p (E.14)where k = 0 , ..., N − , and m, n, n , ..., n N − ∈ Z . By definition of abelian universalcover, each fiber is a copy of the homology group.We can also provide the corresponding critical values by computing the integralsin E.11. Let us take again the curve p . By the relation A = σ N and the residueformula we have A ∗ W ( N,l ) ( p ; τ ) = W ( N,l ) ( p ; τ ) ,L ∗ k W ( N,l ) ( p ; τ ) = W ( N,l ) ( p ; τ ) + 2 πi e πilkN . (E.15)Moreover, from the first algebraic identity in E.13, we easily get B ∗ W ( N,l ) ( p ; τ ) = W ( N,l ) ( p ; τ ) + 2 πie πilN − . (E.16)Finally, by acting with the generators σ, A, B, ( L k ) k =0 ,...,N − on W ( N,l ) ( p ; τ ), weobtain the whole set of critical values W ( N,l ) ( p k ; m,n,n ,...,n N − ; τ ) = A ∗ m B ∗ n (cid:89) ≤ r ≤ N − L ∗ n r r σ ∗ k W ( N,l ) ( p ; τ ) = e πilkN W ( N,l ) ( p ; τ ) + n πie πilN − N − (cid:88) r =0 n r πi e πilrN . (E.17) E.4 tt ∗ Equations E.4.1 General Case Our next aim is to derive the tt ∗ equations for this class of models. We first introducethe point basis Φ k ; m,n,n ,...,n N − defined byΦ k ; m,n,n ,...,n N − ( p k (cid:48) ; m (cid:48) ,n (cid:48) ,n (cid:48) ,...,n (cid:48) N − ; τ ) = δ k,k (cid:48) δ m,m (cid:48) δ n,n (cid:48) N − (cid:89) r =0 δ n r ,n (cid:48) r . (E.18)166he isomorphism between R and C is realized by the identificationΦ k ; m,n,n ,...,n N − −→ | p k ; m,n,n ,...,n N − (cid:105) = A m B n N − (cid:89) r =0 L n r r σ k | p (cid:105) , where the operatorial action of the symmetry algebra is naturally transported onthe Hilbert space. The vacuum space of the theory on H decomposes in a directsum of irreducible representations of the homology group: V H (cid:39) L (cid:0) Hom (cid:0) Z N +1 , U (1) (cid:1)(cid:1) ⊗ C N , provided that N + 1 is the rank of H ( S , Z ) and N the number of vacua on S . Apoint basis of ‘theta-vacua’ for V H is | k ; α, β, λ , ..., λ N − (cid:105) = e − iαkN (cid:88) m,n,n r ∈ Z e − i ( mα + nβ + (cid:80) N − r =0 n r λ r ) A m B n N − (cid:89) r =0 L n r r σ k | p (cid:105) , (E.19)where the angles α, β, λ , ..., λ N − ∈ [0 , π ] label representations of H ( S , Z ). Sincethe homology is abelian, the angles are defined simultaneously in this common basisof eigenstates: A | k ; α, β, λ , ..., λ N − (cid:105) = e iα | k ; α, β, λ , ..., λ N − (cid:105) ,B | k ; α, β, λ , ..., λ N − (cid:105) = e iβ | k ; α, β, λ , ..., λ N − (cid:105) ,L r | k ; α, β, λ , ..., λ N − (cid:105) = e iλ r | k ; α, β, λ , ..., λ N − (cid:105) . Using the symmetries A, B, L r , one finds that in the point basis the ground statemetric diagonalizes with respect to the angles:167 j ; α (cid:48) , β (cid:48) , λ (cid:48) , ..., λ (cid:48) N − | k ; α, β, λ , ..., λ N − (cid:105) = δ ( α − α (cid:48) ) δ ( β − β (cid:48) ) N − (cid:89) r =0 δ ( λ r − λ (cid:48) r ) g k, ¯ j ( α, β, λ , ..., λ N − ) , where the matrix coefficients g k, ¯ j ( α, β, λ , ..., λ N − ) have the Fourier expansion: g k, ¯ j ( α, β, λ , ..., λ N − ) = e iN α ( j − k ) (cid:88) r,s,t q ∈ Z e i ( rα + sβ + (cid:80) N − q =0 t q λ q ) g k, ¯ j ( r, s, t , ..., t N − ) ,g k, ¯ j ( r, s, t , ..., t N − ) = (cid:104) p j ; m,n,n ,...,n N − | p k ; m (cid:48) ,n (cid:48) ,n (cid:48) ,...,n (cid:48) N − (cid:105)| m − m (cid:48) = r,n − n (cid:48) = s,n q − n (cid:48) q = t q . The metric cannot be diagonalized also as N × N matrix, since σ do not commutewith B and L r . The pull-back of the Landau-Ginzburg model on the universalcover is a necessary operation to write the tt ∗ equations, but the price to pay is tohave more vacua and a larger space of solutions. We are not interested in the mostgeneral solution to the tt ∗ equations, but in the subclass reproducing the abelianphysics of the punctured plane. Moreover, the importance of an abelian symmetrygroup with a transitive action on the vacua is that we can completely diagonalizethe ground state metric. Without this property, even writing the tt ∗ equationsbecomes an hopeless problem. Therefore, we have to truncate the tt ∗ equationsin such a way that the solutions are compatible with the abelian representation ofthe symmetry group. We cannot just set the loop angles to be 0, since these aredifferential variables which enter in the equations. So, we have to check that theansatz of a solution with constant vanishing λ r is consistent with all the equations.From the equality2 πie πilN − πiN N − (cid:88) k =0 ke πilkN = 2 πiN N − (cid:88) k =0 ( k − N + 1) e πilkN , we learn that the combination ˜ B = B N N − (cid:81) k =0 L N − k − k leaves invariant the superpoten-tial. Moreover, one can see that it commutes with σσ ˜ B = σB N N − (cid:89) k =0 L N − k − k = B N L N N − (cid:89) k =0 L N − k − k +1 N − (cid:89) k =0 L − k σ = BL N − N − (cid:89) k =0 L N − ( k +1) − k +1 σ = B N N − (cid:89) k =0 L N − k − k σ = ˜ Bσ A, ˜ B, L r as generators of the homology, we can find a basis in which the eigenvalues of σ are simultaneously defined with the two angles α, ˜ β = N β + (cid:80) N − k =0 ( N − k − λ k corresponding to A, ˜ B .The couplings we can vary in these models are the lattice parameter τ and the usualoverall scale multiplying the superpotential. To avoid clash of notations, we denotewith µ this coupling. The corresponding chiral ring coefficients in the point basis ofthe covering model read( C τ ) k (cid:48) ; m (cid:48) ,n (cid:48) ,n (cid:48) ,...,n (cid:48) N − k ; m,n,n ,...,n N − = µ∂ τ W ( N,l ) Φ k (cid:48) ; m (cid:48) ,n (cid:48) ,n (cid:48) ,...,n (cid:48) N − ( p k ; m,n,n ,...,n N − ; τ )= δ k,k (cid:48) δ m,m (cid:48) δ n,n (cid:48) N − (cid:89) r =0 δ n r ,n (cid:48) r µ∂ τ W ( N,l ) ( p ; τ ) e πilkN , ( C µ ) k (cid:48) ; m (cid:48) ,n (cid:48) ,n (cid:48) ,...,n (cid:48) N − k ; m,n,n ,...,n N − = W ( N,l ) Φ k (cid:48) ; m (cid:48) ,n (cid:48) ,n (cid:48) ,...,n (cid:48) N − ( p k ; m,n,n ,...,n N − ; τ )= δ k,k (cid:48) δ m,m (cid:48) δ n,n (cid:48) N − (cid:89) r =0 δ n r ,n (cid:48) r x (cid:32) W ( N,l ) ( p ; τ ) e πilkN + N − (cid:88) r =0 n r πi e πilrN (cid:33) , (E.20)where we have replaced B with ˜ B in the computation of the critical values in E.17.The action of C τ and C µ is naturally projected on the theta-sectors of V H : C point basis τ ( α, ˜ β, λ , ..., λ N − ) j,k = µ∂ τ W ( N,l ) ( p ; τ ) e πilkN δ j,k ,C point basis µ ( α, ˜ β, λ , ..., λ N − ) j,k = (cid:32) W ( N,l ) ( p ; τ ) e πilkN − π N − (cid:88) r =0 e πilrN ∂∂λ r (cid:33) δ j,k . (E.21)The ground state metric satisfy the following set of equations ∂ ¯ τ ( g∂ τ g − ) = (cid:2) C τ , gC † τ g − (cid:3) ,∂ ¯ µ ( g∂ µ g − ) = (cid:2) C µ , gC † µ g − (cid:3) ,∂ ¯ µ ( g∂ τ g − ) = (cid:2) C τ , gC † µ g − (cid:3) , (E.22)169s well as the complex conjugates. If we demand that λ = ... = λ N − = 0 and ∂g∂λ r = 0 for each r , the derivatives ∂∂λ r do not contribute to the tt ∗ equations andthe F-term deformations can be truncated at the non trivial part C point basis τ ( α, ˜ β ) j,k = µ∂ τ W ( N,l ) ( p ; τ ) e πilkN δ j,k ,C point basis µ ( α, ˜ β ) j,k = W ( N,l ) ( p ; τ ) e πilkN δ j,k , (E.23)with ˜ β = N β . It is clear that with this ansatz the equations given above canbe written universally as a unique equation with respect to the critical value w = W ( N,l ) ( p ; τ ): ∂ ¯ w ( g∂ w g − ) = (cid:2) C w , gC † w g − (cid:3) , (E.24)where C point basis w ( α, ˜ β ) j,k = e πilkN δ j,k . (E.25)The equations in the parameters τ and µ can be obtained from this one by specifyingthe variation of w and ¯ w . If the equation in the canonical variable is solved with theboundary conditions given by the cusps, the solution automatically satisfies all theequations in E.22. Hence, it is convenient to pull-back the equation on the spectralcover of the model where one can use w = W ( N,l ) ( p ; τ ) as unique complex coordinateto parametrize models and vacua. By consistency, the cusps should reproduce thecorrect regularity conditions for the truncated equations. We are going to discussthe boundary conditions for the tt ∗ equations in section E.6 and we will see thatthis is the case.Now that we have recovered the abelian representation of the symmetry group, wecan construct a common basis of eigenstates for σ, A, B in which the ground statemetric diagonalizes completely. Given the action of σ on the point basis σ | k ; α, β (cid:105) = e iαN | k + 1; α, β (cid:105) , we can define a set of σ -eigenstates as 170 j ; α, β (cid:105) = N − (cid:88) k =0 e − πilkjN | k ; α, β (cid:105) ,σ | j ; α, β (cid:105) = e iN ( α +2 πlj ) | j ; α, β (cid:105) . (E.26)With this normalization we have the periodicity | k + N ; α, β (cid:105) = | k ; α, β (cid:105) (E.27)and we note that the given definitions are consistent with the relation σ N = A .In the σ -basis the metric becomes g k, ¯ j ( α, β ) = δ k,j e ϕ k ( α,β ) , (E.28)where the ϕ k ( α, β ) , k = 0 , .., N − , are real functions of the angles. It is straight-forward to derive the tt ∗ equations satisfied by these functions. Using C σ − basis w ( α, β ) = ... ... . . . . .. . . . . ... ... , one finds the well known ˆ A N − Toda equations: ∂ ¯ w ∂ w ϕ + e ϕ − ϕ − e ϕ − ϕ N − = 0 ,∂ ¯ w ∂ w ϕ i + e ϕ i +1 − ϕ i − e ϕ i − ϕ i − = 0 , i = 1 , ..., N − ∂ ¯ w ∂ w ϕ N − + e ϕ − ϕ N − − e ϕ N − − ϕ N − = 0 , (E.29)where the dependence on the angles and w is understood. These equations appeartipically in the models with a Z N symmetry group acting transitively on the vacua.We are going to discuss the solution to these equations in section E.6. In particular,171e will see that the RG fixed points reproduce the appropriate boundary conditionsfor the Toda equations, providing a consistency check for the abelian truncation.We can also provide an expression for the symmetric pairing of the topologicaltheory. The operators in the chiral ring corresponding to the basis in E.19 with λ r = 0 and E.26 are respectively χ k ( α, β ) = e − iαkN (cid:88) m,n ∈ Z e − i ( mα + nβ ) Φ k ; m,n , Ψ j ( α, β ) = N − (cid:88) k =0 e − πilkjN χ k ( α, β ) . The topological metric can be computed with the formula 3.80 for one chiral super-field. In the point basis the expression isRes W [ χ k ( α, β ) , χ m ( α (cid:48) , β (cid:48) )] = δ ( α − α (cid:48) ) δ ( β − β (cid:48) ) η point basis k,m ( α, β ) (E.30)where η point basis k,m ( α, β ) = (cid:0) ∂ z W ( N,l ) (0; τ ) (cid:1) − e − ikN (2 α +2 πl ) δ k,m . (E.31)In the basis of σ -eigenstates one getsRes W [Ψ n ( α, β ) , Ψ m ( α (cid:48) , β (cid:48) )] = δ ( α − α (cid:48) ) δ ( β − β (cid:48) ) η σ − basis n,m ( α, β ) , (E.32)where η σ − basis n,m ( α, β ) = (cid:0) ∂ z W ( N,l ) (0; τ ) (cid:1) − N − (cid:88) k =0 e − ikN (2 πl ( m + n +1)+2 α ) . (E.33)We can choose a basis and use one of these expressions to impose the tt ∗ realityconstraint, where the complex conjugates g ∗ , η ∗ in terms of the matrices g ( α, β ), η ( α, β ) are respectively 172 ∗ ( α, β ) = [ g ( − α, − β )] ∗ ,η ∗ ( α, β ) = [ η ( − α, − β )] ∗ . This condition gives g ( − α, − β ) in terms of g ( α, β ), but an explicit computation ishard in general. In the case of a trivial representation of the homology, the realityconstraint simplifies, becoming the same of the A N models [28]. E.4.2 Peculiarities of the N=2 Level We briefly discuss some peculiar aspects of the class of models of level 2. Thederivative of the superpotential is ∂ z W ( z ; τ ) = ζ ( z − πτ ; τ ) − ζ ( z − πτ − π ; τ ) − η , (E.34)where z ∈ S = C \ { πτ, πτ + π + Λ τ ; Λ τ = 2 π Z ⊕ πτ Z } . The simple poles andsimple zeroes are located respectively in πτ, πτ + π + Λ τ and 0 , π + Λ τ . In this casethe Galois group acts trivially and we have only one co-level.In addition to the generators σ, A, B, L already discussed, the symmetry group ofthe models of level 2 contains also the parity transformation ι : z −→ − z ; ∂ z W ( − z ; τ ) = − ∂ z W ( z ; τ ) , which follows from E.7 for N = 2 and l = 1.This operator satisfies the following commutation relations with the generators ofthe abelian group: ιA = A − ι,ιB = B − ι,ιL = Lι,ισ = σ − ι. The action of ι , as well as its algebraic relations, can be extended to the abelianuniversal cover. There are inequivalent choices which differ by compositions with theloop generator L . Indipendently from the definition, the set of curves with p (1) = 0is left invariant by the action of ι . 173he presence of an extra symmetry, as well as the possibility of working with 2 × tt ∗ equations. Apoint basis of theta-vacua for this family of theories is | α, β, λ (cid:105) = (cid:88) m,n,j ∈ Z e − i ( mα + nβ + jλ ) A m B n L j | p (cid:105) , | α, β, λ (cid:105) = (cid:88) m,n,j ∈ Z e − i ( mα + nβ + jλ ) A m B n L j σ | p (cid:105) . Setting λ = 0, the operators σ and ι act on these states as: σ | α, β (cid:105) = | α, β (cid:105) , σ | α, β (cid:105) = e iα | α, β (cid:105) ,ι | α, β (cid:105) = | − α, − β (cid:105) , ι | α, β (cid:105) = e − iα | − α, − β (cid:105) . The ground state metric in this basis is represented by the 2 x 2 hermitian matrix g ( α, β ) = (cid:18) g ( α, β ) g ( α, β ) g ( α, β ) g ( α, β ) (cid:19) . The symmetries σ and ι imply respectively g ( α, β ) = g ( α, β ) , g ( α, β ) = e iα g ( α, β ) , and g ( α, β ) = g ( − α, − β ) , g ( α, β ) = e − iα g ( − α, − β ) . Therefore the metric can be written as g ( α, β ) = (cid:32) A ( α, β ) e iα B ( α, β ) e − iα B ( α, β ) A ( α, β ) (cid:33) , where A ( − α, − β ) = A ( α, β ) and B ( − α, − β ) = B ( α, β ). The transpose and complexconjugate of g read 174 T ( α, β ) = [ g ( − α, − β )] T ,g ∗ ( α, β ) = [ g ( − α, − β )] ∗ , and therefore g † ( α, β ) = [ g ( − α, − β )] † . The hermiticity of g implies that A ( α, β ) and B ( α, β ) are real functions, while thepositivity requires that A ( α, β ) > 0. The matrix C w and the topological metric inthis basis are C w ( α, β ) = (cid:18) − (cid:19) , η ( α, β ) = (cid:0) ∂ z W (0; τ ) (cid:1) − (cid:18) − (cid:19) . (E.35)We see that, by the parity properties of A ( α, β ) and B ( α, β ) implied by ι , the realityconstraint ?? reduces to | ∂ z W (0; τ ) | ( A ( α, β ) − B ( α, β ) ) = 1 . (E.36)As a conquence, the metric can be parametrized in term of a single function of theangles A ( α, β ) = 1 | ∂ z W (0; τ ) | cosh [ L ( α, β )] ,B ( α, β ) = 1 | ∂ z W (0; τ ) | sinh [ L ( α, β )] . (E.37)Finally, by plugging the given expressions of g ( α, β ) and C w in E.24 we find theSinh-Gordon equation ∂ ¯ w ∂ w L ( α, β ) = 2 sinh [2 L ( α, β )] . (E.38)175 .5 Modular Properties of the Models E.5.1 Modular Transformations of ∂ z W ( N,l ) ( z ; τ )The underlying structure of this class of models is the theory of the modular curves.In particular, as discussed in E.2.2, the space of models and its spectral cover are de-scribed respectively by the modular curves for Γ ( N ) and Γ( N ). Now that we haveprovided an explicit description of these systems, we want to investigate the modularproperties of the superpotential, as well as the ground state metric, with respect tothe relevant congruence subgroups of SL (2 , Z ). Let us first consider ∂ z W ( N,l ) ( z ; τ ).Given a modular transformation γ = (cid:18) a bc d (cid:19) ∈ Γ ( N ), the derivative of the super-potential transforms as ∂ z W ( N,l ) (cid:18) zcτ + d ; aτ + bcτ + d (cid:19) = (E.39)( cτ + d ) N − (cid:88) k =0 e πilkN (cid:20) ζ (cid:18) z − πN (( aτ + b ) l + k ( cτ + d )); τ (cid:19) + 2 dη + cη N k (cid:21) =( cτ + d ) N − (cid:88) k =0 e πilkN (cid:20) ζ (cid:18) z − πN bl − πN ( lτ + k ); τ (cid:19) + 2 η kN (cid:21) =( cτ + d ) ∂ z W ( N,l ) (cid:18) z − πN bl ; τ (cid:19) , where we have used the fact that a, d = 1 mod N , c = 0 mod N and the modularproperties of the Weierstrass function: ζ (cid:18) zcτ + d ; aτ + bcτ + d (cid:19) = ( cτ + d ) ζ ( z ; τ ) ,η (cid:18) aτ + bcτ + d (cid:19) = ( cτ + d )( dη + cη ) . (E.40)We see that Γ ( N ) preserves only one of the torsion point. The transformation πτN → π ( aτ + b ) N results in a shift of poles and vacua by πN bl . If we set b = 0 mod N ,176.e. γ ∈ Γ( N ), also the order of vacua is preserved modulo periodicity.On the other hand, a matrix γ = (cid:18) a bc d (cid:19) ∈ Γ ( N ) acts on ∂ z W ( N,l ) ( z ; τ ) as ∂ z W ( N,l ) (cid:18) zcτ + d ; aτ + bcτ + d (cid:19) =( cτ + d ) N − (cid:88) k =0 e πilkN (cid:20) ζ (cid:18) z − πN (( aτ + b ) l + k ( cτ + d )); τ (cid:19) + 2 dη + cη N k (cid:21) =( cτ + d ) N − (cid:88) k =0 e πialkN (cid:20) ζ (cid:18) z − πN bl − πN ( alτ + k ); τ (cid:19) + 2 η kN (cid:21) =( cτ + d ) ∂ z W ( N,al ) (cid:18) z − πN bl ; τ (cid:19) , (E.41)where in the third line we used the fact that a = d − mod N . As discussed in sectionE.2.2, Γ ( N ) changes the co-level with the map l → al , acting as the Galois groupof the real cyclotomic extension. E.5.2 Modular Transformations of the Physical Mass It turns out that the superpotential is not invariant under a transformation of Γ( N ),but it shifts by a constant. On one side, it is not wrong to say that the model is leftinvariant, since a constant is irrilevant in determining the physics of a system. But,at the same time, the ‘physical mass’ W ( N,l ) ( p ; τ ) is the parameter that we haveused to write the tt ∗ equations. This means that Γ( N ) changes the coordinate of themodel on the spectral cover, with a consequent transformation of the ground statemetric. An analogous constant shift is induced on the superpotential also by Γ ( N )and Γ ( N ), in addition to the effects discussed in the previous paragraph. Beforegiving an interpretation of such phenomena and solving the apparent contraddiction,we want to compute these constants.The fact that W ( N,l ) ( p ; τ ) transforms non trivially under Γ( N ) is connected to thegeometry of the modular curves. These spaces are not simply connected and acritical value can be defined only on the universal cover, i.e. the upper half plane,where Γ( N ) plays the role of deck group. This is also strictly related to the factthat the superpotential is defined on the universal cover of the target space, wherewe have more than N vacua. On the other hand, variations of W ( N,l ) ( p ; τ ) with177espect to some coordinate on the modular curve must be modular functions ofΓ( N ). Let us give some examples. Among the curves X ( N ), the cases with N ≤ N ). Moreover, it is the projective coordinate whichrealizes the isomorphism between X ( N ) and the Riemann sphere punctured withthe cusps. Denoting with x the coordinate on the sphere, we can define the one-form dW ( x ) = C ( x ) dx , where C ( x ) = ∂ x W ( p ; x ) is the chiral ring coefficient whichdescribes variations of the critical value with respect to the Hauptmodul. Exceptthe case of N = 5, for these curves we have only the co-level 1 (and the inverse),so we suspend the notation ( N, l ). From the theory of projective algebraic curves, C ( x ) must be a rational function with coefficients in the cyclotomic field. Moreover,the poles must be located in the free IR cusps, where the physical mass is expectedto diverge. Thus, C ( x ) can be fixed up to a multiplicative constant by requiring thecorrect transformation properties under Γ ( N ) and Γ ( N ). In particular, from thetransformations of ∂ z W ( N,l ) ( z ; τ ) under Γ ( N ), we learn that T : τ → τ + 1 acts onthe one form as an automorphism of the sphere with the formula T ∗ dW ( x ) = dW ( p ; T ∗ x ) = dW ( σ − ( p ); x ) = e − πiN dW ( x ) , where l = 1 is understood.Starting from X (2), we have 3 cusps which can be viewed as the 3 vertices ofthe equatorial triangle of a double triangular pyramid inscribed in P . A set ofrepresentatives for the cusps is given by C Γ(2) = { , , ∞} where ∞ , being a fixed point of τ → τ + 1, is a UV cusp and 0 , P SL (2 , Z ) / Γ(2) is isomorphic to S , which is the symmetry group of a triangle. We can use x ( τ ) = 1 − λ ( τ ) ascoordinate on the sphere, where λ ( τ ) : H / Γ(2) ∼ −→ P / { , , ∞} is the modularlambda function defined by λ ( τ ) = θ (0; τ ) θ (0; τ ) . The 3 cusps fall in the point x ( ∞ ) = 1 , x (0) = 0 , x (1) = ∞ and T acts on x by thetransformation of P SL (2 , C ) 178 ∗ x = 1 x . From what we have said C ( x ) must have a simple pole in x = 0 and x = ∞ . Askingthat T ∗ dW ( x ) = − dW ( x ), we find dW ( x ) = α dxx (E.42)where α is a complex constant. We see that there is a unique generator of Γ(2)which acts non trivially on the critical value. This generates loops around the cuspin 0 (as well as in ∞ ) and produces the constant shift of the superpotential.In the case of N = 3 the cusps are located at the 4 vertices of a regular tetrahedroninscribed in the Riemann sphere. It is not a coincidence that P SL (2 , Z ) / Γ(3) isisomorphic to the symmetry group of this solid figure, i.e. A . The set of cusps is C Γ(3) = { , / , , ∞} . Also in this case ∞ is the only UV cusp, while the other rationals are in the sameorbit of T and identify free theories. The integers coprime with 3 are 1 and 2 = − J ( τ ) = 1 i √ q − τ (cid:32) ∞ (cid:89) n =1 − q n/ τ − q nτ (cid:33) = 1 i √ (cid:18) η ( τ / η (3 τ ) (cid:19) where q τ = e πiτ and η ( τ ) = q / τ ∞ (cid:89) n =1 (1 − q nτ ) (E.43)is the Dedekind eta function. The generator of Γ (3) / Γ(3) acts on the Hauptmodulas T ∗ J = ζ + ζ J where we use the notation ζ k = e πi/k . Denoting x = J , the one-form with valuesin the chiral ring is 179 W ( x ) = α (cid:88) k =0 ζ − k dxx − x k where x = J (0) = 0 , x = J (1) = ζ and x = J (1 / 2) = 1 + ζ , with x k +1 = T ∗ x k .The cusp at τ = ∞ is instead sent to x = ∞ on the sphere. It is straightforward tocheck that dW ( x ) satisfies T ∗ dW ( x ) = ζ − dW ( x ).Another spherical version of a platonic solid appears for N = 4. In this curvethe cusps are the 6 vertices of a regular octrahedron in P with symmetry group P SL (2 , Z ) / Γ(4) ∼ S . Also in this case the co-levels are just ± 1. The critical pointsare given by C Γ(4) = { , / , / , / , , ∞} . In this curve we have two UV cusps, i.e. ∞ and 1 / 2, which are both fixed points of T . The remaining ones represent instead free IR critical points. The Hauptmodulof level 4 is J ( τ ) = ζ √ q / τ ∞ (cid:89) n =1 (1 − q nτ ) (1 − q n/ τ )(1 − q n/ τ ) (1 − q nτ ) , which transforms under T as T ∗ J = ζ J − J . The fixed points of this map are 0 and 1 − ζ , which correspond respectively to τ = ∞ , . The 4 IR cusps are all in the same orbit of T and fall in the points J (0) = ∞ , J (1) = − ζ , J (2 / 3) = 1 / (1 + ζ ) , J (1 / 3) = 1. These must be simplepoles for dW ( x ), with x = J , which reads dW ( x ) = α (cid:18) x − − x + ζ + ζ x − / (1 + ζ ) (cid:19) dx, (E.44)and satisfies T ∗ dW ( x ) = ζ − dW ( x ). X (5) is the last case of genus 0. This modular curve has 12 cusps which identify thevertices of a regular icosahedron. The quotient P SL (2 , Z ) / Γ(5) acts on the Riemannsphere as A , the symmetry group of this platonic solid. Among the 12 cusps C Γ(5) = { , / , / , / , / , / , / , / , / , / , , ∞} / ∞ , while { , / , / , / , } and { / , / , / , / , / } represent the 5 decoupled vacua in two inequivalent IR limits. This curve has twoinequivalent co-levels, i.e. l = 1 , P ( C ) the Hauptmodul J ( τ ) = ζ q − / τ ∞ (cid:89) n =1 (1 − q n − τ )(1 − q n − τ )(1 − q n − τ )(1 − q n − τ )which transforms under T as T ∗ J = ζ − J . The UV cusps 2 / , ∞ are sent by J respectively to x = 0 , ∞ , which are the fixedpoints of T . We know from E.41 that a transformation γ ∈ Γ ( N ) with b = 0 mod N acts on the cyclotomic units through the Galois group and permutes the residueof the poles in dW ( x ). It follows that if two IR cusps are in the same orbit of such γ , their residue must be related by the correspondent Galois transformation. It isthe case for instance of 0 and 5 / b = 5 and c = 8. Imposing also thecorrect transformation property under T for l = 1, one can repeat the procedureand fix all the coefficients up to an overall constant. Setting x k = ζ k J (0) and x k = ζ k J (5 / J (0) = 1 + ζ + ζ and J (5 / 8) = ( ζ − ζ ) / (1 + ζ − ζ − ζ ),we find dW ( x ) = α (cid:88) k =0 (cid:18) x − x k + 1 x − x k (cid:19) ζ k dx. where the residue of J (0) and J (5 / 8) are both normalized to 1.By summarizing, the generators of loops around the IR cusps on the sphere cor-respond to the subset of generators of Γ( N ) which act non trivially on the criticalvalue. Once the normalization is fixed, the constant generated by modular trans-formations can be computed with the residue formula. Moreover, the free IR cuspsare all in the same orbit of Γ ( N ), which acts by multiplication on the coefficientsof dW ( x ). This implies in particular that all the poles have the same order, whichmust be 1 from the non trivial monodromy of W ( p ; x ), and their residue are relatedby Galois transformations.A similar procedure could be carried on in principle also for modular curves ofhigher genus, but it is more complicated. It is instead much more convenient to finda general expression of the critical value as function on the upper half plane and181tudy its modular properties. Let us take the multi-valued function in E.8. Sincethe constants generated by the modular transformations are indipendent from thepoint, we can set z = 0. The expression of W ( N,l ) (0; τ ) reads W ( N,l ) (0; τ ) = N − (cid:88) k =0 e πilkN log (cid:20) Θ (cid:20) − lN − kN (cid:21) (0; τ ) e − πi ( lN − )( kN − ) (cid:21) . (E.45)where the phase e − πi ( lN − )( kN − ) is a convenient normalization constant. Thisfunction remains ill defined as long as we do not specify the determination of thelogarithm. This is equivalent to choose, for a fixed τ , a representative of z = 0 anddetermine its critical value. We first set the notations q τ = e πiτ , q z = e πiz ,z = u τ + u , with u , u ∈ Z /N . Then, we recall the definition of Siegel functions: g u ,u ( τ ) = − q B ( u ) / τ e πiu ( u − / (1 − q z ) ∞ (cid:89) n =1 (1 − q nτ q z )(1 − q nτ /q z ) , (E.46)where B ( x ) = x − x + is the second Bernoulli polynomial. Because of theirmodular properties, these objects are a sort of ‘building blocks’ for the modularfunctions of level N . In particular, all the Hauptmoduls defined previously can beexpressed in terms of g u ,u ( τ ) and the Dedekind eta function [58, 59, 60]. The thetafunctions Θ (cid:20) − u − u (cid:21) (0; τ ) have the q -product representationΘ (cid:20) − u − u (cid:21) (0; τ ) = − q B ( u ) / τ q / τ e πi ( u − / u − / (1 − q z )x ∞ (cid:89) n =1 (1 − q nτ )(1 − q nτ q z )(1 − q nτ /q z ) (E.47)and can be written in terms of g u ,u ( τ ) and η ( τ ) asΘ (cid:20) − u − u (cid:21) (0; τ ) = ig u ,u ( τ ) η ( τ ) e πiu ( u − / . (E.48)182ince the Siegel and Dedekind functions have neither zeroes nor poles, there is asingle-valued branch of log Θ (cid:20) − u − u (cid:21) (0; τ ) on the upper half plane. Therefore, thecritical value can be consistently defined as holomorphic function of τ ∈ H . Providedthe above relations, we can rewrite W ( N,l ) (0; τ ) as W ( N,l ) (0; τ ) = N − (cid:88) k =0 e πilkN log E lN , kN ( τ ) , (E.49)where E lN , kN ( τ ) is the Siegel function of characters u = lN , u = kN without theroot of unity e πiu ( u − / . Under an integer shift of the characters, these functionssatisfy [58] E u +1 ,u ( τ ) = − e − πiu E u ,u ( τ ) , E u ,u +1 ( τ ) = E u ,u ( τ ) . (E.50)Moreover, being the Siegel functions up to a multiplicative constant, they have goodmodular properties. For γ = (cid:18) a bc d (cid:19) ∈ SL (2 , Z ), they transform with a phase: E u ,u ( τ + b ) = e πibB ( u ) E u ,u + bu ( τ ) , for c = 0 ,E u ,u ( γ ( τ )) = ε ( a, b, c, d ) e πiδ E u (cid:48) ,u (cid:48) ( τ ) , for c (cid:54) = 0 , (E.51)where ε ( a, b, c, d ) = (cid:40) e iπ ( bd (1 − c )+ c ( a + d − / , if c is odd , − ie iπ ( ac (1 − d )+ d ( b − c +3)) / , if d is odd ,δ = u ab + 2 u u bc + u cd − u b − u ( d − , (E.52)and u (cid:48) = au + cu , u (cid:48) = bu + du . (E.53)In order to compute the constants generated by the modular transformations, weneed to evaluate the difference 183 u ,u ( γ ) = log E u ,u ( γ ( τ )) − log E u (cid:48) ,u (cid:48) ( τ ) , (E.54)for γ ∈ SL (2 , Z ). Here we assume the characters of the Siegel functions to benormalized such that 0 < u , u , u (cid:48) , u (cid:48) < 1. This can always be achieved by theproperty E.50. The computation for the case of Γ( N ) has already been done in [59].In F we follow closely that derivation, adapting it to the general case. For c = 0 thetransformations belong to the coset group Γ ( N ) / Γ( N ) (cid:39) Z N and we obtain χ u ,u ( γ ) = 2 πi B ( u ) . (E.55)It is clear that in this case we cannot appreciate a modular shift of the critical value.Indeed, these transformations simply translate the vacua: W ( N,l ) (0; τ + b ) = e − πibl N W ( N,l ) (0; τ ) . (E.56)On the other hand, for c (cid:54) = 0, we get the formula χ u ,u ( γ ) = 2 πi (cid:18) B ( u ) ac + B ( u (cid:48) ) dc − c B ( u (cid:48) ) B ( (cid:104) du (cid:48) − u (cid:48) c (cid:105) ) (cid:19) − πic (cid:88) x ∈ Z /c Z , x (cid:54) =0 [ x, u (cid:48) , u (cid:48) ] d,c , (E.57)where B ( x ) = x − / (cid:104) x (cid:105) represents the fractionalpart of x and the symbol [ x, u (cid:48) , u (cid:48) ] d,c denotes[ x, u (cid:48) , u (cid:48) ] d,c = e πix (cid:0) (cid:104) du (cid:48) − cu (cid:48) (cid:105)− du (cid:48) c + u (cid:48) (cid:1) (1 − e − πixd/c )(1 − e πix/c ) . (E.58)This result turns out to be indipendent from the branch of the logarithm and inparticular from τ . This is consistent with the fact that the modular shift of thecritical value is indipendent from the vacuum that we choose. From the generalformula we can reduce to the cases of the congruence subgroups. Let us consider γ ∈ Γ ( N ). Using the fact that c = 0 mod N and ad = 1 mod N , we have to plugin the above expression: 184 (cid:48) = (cid:104) au (cid:105) , u (cid:48) = (cid:104) du + bu (cid:105) , (cid:104) du (cid:48) − u (cid:48) c (cid:105) = u . (E.59)If γ ∈ Γ ( N ), these becomes u (cid:48) = u , u (cid:48) = (cid:104) u + bu (cid:105) , (cid:104) du (cid:48) − u (cid:48) c (cid:105) = u . (E.60)The case of γ ∈ Γ( N ) follows from this by requiring further b = 0 mod N .Setting u = lN , u = kN and summing over k with the residue e πilkN , we find themodular transformations of the physical mass. In sequence γ ∈ Γ ( N ) : W ( N,l ) (0; γ ( τ )) = e − πial bN W ( N,al ) (0; τ ) + ∆ W ( N,l )Γ ( N ) ( γ ) ,γ ∈ Γ ( N ) : W ( N,l ) (0; γ ( τ )) = e − πil bN W ( N,l ) (0; τ ) + ∆ W ( N,l )Γ ( N ) ( γ ) ,γ ∈ Γ( N ) : W ( N,l ) (0; γ ( τ )) = W ( N,l ) (0; τ ) + ∆ W ( N,l )Γ( N ) ( γ ) , (E.61)with∆ W ( N,l )Γ ( N ) ( γ ) = − πiNc e − πial bN (cid:88) x ∈ Z /c Z , x = − al mod N e πix (cid:0) l/N − d (cid:104) al/N (cid:105) c (cid:1) (1 − e − πixd/c )(1 − e πix/c ) , ∆ W ( N,l )Γ ( N ) ( γ ) = − πiNc e − πil bN (cid:88) x ∈ Z /c Z , x = − l mod N e πix (cid:0) lN − dc (cid:1) (1 − e − πixd/c )(1 − e πix/c ) , (E.62)185 W ( N,l )Γ( N ) ( γ ) = − πiNc (cid:88) x ∈ Z /c Z , x = − l mod N e πix (cid:0) lN − dc (cid:1) (1 − e − πixd/c )(1 − e πix/c ) , where the constraints on x follow from the summation over k . These formulas arecoherent with the transformations of ∂ z W ( N,l ) ( z ; τ ) that we found in the previousparagraph.We can check in the simple case of N = 2 that the formulas above give the resultsobained with the geometrical approach. The physical mass has the expression W (0; τ ) = log Θ (0; τ )Θ (0; τ ) = 14 log(1 − λ ( τ )) , (E.63)where 1 − λ ( τ ) = (Θ (0; τ ) / Θ (0; τ )) is the Hauptmodul of level 2 that we havedefined previously. Although 1 − λ ( τ ) is invariant under transformation of Γ(2),the logarithm is not. We can use the expression for ∆ W Γ(2) to derive the modulartransformations of log(1 − λ ( τ )). Γ(2) is freely generated by the matrices T = (cid:20) (cid:21) , T = (cid:20) − (cid:21) . (E.64)Using respectively the E.56 and the E.61, E.62 with N = 2 , l = 1, one findslog(1 − λ ( τ + 2)) = log(1 − λ ( τ )) , log (cid:18) − λ (cid:18) τ − τ (cid:19)(cid:19) = log(1 − λ ( τ )) + 2 πi. (E.65)It is clear that T is the generator of anticlockwise loops around the IR cusp in τ = 0. Indeed, the constant is the same we obtain with the residue formula. E.5.3 Modular Transformations of the Ground State Metric The modular shift of the superpotential seems to contraddict the statement that themodel is invariant under transformations of Γ( N ). But, if we assume the perspectiveof the universal cover, there is no contraddiction at all. Indeed, the physical massparametrizes not only models, but also vacua. The modular transformation simply186hanges the initial choice of the vacuum p with another one of the same fiber inthe universal cover. Therefore, the new coordinate on the spectral curve describesthe same model, but a different vacuum.The tt ∗ equations for these class of theories are manifestly covariant under trans-formations of the congruence subgroups. In particular, the covariance under Γ ( N )implies that the equation naturally descends on the space of models. However, asa consequence of the modular shift, the ground state metric is not left invariant.Indeed, matrices of Γ( N ) and Γ ( N ) change the basis of lattice generators andconsequently the representation of the homology group. Besides this effect, a trans-formation of Γ ( N ) changes also the torsion point of the Z N symmetry, resulting ina permutation of the metric components. Let us consider this more general case.From the transformation of ∂ z W ( N,l ) ( z ; τ ) under Γ ( N ) one can read how the gen-erators of the symmetry group are modified. We note that the new function is stillperiodic of 2 πτ . Therefore, we can consider again B as a generator of H ( S ; Z ) inthe new model. The operators L k are left invariant by the transformation as well,since the corresponding homology cycles have the same definition in the model ofco-level al . Differently, σ and A change in relation to the transformation of thetorsion point of the vacua. For a γ ∈ Γ ( N ), we can write γ ∗ B = B,γ ∗ L k = L k πN −→ πN ( cτ + d ) = ⇒ (cid:40) γ ∗ σ = σ d B cN = ˜ σB cN γ ∗ A = A d B c = ˜ AB c , where we denote with ˜ σ = σ d the operator associated to the torsion point of co-level al , and with ˜ A = A d the homology cycle satisfying ˜ A = ˜ σ N . The fact that thegenerator of loops are not involved in the modular transformations is consistencewith the truncation of the tt ∗ equation that we discussed in section 3.We have also to include the shift z → z − πN bl , which implies the vacuum transfor-mation p −→ σ − bl ( p ) = ⇒ | p (cid:105) (cid:55)→ σ − bl | p (cid:105) . Let us study what these transformations mean at the level of vacuum states. Weconsider again trivial representations of L k . The action of γ on the point basis is γ | k ; α, β (cid:105) = e − iαkN (cid:88) m,n ∈ Z e − i ( mα + nβ ) ( A d B c ) m B n ( σ d B cN ) k σ − bl | p (cid:105) e − i kdN ( α − βcd ) (cid:88) m,n ∈ Z e − i ( m ( α − βcd ) + nβ ) A m B n σ kd − bl | p (cid:105) = e − i blN ( α − βcd ) | kd − bl ; ( α − βc ) /d, β (cid:105) . We note that, in the case of b = 0 mod N and d = 1 mod N , the fiber index k isleft invariant. This follows from the fact that a transformation of Γ( N ) preservesthe torsion point up to a shift of a lattice vector, which moves the base point p without changing the fiber. The σ -eigenstates transform consequently as γ | j ; α, β (cid:105) = e − i blN ( α − βcd ) N − (cid:88) k =0 e − πilkjN | kd − bl ; ( α − βc ) /d, β (cid:105) = e − i blN ( πalj + α − βcd ) N − (cid:88) k =0 e − πialkjN | k ; ( α − βc ) /d, β (cid:105) = e − i blN ( πalj + α − βcd ) | aj ; ( α − βc ) /d, β (cid:105) . The action of the symmetry group generators ˜ σ, ˜ A, B on the transformed states isgiven by ˜ σ | aj ; ( α − βc ) /d, β (cid:105) = e πiljN e iN ( α − βc ) | aj ; ( α − βc ) /d, β (cid:105) , ˜ A | aj ; ( α − βc ) /d, β (cid:105) = e i ( α − βc ) | aj ; ( α − βc ) /d, β (cid:105) ,B | aj ; ( α − βc ) /d, β (cid:105) = e iβ | aj ; ( α − βc ) /d, β (cid:105) . From the eigenvalues of the new basis, we learn that the ground state metric trans-forms in the following way: γ ∗ ϕ j ( w ; α, β ) = ϕ j ( γ ∗ w ; α, β ) = ϕ aj ( w ; α − βc, β ) . (E.66)As anticipated, we see that a transformation of Γ( N ) changes the representation ofthe homology, resulting in the character shift α → α − βc . In the case of Γ ( N ),188ince the operators ˜ σ and σ are inequivalent and have a different set of eigenstates,we appreciate also the exchange of the metric components along the diagonal. Thiseffect takes place specifically for N > 2, where we have a non trivial co-level struc-ture. We note further that transformations of the coset Γ ( N ) / Γ( N ), i.e. with c = 0, leave the metric completely invariant. E.6 Physics of the Cusps E.6.1 Classification of the Cusps Now that we have discussed the modular properties of the solution, we want todescribe its behaviour around the boundary regions of the domain. These are rep-resented by the cusps of the modular curve H / Γ( N ), i.e. the equivalence classes ofΓ( N ) in Q ∪ {∞} . First of all, we have to understand which type of model eachcusp corresponds to. Let us begin with the cusp at τ = i ∞ . It is convenient to comeback to the initial lattices of poles and vacua with the shift z → z + πlτN . Moreover,we rewrite the derivative of the superpotential in terms of Θ ( z ; τ ) as ∂ z W ( N,l ) ( z ; τ ) = 12 N − (cid:88) k =0 e πilkN Θ (cid:48) (cid:0) (cid:0) z − πkN (cid:1) ; τ (cid:1) Θ (cid:0) (cid:0) z − πkN (cid:1) ; τ (cid:1) . (E.67)Using the relation [57]Θ (cid:48) ( z, τ )Θ ( z ; τ ) = cot z + 4 ∞ (cid:88) n =1 q nτ − q nτ sin 2 nz, with q τ = e πiτ , the expression above becomes ∂ z W ( N,l ) ( z ; τ ) = 12 N − (cid:88) k =0 e πilkN (cid:20) cot (cid:18) (cid:18) z − πkN (cid:19)(cid:19) +4 ∞ (cid:88) n =1 q nτ − q nτ sin (cid:18) n (cid:18) (cid:18) z − πkN (cid:19)(cid:19)(cid:19)(cid:21) . The theta function in these expressions is normalized with quasi-periods π, πτ . Letus denote with S the infinite sum in n . Manipulating the expression, we get189 = 2 N − (cid:88) k =0 e πilkN ∞ (cid:88) n =1 q nτ − q nτ sin (cid:18) n (cid:18) z − πkN (cid:19)(cid:19) = − i ∞ (cid:88) n =1 q nτ − q nτ (cid:32) e inz N − (cid:88) k =0 e πikN ( l − n ) − e − inz N − (cid:88) k =0 e πilkN ( l + n ) (cid:33) . The two sums over k are not 0 if and only if n satisfies respectively n = l mod N and n = − l mod N . Therefore, the derivative of the superpotential becomes ∂ z W ( N,l ) ( z ; τ ) = 12 N − (cid:88) k =0 e πilkN cot (cid:18) (cid:18) z − πkN (cid:19)(cid:19) − iN ∞ (cid:88) n =1 n = l mod N q nτ − q nτ e inz − ∞ (cid:88) n =1 n = − l mod N q nτ − q nτ e − inz . Taking the limit τ → i ∞ , the series in q τ are truncated at the leading order. More-over, since the vacua − πlτN + πkN + Λ τ escape to infinity for large τ , we have also totake z → i ∞ . Thus, we obtain ∂ z W ( N,l ) ( z ; τ ) τ → i ∞ −−−→ − iN (cid:0) q lτ e ilz − q ( N − l ) τ e − i ( N − l ) z (cid:1) . Integrating this expression, we find W ( N,l ) ( z ; τ ) τ → i ∞ −−−→ − N (cid:18) q lτ e ilz l + q ( N − l ) τ e − i ( N − l ) z N − l (cid:19) , (E.68)wich we recognize as the superpotential of a ˆ A N − model of co-level l . In particular,the case of N = 2 corresponds to the Sinh-Gordon model W ( z ; τ ) ∼ q τ cos z. (E.69)Now let us consider the rationals. We associate to a cusp ac with gcd( a, c ) = 1 amodular transformation γ ac = (cid:18) a bc d (cid:19) which sends i ∞ to ac . In this definition b, d γ ac ∈ SL (2 , Z ) and clearly the case of c = 0 corresponds totake again τ = i ∞ . One can study the behaviour of the model around τ = ac byacting on ∂ z W ( N,l ) ( z ; τ ) with γ ac and then taking the limit τ → i ∞ . Using themodular properties E.40 of the zeta function, we have ∂ z W ( N,l ) (cid:18) zcτ + d ; aτ + bcτ + d (cid:19) =( cτ + d ) N − (cid:88) k =0 e πilkN (cid:20) ζ (cid:18) z − πN k ( cτ + d ); τ (cid:19) + 2 dη + cη N k (cid:21) , where the torsion point of the poles is now πN ( cτ + d ). Let us introduce the integers Q = gcd( c, N ), with 1 ≤ Q ≤ min { c, N } , j = NQ and r = cQ . Clearly we havegcd( r, j ) = 1. By these definitions, we can split the sum over k by writing k = m + jp ,with m = 0 , ..., j − p = 0 , ..., Q − 1. Let us consider for the moment the cuspswith divisor Q > 1. One obtains ∂ z W ( N,l ) (cid:18) zcτ + d ; aτ + bcτ + d (cid:19) =( cτ + d ) j − (cid:88) m =0 e πilmN Q − (cid:88) p =0 e πilpQ (cid:20) ζ (cid:18) z − (cid:18) πrj τ + 2 πdN (cid:19) m − πQ dp ; τ (cid:19) + 2 (cid:18) rη j + dη N (cid:19) m + 2 dη Q p (cid:21) . As τ becomes very large, for m (cid:54) = 0 the torsion point approaches πrj τ .Moreover, by the formulas [63]2 η = G ( τ )2 π , η = τ G ( τ ) − πi π , where G ( τ ) is the Eisenstein series G ( τ ) = (cid:88) c,d ∈ Z \{ } cτ + d ) , G ( τ ) τ → i ∞ −−−→ ζ (2) , (E.70)where ζ ( z ) is the Riemann zeta function, one has η η τ → i ∞ −−−→ τ. (E.71)Thus, we get the limit ∂ z W ( N,l ) (cid:18) zcτ + d ; aτ + bcτ + d (cid:19) τ → i ∞ −−−→ ( cτ + d ) Q − (cid:88) p =0 e πilpQ (cid:20) ζ (cid:18) z − πQ dp ; τ (cid:19) + 2 dη Q p (cid:21) τ → i ∞ +( cτ + d ) j − (cid:88) m =1 e πilmN Q − (cid:88) p =0 e πilpQ (cid:20) ζ (cid:18) z − πrj τ m ; τ (cid:19) + 2 rη j m (cid:21) τ → i ∞ =( cτ + d ) Q − (cid:88) p =0 e πilpQ (cid:20) ζ (cid:18) z − πQ dp ; τ (cid:19) + 2 dη Q p (cid:21) τ → i ∞ =( cτ + d ) Q − (cid:88) p =0 e πialpQ (cid:20) ζ (cid:18) z − πQ p ; τ (cid:19) + 2 η Q p (cid:21) τ → i ∞ , where in the last line we have used the fact that ad = 1 mod Q . So, we learnfrom this expression and the limit E.68 that the cusp ac with divisor Q = gcd( c, N )correspond to an ˆ A Q − model of co-level al : W ( N,l ) ( z ; τ ) τ → ac −−−→ − Q (cid:18) q alτ e ialz al + q ( Q − al ) τ e − i ( Q − al ) z Q − al (cid:19) , (E.72)which is a theory with Q vacua up to periodicity z ∼ z + 2 π .In the case of Q = 1, the poles πkN ( cτ + d ) + Λ τ are all pushed to infinity when τ becomes large except for k = 0. Thus, using the E.67 we can write symbolically W ∼ “ log Θ ( z/ τ → i ∞ )” , ( z ; τ ) τ → i ∞ −−−→ q τ sin z, one finds that these cusps are decribed by the multi-valued superpotential W ( z ) = “ log sin ( z/ 2) ” . (E.73)This model represents the free version of our class of theories, with a 1 dimensionallattice of poles and one of vacua.We know that the cusps of X ( N ) are ramification points of the cover X ( N ) → X ( N ). Denoting with Γ ac the stability group of the cusp a/c in Γ ( N ), one canwrite the equality [55] γ − ac Γ ac γ ac = (cid:28) ± (cid:18) h (cid:19)(cid:29) , (E.74)which is satisfied with one of the two signs. This relation means that the generatorof Γ ac is conjugated to ± (cid:18) h (cid:19) in γ − ac Γ ( N ) γ ac .The number h is called width of the cusp and represents the minimal integer suchthat a/c + h ∼ a/c in Γ( N ). The absolute value can be equal or less than N andcount the number of degenerate vacua of the model labelled by the cusp (cid:2) ac (cid:3) ofΓ ( N ). The cusps which satisfy the relation with the plus sign are called regular.From the theorems E.3, E.4, the stability condition can be written as: (cid:20) a + chc (cid:21) = (cid:20) ac (cid:21) mod N. It is clear that the minimal integer h satisfying this condition is h = j = NQ . Thus,for the cusps with divisor Q , the N vacua split in j decoupled theories which appearon X ( N ) as Q -degenerate points. These are described by the ˆ A Q − models E.72 for1 < Q ≤ N , or by the free theories E.73 if Q = 1. In particular, the cusps with Q = N , or equivalently c = 0 mod N , are the UV limits, since the vacua tend to aunique point on the spectral curve.The exceptions to this picture are represented by the so called irregular cusps, i.e.those which satisfy the E.74 with the mignus sign. In this case the stabilizer of thecusp belongs to − Γ ( N ) and we have 193 a + chc (cid:21) = − (cid:20) ac (cid:21) mod N. If we exclude the trivial case of N = 2 where 1 ∼ − a and c to becoprime, we find that the stability condition is satisfied only by cusp 1 / ( N ). Despitewe have Q = N/Q = 2, the width of the cusp is h = 1, and the correspondingtheory is actually a ˆ A model with 4 vacua. The superpotential is the Sinh-Gordonone in E.69 as for the cusps with divisor 2, but we have to impose the identification z ∼ z + 4 π .Now we want to determine the positions of the cusps on the W-plane. Using theexpression E.49 for the critical value, we have W ( N,l ) (0; τ → ac ) = W ( N,l ) (0; γ ac ( τ → i ∞ )) = N − (cid:88) k =0 e πilkN log E lN , kN ( γ ac ( τ → i ∞ ))= N − (cid:88) k =0 e πilkN log E (cid:10) al + ckN (cid:11) , (cid:10) dk + blN (cid:11) ( τ → i ∞ ) + N − (cid:88) k =0 e πilkN χ lN ; kN ( γ ac ) , where χ lN ; kN ( γ ac ) is given by the formula in E.57.Let us consider the limit of the first piece. The leading order of E u ,u ( τ ) for τ → i ∞ is ord i ∞ E u ,u ( τ ) = 12 B ( (cid:104) u (cid:105) ) . Therefore, we find W ( N,l ) (0; τ → ac ) = N − (cid:88) k =0 e πilkN log E (cid:10) al + ckN (cid:11) , (cid:10) dk + blN (cid:11) ( τ ) τ → i ∞ −−−→ N − (cid:88) k =0 e πilkN log q B (cid:18) (cid:10) al + ckN (cid:11) (cid:19) τ = log q K ( N,l ) ac τ , where 194 ( N,l ) ac = 12 N − (cid:88) k =0 e πilkN B (cid:18)(cid:28) al + ckN (cid:29)(cid:19) . Let us develop this expression. Using the Fourier expansion of the second Bernoulliperiodic polynomial B ( (cid:104) x (cid:105) ) = − πi ) ∞ (cid:88) m = −∞ m (cid:54) =0 e πimx m , we get K ( N,l ) ac = − πi ) N − (cid:88) k =0 e πilkN ∞ (cid:88) m = −∞ m (cid:54) =0 e πim ( al + ckN ) m = − πi ) ∞ (cid:88) m = −∞ m (cid:54) =0 e πim alN m N − (cid:88) k =0 e πikN ( l + mc ) . The sum (cid:80) N − k =0 e πikN ( l + mc ) is not vanishing if and only if l + mc = 0 mod N , whichadmits solution only when c is coprime with N . Thus, we have K ( N,l ) ac = − N (2 πi ) − πial rN ∞ (cid:80) m = −∞ m = − lr mod N m (cid:54) = 0 , if gcd( c, N ) = 1 , , otherwise , where r = c − mod N . So, we learn that if gcd( c, N ) = 1 the cusp order is notvanishing and therefore the critical value is divergent. Coherently with our analy-sis, these are the IR fixed points described by free theories. In this limits all thevacua decouple and the solitons connecting them become infinitely massive. On thecontrary, the cusps with 1 < Q ≤ N have a finite coordinate on the W-plane: W ( N,l ) (cid:16) ac (cid:17) = N − (cid:88) k =0 e πilkN χ lN ; kN ( γ ac ) . c = 0 mod N this becomes W ( N,l ) (cid:16) ac (cid:17) UV = ∆ W ( N,l )Γ ( N ) ( γ ac ) . In particular, the cusp τ = i ∞ is located at the origin and provides the boundarycondition for the critical limit µ → ( N ), thesemodels are labelled by the co-levels ± l and their number is equal to φ ( N ) / 2. Fromthis point of view, choosing the co-level is equivalent to pick which UV cusp to putin the origin. Also the free IR cusps are all in the same orbit of the Galois group,since we can map τ = 0 to a generic rational a/c such that gcd( c, N ) = 1 with amatrix of Γ ( N ). Instead, the other IR cusps with divisor 1 < Q < N can splitin different equivalence classes of Γ ( N ). In general the Galois group maps a cusp a/c with gcd( c, N ) = Q and gcd( a, Q ) = 1 to another cusp a (cid:48) c (cid:48) with gcd( c (cid:48) , N ) = Q and gcd( a (cid:48) , Q ) = 1. A computation in [55] shows that for a given divisor Q we have φ (gcd( Q, N/Q )) cusps of Γ ( N ). E.6.2 Boundary Conditions In this last section we provide the boundary conditions for the tt ∗ equations anddescribe the solution around the cusps. Approaching a critical point, the solutionhas to match the asymptotic behavour required by the physics of the correspondingcusp. The deformations in the space of couplings near these points regard the overallparameter µ rescaling the superpotential. Near the UV fixed point the ground statemetric can be diagonalized in a basis of vacua with definite Ramond charge: g i ¯ i µ → −−→ ( µ ¯ µ ) − q Ri − n/ , where n is the complex dimension of the target manifold. In the present case wehave n = 1. So, the solution to the tt ∗ equations near the critical point is givenin terms of the Ramond charges of the vacua, or equivalently the charges of thevector R-symmetry. These are given by the scaling dimensions of the chiral primaryoperators of the CFT.Let us start with the cusps corresponding to ˆ A N − models. These are Landau-Ginzburg theories with superpotential [30] W ( N,l ) ( z ; t ) = µ (cid:18) e − lz l + e ( N − l ) z N − l (cid:19) , l, N ) = 1. These are integrable models with a Z N symmetry generated by σ : z → z + πiN and N vacua, provided the periodic identification z ∼ z + 2 πi . Thesymmetry acts transitively on the critical points, which are given by the condition e Nz = 1. In the UV limit these theories tend to σ -models over abelian orbifoldsof CP . These are known to be asymptotically free theories with central chargeˆ c = 1 [30]. The tt ∗ equations in canonical form are the Toda equations in E.29with vanishing β . Indeed, in this limit the unique non trivial homology operator is A = σ N and the U (1) charges defining the solution can depend only on α . Since U (1) is broken by the superpotential to Z N , it is clear that the generators of thetwo symmetry groups have a common basis of eigenstates. Let us first consider thecase of α = 0. A basis of U (1) eigenstates in the chiral ring can be generated withthe operators e − z , e z . Since the superpotential must have R-charge 1, these haverespectively charge l and N − l . Given that e − lz = e ( N − l ) z in the chiral ring from thevacua condition, we find that the set of eigenstates split in two ‘towers’ e − z e − z · · · · · e − ( l − z e z e z · · · · · e ( N − l − z with U (1) V charges1 l l · · · · · l − l N − l N − l · · · · · N − l − N − l . Approximately, we can say that the theory splits in two, with a set of operatorsdominant on the other one according to how we take the limit. We complete thebasis by adding the identity I and e − lz , which have respectively charge 0 and 1.Near the critical point these two operators correspond to a unique marginal degreeof freedom which gets a logarithmic correction to the scaling [30]. We point out thatin this language the Galois group acts directly on the U (1) charges with the map l → al and puts in relation the solutions of the different ˆ A N − models. To see thatthe set of charges is invariant under this map we have to use the chiral ring condition e Nz = 1. The operatorial equality e − kz = e ( N − k ) z for a generic k ∈ Z implies theequivalence kl ∼ N − kN − l at the level of corresponding charges. In general, the integer k and the co-level l are periodic of N in the chiral ring. So, one can recast all thecharges above as q k = kl , k = 0 , ..., N − q k → kal . The relation kl = akal ∼ k (cid:48) al , with k (cid:48) = ak mod N , shows that the set of197harges is left invariant by this map.We can include the dependence from the angle α by multiplying the basis above by e α π z . In this way the operators have the correct eigenvalues under Z N when α (cid:54) = 0.The U (1) charges as functions of the angle are1 l (cid:16) k − α π (cid:17) , k = 1 , ..., l − , N − l (cid:16) k + α π (cid:17) , k = 1 , ..., N − l − ,α π ( N − l ) , − α πl . It is clear from E.66 that for β = 0 a transformation of Γ ( N ) does not changethe dependence on α of the metric components. This can be seen at the level ofcharges by the fact that the map l → al is compensated by the rescaling of the angle α → α/d .We note further that, since β is vanishing, the UV cusps turn out to be fixed pointsof Γ( N ). This is consistent with the fact that A is the unique generator of thehomology in this regime.The irregular cusp 1 / W ( z ) = µ (cid:0) e z + e − z (cid:1) with the identification z ∼ z + 2 πi . This theory has a Z symmetry generated by σ : z → z + iπ , but 4 vacua determined by the condition e z = e − z . This modelbelongs to ˆ A family and is asymptotically a σ -model on the CP / Z orbifold. The tt ∗ equations are the Toda ones with N = 4 and a basis of U (1) eigenstates is givenby e α π z e ( α π ) z e − ( − α π ) z e − ( − α π ) z with charges respectively α π (cid:16) α π (cid:17) (cid:16) − α π (cid:17) (cid:16) − α π (cid:17) . tt ∗ equation is singular in the UVcusps: ϕ i ( t ; α ) t → −−→ − (cid:18) q i ( α ) − (cid:19) log t. (E.75)A solution in terms of regular trascendents can be given only on the upper halfplane, which is a simply connected space.The discussion for the ˆ A Q − models for 1 < Q < N is pretty much the same ofthe previous paragraph. So, we focus on the free massive theories corresponding tothe case of Q = 1. These IR cusps are Landau-Ginzburg models described by thederivative ∂ z W ( z ; τ ) = µ cot (cid:16) z (cid:17) . (E.76)This function is periodic of 2 π and has simple poles and simple zeroes respectively in2 kπ and π +2 kπ , κ ∈ Z . Moreover, it is odd with respect to the parity transformation ι : z → − z . Since the target space is not simply connected we need to pull-back themodel on the abelian universal cover. A natural basis for the homology is given bythe cycles B, B (cid:48) in figure 6. From the residue formula and the parity properties of ∂ z W ( z ) one gets the transformations of the superpotential B ∗ W ( p ) = W ( p ) − πiµ,B (cid:48)∗ W ( p ) = W ( p ) + 2 πiµ. (E.77)Proceeding as in D.4 we can construct the unique theta-vacua of this theory: | φ, ψ (cid:105) = (cid:88) n,m ∈ Z e − i ( mφ + nψ ) B m B (cid:48) n | (cid:105) , (E.78)where we denote with | (cid:105) some vacuum state of the covering model. Setting to 0the corresponding critical value, the whole set is simply W n,m = 2 πiµ ( n − m ) . (E.79)We want to derive the tt ∗ equation in the parameter µ . The chiral ring operator C µ ( φ, ψ ) acts on the theta-vacuum as differential operator in the angles199 µ | φ, ψ (cid:105) = (cid:88) n,m ∈ Z e − i ( mφ + nψ ) πi ( n − m ) B m B (cid:48) n | (cid:105) = 2 π (cid:18) ∂∂φ − ∂∂ψ (cid:19) | φ, ψ (cid:105) . (E.80)We define the ground state metric g ( t, φ, ψ ) = (cid:104) φ, ψ | φ, ψ (cid:105) = e L ( t,φ,ψ ) , (E.81)where L ( t, φ, ψ ) is a real function of the angles and the RG scale t = | µ | . We cannormalize the state so that the topological metric is 1. Thus, the reality constraintimplies L ( − φ, − ψ ) = − L ( φ, ψ ) . (E.82)Moreover, by the commutation relations ιB = B (cid:48)− ι,ιB (cid:48) = B − ι, (E.83)we have also L ( − ψ, − φ ) = L ( φ, ψ ) . (E.84)The tt ∗ equation for g ( t, φ, ψ ) reads (cid:32) ∂ µ ∂ ¯ µ + 4 π (cid:18) ∂∂φ − ∂∂ψ (cid:19) (cid:33) L ( t, φ, ψ ) = 0 . (E.85)We recognize in this expression the equation of a U (1) Bogomolnyi monopole on R × T . Abelian tt ∗ monopoles have been studied in [31, 41]. The solution can beexpanded in Bessel-MacDonald functions as L ( t, φ, ψ ) = (cid:88) m ,m ∈ Z \{ } A ( m , m ) K (4 πt | m + m | ) exp ( i ( m φ − m ψ )) , (E.86)200here the coefficients A ( m , m ) can be determined by imposing appropriate bound-ary conditions. One can easily see that the tt ∗ reality constraint E.82 implies A ( − m , − m ) = − A ( m , m ) , A ( m , m ) ∈ i R , (E.87)while the parity condition E.84 requires A ( m , m ) = A ( m , m ) . (E.88)Combining these two conditions one gets the further constraint L ( t, ψ, φ ) = − L ( t, φ, ψ ) . (E.89)According to the discussion in section E.4, in order to have the abelianity of thesolution one should consider trivial representations of the loop generator. If wedemand the loop angle to vanish, namely φ = ψ , we simply find the trivial solution g ( t, φ, φ ) = 1 . (E.90) E.7 Conclusions In this chapter we have shown how the tt ∗ geometry of the modular curves is richof interesting phenomena and outstanding connections between geometry, numbertheory and physics. These Riemann surfaces parametrize a family of supersymmetricFQHE models in which the usual setting degenerates in a doubly periodic physicson the complex plane. In the subclass of theories of level N , the elliptic functionsplaying the role of superpotentials have N vacua and N poles in the fundamental cell,with the corresponding residues which add up to zero by definition. The cancellationof the total flux between the magnetic field and the quasi-holes guarantees theenhancement of symmetry that makes possible to face analitically these models.In particular, the presence of an abelian symmetry group with a transitive actionon the vacua allows to diagonalize the ground state metric, as well as to find thenecessary topological data to write the tt ∗ equations. This requires to pull-backthe model on the abelian universal cover of the target manifold, where we haveseen that the physics is non-abelian. On this space the symmetry group is enlargedwith the generators of loops around the poles, which are responsible for the nontrivial commutation relations between the generators of the algebra. Hovewer, the201belianity that we have required in the classification can be recovered at the quantumlevel. In particular, the ansatz of a solution with vanishing loop angles is consistentwith all the tt ∗ equations, which can be recasted as Toda equations in the canonicalcoordinates.Studying the modular properties of these models, we have underlined that the nontrivial modular transformations of the superpotential are a natural consequence ofthe geometry of the modular curves. A critical value as coordinate on the spectralcover can be defined only on the upper half plane, since the F-term variations arerational functions in projective coordinates on the modular curves. This has beenstudied in the easiest cases of the platonic solids inscribed in the Riemann sphere,but for surfaces of higher genus it is more convenient to parametrize the critical valuein terms of the fundamental units of the modular function field. The congruencesubgroups have a not trivial effect also on the components of the ground state metric,since they change the representation of the abelian symmetry group.The known results and theorems about the cusps counting and classification havebeen recovered in a physical language when we have classified the critical limits ofthis family of theories. One of the main point is that the width of a cusp allows todetermine the UV or IR nature of the corresponding RG fixed point.Our investigation has also revealed the algebraic properties of the modular curves.As we pointed out, the most remarkable connection with number theory is that theGalois group of the real cyclotomic extensions acts on the regularity conditions ofthe ˆ A N − Toda equations. This follows from the fact that the ˆ A N − models play therole of UV critical limits and belong to the same orbit of the Galois group. F Modular Transformations of log E u ,u ( τ ) In section E.5.2 we have setted the notations q τ = e πiτ , q z = e πiz ,z = u τ + u , with u , u ∈ Z /N , and defined the modular units E u ,u ( τ ) = q B ( u ) / τ (1 − q z ) ∞ (cid:89) n =1 (1 − q nτ q z )(1 − q nτ /q z ) , (F.1)which are the Siegel functions up to the root of unity e πiu ( u − / . These objects202atisfy [58] E u +1 ,u ( τ ) = − e − πiu E u ,u ( τ ) , E u ,u +1 ( τ ) = E u ,u ( τ ) , (F.2)and transform under γ = (cid:18) a bc d (cid:19) ∈ SL (2 , Z ) as E u ,u ( τ + b ) = e πibB ( u ) E u ,u + bu ( τ ) , for c = 0 ,E u ,u ( γ ( τ )) = ε ( a, b, c, d ) e πiδ E u (cid:48) ,u (cid:48) ( τ ) , for c (cid:54) = 0 , (F.3)where ε ( a, b, c, d ) = (cid:40) e iπ ( bd (1 − c )+ c ( a + d − / , if c is odd , − ie iπ ( ac (1 − d )+ d ( b − c +3)) / , if d is odd , (F.4) δ = u ab + 2 u u bc + u cd − u b − u ( d − , and u (cid:48) = au + cu , u (cid:48) = bu + du . (F.5)With these definitions, we want to compute the difference χ u ,u ( γ ) = log E u ,u ( γ ( τ )) − log E u (cid:48) ,u (cid:48) ( τ ) , (F.6)for γ ∈ SL (2 , Z ) and generic characters u , u ∈ Z /N . From F.3 we know thatthere is a power of E u ,u ( γ ( τ )) /E u (cid:48) ,u (cid:48) ( τ ) which is equal to one. This number is12 N for Γ( N ) and 12 N for Γ ( N ) , Γ ( N ) and the whole SL (2 , Z ). Therefore, thedifference χ u ,u ( γ ) must be equal to 2 πi times a rational number. Given that theupper half plane is simply connected, this number is independent of τ . Moreover,since log E u ,u ( τ ) is single-valued on the upper half plane, it is also indipendentfrom the branch of the logarithm. A natural choice, suggested by the q -expansion ofthe Siegel functions, is the principal branch on C with the negative real axis deleted.From now on we will use this determination. Because E u ,u ( τ ) changes by a phaseunder an integer shift of the characters, we can assume without loss of generalitythe canonical normalization 0 < u , u , u (cid:48) , u (cid:48) < c = 0. These transformations belong to the cosetgroup Γ ( N ) / Γ( N ) (cid:39) Z N and are generated by γ ( τ ) = τ + 1. Using the expansionof the Siegel function in F.1 we easily obtain χ u ,u ( γ ) = 2 πi B ( u ) . (F.7)From now on we assume c (cid:54) = 0 and write γ ( τ ) = aτ + bcτ + d = ac − c τ + cd .Using again the E.46 we havelog E u ,u ( τ ) = 2 πiB ( u ) τ + log(1 − q z ) + ∞ (cid:88) n =1 (log(1 − q nτ q z ) + log(1 − q nτ /q z )) . With τ in the upper half plane and the characters canonically normalized, the con-ditions of absolute convergence for the standard series of the principal logarithm aresatisfied. Therefore, using series expansions likelog(1 − q z ) = − ∞ (cid:88) m =1 q mz m for the logarithms in the expression, we obtainlog E u ,u ( τ ) = 2 πi B ( u ) τ − Q ( z ; τ ) , where Q ( z ; τ ) = ∞ (cid:88) m =1 m q mz + ( q τ /q z ) m − q mτ . Then, let us put τ = − dc + iy, with y > , γ ( τ ) = ac + ic y . Since it is indipendent of τ , we can calculate χ u ,u ( γ ) in the limit y → 0, i.e. τ → − dc and γ ( τ ) → i ∞ , by applying the Abel limit formula. Setting z γ = u γ ( τ ) + u , z (cid:48) = u (cid:48) τ + u (cid:48) , χ u ,u ( γ ) is pure immaginary, we haveto evaluate the expression χ u ,u ( γ ) = 2 πi (cid:18) B ( u ) ac + B ( u (cid:48) ) dc (cid:19) − lim τ →− dc (Im Q ( z γ ; γ ( τ )) − Im Q ( z (cid:48) ; τ )) . (F.8)Let us start with Im Q ( z γ ; γ ( τ )). As γ ( τ ) → i ∞ , q z γ and q γ ( τ ) /q z γ approach 0,therefore lim τ →− dc Im Q ( z γ ; γ ( τ )) = 0 . Now it is the turn of Q ( z (cid:48) ; τ ). We can decompose it in two pieces :lim τ →− dc Im Q ( z (cid:48) ; τ ) = lim τ →− dc Im (cid:88) c (cid:45) m m Q m ( z (cid:48) ; τ ) + lim τ →− dc Im (cid:88) c | m m Q m ( z (cid:48) ; τ )= L (cid:48) + L (cid:48)(cid:48) , where Q m ( z (cid:48) ; τ ) = q mz (cid:48) + ( q τ /q z (cid:48) ) m − q mτ . The symbols L (cid:48) and L (cid:48)(cid:48) denote the sum respectively for c (cid:45) m and c | m . Weintroduce r = e − πy , M = N | c | , ζ = e − πid/c , λ = e πi ( − dc u (cid:48) + u (cid:48) ) . It is shown in [59, 60] that the partial sums of these series are uniformly bounded.Therefore, we are allowed to take the limit under the sign of summation. Let usconsider first L (cid:48)(cid:48) . Using the notation above and taking the immaginary part, wehave 205 (cid:48)(cid:48) = lim r → (cid:88) c | m r u (cid:48) m − r (1 − u (cid:48) ) m − r m m ( λ m − λ − m ) . Taking the limit under the summation sign, one gets L (cid:48)(cid:48) = (cid:88) c | m (1 − u (cid:48) ) λ m − λ − m m = ∞ (cid:88) m =1 (1 − u (cid:48) ) 12 | c | m ( λ | c | m − λ −| c | m )= (1 − u (cid:48) ) 12 | c | ∞ (cid:88) m =1 m (cid:16) e πi ( − dε ( c ) u (cid:48) + u (cid:48) | c | ) m − e − πi ( − dε ( c ) u (cid:48) + u (cid:48) | c | ) m (cid:17) , where ε ( c ) = | c | /c . If t is real and not integer, it holds the Fourier expansion ∞ (cid:88) m =1 m ( e πimt − e − πimt ) = − πiB ( (cid:104) t (cid:105) ) , where B ( x ) = x − is the first Bernoulli polynomial. Thus L (cid:48)(cid:48) = − πi (1 − u (cid:48) ) 12 | c | B ( (cid:104)− dε ( c ) u (cid:48) + u (cid:48) | c |(cid:105) ) = − πic B ( u (cid:48) ) B ( (cid:104) du (cid:48) − u (cid:48) c (cid:105) ) . Now we turn to the last piece L (cid:48) . Taking the limit under the summation sign, weobtain L (cid:48) = lim τ →− dc Im (cid:88) c (cid:45) m m Q m ( z (cid:48) ; τ ) = (cid:88) c (cid:45) m m ϕ ( m ) , where ϕ ( m ) = Q m ( z (cid:48) ; τ ) | τ = − d/c = λ m + ( ζ/λ ) m − ζ m . Since ϕ ( − m ) = − ϕ ( m ) = ϕ ( m ), we note that ϕ ( m ) is pure immaginary and an oddfunction of m mod M = N | c | . Now, for each class x ∈ Z /M Z and 2 x (cid:54)∈ M Z , wedefine 206 ( x ) = ∞ (cid:88) m =1 a ( m, x ) m (F.9)where a ( m, x ) = m (cid:54) = ± x mod M m = x mod M − m = − x mod M. (F.10)Then, L (cid:48) can be rewritten as L (cid:48) = 12 (cid:88) x ∈ Z /M Z ,x (cid:54) =0 mod c Z , x (cid:54)∈ M Z ϕ ( x ) f ( x ) . (F.11)In [59] is shown that f ( x ) = − iπM (cid:20) − e πix/M − − e − πix/M (cid:21) . (F.12)Let ω = e πi/N | c | . Using this expression L (cid:48) becomes L (cid:48) = − πi M (cid:88) c (cid:45) x λ x + ( ζ/λ ) x − ζ x (cid:20) − ω x − − ω − x (cid:21) = − πi M (cid:88) c (cid:45) x (cid:20) λ x (1 − ζ x )(1 − ω x ) + ( ζ/λ ) x (1 − ζ x )(1 − ω x ) − λ x (1 − ζ x )(1 − ω − x ) − ( ζ/λ ) x (1 − ζ x )(1 − ω − x ) (cid:21) . Changing x to − x in the last two terms, we find L (cid:48) = − πiM (cid:88) c (cid:45) x λ x (1 − ζ x )(1 − ω x ) + (cid:88) c (cid:45) x ( ζ/λ ) x (1 − ζ x )(1 − ω x ) . x = y + k | c | , < y < | c | , (cid:54) k (cid:54) N − . Let us denote with S the partial sum in the variable k of the first term in L (cid:48) . Onegets S = λ y − ζ y N − (cid:88) k =0 λ k | c | − ω y + k | c | = − M λ y − ζ y N − (cid:88) r =0 rω ry N − (cid:88) k =0 ( λω r ) k | c | . The sum on the right is 0 unless ( λω r ) | c | = 1. Using the definitions of λ and ω interms of u (cid:48) , u (cid:48) , d, c , we see that ( λω r ) | c | = 1 if and only if r = N du (cid:48) − N cu (cid:48) mod N. Letting consequently r = N (cid:104) du (cid:48) − cu (cid:48) (cid:105) + sN with 0 ≤ s ≤ | c | − 1, we have S = − | c | λ y − ζ y (cid:88) ≤ r ≤ N − , r = Nu (cid:48) d − Ncu (cid:48) mod N rω ry = − | c | λ y − ζ y | c |− (cid:88) s =0 ( N (cid:104) du (cid:48) − cu (cid:48) (cid:105) + sN ) e πi yNc ( N (cid:104) du (cid:48) − cu (cid:48) (cid:105) + sN ) = − | c | λ y − ζ y e πi yc (cid:104) du (cid:48) − cu (cid:48) (cid:105) | c |− (cid:88) s =0 sN e πiys/c N λ y − ζ y e πi yc (cid:104) du (cid:48) − cu (cid:48) (cid:105) − e πiy/c = N e πiy (cid:0) (cid:104) du (cid:48) − cu (cid:48) (cid:105)− du (cid:48) c + u (cid:48) (cid:1) (1 − e − πiyd/c )(1 − e πiy/c ) . In order to write the final result in a more compact way, we introduce the symbol[ x, u (cid:48) , u (cid:48) ] d,c = e πix (cid:0) (cid:104) du (cid:48) − cu (cid:48) (cid:105)− du (cid:48) c + u (cid:48) (cid:1) (1 − e − πixd/c )(1 − e πix/c ) . (F.13)Noting that the second sum in L (cid:48) can be obtained from the first one with thesubstitution u (cid:48) → − u (cid:48) , u (cid:48) → − u (cid:48) , we get L (cid:48) = − πic (cid:88) x ∈ Z /c Z , x (cid:54) =0 [ x, u (cid:48) , u (cid:48) ] d,c + (cid:88) x ∈ Z /c Z , x (cid:54) =0 [ x, − u (cid:48) , − u (cid:48) ] d,c . From the property [ − x, − u (cid:48) , − u (cid:48) ] d,c = [ x, u (cid:48) , u (cid:48) ] d,c , one obtains further L (cid:48) = − πic (cid:88) x ∈ Z /c Z , x (cid:54) =0 [ x, u (cid:48) , u (cid:48) ] d,c . (F.14)Putting all the pieces together, we finally have χ u ,u ( γ ) = 2 πi (cid:18) B ( u ) ac + B ( u (cid:48) ) dc − c B ( u (cid:48) ) B ( (cid:104) du (cid:48) − u (cid:48) c (cid:105) ) (cid:19) − πic (cid:88) x ∈ Z /c Z , x (cid:54) =0 [ x, u (cid:48) , u (cid:48) ] d,c . (F.15)209 eferences N = 2 Gauge Systems”, arXiv:0909.2453v1 [hep-th] 13 Sep 2009.[28] S. Cecotti, C. Vafa, “ Topological-anti-topological fusion”, nuclear Physics B367 (1991) 359-461. 21129] K.Hori, A. Iqbal and C.Vafa, “D-branes and Mirror symmetry ”, hep-th/0005247.[30] S.Cecotti, C.Vafa, “ On classification of N=2 Supersymmetric Theories ”,hep-th/9209085, November 1992.[31] S.Cecotti, D.Gaiotto, C.Vafa, “ tt ∗ geometry in 3 and 4 dimensions ”,arXiv:1312.1008, Dec 3, 2013.[32] S.Cecotti, L.Girardello, A.Pasquinucci, “ Singularity-Theory and N tt ∗ Geometry for N = 2 Theories in 4 d ” arXiv:1412.4793 [hep-th], 15 Dec 2014.[42] N.A. Nekrasov, “Seiberg-Witten prepotential from instanton counting” Adv.Theor. Math. Phys. /, 831 (2004) [hep-th/0206161][43] R.Dijkgraaf and C.Vafa, “ On geometry and matrix models” Nucl. Phys. B644, 21 (2002) [hep-th/0207106][44] M. Aganagic, M. C. N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, “Quantum Geometry of Refined Topological Strings”, JHEP 1211, 019 (2012)212arXiv:1105.0630 [hep-th]].[45] D.Gaiotto, “Asymptotically free N = 2 theories and irregular conformalblocks”, arXiv:0908.0307 [hep-th].[46] C.Rim and H.Zhang, “ Classical Virasoro irregular conformal block”,arXiv:1504.07910v2 [hep-th] 24 Jun 2015[47] D. Gaiotto and J. Teschner, “Irregular singularities in Liouville theory andArgyres-Douglas type gauge theories, I”, arXiv:1203.1052v1 [hep-th] 5 Mar2012.[48] S.Cecotti, C.Vafa, “Ising Model and N=2 Supersymmetric Theories ”,arXiv:hep-th/9209085.[49] B. Dubrovin, “Geometry and integrability of topological antitopological fu-sion”, Commun. Math. Phys. 152, 539-564 (1993).[50] S.Cecotti and C.Vafa, “ 2d Wall-Crossing, R-twisting, and a SupersymmetricIndex”, arXiv:1002.3638 [hep-th].[51] N.A. Nekrasov and S.L. Shatashvili, “ Quantization of Integrable Systems andFour Dimensional Gauge Theories”, arXiv:09008.4052 [hep-th].[52] B. Estienne, V. Pasquier, R. Santachiara and D. Serban, “ Conformal blocksin Virasoro and W theories: Duality and the Calogero-Sutherland model,”Nucl. Phys. B 860, 377 (2012) [arXiv:1110.1101 [hep-th]].[53] M. Bershtein and O. Foda, “AGT, Burge pairs and minimal models,” JHEP1406, 177 (2014) [arXiv:1404.7075 [hep-th]].[54] K. B. Alkalaev and V. A. Belavin,“ Conformal blocks of WN minimal modelsand AGT correspondence,” JHEP 1407, 024 (2014) [arXiv:1404.7094 [hep-th]][55] F. Diamond and J. Shurman,“ A First Course in Modular Forms”, Springer,2005.[56] J. Silvermann, “ The Arithmetic of Elliptic Curves”, Springer-Verlag, 1986.[57] E. T. Whittaker and G. N. Watson, “ A Course of Modern Analysis”, Cam-bridge University Press, 1902.[58] Y. Yang, “ Transformation Formulas for Generalized Dedekind Eta Func-tions”, Bull. London math. Soc. 36 (2004) 671-682.[59] D.S. Kubert, S. Lang, “ Modular Units ”, Springer Verlag, New York 1981.21360] S. Lang, “ Introduction to Modular Forms ”, Springer Verlag, 1977.[61] B.R. Cais, supervised by N.D. Elkies, “ Riemann Surfaces and Modular Func-tion Field Extensions ”, Bachelor thesis, Harward University Cambridge, Mas-sachusetts, April 2002.[62] E.Witten, “ Topological Sigma Models ”, Commun. Math. Phys. 118 (1988)411; E. Witten, Nucl. Phys. “ On the structure of the Topological Phase ofTwo-dimensional Gravity ”, B 340 (1990) 281-332.[63] S. Zemel, “A Direct Evaluation of the Periods of the Weierstrass Zeta Function”, arXiv:1304.7194 [math.CV], 25 April 2013.[64] R.Bergamin, “ tt ∗∗