Duality and Modularity in Elliptic Integrable Systems and Vacua of N=1* Gauge Theories
DDuality and Modularity in Elliptic Integrable Systemsand Vacua of N = 1 ∗ Gauge Theories
Antoine Bourget and Jan Troost
Laboratoire de Physique Th´eorique Ecole Normale Sup´erieure24 rue Lhomond, 75005 Paris, France
Abstract :
We study complexified elliptic Calogero-Moser integrable systems. We determine the value of thepotential at isolated extrema, as a function of the modular parameter of the torus on which the integrablesystem lives. We calculate the extrema for low rank
B, C, D root systems using a mix of analytical andnumerical tools. For so (5) we find convincing evidence that the extrema constitute a vector valuedmodular form for the Γ (4) congruence subgroup of the modular group. For so (7) and so (8), the extremasplit into two sets. One set contains extrema that make up vector valued modular forms for congruencesubgroups (namely Γ (4), Γ(2) and Γ(3)), and a second set contains extrema that exhibit monodromiesaround points in the interior of the fundamental domain. The former set can be described analytically,while for the latter, we provide an analytic value for the point of monodromy for so (8), as well as extensivenumerical predictions for the Fourier coefficients of the extrema. Our results on the extrema provide arationale for integrality properties observed in integrable models, and embed these into the theory ofvector valued modular forms. Moreover, using the data we gather on the modularity of complexifiedintegrable system extrema, we analyse the massive vacua of mass deformed N = 4 supersymmetricYang-Mills theories with low rank gauge group of type B, C and D . We map out their transformationproperties under the infrared electric-magnetic duality group as well as under triality for N = 1 ∗ withgauge algebra so (8). We compare the exact massive vacua on R × S to those found in a semi-classicalanalysis. We identify several intriguing features of the quantum gauge theories. Unit´e Mixte du CNRS et de l’Ecole Normale Sup´erieure associ´ee `a l’universit´e Pierre et Marie Curie 6, UMR 8549. a r X i v : . [ h e p - t h ] A p r ontents A r = su ( r + 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 The B, C, D
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 The Case C = sp (4) = so (5) and Vector Valued Modular Forms . . . . . . . . . . . . . . 82.7 The Case D = so (8) and the Point of Monodromy . . . . . . . . . . . . . . . . . . . . . . 122.8 The Dual Cases B = so (7) and C = sp (6) . . . . . . . . . . . . . . . . . . . . . . . . . . 162.9 Limiting Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.10 Partial Results for Other Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 N = 1 ∗ gauge theories 20 C.1 Theta and Eta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27C.2 Modular Forms and Sublattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
D The List of Extrema 29
D.1 The List of Extrema for so (8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29D.2 The List of Extrema for so (7) and sp (6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Four-dimensional gauge theories accurately describe forces of nature. Since solving them is hard, we mayrevert to studying supersymmetric four-dimensional gauge theories, in which the power of holomorphylends a helping hand. Twenty years ago, we realised how to solve for the low-energy effective action onthe Coulomb branch of N = 2 gauge theories in four dimensions [1, 2]. The solution techniques were soonrecognised to lie close to those studied in integrable systems [3, 4]. It is the bridge between integrablemodels and supersymmetric gauge theories that we will further explore in this paper. We also attemptto reinforce both sides separately, and present results in a manner such that the contributions to thesetwo domains may be read independently.The link between supersymmetric gauge theories and integrable systems was useful in writing downthe low-energy effective action for N = 2 ∗ gauge theory, namely N = 4 super Yang-Mills theory withgauge group G , broken to N = 2 supersymmetry by adding a mass term for one hypermultiplet. Forthe gauge group G = SU ( N ) this program was completed in terms of a Hitchin integrable system with SL ( N, C ) bundle over a torus with puncture [5]. The associated elliptic Calogero-Moser system permitsgeneralisations to any root system, and allows for twists, which were used to provide Seiberg-Wittencurves and differentials for N = 2 ∗ theory with general gauge group G [6]. The generalisation was non-trivial since the elegant technique of lifting to M-theory [7] is difficult to implement in the presence oforientifold planes (see e.g. [8, 9]), while the relevant generalised Hitchin integrable system has a gaugegroup which is related to the gauge group of the Yang-Mills theory in an intricate manner [10]. For areview of part of the history, see the lectures [11].We will be interested in breaking supersymmetry further, from N = 2 to N = 1 by adding anothermass term for the remaining chiral multiplet (providing us with three massive chiral multiplets of arbitrarymass). We will study this N = 1 ∗ gauge theory with generic gauge group G . With N = 1 supersym-metry, we hope to calculate the effective superpotential W at low energies exactly. For an adjoint massdeformation from N = 2 to N = 1 this was done in the original work [1] in certain cases. For N = 1 ∗ and gauge group G = SU ( N ), the exact superpotential was proposed in [12] following the techniques of21, 13]. The superpotential is the potential of the complexified elliptic Calogero-Moser integrable systemassociated to the root lattice of type A N − . In [14] the exact superpotential for N = 1 ∗ with moregeneral gauge algebra was argued to be the potential of the twisted elliptic Calogero-Moser system withroot lattice associated to the Lie algebra of the gauge group G . See [15] for further generalizations to N = 1 ∗ theories with twisted boundary conditions on R × S .In this paper, we wish to analyse the proposed exact superpotential in more detail. This involves astudy of the properties of the isolated extrema of the complexified and twisted elliptic Calogero-Moserintegrable system. The results are of independent interest, and we have therefore dedicated a first partof this paper to the study of the integrable systems per se.The paper is structured as follows. In section 2, we review the relevant elliptic Calogero-Moser models.We pause to demonstrate a Langlands duality between the B and C type integrable systems. We thenanalyse the isolated extrema of the complexified potential of low rank integrable systems of B, C and D type, and their modular properties. We observe the strong connection to vector valued modular forms.The latter in turn provide a natural backdrop for integrality properties of integrable systems (see e.g.[16, 17, 18]). Section 2 is the technical heart of the paper, and we will lay bare many properties of thevector valued modular forms, using a combination of analytical work and extensive numerics. We willanalytically describe the potential in certain classes of extrema. We also find sets of extrema that exhibita monodromy in the interior of the fundamental domain. In these cases we are able to calculate themonodromy, as well as to provide extensive numerical data for the integer valued coefficients describingthe value of the potential at the extrema.Finally, in section 3, we reinterpret the results we obtained in terms of the physics of massive vacua of N = 1 ∗ theories. We compare our results for the quantum theory on R × S to semi-classical results formassive vacua and discuss electric-magnetic duality properties in the infrared under the modular groupas well as the Hecke group. For so (8), we also detail the action of the global triality symmetry on themassive vacua. We will encounter several interesting phenomena. We conclude in section 4 and arguethat we have only scratched the surface of a broad field of open problems. It is interesting to identify and study dynamical systems that are integrable. Often they form solvablesubsectors of more complicated theories of even more physical interest. There exist one-dimensionalmodels of particles with interactions that are integrable, and the Calogero-Moser models of our interestare one such class [19, 20, 21]. These models are associated to root systems of Lie algebras (amongstothers). See e.g. [22, 23] for a review. Integrable systems are also known to have certain integralityproperties. Namely, their minimal energy, frequencies of small oscillations as well as eigenvalues of Laxmatrices are often expressible in terms of a series of integers [16, 17, 18].In this section, we study properties of (twisted) elliptic Calogero-Moser systems. We analyse thecomplexified model, defined on a torus with modular parameter τ . In particular, we examine the extremaof the complexified potential, and exhibit their curious characteristics. The member of the pyramid of Calogero-Moser integrable systems we concentrate on is the ellipticCalogero-Moser model. We concentrate on the models associated with a root system ∆, as well as theirtwisted counterparts. These models have a Hamiltonian with rank r variables, with canonical kineticterm, and a potential of the form: V ∆ = g (cid:88) α ∈ ∆ ℘ ( α ( X ); ω , ω ) , (2.1)where ℘ is the Weierstrass elliptic function on a torus with periods 2 ω and 2 ω and g is a couplingconstant. We choose the half-periods such that the imaginary part of the modular parameter τ = ω /ω is positive. The vector X lives in the space dual to the root lattice of rank r and the sum in the potentialis over all the roots α of the root system ∆. The model is integrable for all Lie algebra root systems.The twisted elliptic Calogero-Moser model is defined in terms of twisted Weierstrass functions: ℘ n ( x ; ω , ω ) = (cid:88) k ∈ Z n ℘ ( x + kn ω ; ω , ω ) , (2.2)which are summed over shifts by fractions of periods (thus in effect modifying that period). We have atwisted elliptic Calogero-Moser model for all non-simply laced root systems and the value of n is then See appendix B for more on our conventions for elliptic functions. See appendix A for our conventions and a compendium of properties of Lie algebras and Lie groups. V ∆ ,tw = g l (cid:88) α l ∈ ∆ l ℘ ( α l ( X ); ω , ω ) + g s (cid:88) α s ∈ ∆ s ℘ n ( α s ( X ); ω , ω ) , (2.3)where α l denote the long and α s the short roots in the root system ∆ = ∆ l ∪ ∆ s , and g l and g s are twocoupling constants. We will concentrate on the root systems A r , B r , C r and D r corresponding to theclassical algebras su ( r + 1), so (2 r + 1), sp (2 r ) and so (2 r ). We allow complex values for the componentsof the vector X (i.e. X ∈ C r ). The symmetries of the potential
Let us discuss in detail the symmetries of the twisted elliptic Calogero model that act on the set of variables X . We first observe that the Weyl group action leaves invariant the scalar product α ( X ) = ( α, X ) and thatthe root system is Weyl invariant. This implies that the Weyl group action on X leaves the potentialinvariant. Secondly, we note that the outer automorphisms of the Lie algebra, which correspond tosymmetries of the Dynkin diagram, also leave the set of roots and the scalar product invariant. Therefore,outer automorphisms as well form a symmetry of the model.Moreover, the periodicities of the model in the two directions of the torus are as follows. By thedefinition of the dual weight, or co-weight lattice, we have that α ( λ ∨ ) ∈ Z for all roots α . This impliesthat shifts of X by 2 ω P ∨ , namely shifts by periods times co-weights, leave the potential invariant.To discuss the periodicity in the ω direction, we concentrate for simplicity on the algebras A, B, C and D , and normalize their long roots to have length squared two. We then have that for a long root α l and a weight λ , the equation ( α l , λ ) ∈ Z holds while for a short root α s of the B or C algebras wehave ( α s , λ ) ∈ Z , for all weights λ . As a consequence, the periodicity in the (twisted) ω direction isthe lattice 2 ω P where P is the weight lattice. The group of all symmetries is a semi-direct product ofthe lattice shifts, the Weyl group as well as the outer automorphism group. Beyond the many features of these integrable systems already discussed in the literature, the first sup-plementary property that will be pertinent to our study of isolated extrema, is their behaviour under aninversion of the modular parameter τ . We therefore briefly digress in this subsection to discuss a fewof the details of the duality. Models associated to simply laced Lie algebras map to themselves underthe modular S-transformation S : τ → − /τ . This is easily confirmed using the transformation rule(B.3) of the Weierstrass ℘ function under modular transformations. We do have a non-trivial Langlandsor short-long root duality between the twisted elliptic Calogero-Moser model of B-type and the twistedmodel of C-type. In order to exhibit the duality, we make the potential for the (twisted) B r = so (2 r + 1)theory more explicit: V B = b l (cid:88) i 5f we parameterise the potential in terms of the modular parameter τ = ω /ω , the duality transformationfor so (5) reads: V so (5) ( x , x , τ ) = 12 τ V so (5) (cid:18) x + x τ , x − x τ , − τ (cid:19) + 4 π E (2 τ ) − E ( τ )] . (2.10)In summary, we derived a Langlands duality between B and C type (twisted) elliptic Calogero-Mosermodels. The resulting identities captured in equations (2.8) and (2.10) and the shifts appearing in theseduality transformations will be useful. We return to the more general discussion of the integrable systems,and in particular their extrema. There have been many studies of classical integrable models at equilibrium. These have uncoveredremarkable properties, like the integrality of the minimum of the potential and of the frequencies of smalloscillations around the minimum, amongst others (see e.g. [16, 17, 18]). We will analyse the potential ofcertain elliptic integrable systems evaluated at generalised equilibrium positions. We show that they giverise to interesting vector valued modular forms as well as more general non-analytic modular vectors.Modularity provides a more conceptual way of understanding the integrality properties of the integrablesystem. This rationale then continues to hold for the integrable systems that can be obtained from theelliptic Caloger-Moser systems by limiting procedures (e.g. the trigonometric models). Thus, studying elliptic integrable systems, depending on a modular parameter, is found to have an additional pay-off.It is known that A -type integrable systems often have simpler properties than do the integrablesystems associated with other root systems. As a relevant example, let us quote the fact that the(real) Calogero-Moser (Sutherland) system with trigonometric potential of A -type has equally spacedequilibrium positions along the real axis, while the B, C, D -type potentials have minima associated tozeroes of Jacobi polynomials [17], which satisfy known relations [25], but are not known explicitly ingeneral. The elliptic Calogero-Moser systems that we examine show a similar dichotomy. Extrema of the(complex) elliptic A -model are equally spaced. This fact leads to relatively easily constructable valuesfor the potential at extrema, for any rank [5, 12, 26]. For the B, C, D -type models that we study inthis paper, much less is known, and we need to combine numerical searches with analytic approaches todetermine the extremal values of the potential, for low rank cases.To be more precise, we will be interested in extrema of the complexified potential, satisfying: ∂ X i V ( X j ) = 0 ∀ i , (2.11)and we moreover demand that at the extremum (2.11) the function r (cid:88) i =1 | ∂ X i V ( X j ) | (2.12)not posses any flat directions. Recall that the group of symmetries acting on the variables X were a lattice group of translations,the Weyl group as well as the outer automorphisms of the Lie algebra. Using these symmetries, we willintroduce a notion of equivalence on the variables X . We will consider the vector X to be identified bythe periodicities of the model. The periodicity in the ω direction is given by the weight lattice P , whilein the ω direction it is the co-weight lattice P ∨ . Furthermore, we will consider extrema that are relatedby the action of the Weyl group of the Lie algebra to be equivalent. By contrast, outer automorphismsare taken to be global symmetries of the problem. When the global symmetry group is broken by a givenextremum, the global symmetries will generate a set of degenerate extrema. A r = su ( r + 1) The extrema of the elliptic Calogero-Moser model of type A r have been studied in great detail, mostlyin the context of supersymmetric gauge theory dynamics (see e.g. [5, 12, 26]). Firstly, we remark that inthis case, the equivalence relations that follow from the periodicity of the potential as well as the Weylsymmetry group of the Lie algebra are straightforwardly implemented. We use the parameterisationof simple roots in terms of orthogonal vectors α i = e i − e i +1 , and the fundamental weights then read π i = (cid:80) ij =1 e j , with weight lattice spanned by the vectors e i . We can parameterise the coordinates of our This will correspond, in section 3, to a supersymmetric vacuum in the N = 1 ∗ gauge theory, where the effectivesuperpotential W is identified with the potential V of the integrable system. This condition implies that the vacuum is massive in the supersymmetric gauge theory. We briefly comment on masslessvacua later on. X j e j living in the dual to the root space (and e j ( e i ) = δ ji ). The Weylgroup S n acts by permuting the components X j . We can shift one of the components X j to zero byconvention. The equivalence under shifts by fundamental weights is identical to the toroidal periodicityrelations for the individual coordinates X j . The inequivalent extrema of the su ( n ) potential (satisfyingthe additional condition (2.12) of non-flatness) are then argued to correspond one-to-one to sublatticesof order n of the torus with modular parameter τ [5, 12]. These extrema are classified by two integers p and k satisfying that p is a divisor of n and k ∈ { , , . . . , np − } . The number of extrema is equal tothe sum of the divisors of n . The Z outer automorphism of A r> acts trivially on the minima, since itacts by permutation, combined with a sign flip for all X j , which leaves a sublattice ankered at the origininvariant.The value of the potential at one of these extrema is (with a given choice of coupling constant): V A n − ( τ ) = n (cid:18) E ( τ ) − pq E (cid:18) pq τ + kq (cid:19)(cid:19) . (2.13)Under the SL (2 , Z ) action on the torus modular parameter τ , the sublattices of order n of the torus arepermuted into each other (in a way that depends intricately on the integer n ). The permutation of thesublattices also entails the permutation of the values (2.13) at these extrema under SL (2 , Z ). The list ofextremal values of the elliptic Calogero-Moser model therefore form a vector valued modular form (seee.g. [27, 28, 29]) of weight two under the group SL (2 , Z ). The associated representation of the modulargroup is a representation in terms of permutations specified by the SL (2 , Z ) action on sublattices of order n . One can identify a subgroup of the modular group under which a given component of the vector-valuedmodular form is invariant, and then use minimal data to fix it [30].In summary, the extrema of the Calogero-Moser model of type A r = su ( r + 1) are under analyticcontrol. The positioning of the extrema can be expressed linearly in terms of the periods of the model,and the vector valued modular form of extremal values for the potential has an automorphy factor thatcan be characterised by sublattice permutation properties. The extremal values are generalised Eisensteinseries of weight two under congruence subgroups of the modular group. B, C, D Models For other algebras, we are at the moment only able to study low rank cases. From the analysis, it isclear that crucial simplifying properties of the A r case are absent. Nevertheless, generic features of the A r case persist in a subclass of extrema, in that we find vector-valued modular forms as extremal valuesfor the potential. We also find a class of extremal values that exhibit new features.To describe in detail which extrema are considered to be equivalent, we must discuss the equivalencerelations that we mod out by for the B, C and D root systems individually. D r = so (2 r )For the D r case, we can parameterise the roots as α i = e i − e i +1 (for i ∈ { , , . . . , r − } ) and α r = e r − + e r . We put X = X j e j and imply that the relation e i ( e j ) = δ ij holds. The equivalence of thevector X under shifts proportional to the weight lattice implies that each variable X j lives on a toruswith modular parameter τ . It moreover identifies the vector X with the vector X shifted by a half-periodin each variable simultaneously. The Weyl group is W ( so (2 r )) = S r (cid:110) Z r − , and acts by permutationof the components X j , as well as the sign change of an even number of them. The outer automorphismgroup (for r (cid:54) = 4) is equal to Z and acts as X r → − X r . For r = 4, the global symmetry group is S triality. B r = so (2 r + 1)For B r , the roots are α i = e i − e i +1 (for i ∈ { , , . . . , r − } ) and α r = e r . We recall that the periodicityis the weight lattice in the ω direction (due to the twist), and the co-weight lattice in the ω direction.Thus, we can shift components of the vector X = X j e j by periods, or all components simultaneously bya half period in the ω direction. In the ω direction, we allow shifts of the individual components byperiods. The Weyl group acts by combinations of permutations and any sign flip of the coordinates. C r = sp (2 r )The roots are α i = ( e i − e i +1 ) / √ i ∈ { , , . . . , r − } ) and α r = √ e r . We can shift components X j of X = √ X j e j by half-periods in the ω direction, while in the ω direction, we can allow shifts by anyperiod, as well as a half-period shift of all X j simultaneously. The Weyl group allows any permutation By our conventions, we normalise the long roots such that they have length squared two. B, C and D cases,beyond permutation symmetries and toroidal periodicity, are summarized in the table: B r Individual X i → − X i Collective X i → X i + ω C r Individual X i → − X i and X i → X i + ω Collective X i → X i + ω D r Even number of sign flips X i → − X i Collective X i → X i + ω and X i → X i + ω Global symmetries : Z generically and S for D .Armed with this detailed knowledge about the equivalence of configurations, we programmed a nu-merical search for isolated extrema. In the following subsections, we list the results we found by rootsystem. For simply laced root systems we studied the elliptic Caloger-Moser model, while results for non-simply laced root systems correspond to the twisted elliptic Calogero-Moser model with a coefficient forthe short root term which is equal to one half the coefficient in front of the long root terms (as describedbelow equation (2.7)). C = sp (4) = so (5) and Vector Valued Modular Forms Since the root system C is the first example of our series, we provide a detailed discussion. We discussthe positions of the isolated extrema, the series expansions relevant to the potential at these extrema,the action of the duality group, as well as the identification of the relevant vector valued modular forms. For the Lie algebra so (5) = sp (4) we found 7 isolated extrema of the potential. We provide theirpositioning at τ = i in figure 1. We have drawn in bold the positions of the extrema as well as theiropposites, in a fundamental cell of the torus. These numerical results were found using a Mathematica program, which was written around thebuilt-in function FindMinimum . Careful programming augments the precision of the algorithm to at leasttwo hundred digits. The most costly part of the algorithm is the random search for extrema. Indeed,the intricate landscape drawn by the potential can hide extrema. We gave a drawing of the positionof the numbered extrema on the torus with modular parameter τ = i . The positions of the extremafor other values of the modular parameter can be reached by interpolation. We have analytic controlover a few extra properties of the extrema. E.g. if we follow extremum 1 to τ = i ∞ , we find thatthe equilibrium positions are given by π arccos( ± / √ 3) where ± / √ P (0 , . The first extremum, which we label 1, lies on the real axis and is the equilibriumposition of the real integrable system. The extremum 2 lies on the imaginary axis, while extrema 3 and4 are then approximately obtained by applying the transformation τ → τ + 1. The extrema 5 and 6 are S Langlands duals of extrema 3 and 4. It is easy to deduce from the potential that the positions of theextrema generically behave non-linearly as a function of τ . By numerically evaluating the extrema of the potential for a range of values of the modular parameter τ ,we are able to write the extrema as an expansion in terms of a power of the modular parameter q = e πiτ .The extremal values can be written in terms of the series: A ( q ) = 124 + q + q + 4 q + q + 6 q + 4 q + 8 q + q + 13 q + 6 q + 12 q +4 q + 14 q + 8 q + . . . (2.14) A ( q ) = 1 + 48 q + 828 q + 8064 q + 109890 q + 1451520 q + 11198088 q + 141212160 q +1666682811 q + 9413050176 q + 145022264892 q + 1838450006784 q +11103941590326 q + 138638111404032 q + . . .A ( q ) = 2 + 48 q + 576 q + 9792 q + 99576 q + 743904 q + 13146624 q + 115737984 q We have indicated reflections over other half-periods in grey, to illustrate that the minima are close to forming sublatticestructures. τ = i for the Lie algebra so (5) 14 12 34 Extremum 1 14 12 34 Extremum 2 14 12 34 Extremum 3 14 12 34 Extremum 4 14 12 34 Extremum 5 14 12 34 Extremum 6 14 12 34 Extremum 7 91015727364 q + 14338442448 q + 102050482176 q + 935515738944 q +12532363069968 q + 122390111091744 q + . . .A ( q ) = 13216 + 7 q + 541 q + 24508 q + 939669 q + 19944842 q + 764752180 q +21016537080 q + 905672825157 q + 38827071780859 q + 827503353279726 q + . . .A ( q ) = 1 + 148 q + 7446 q + 154344 q + 5100349 q + 352720380 q + 10627587582 q +166124184888 q + 5419843397586 q + 294399334337124 q + . . .A ( q ) = − q + 431 q + 80468 q − q + 94846414 q + 1301490428 q +90560563752 q − q + 93349951292249 q + . . . . (2.15)The integer coefficients have been determined up to an accuracy of at least 10 − . For the first orderterms, the accuracy can be up to 10 − . In terms of these series, the potential in extremum number 1,on the real axis is (with a given choice of normalisation): V = 144 π A (cid:16) q (cid:17) . (2.16)The potential in the other extrema are: V = − π (cid:18) A ( q ) + (2 q ) / A ( q/ 9) + (2 q ) / A ( q/ (cid:19) V = − π (cid:18) A ( q ) + (2 q ) / e πi/ A ( q/ 9) + (2 q ) / e πi/ A ( q/ (cid:19) V = − π (cid:18) A ( q ) + (2 q ) / e πi/ A ( q/ 9) + (2 q ) / e πi/ A ( q/ (cid:19) , (2.17)and V , = 72 π (cid:18) A (cid:16) q (cid:17) ± i (cid:114) q A (cid:16) − q (cid:17)(cid:19) V = 483 π A ( q ) . (2.18)The growth properties of these series, as well as the fact that we are dealing with a physical system livingon a torus suggests turning these numerical data into an analytic understanding, based on the theory ofmodular forms. In the following, we show that this is possible for the rank 2 root system B . Γ (4) subgroup We need to introduce a few groups related to the modular group. We already noted the duality transformfor the B, C -type twisted Calogero-Moser system under the map S : τ → − / (2 τ ) (see equation (2.7)).For the so (5) Lie algebra, which is identical to the sp (4) Lie algebra, this transformation maps theintegrable system to itself (up to a τ dependent shift of the potential and an overall factor – see equation(2.10)). The map T : τ → τ + 1 also maps the integrable system to itself. Together, these transformationsgenerate the action of a Hecke group dubbed Γ ∗ (2) on the modular parameter τ . This group contains asubgroup Γ (4) which is a congruence subgroup of the modular group SL (2 , Z ). Generators of the groupΓ (4) can be chosen to be the 2 × T : (cid:18) (cid:19) U : (cid:18) (cid:19) . (2.19)The action of these matrices on τ coincides with the action of the elements T and U = S T − S of theHecke group. For more information on Hecke groups and associated modular forms see e.g. the lectures[31].The extremal values of the potential may therefore form a vector valued modular form with respectto the Hecke group Γ ∗ (2), and as a consequence also with respect to the congruence subgroup Γ (4) ofthe modular group SL (2 , Z ), since we expect extrema to be at most permuted and/or rescaled under thegroup. Here, we assume analyticity in the interior of the fundamental domain. We will mostly exploitthe group Γ (4) in the following, since the literature on the subject of modular forms with respect to10 Figure 2: The diagram of the action of dualities on the extrema for B = so (5). In red, we draw theaction of Langlands S -duality, and in green, T -duality (when the action is non-trivial).congruence subgroups is abundant. For starters, we determine the action of the operations T and S onthe vector V i of extremal values of the twisted Calogero-Moser potential: T : ,S : − 21 0 0 0 0 0 − 20 0 0 0 1 0 − 20 0 0 0 0 1 − 20 0 1 0 0 0 − 20 0 0 1 0 0 − 20 0 0 0 0 0 − . (2.20)See figure 2 for a summary of the action of the duality group. To this information, we add the lastcolumn in the matrix S , which originates in the shift of the potential under Langlands duality. Fromthese data, we easily calculate the action of the generator U = S T − S on the vector valued modularform: U : . (2.21)We thus find the action of Γ (4) on the vector valued modular form, and we observe the following pattern:there is one entry (the seventh) which is an ordinary modular form of weight 2 under Γ (4), and thereare two sets of three components (namely { , , } and { , , } ) that mix under Γ (4). Thus, our vectorvalued modular form of dimension seven splits into a singlet and a sextuplet. Concentrating on theordinary modular form of weight 2, we have that it is a linear combination of Eisenstein series E ,N defined by: E ,N ( τ ) = E ( τ ) − N E ( N τ ) . (2.22)Indeed, the dimension of the space M (Γ (4)) of modular forms of Γ (4) is two, and it is spanned by E , and E , . We thus only need two Fourier coefficients to fix the entire modular form, and we findthat: A ( q ) = − E , ( τ ) = 148 ( θ + θ )( τ ) (2.23) V = π θ + θ )( τ ) . (2.24)We then have a slew of consistency checks on all the other integers that we determined numerically (see(2.14)). These thirteen checks work out. We do therefore claim that the result (2.24) is exact. This is asimple example illustrating our methodology.Next, we consider the triplet consisting of the components { , , } . We find three eigenvectors of T , with eigenvalues corresponding to the cubic roots of unity. The eigenvector with eigenvalue 1 is also11apped to itself under the U transformation, and forms again a modular form of weight 2 under Γ (4).It is indeed proportional to E , : V + V + V = − π ( θ + θ )( τ ) . (2.25)The other two eigenvectors, we raise to the power three, such that they become invariant under the T -transformation. These forms belong to the space M (Γ (4)) of weight six modular forms. The dimensionof this vector space is 4 (see theorem 3.5.1 in [32] with g = ε = ε = 0 and ε ∞ = 3), and it consists ofthree Eisenstein series, and one cusp form. A basis for these vector spaces is given by: E = − E ( τ ) (2.26) E = − E (2 τ ) (2.27) E = − E (4 τ ) (2.28) S = η ( q ) , (2.29)where E is the Eisenstein series of weight six, and η is the η -function, also recorded in appendix C.We need four coefficients to fix the eigenvectors in terms of this basis and we find (using the notation ω = exp(2 πi/ V + ω V + ω V ) = − π ( E − E − S ) (2.30)( V + ω V + ω V ) = − π ( E − E + 2 S ) . (2.31)The consistency checks using the numerics work out.For the second triplet, we diagonalise U first, and proceed very analogously as above, except that wehave to take a higher power for the second combination to find a modular form of weight 12 with respectto Γ (4). We find the relations:( V + ω V + ω V ) + ( V + ω V + ω V ) = 5832 π ( E ( q ) − E )(( V + ω V + ω V ) − ( V + ω V + ω V ) ) = 136048896 π η ( q ) . (2.32)Note that the sum of all potentials is necessarily a modular form with weight 2 of Γ (4). Indeed, thissum is equal to 112 π A ( q ) (as follows from the identity A ( q/ 27) + A ( q/ 27) = A ( q )). There are also branches of extrema, namely, non-isolated extrema. These too, we expect to behave wellunder a modular subgroup. Although this was not the focus of our investigation, we did find numericalevidence for a manifold of extrema at which the potential takes the Γ (4) covariant value − π E , . Summary In summary, we have full analytic control over the value of the potential for all isolated extrema of the so (5) twisted Calogero-Moser integrable system. We have found a vector valued modular form of weighttwo of Γ (4), and we were able to explicitly express its seven components in terms of ordinary modularforms of Γ (4). The vector valued septuplet splits into a singlet modular form and a sextuplet vectorvalued modular form. The plot will thicken at higher rank. D = so (8) and the Point of Monodromy At this stage, we choose to present our results on the rank four D = so (8) model first, since they aresimpler than those on the non-trivial rank three cases to be presented in subsection 2.8. The so (8) modelis simply laced and we therefore expect the ordinary modular group SL (2 , Z ) to play the leading role.The integrable system exhibits a global symmetry group S that permutes the three satellite simple rootsof the Dynkin diagram of so (8). We will refer to the S permutation group as triality. We turn to theenumeration and classification of the extrema of the potential. We found 34 extrema. These are listedand labelled in appendix D.1. If we mod out by the global symmetry group, we are left with 20 extrema.The latter fall into multiplets of the duality group of size 1 , , There is a singlet under S and T duality as well as triality. It has zero potential: V = 0.12 .7.2 The triplet There is also a triplet under the duality group, labelled { , , } , and the dualities act as: T = S = . The relations S = 1 and ( ST ) = 1 are satisfied. We note that in these extrema, the positions belongto the lattice generated by ω / ω / 2. For this multiplet,T-duality acts geometrically.We would like to deduce again from the S and T matrices and from the known first coefficients ofthe series expansions (see appendix D.1) the exact expressions of the potentials in these extrema. Thefunctions are expected to transform well under some congruence subgroup of the modular group. Notethat the sum of the three functions must be a full-fledged modular form – indeed, the sum V ( q ) + V ( q ) + V ( q ) vanishes. A brute force strategy leading to the identification of the appropriate congruencesubgroup is the following. We decompose the generators of congruence subgroups in terms of a productof S and T operations. We evaluate the product using the representation at hand (here 3 × V , V and V belong to M (Γ(2)). This space has dimension 2, and it is the set of linear combinations ofthe three Eisenstein functions associated to the three vectors of order 2 in ( Z ) which have the propertythat the sum of the three coefficients vanishes. (See appendix C for details and conventions). Matchinga few coefficients, we find that V = 12 (cid:18) G , (cid:20) (cid:21) − G , (cid:20) (cid:21) − G , (cid:20) (cid:21)(cid:19) V = 12 (cid:18) − G , (cid:20) (cid:21) − G , (cid:20) (cid:21) + 2 G , (cid:20) (cid:21)(cid:19) V = 12 (cid:18) − G , (cid:20) (cid:21) + 2 G , (cid:20) (cid:21) − G , (cid:20) (cid:21)(cid:19) . This can also be written in terms of the Weierstrass ℘ function : V ( τ ) = 3 (cid:18) ℘ (cid:18) 12 ; τ (cid:19) − ℘ (cid:18) τ + 12 ; τ (cid:19) − ℘ (cid:16) τ τ (cid:17)(cid:19) V ( τ ) = 3 (cid:18) − ℘ (cid:18) 12 ; τ (cid:19) − ℘ (cid:18) τ + 12 ; τ (cid:19) + 2 ℘ (cid:16) τ τ (cid:17)(cid:19) V ( τ ) = 3 (cid:18) − ℘ (cid:18) 12 ; τ (cid:19) + 2 ℘ (cid:18) τ + 12 ; τ (cid:19) − ℘ (cid:16) τ τ (cid:17)(cid:19) . These two ways of writing the potentials make the action of dualities manifest. For instance, the transfor-mation properties (B.3) show that under S -duality, ℘ ( , τ ) becomes ℘ ( , − τ ) = τ ℘ ( τ , τ ) while ℘ ( τ +12 , τ )becomes τ ℘ ( τ +12 , τ ), so that V and V are S -dual, et cetera. The result can also be written using perhapsmore familiar modular forms V ( q ) = − π E , ( q ) V ( q ) = 32 π (cid:0) E , ( q ) − θ ( q ) (cid:1) V ( q ) = 32 π (cid:0) E , ( q ) + 3 θ ( q ) (cid:1) . The action of T -duality is again clear from these expressions. For S -duality it is slightly more intricate.Given that E , ( q ) = − θ ( q ) − θ ( q ), it relies on the identities2 θ (2 τ ) + 2 θ (2 τ ) + 3 θ ( τ ) = − θ ( τ / 2) + 2 θ ( τ / θ ( τ / 2) + θ ( τ / − θ ( τ ) = − θ (2 τ ) − θ (2 τ ) + 6 θ ( τ ) , for S -duality between extrema 2 and 3, and self- S -duality for extremum 4, respectively. There exist algorithms to find the generators. These are for instance implemented in Sage. .7.3 The quadruplet We move on to discuss the extremal values of the potential in the quadruplet. We can arrive at thefollowing closed form for the potential in extremum 6: V ( q ) = − π ( − E , ( q ) + ( η ( q ) + 9 η ( q ) ) η ( q ) /η ( q ) + 3( η ( q ) /η ( q )) ) . Note that this can alternatively be written as V ( q ) = − π ( g ( q ) + q / g ( q ) + 3 q / g ( q )) , where the g i are functions that can be expanded into series with only integer powers of q (and the threesummands in this expression correspond to the same summands in the expression above). Thus we knowhow the operation τ → τ + 1 acts on the extremum, and it generates two other extrema, whose potentialwe also know exactly. These are extrema 7 and 8: V ( q ) = − π ( g ( q ) + e iπ/ q / g ( q ) + 3 e − iπ/ q / g ( q )) V ( q ) = − π ( g ( q ) + e − iπ/ q / g ( q ) + 3 e iπ/ q / g ( q )) . The potential for the extremum 5 is: V ( q ) = − π E , ( q ) . In the basis { , , , } the matrices for S - and T -dualities are : T = S = . We can also apply the same method as above. The generators of Γ(3) are all trivial in this basis. Thusthe potentials are weight 2 modular forms of this congruence subgroup. The latter form a 3-dimensionalspace, generated by the zero-sum linear combinations of the 4 Eisenstein series associated to the order 3vectors in ( Z ) (there are 8 such vectors, but the Eisenstein series are invariant under v → − v , leavingonly 4 distinct functions, see appendix). We find V = 272 (cid:18) G , (cid:20) (cid:21) − G , (cid:20) (cid:21) − G , (cid:20) (cid:21) − G , (cid:20) (cid:21)(cid:19) V = 272 (cid:18) − G , (cid:20) (cid:21) + 3 G , (cid:20) (cid:21) − G , (cid:20) (cid:21) − G , (cid:20) (cid:21)(cid:19) V = 272 (cid:18) − G , (cid:20) (cid:21) − G , (cid:20) (cid:21) + 3 G , (cid:20) (cid:21) − G , (cid:20) (cid:21)(cid:19) V = 272 (cid:18) − G , (cid:20) (cid:21) − G , (cid:20) (cid:21) − G , (cid:20) (cid:21) + 3 G , (cid:20) (cid:21)(cid:19) , or alternatively, V ( τ ) = 32 (cid:18) ℘ (cid:18) 13 ; τ (cid:19) − ℘ (cid:16) τ τ (cid:17) − ℘ (cid:18) τ + 13 ; τ (cid:19) − ℘ (cid:18) τ + 23 ; τ (cid:19)(cid:19) V ( τ ) = 32 (cid:18) − ℘ (cid:18) 13 ; τ (cid:19) + 3 ℘ (cid:16) τ τ (cid:17) − ℘ (cid:18) τ + 13 ; τ (cid:19) − ℘ (cid:18) τ + 23 ; τ (cid:19)(cid:19) V ( τ ) = 32 (cid:18) − ℘ (cid:18) 13 ; τ (cid:19) − ℘ (cid:16) τ τ (cid:17) + 3 ℘ (cid:18) τ + 13 ; τ (cid:19) − ℘ (cid:18) τ + 23 ; τ (cid:19)(cid:19) V ( τ ) = 32 (cid:18) − ℘ (cid:18) 13 ; τ (cid:19) − ℘ (cid:16) τ τ (cid:17) − ℘ (cid:18) τ + 13 ; τ (cid:19) + 3 ℘ (cid:18) τ + 23 ; τ (cid:19)(cid:19) . The dualities act on the vectors characterising the modular forms as follows T : (cid:20) (cid:21) → (cid:20) (cid:21)(cid:20) (cid:21) → (cid:20) (cid:21) → (cid:20) (cid:21) → (cid:20) (cid:21) , : (cid:20) (cid:21) ↔ (cid:20) (cid:21)(cid:20) (cid:21) ↔ (cid:20) (cid:21) . This reproduces the action of the dualities on the associated extrema. Thus, while the pattern of thepositions of the extrema is non-linear, the arguments of the values of the potential at certain extrema doprovide a linear realisation of the duality group.Finally, we note that triality generates three copies of the triplet as well as of the quadruplet. Indeed,each of these extrema is left invariant by a Z subgroup of S (as described in appendix D.1).Up to now, we have discussed the singlet, triplet and quadruplet whose duality diagrams are sum-marised in figure 3. In the multiplet of size twelve, also depicted in figure 3, a new feature appears. We find that the extremaexhibit a monodromy around a point in the interior of the fundamental domain of the parameter τ . Thus,to be able to describe the multiplet structure in this case we must first discuss the monodromy. The point of monodromy We find a single point in the interior of the fundamental domain around which there is monodromyamongst extrema. It is possible to determine this point numerically and its value is close to τ M ∼ . i . In particular, the extrema 13 and 16 are exchanged when we follow a loop in the τ -plane thatclosely circles the value τ M . Moreover, using the geometry of the positions of the extrema 13 and 16, onecan show that τ M is a solution of the system of equations (cid:26) ℘ ( z ; τ ) + ℘ ( z − ω ; τ ) + ℘ (2 z − ω ; τ ) = π E ( τ )2 ℘ (cid:48) ( z ; τ ) + 2 ℘ (cid:48) ( z − ω ; τ ) + ℘ (cid:48) (2 z − ω ; τ ) = 0 , (2.33)where ω = ω + ω , which gives the numerical result τ M = 2 . ...i . Using the large accuracy of the value of the point of monodromy τ M , we find the corresponding rationalKlein invariant (with the normalisation (C.1)): j ( τ M ) = 488095744125 = 1728 × . This can be considered as an exact statement – the uncertainty is as low as 10 − . Elliptic curves withrational Klein invariant have interesting arithmetic properties (see e.g. [32]). The extended duality group We can add the monodromy group to the set of generators S and T that act on our vector of extrema.The resulting diagram of dualities then becomes the one in figure 3. The generators satisfy the relations: • S = M = 1 and T = 1, while ( T M ) = 1 • SM = M S • ( M ST ) = 1.Once we are underneath the point of monodromy in the canonical fundamental domain, the matrix M T plays the role usually taken by the matrix T in SL (2 , Z ). In particular, relations like ( ST ) = 1 impliedby the geometry of the fundamental domain of the modular group take on the form ( SM T ) = 1, etcetera. Triality leaves each extremum invariant.In appendix D.1, we give terms in the Fourier expansion of the extremal values of the potential inthe duodecuplet. We note that a consistency and exhaustivity check on all multiplets is provided by thefact that the sum of all extrema in a given multiplet of SL (2 , Z ) has to be a weight 2 modular form. Thecheck works out: the sum equals zero in each multiplet separately, as it must. An analytic understandingof the duodecuplet extrema remains desirable. The most immediate manifestation of the monodromy phenomenon can be seen as a symmetry breaking in the equi-librium positions for extrema 13 and 16 when moving on the imaginary axis across the point of monodromy τ M (which ispurely imaginary). Below this critical value, as can be seen in the diagrams drawn at τ = i (in appendix D.1), the twoextrema are exchanged by the Z action X i ↔ − ¯ X i , while above the critical value, they are both invariant with respect tothis action. This makes it possible to determine 2 . ≤ Im τ M ≤ . Figure 3: The diagram of the action of dualities on the D = so (8) extrema. In red we exhibit the actionof S -duality, in green, T -duality, and in dotted blue, the monodromy. B = so (7) and C = sp (6) For the twisted elliptic integrable models associated to the dual Lie algebra root systems so (7) and sp (6),we present our results succinctly. We have found 17 isolated extrema for each, and they are Langlandsdual. We have therefore 34 extrema in total. We identified two quadruplets of the full duality groupfor which we found analytic expressions for the potential at the extrema. The list of the correspondingextrema is given in appendix D.2. We find the following duality properties and analytic values for theextrema of the potential. The extrema labelled { , } have extremal values for the so (7) potential equalto V ( τ ) and V ( τ ). From the diagram of dualities (figure 4), we read off that these extremal valuesare modular forms of Γ (4) with weight 2. Moreover, Langlands duality then implies that V ∨ (2 τ ) and V ∨ (2 τ ) are also of that ilk. The space M (Γ (4)) of these weight 2 forms has the two generators − E , ( τ ) = θ (2 τ ) + θ (2 τ ) = 1 / θ ( τ ) + θ ( τ )) − E , ( τ ) = 3 θ (2 τ ) = 3 / θ ( τ ) + θ ( τ )) . In terms of the generators, the extrema are: V ( τ ) = π ( − E , ( τ ) − E , ( τ )) V ( τ ) = π ( − E , ( τ ) + 2 E , ( τ )) V ∨ (2 τ ) = π (+ E , ( τ ) + 0 E , ( τ )) V ∨ (2 τ ) = π ( − E , ( τ ) + 1 E , ( τ )) . For the other quadruplet under the full duality group, we have a similar story, with the happy ending: V (2 τ ) = π / − E , ( τ ) + 7 E , ( τ )) V (2 τ ) = π / E , ( τ ) − E , ( τ )) V ∨ ( τ ) = 8 π / − E , ( τ ) + 1 E , ( τ )) V ∨ ( τ ) = 8 π / E , ( τ ) − E , ( τ )) . S duality as well as T-duality can be found explicitly using these exact expres-sions, for instance by exploiting properties of θ functions. As an example, we note that the action of T -duality is summarised in the equalities: E , (cid:18) τ + 12 (cid:19) = − E , ( τ ) + E , ( τ ) E , (cid:18) τ + 12 (cid:19) = − E , ( τ ) + 2 E , ( τ ) . Moreover, on the extrema, the Langlands duality S acts as12 τ V (cid:18) − τ (cid:19) = V ∨ ( τ ) + 3 π E , ( τ ) , and similar relations hold for the other S -dual couples, as predicted by the duality formula (2.8). We further identified a duodecuplet and a quattuordecuplet under the duality group (for a total of(4 + 4 + 12 + 14) / B = so (7)). Sufficient data to reproduce them is provided inappendix D.2. These multiplets exhibit points of monodromy, and the full duality diagram is capturedin figure 4. It should be understood that we only represent points of monodromy that are inequivalent(where two monodromies are taken to be equivalent when they are equal up to conjugation by otherelements of the duality group). For instance S M τ S is the monodromy around − / (2 τ ).We draw attention to a few features of the diagram. There are 5 extrema that form a quintuplet underT-duality (around τ = i ∞ ), labelled 5 , , , , 9. When we also turn around the point of monodromy,the quintuplet enhances to a septuplet. This is reminiscent of a feature of the duality diagram for theduodecuplet of so (8).Finally, we performed an exhaustivity check on the extrema by summing the extremal values of thepotential. We found (cid:88) i ∈ V i ( τ ) = − π E , ( τ ) (cid:88) i ∈ V i ( τ ) = 2 π E , ( τ ) (cid:88) i ∈ V i ( τ ) = − π E , ( τ ) (cid:88) i ∈ V i ( τ ) = 19 π E , ( τ ) , showing again that the sum of potentials over every multiplet is a modular form of Γ (4).This concludes our systematic case-by-case discussion of the low rank B, C, D isolated extrema of(twisted) elliptic Calogero-Moser models. We finish the section with a few further remarks on generalfeatures of the problem of identifying isolated extrema. We wish to make a remark on the limiting behaviour of the integrable models near an extremum. Asan example, consider extremum number 7 for B = so (5) which has its extremal positions equal to areal number plus τ / 2. We can take the limit of the potential as τ → i ∞ while keeping the differencebetween the extremal positions and τ / D = so (4), and indeed, the real part of the extremal positions agrees withthose of the Sutherland system. This is but one example of the limiting behaviour of the models nearthe extrema. In this subsection, we discuss very partial results for some higher rank Lie algebras. We think of theelliptic integrable model as a perturbation of the Sutherland model, with trigonometric potential. The We evaluated the sum of the extrema numerically at two different values of τ to identify the linear combination of E , and E , that equals the sum. We can then perform arbitrary many numerical checks at other values of τ , and these workout. v 312 42 v v v v v v v v v 105 6789 5 v v v v v v v Figure 4: The diagram of dualities for so (7) and sp (6) extrema. In red, we show the action of Langlands S -duality on the extrema, in green, T -duality, and in dotted blue, monodromies, with the correspondingapproximate values of the points of monodromy τ . As discussed in the text, monodromies relating sp (6)extrema exist but are not represented here as they are equivalent to those already depicted.18igure 5: The positions of the monodromies (red dots) inside the fundamental domain of Γ (4) (shaded)Sutherland model has a ground state with all particles sprinkled on the real circle. We can perturbthis traditional ground state by turning on the elliptic deformation by powers of the small parameter q ,and follow the ground state under perturbation. In this way, we can reconstruct the extremum of thecomplexified elliptic potential associated to the Sutherland extremum on the real line. To take the limitfrom the elliptic integrable system towards the Sutherland model, it is sufficient to use the expansionformula: ℘ ( x ; ω , ω ) = − π ω E ( q ) + π ω csc (cid:18) πx ω (cid:19) − π ω ∞ (cid:88) n =1 nq n − q n cos nπxω , (2.34)valid when the imaginary part of the modular parameter τ is sufficiently large. The first term in theformula (2.34) is constant from the perspective of the integrable system dynamics, while the secondterm gives rise to the leading Sutherland potential. The minimum at the equilibrium of the Sutherlandpotential on the real line can be computed analytically [17] – it is related to the norm of the Weyl vectorof the Lie algebra. The positions of the equilibria are given in terms of zeroes of the Jacobi polynomials.We can perform perturbation theory around these extrema (numerically), and we find the following seriesin q for the potential at perturbed Sutherland extrema, for various gauge algebras: V so (5) π = 263 + 112 q q 81 + 392128 q q q q q . . .V so (6) π = 8 + 64 q + 192 q + 256 q + 192 q + 384 q + 768 q + 512 q + 192 q + . . .V so (7) π = 25 + 408 q q 625 + 23730528 q q q q q . . .V so (8) π = 24 + 576 q q 625 + 39538944 q q q q q q . . .V so (9) π = 1643 + 992 q q q q N Table 1: The integer N for gauge algebra so ( k ) rendering the Fourier expansion integral+ 11307570247017024 q q q q q ...V so (10) π = 1603 + 1280 q q q q q q q q ...V so (11) π = 3053 + 1960 q q q q q q q q ...V so (12) π = 100 + 800 q q q q q q ... We see that at least for some extrema, it is fairly straightforward to generate interesting data on the valueof the potential at these extrema at higher rank. We note a first pattern, valid at the order to which wehave worked, in both the rank of the gauge group, and the power of the modular parameter q . Table 1gives the conjectured smallest integer N such that for gauge algebra g , the potential V g ( N q ) /π has aFourier expansion with only integer coefficients in the following sense: the expansion can be written as n ( r + n q + n q + ... ) where the n i are integers, and the first term r is rational.As an example of this pattern, let us quote the formula:166679200 π V so (12) (cid:0) × q (cid:1) = 1666792 + q + 745143 q + 252572301828 q + 108583732036588599 q +25066769592690393853446 q + 11087973934403204342320752348 q +1966652180387341854168182867614728 q + ... As a final remark, we note that our numerical searches in this and previous subsections are far fromexhausting the capabilities of present day computers. N = 1 ∗ gauge theories In this section, we first briefly review properties of the infrared physics of the N = 1 ∗ supersymmet-ric gauge theory, and then show how the data we gathered on elliptic integrable systems in section 2elucidates the physics of this gauge theory further. We obtain the N = 1 ∗ gauge theory from N = 4supersymmetric Yang-Mills theory with gauge group G by adding three masses m i for the three chiral N = 1 supermultiplets. We can then go to the Coulomb branch of the gauge theory, and compactify thetheory on a circle [1, 13]. Two massless scalars remain in the theory for each U (1) in the unbroken U (1) r gauge group, namely the Wilson lines φ = (cid:82) S A µ dx µ and the scalar duals σ of the photons. Since thereare no fields in the theory which are charged under the center of the gauge group, we may choose thegauge group such that we allow for gauge transformations that twist around the circle by an element ofthe center. This lends a periodicity to the Wilson line under shifts taking values in the co-weight lattice P ∨ . This reasoning corresponds to a choice of gauge group G = ˜ G/C where ˜ G is the universal cover, and C its center. Note that the dual theory to the one with gauge group ˜ G/C has gauge group ˜ G , for a simply laced group. The twoscalars are interchanged under S-duality. Thus, the duality symmetries mix various global choices of gauge groups. Dualityalso acts on the twist direction of the twisted elliptic potential. Q ∨ . The scalar dualsof the photons have as a result a smallest possible periodicity equal to the weight lattice P . We chooseto classify extrema of the superpotential with respect to these identifications. We should mention thatother choices will be physically relevant. Since in deriving the effective superpotential we compactifiedthe theory on R × S , the resulting effective theory is influenced by the choice of the spectrum of lineoperators that probe the phases of our four-dimensional theory [33, 5, 34]. These determine the set ofallowed monopole operators in three dimensions, and this set may be larger than the collection allowed bythe minimal periodicity relation chosen above. Depending on the choice of the spectrum of line operators,this can lead to an increase of the number of inequivalent solutions, and therefore to an increase in theWitten index. This was analyzed carefully in [34, 35].We have identified the shift symmetries acting on the Wilson line and the dual photon. We furtherdivide out the configuration space by the Weyl group, which is the remnant of gauge invariance. Thisclassification of supersymmetric vacua agrees with the classification we did in section 2 in the ellipticintegrable systems, on the condition that we identify the ω direction with the Wilson line.Our N = 1 ∗ theory is a deformation of N = 4 theory, and it inherits some of its properties. Inparticular, the electric-magnetic duality group of N = 4 gauge theories in four dimensions [36, 37, 38]plays a crucial role. The duality symmetry was determined to be the group SL (2 , Z ) for simply lacedgauge groups and Γ (4) for the B and C type gauge groups [39, 40, 41]. Moreover, the S generator ofthe Hecke group exchanges the B and C type systems. An infrared counterpart to these duality groupsare present in our integrable systems, which allow for a (generalized) duality group action on the infraredmodular parameter τ [26], inherited after mass deformation from the N = 4 duality. Note in particularthat the requirement of the B type and C type exchange is implemented in our integrable system by theLanglands duality we discussed in subsection 2.2. This duality provides a further consistency check onthe relative weight of the short and long root contributions, fixed in [14] through consistency with thesuperpotential of the pure N = 1 super Yang-Mills theory.In [14], following the reasonings in [1, 13, 6, 12], an exact effective superpotential for N = 1 ∗ wasproposed for any gauge group, equal to the potential of the twisted elliptic Calogero-Moser model. Thearguments were based on holomorphy, uniqueness of the deformation from N = 2 ∗ , the form of non-perturbative contributions, and integrability. We have added to these reasonings the test of S-duality insubsection 2.2. We wish to further strengthen the arguments for the superpotential by comparing theresults for the exact quantum vacua for the theory on R × S with semi-classical results. A semi-classical analysis of the massive vacua of N = 1 ∗ on R proceeds in several steps. First one solvesthe equations of motion for constant scalar field configurations which are equivalent to the statement thatthe three complex scalars satisfy a su (2) algebra. The enumeration of inequivalent embeddings of su (2) inthe gauge algebra then provides the set of classical solutions. In a second step, one analyzes the unbrokengauge group for each classical vacuum, and then counts the number of vacua that the corresponding pure N = 1 quantum theory gives rise to in the infrared (using e.g. the index calculation [42]). For gaugealgebra su ( n ) the number of classical vacua was thus argued to be equal to the sum of the divisors of n [5], and this number coincides precisely with the number obtained from the exact superpotential [5, 12]for the theory on R × S (where one classifies vacua in the manner described above). For other gaugealgebras, the semi-classical counting of vacua was performed in [43]. For gauge algebra so ( n ) it wasargued to be: Z semi − class ( x, y ) = 1 + x + x y + 3 x + 6 x + x (6 + y ) + x (7 + 3 y ) + x (15 + 2 y )+ x (26 + y ) + x (31 + 5 y ) + . . . (3.1)where the power of x is equal to n and the power of y is the number of massless U (1)’s in a given masslessbranch of vacua. Although we will concentrate on massive vacua, let us remark that the semi-classicalcounting of massless vacua may well be futile in the full quantum theory, where there may be a singlemanifold of massless vacua of given rank [51] (albeit with different branches). Thus, we will only furtherconsider the semi-classical formula (3.1) for y = 0. For low rank then, the formula gives the followingnumber of massive vacua, semi-classically: so (5) : sc so (6) : sc so (7) : sc so (8) : sc . (3.2)We know the result for so (6) = su (4) to be in agreement with the number of massive vacua of the exactsuperpotential for the theory on R × S . We moreover have that this counting of so (2 r + 1) vacua agreeswith the semi-classical counting of vacua of sp (2 r ) [45], both on R . In the following, we compare thepredictions for the number of vacua some low rank gauge theories on R to the results we obtained for21he massive vacua coded in the superpotential on R × S . To make the comparison, we need to go intoa little more detail of the semi-classical analysis. In this subsection, we compare the analysis of integrable system extrema to the semi-classical analysisof massive vacua of N = 1 ∗ gauge theory on R case by case. We will moreover refine the counting atsome stages by taking into account the transformation properties of the vacua under remaining globalsymmetries. This will also be the occasion to interpret the many duality properties that we found for theintegrable systems in section 2. We also briefly comment on the monodromies.To wrap up a loose end first, let us note that the minimal mass M i of a given vacuum i can becomputed using the equation M k = min (cid:2) Spec( M Tk M k ) (cid:3) , (3.3)where M k is the matrix of second derivatives of the potential in vacuum k :( M k ) ij = ∂ V k ( X ) ∂X i ∂X j . This clarifies the logic behind our definition of isolated extrema of the integrable system (see equation(2.12) and the corresponding footnote). The case so (5)Semi-classically, we expect six vacua for the gauge theory on R . Let’s recall in a little more detailhow this counting arises. We allow for various five-dimensional representations of su (2) as vacuumexpectation values for the three complex scalars of N = 1 ∗ . Even-dimensional representations mustappear in even numbers. They need to take values in the gauge Lie algebra, and we classify them up togauge equivalence. One then finds the following allowed representations [43] – we indicate the dimensionsof the su (2) representations, the unbroken part of the gauge group, and then the number of massivevacua they give rise to in the infrared: 5 : 1 : 13 + 1 + 1 : so (2) : 02 + 2 + 1 : sp (2) : 21 + 1 + 1 + 1 + 1 : so (5) : 3 . (3.4)For instance, the 2+2+1 dimensional representation breaks the gauge algebra down to sp (2). Classically,this gives rise to a pure N = 1 theory with sp (2) gauge algebra at low energies, which gives rise to twoquantum vacua. Summing all the resulting numbers of semi-classical vacua, we find six massive vacua intotal.When we compare this analysis to the exact quantum vacua that we found for the so (5) gauge theoryon R × S , we remark that we have a neat correspondence. In particular, there is one vacuum, on thereal axis, that we can identify in the exact quantum regime as the fully Higgsed vacuum (correspondingto the 5-dimensional irreducible representation of su (2) in the list (3.4)). Its S-dual we interpret as aconfining vacuum, and it is a triplet under the T-transformation, agreeing neatly with the so (5) confiningvacua (corresponding to the trivial representation of su (2) in (3.4)). We moreover found a doublet underT-transformation, again in agreement with the two vacua corresponding to the sp (2) classical vacuum.Thus, at this level, we find excellent agreement. We note that the analysis of section 2 demonstratesthat the six vacua are in a single SL (2 , Z ) sextuplet and that their transformation properties are incorrespondence with the transformation properties of sublattices of the torus lattice. Their (generalized)S-duality and T-duality properties are now entirely known.The exact analysis has revealed a seventh vacuum on R × S . Its origin lies in the massless vacuum on R . After compactication, we can render the massless vacuum massive. Indeed, we can turn on a Wilsonline that commutes with the semi-classical configuration for the adjoint scalars, and that simultaneouslybreaks the abelian gauge group. The necessary Wilson line is precisely the one we found in the seventhquantum vacuum. We have thus found its semi-classical origin.One can wonder whether our identification used for the dual of the photon (mentioned in the intro-duction to section 3), and therefore of the parameterization of the Coulomb branch moduli space reducedthe number of physical vacua on R × S . Indeed, identifying our model as the one corresponding to gaugegroup SO (5) + (in the nomenclature of [34, 35]), leads to a doubling of the triplet in the semi-classicalanalysis, while the other multiplets remain unchanged. For the 1 + 2 + 2 semi-classical split, this is the22ase because the commutant is a SU (2) ⊂ SO (5) gauge group (corresponding to a long root in SO (5)),and thus the pure N = 1 gauge theory gives rise to only a doublet of vacua upon compactification.In the integrable system, this more careful analysis corresponds to the rule that solutions can only beidentified under shifts by 2 ω (and not ω ). A look at the so (5) extrema in the diagrams in subsection2.5 shows that this relaxed equivalence relation adds precisely three vacua, namely each of the confiningvacua (labelled 2 , N = 1 [34, 35]. Thus,in this more careful treatment, we increase the number of vacua by three on both sides of the analysis.We have computed the masses of the vacua. They are all roughly of the same order, and much abovethe accuracy of our numerical approximations, thus guaranteeing that our vacua are indeed massive.Moreover, for a given massive vacuum, the values of the masses are all approximately within a factor of100 from each other. Interesting patterns in the (ratios) of masses (of various vacua) exist – it should befruitful to study them systematically. The case so (8)In the case of the gauge algebra so (8), we find a further set of subtleties. First, let’s compare the quantumvacua on R × S to the semi-classical analysis on R . The semi-classical analysis yields [43]:7 + 1 : H : 1 s H : 1 s so (3) : 24 + 4 : sp (2) : 2 ∗ so (2) × so (2) : 03 + 2 + 2 + 1 : sp (2) : 2 s so (5) : 32 + 2 + 2 + 2 : sp (4) : 3 ∗ sp (2) × so (4) : 6 s so (8) : 6 s (3.5)for a total of 26 massive vacua. The semi-classical analysis was done under the assumption that the Z outer automorphism of so (2 N ) is a gauge symmetry [43], in contrast to our analysis in section 2. If weadopt this point of view, we are left with a single Z global symmetry, and we have indicated in thecounting above whether a set of vacua are a singlet ( s ) or are conjugate ( ∗ ) under that remaining Z .Using this analysis, and the T-duality transformation properties of the integrable system extrema,we can partially match the list of semi-classical and quantum vacua on R and R × S respectively.The 6 under the T-duality group makes for a match between extrema 10 , , , , , 15 and the su (2)representation 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. These correspond to the confining vacua. The doublets whichare conjugate under the remaining global Z match extrema 3 and 4 (as well as their Z reflections) tothe representations 4 + 4 and 5 + 1 + 1 + 1. The conjugate triplets match 6 , , Z reflections) onto 3 + 1 + 1 + 1 + 1 + 1 and 2 + 2 + 2 + 2. The smaller representations of the T-dualitygroup are harder to match. We can still identify the Higgs vacuum with the extremum number 9, whichlies on the real axis and which we can therefore follow all the way to weak coupling. For other extrema,it is harder to follow the change of effective description of the gauge theory dynamics from the ultravioletto the infrared. There is again a seeming mismatch of one in the total number of vacua. The origin isthe same as before: one extra quantum vacuum arises from the massless vacuum in R by turning on theappropriate Wilson line after compactification on S .Of course, our modular analysis of extrema again obtains a gauge theory interpretation. Recall thatwe found a singlet, triplet and quadruplet under the modular group. The modular group plays the roleof a generalized duality group [26], acting on the effective gauge coupling in the infrared.Note that we also found a more surprising feature: a new duality group, with a generator correspondingto a monodromy around a point in the fundamental domain of the effective coupling that we used todescribe our theory. We found a duodecuplet of vacua transforming under this new duality group. It couldbe very interesting to understand this group better in terms of gauge theory physics, or, as associated tothe choice of parameterisation in the infrared.Again, the masses all lie very amply above our numerical accuracy, such that we can claim that weindeed identified massive vacua. Masses are again within a factor of 100 or so from each other, andexhibit interesting patterns that could be explored.23 he cases so (7) and sp (6)For gauge algebra so (7), the semi-classical analysis on R predicts fifteen massive vacua, that arise asfollows [43, 45]: 7 : 1 : 15 + 1 + 1 : so (2) : 03 + 3 + 1 : so (2) : 03 + 2 + 2 : sp (2) : 23 + 1 + 1 + 1 + 1 : so (4) : 32 + 2 + 1 + 1 + 1 : sp (2) × so (3) : 41 + 1 + 1 + 1 + 1 + 1 + 1 : so (7) : 5 . (3.6)In the quantum theory on R × S , we do find a quintuplet under T-duality associated to the confiningvacuum on the imaginary axis, dual to the Higgs vacuum on the real axis, near weak effective coupling. Itenhances to a septuplet under T-duality at stronger effective coupling. From our analysis of the quantumtheory on R × S we see that the theory permits two more quantum vacua, labelled 1 and 2. Again,we checked explicitly that these arise from turning on Wilson lines in the massless vacua on R . On the sp (6) side of the duality, the two extra vacua 1 ∨ and 2 ∨ arise from an unbroken sp (2) gauge group (afterbreaking the abelian group that commutes with the 2 + 2 + 1 + 1 representation). We note again thatthe multiplet structure under T-duality is blurred at strong effective coupling. The point in the fundamental domain around which we have found a monodromy in the case of the so (8) gauge algebra, corresponds to a point at which two massive vacua have equal superpotential. Atthis point, a supersymmetric domain wall between the vacua becomes tensionless [46, 47]. The physicsassociated to such a situation is hard to discuss in detail, because of the difficulty of controlling theK¨ahler potential in gauge theories with N = 1 supersymmetry only. Explorations of the physics in thisregime can be found in [48, 44, 49]. We note in particular that in a mass and cubically deformed N = 1 U ( N ) theory in [48, 49], an extension of the Z N action associated to shifts in the θ angle of the gaugetheory to Z N was observed due to the presence of a point of monodromy in an effective coupling. TheT-operation (shifting the θ angle of the gauge theory) in our situation is also crucially influenced by thepresence of the point of monodromy : above the point of monodromy (at weak effective coupling), wefind a Z N − action, while below (at strong effective coupling), we find a Z N action (for the case N = 8as well as for the case of N = 7). We also note that the collision of the extrema of the superpotentialindicates the existence of an effectively massless excitation since there will be a zero mode for the matrixof second derivatives. The physics, or at least the properties of the effective description, seem close tothe discussion in e.g. [48]. It would be interesting to elucidate this point further. We studied the isolated extrema of complexified elliptic Calogero-Moser models, and encountered aplethora of beautiful phenomena. The values of the integrable interparticle potential at the extrema aretrue vector-valued modular forms in some cases, allowing for an analytic determination of the extrema interms of modular forms of congruence subgroups of the modular group. This gives rise to webs of extremathat form representations under the duality group of the model. The latter can either be a modular or aHecke group. A more intricate phenomenon is the appearance of monodromies amongst a second class ofextrema as we loop around a point in the fundamental domain of the modular group. The duality groupis then enlarged to include the monodromy generator. We determined the action of these generators onextrema. Moreover, we provided a wealth of Fourier coefficients of the extremal potential. These analysescan be viewed as a considerable widening of the observation of the integrality of observables in equilibriaof integrable systems.Secondly, we interpreted the results on extrema of Calogero-Moser systems in terms of mass-deformed N = 4 supersymmetric gauge theory in four dimensions. We compared our results based on a low-energyeffective action for the quantum theory on R × S to semi-classical predictions for the theory. The totalnumber of quantum vacua matched the number resulting from the semi-classical analysis, provided wetook into account massive vacua that originate in massless vacua on R . A Wilson line on the circle cancommute with semi-classical expectation values for the adjoint scalars, yet break the remaining abelianfactors in the gauge group to give rise to massive quantum vacua, of either Higgs or confining type. Wenote that the appearance of extra vacua after compactification is typical of the B, C and D series. The24ompactified theory manifestly differs from the theory on R . We also performed a more refined matchingof quantum vacua in certain cases, thus providing further evidence that the superpotential gives a correctdescription of the physics of N = 1 ∗ theory on R × S . Furthermore, we noted that the precise multipletstructures in the quantum theory showed surprising features, including monodromy properties of thequantum vacua.It should be clear that we only scratched the surface of this intriguing domain at the intersection ofintegrable systems, modularity and gauge theory. The new features of the extrema that we laid bare inthe B, C, D -type integrable models (compared to the A -type theories) prompts the question of the genericcounting of the extrema, the relevant duality group and modular structure (including monodromies) aswell as their representation and number theoretic content. Clearly, our analysis begs to be extended toexceptional algebras of low rank, to higher rank root systems, to models with different choices of couplingconstants, as well as to the integrable generalizations of the elliptic Calogero-Moser models, including forinstance the Ruijsenaars model with spin. Moreover, our study can be extended to other observables,like the ratio of the frequencies of fluctuations. It would also be interesting to attempt to characterizethe positions of the extrema through e.g. a generalization of zeroes of orthogonal polynomials [25]. Allindications are that similarly intriguing phenomena as the ones we uncovered will appear in this broaderfield.In gauge theory, one would like to analyze more closely the Seiberg-Witten curves of the N = 2 ∗ theories that underlie our models, and in particular, locate the points in Coulomb moduli space wherethe curve develops a number of nodes equal to the rank of the gauge group, and where the vanishingcycles are mutually local (indicating the existence of massive vacua after mass deformation). The Seiberg-Witten curves are defined by equations of higher order, rendering this analysis harder than in the casestreated in detail so far [5].One would also like to have access to the large rank generalization of our results, to connect toholographic dual backgrounds with orientifold planes [50, 26, 43]. For these purposes it might suffice tohave access to the large rank generalization of a Higgs and confining vacuum, which one may hope tocharacterize analytically. It would also be useful to perform a more careful analysis of the discrete choicesof gauge groups and line operators in our model [34, 35], and to classify various supersymmetric vacuafurther [33, 34], e.g. by understanding a single phase, then chasing it through the duality chains.Finally, we already noted in passing that the branches of massless vacua predicted by the semi-classicalanalysis may turn out to be connected in the quantum theory, giving rise to a single massless vacuummanifold, consisting of branches that can be characterized by differing algebraic equations [51]. Ournumerical explorations up to now are consistent with the fact that all massless vacua have the same valueof the superpotential. It would be interesting to clarify the structure of these vacuum manifolds furtherfor the models at hand. Acknowledgments We thank Antonio Amariti as well as the JHEP referee for encouraging comments. We would like toacknowledge support from the grant ANR-13-BS05-0001-02, and from the ´Ecole Polytechnique and the´Ecole Nationale Sup´erieure des Mines de Paris. A Lie Algebra We briefly review Lie algebra concepts that are useful to us in discussing the symmetries of both theintegrable systems and gauge theories we discuss in the bulk of the paper. See e.g. [24] for a detailedexposition of the following facts. Let us consider a (compact simple) Lie group G with maximal torus T . They have corresponding tangent algebras g and t . We can then identify T as a linear group, and itsspace of characters χ ( T ) is in bijection with a lattice in the space t ∗ ( R ) dual to the tangent algebra t , anddefined over the real numbers R . To the Lie algebra, we can associate its space of weights in the adjointrepresentation, which is the set of roots ∆. Again, these roots are elements of the Euclidean space t ∗ ( R ).The space t ( R ) comes equipped with a non-degenerate scalar product, which we will denote ( · , · ). Thisscalar product allows us to identify a function λ on the space t ( R ) with an element u λ of the space t ( R )through the relation: λ ( x ) = ( u λ , x ) , (A.1)valid for all elements x of t ( R ). We will occasionally abuse notation and write λ ( x ) = ( λ, x ), andalso ( u λ , u λ (cid:48) ) = ( λ, λ (cid:48) ), which defines a dual scalar product on t ∗ ( R ). The bijection between the spacegenerated by the roots and its dual allows us to define the dual roots (i.e. the co-roots) through the25elation: α ∨ = 2 u α ( α, α ) . (A.2)The root lattice Q is the lattice generated by the roots. Any set of simple roots α i spans the space t ∗ .The weight lattice P also sits inside t ∗ and is defined to be generated by a basis π j such that:2 ( α i , π j )( α i , α i ) = δ ij , (A.3)for all i and j that run from 1 to the rank of the group G . We moreover define the dual root lattice Q ∨ to be the lattice generated by the dual roots, and the dual weight lattice P ∨ to be the weight latticecorresponding to the dual root lattice. The dual of the lattice generated by the characters of a givengroup G will be denoted t ( Z ). We have the following properties. The center C ( G ) of the group G is givenby: C ( G ) ≡ P ∨ / t ( Z ) ≡ χ ( T ) /Q . (A.4)Moreover, when G is simply connected it is equal to its universal cover ˜ G . We then have that the space ofcharacters is bijective to the whole of the weight lattice χ ( T ) = P , and that t ( Z ) = Q ∨ , such that C ( G )is maximal and C ( ˜ G ) = P/Q = P ∨ /Q ∨ . The group with minimal center C ( G ) = 1 is the universal cover˜ G divided by its center C ( ˜ G ). In this case we have that the set of weights is the set of roots χ ( T ) = Q and that t ( Z ) = P ∨ .Our definitions imply that the fundamental weights π ∨ j that generate the dual weight lattice P ∨ satisfy: ( π ∨ j , u α i ) = δ ij , (A.5)and therefore that: α ( X ) = ( u α , X ) = ( α, X ) (A.6)is integer for X in the dual weight lattice, i.e. for X a co-weight.We summarize inclusions and dualities in the diagram below. The arrows indicate that the latticesare dual, i.e. that the contractions give integers. t ∗ ( R ) ⊃ P ⊃ χ ( T ) ⊃ Q (cid:108) (cid:108) (cid:108) Q ∨ = P ∗ ⊂ t ( Z ) ⊂ P ∨ = Q ∗ ⊂ t ( R )We end this review on Lie group and Lie algebra theory with table 2 which exhibits useful data on theWeyl group, the outer automorphisms, the dual Coxeter number and the center of the universal coveringgroup corresponding to the classical Lie algebras:Algebra Weyl group Outer Automorphisms Dual Coxeter number Center Univ. Cover A r , r > S r +1 Z r + 1 Z r +1 A Z Z B r S r (cid:110) Z r r − Z C r S r (cid:110) Z r r + 1 Z D r , odd r S r (cid:110) Z r − Z r − Z D r , even r > S r (cid:110) Z r − Z r − Z × Z D S (cid:110) Z S Z × Z Table 2: Lie Algebra Data B Elliptic Functions Our conventions for the elliptic Weierstrass function are: ℘ ( x ; ω , ω ) = 1 x + (cid:88) ( m,n ) (cid:54) =(0 , (cid:18) x + 2 mω + 2 nω ) − mω + 2 nω ) (cid:19) ℘ ( z ; τ ) = 1 z + (cid:88) ( m,n ) (cid:54) =(0 , (cid:18) z + m + nτ ) − m + nτ ) (cid:19) (B.1)26hich entails the equality ℘ ( z ; τ = ω /ω ) = 4 ω ℘ (2 ω z ; ω , ω ) . (B.2)We impose the convention that (cid:61) ( ω /ω ) = (cid:61) ( τ ) > 0. The Weierstrass function is a Jacobi form of level2 and index 0 : ℘ (cid:18) zcτ + d ; aτ + bcτ + d (cid:19) = ( cτ + d ) ℘ ( z ; τ ) . (B.3)It has the following expansion for large imaginary part of τ : ℘ ( x ; ω , ω ) = − π ω E ( q ) + π ω csc (cid:18) πx ω (cid:19) − π ω ∞ (cid:88) n =1 nq n − q n cos nπxω . (B.4)For the twisted Weierstrass functions, we can derive the equalities: ℘ ( x ; ω , ω ) + ℘ ( x + ω ; ω , ω ) = ℘ ( x ; ω , ω ) + π ω (2 E (2 ω ω ) − E ( ω ω )) ℘ ( x ; ω , ω ) + ℘ ( x + ω ; ω , ω ) = ℘ ( x ; ω , ω − π ω ( E ( ω ω ) − E ( ω ω )) . (B.5)These can be proven using the definition of the Weierstrass function ℘ , as well as the definition of thesecond Eisenstein series E . C Modular Forms We present a compendium of modular forms that we put to use in the bulk of our paper. C.1 Theta and Eta Functions We first fix our conventions for the theta-functions with characteristics: θ (cid:20) αβ (cid:21) ( τ ) = (cid:88) n ∈ Z exp (cid:2) iπ ( n + α ) τ + 2 πiβ ( n + α ) (cid:3) . Particular examples of these theta-functions include: θ ( q ) = θ (cid:20) (cid:21) ( q ) = 2 q / + 2 q / + 2 q / + 2 q / + ...θ ( q ) = θ (cid:20) (cid:21) ( q ) = 1 + 2 q / + 2 q + 2 q / + 2 q + ...θ ( q ) = θ (cid:20) (cid:21) ( q ) = 1 − q / + 2 q − q / + 2 q + ... We also make use of the Dedekind eta-function: η ( q ) = q / ∞ (cid:89) n =1 (1 − q n ) , and the Klein invariant j ( q ) = 1728 E ( q ) E ( q ) − E ( q ) = 1 q + 744 + 196884 q + ... (C.1) C.2 Modular Forms and Sublattices In this subsection we recall how to find a basis of the space of modular forms of weight k for the congruencesubgroup ([32]) : Γ( N ) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( Z ) : (cid:18) a bc d (cid:19) ≡ (cid:18) (cid:19) mod N (cid:27) . N ) can be identified with the pairs ± v ∈ ( Z /N Z ) of order N . Thismakes it possible to count the number of such cusps : (cid:15) ∞ (Γ( N )) = N = 2 N (cid:81) p | N (cid:16) − p (cid:17) N ≥ . For any congruence subgroup Γ the space of modular forms M k (Γ) of weight k decomposes into thesubspace of cusp forms and the Eisenstein space: M k (Γ) = S k (Γ) ⊕ E k (Γ). For N = 2 we havedim S (Γ(2)) = 0and for N ≥ 3, dim S (Γ( N )) = 1 + N ( N − (cid:89) p | N (cid:18) − p (cid:19) . In particular, dim S (Γ(3)) = 0 and dim S (Γ(6)) = 1.We want an explicit basis of the Eisenstein space. For any vector v = (cid:20) cd (cid:21) ∈ ( Z /N Z ) of order N ,and for k ≥ 3, we define the (non-normalized) Eisenstein series G k,N [ v ] ( τ ) = G k,N (cid:20) cd (cid:21) ( τ ) = (cid:88) (cid:48) v (cid:48) ≡ v ( N ) c (cid:48) τ + d (cid:48) ) k , and for weight two G ,N [ v ] ( τ ) = G ,N (cid:20) cd (cid:21) ( τ ) = 1 N (cid:20) ℘ (cid:18) cτ + dN , τ (cid:19) + G ( τ ) (cid:21) , where the primed sum runs over those non-vanishing vectors v (cid:48) = (cid:20) c (cid:48) d (cid:48) (cid:21) that equal v modulo N . One canshow that the Fourier expansion of these functions in terms of q = e iπτ is: G k,N (cid:20) cd (cid:21) ( q ) = δ ( c ) ζ dN ( k ) + ( − πi ) k N ( k − ∞ (cid:88) n =1 σ k − ,N (cid:20) cd (cid:21) ( n ) q n/N where σ k − ,N (cid:20) cd (cid:21) ( n ) = (cid:88) m | n and nm ≡ c ( N ) sgn( m ) m k − exp (cid:18) πi dmN (cid:19) and ζ dN ( k ) = (cid:88) (cid:48) d (cid:48) ≡ d ( N ) d (cid:48) ) k . This Fourier expansion is valid for all k ≥ 2, including k = 2 which is the case we are mostly interestedin. For k ≥ 3, any set { G k,N [ v ] } with one v corresponding to each cusp of Γ( N ) represents a basis ofthe space E k (Γ( N )) of Eisenstein series of weight k on Γ( N ) (and in particular dim E k (Γ( N )) = (cid:15) ∞ ). Forthe case k = 2 these statements have to be modified, because of the lack of modularity of the (ordinary)weight 2 Eisenstein series. It turns out that dim E (Γ( N )) = (cid:15) ∞ − 1, and that E (Γ( N )) is the set oflinear combinations of the { G k,N [ v ] } (where v ∈ ( Z /N Z ) is of order N ) whose coefficients sum to 0. The Eisenstein series G k,N [ v ] have good transformation properties under SL (2 , Z ) for k ≥ N ∈ { , } provided the vector v is transformed accordingly: cτ + d ) k G k,N [ v ] (cid:18) aτ + bcτ + d (cid:19) = G k,N (cid:20)(cid:18) a bc d (cid:19) v (cid:21) ( τ ) . For k = 2, we have to take into account a non-holomorphic term, except for linear combinations wherethe sum of the coefficients vanishes, as is the case for the potentials considered in the bulk of the paper. Theorem 4.6.1 in [32] For generic N the relation between the normalized Eisenstein series, which enjoy these good transformation properties,and the series G k,N [ v ] is not simply a proportionality relation (see formula (4.5) in [32]), but it is a simple rescaling for N = 2 and N = 3. E ,N ( τ ) = E ( τ ) − N E ( N τ ) . (C.2)These are weight 2 modular forms of Γ ( N ). We use extensively the fact that M (Γ (4)) has dimension2 and is generated by − E , ( q ) = 1 + 24 q + 24 q + 96 q + 24 q + 144 q + ... − E , ( q ) = 3 + 24 q + 72 q + 96 q + 72 q + 144 q + ... We note the transformation property of the form E , under τ → − / (2 τ ): E , ( − / (2 τ )) = E ( − / (2 τ )) − E ( − /τ ) = − τ E , ( τ ) . (C.3) D The List of Extrema In this appendix, we list the extrema of the complexified (twisted) elliptic Calogero-Moser models withroot systems D = so (8) and B = so (7). We provide a few more details on how we obtained them, howto relate them through dualities, as well as Fourier expansions of the extremal potentials. D.1 The List of Extrema for so (8) The strategy we used to find extrema boils down to finding all the minima (which are also zeros) ofthe (auxiliary, gauge theory) potential (2.12) with non vanishing mass (3.3) using a simple gradientalgorithm with random initial conditions. Then we identify those configurations that are related by oneof the symmetries we quotient by. This procedure is executed at a given value of τ . Once the completelist of extrema is known, we can follow a given extremum along any curve in the τ upper half plane, byadiabatically varying τ . The T -dual extrema and the monodromies are obtained in this way, while theaction of S -duality is known exactly. We thus unfold the whole web of dualities.In order to determine the potential at the extrema, we first make use of our knowledge of T -duality,which dictates the Fourier expansion variable q /n , where n is the smallest positive integer such that T n acts trivially on the extremum under consideration. Then we evaluate the extremal potential at manydifferent values of τ and find recursively the rational Fourier coefficients. D.1.1 The diagrams of the extrema In the diagrams that follow, the black dots represent the values of the components X i , i = 1 , , , S symmetry. For every such extremum, one subgroup Z ⊂ S actstrivially. One of the three extrema is obtained by choosing one of the circled black dots and the threeordinary black dots, a second one is obtained by choosing the other circled black dot and the three blackdots, while the third is determined by the four small black dots. (The pale grey dots show the possibletranslations of this extremum by half-periods.)We note in passing that some exact information on the positioning of the extrema is available. Forinstance, for extremum number 9, some exact information on the positions is the following. At τ → i ∞ ,the system reduces to the Sutherland system (with trigonometric potential). According to [17], thepositions at equilibrium are related to the roots of a Jacobi polynomial. Explicitly in the case of D ,the polynomial is P (1 , ( y ) = ( y − + ( y − 1) + 3, from which we deduce the positions X = 0, X , = π arccos( ± / √ 5) and X = . For τ → 0, the positions converge on X = 0, X = 1 / X = 1 / X = . This numerical convergence is slow.S-duality guarantees that the situation is similar for the extremum on the imaginary axis, with thetwo limits exchanged. Moreover, T-duality then acts in the τ → i ∞ limit as X → X , X → X + 1 / X → X + 1 / X → X + 1 / 2. (These transformations are exact within the precision of the numerics.)This generates the 6-cycle. Et cetera. 29xtrema at τ = i for so (8) Extremum 1 Extremum 2 Extremum 3 Extremum 4 Extremum 5 Extremum 6 Extremum 7 Extremum 830xtrema at τ = i Extremum 9 Extremum 10 Extremum 11 Extremum 12 Extremum 13 Extremum 14 Extremum 15 Extremum 16 Extremum 17 Extremum 18 Extremum 19 Extremum 20 D.1.2 The series for the so (8) extremal potentials We have been able to determine the q -expansions of the potentials in each extremum with great accuracy,in terms of functions with integer coefficients. For extrema 1 to 8, we gave the exact expression in section2.7. To list the series for the remaining extrema, we introduce 11 functions, for which we only reproduce31 Figure 6: The dots show the successive ratios of the coefficients of f , and a line has been drawn, forcomparison, at the value 1 /q M = e − πiτ M .the first few coefficients – more can be obtained – : f ( q ) = − q + 45379 q − q − q − q + ...f ( q ) = 1 − q + 51582918 q − q + 5101142359347277 q + 94300056917523369780 q + ...f ( q ) = + q + 369 q + 68644 q + 11490041 q + 1579638246 q + ...f ( q ) = 1 + 3096 q + 1818378264 q + 2446348866170976 q + 4535490919062930456600 q + ...f ( q ) = 1 − q − q − q + ...f ( q ) = 2 + 780960 q + 18367562372664 q + 762875530342634406144 q + ...f ( q ) = 1 − q − q − q f ( q ) = (14 + 79929712 q + 2425403175787968 q + 111756708524847535116096 q + ... ) f ( q ) = ( − − q − q − q + ... ) f ( q ) = 1 − q − q − q − q + ...f ( q ) = 1 + 110596 q + 110757888006 q + 180011523750912008 q + 367762906594569664954381 q + ... The potentials then read V = 14400 π f (cid:0) q (cid:1) V k = − π (cid:80) j =0 (16 q ) j/ exp (cid:16) πi kj (cid:17) f j (cid:0) q (cid:1) V = − π ( f ( q ) − √ qf ( q )) V = − π ( f ( q ) + 72 √ qf ( q )) V = − π (75 f ( q/ ) + i (cid:112) q/ f ( q/ )) V = − π (75 f ( q/ ) − i (cid:112) q/ f ( q/ )) , where k = 0 , ..., 5. The last series V can then be deduced from the fact that the sum of all potentialsin the duodecuplet vanishes. Note that the coefficients grow rapidly, preventing the functions above tobe modular forms. The monodromy is responsible for this phenomenon, as can be confirmed by theestimation of the convergence radius given by the successive ratios of the coefficients (see figure 6). D.2 The List of Extrema for so (7) and sp (6) Finally, in the case of the algebras B = so (7) and C = sp (6), we only present diagrams of the extremalpositions for the so (7) root system, since the corresponding extrema for sp (6) can be found by Langlandsduality. 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