Recent developments in 2d N=(2,2) supersymmetric gauge theories
IInternational Journal of Modern Physics Ac (cid:13)
World Scientific Publishing Company
Recent developments in2d N “ p , q supersymmetric gauge theories Daniel S. Park
NHETC and Department of Physics and AstronomyRutgers University, Piscataway, NJ 08855-0849, U.S.A.dspark at physics.rutgers.edu
We review recent developments in two-dimensional N “ p , q supersymmetric gaugetheories focusing on the implementation and applications of localization techniques. Keywords : supersymmetry; gauge theory; localization.PACS numbers:
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. 2d N “ p , q gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . 53. Supersymmetric backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . 124. The round sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155. The torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.1. Discrete gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . 266. The Ω-deformed sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297. More backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.1. Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2. Geometric applications . . . . . . . . . . . . . . . . . . . . . . . . . . 418.3. More applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1. Introduction
Two-dimensional gauge theories with N “ p , q supersymmetry became a topicof intensive research following the poineering work Ref. 1. While gauge fields donot have propagating degrees of freedom in two dimensions, there is still some richphysics at play. In particular, a 2d gauge theory becomes strongly coupled in theinfra-red (IR), opening the possibility for it to flow to a non-trivial fixed point. Ref.1 not only explains how certain 2d N “ p , q gauge theories—gauged linear sigmamodels (GLSMs)—flow to non-linear sigma models (NLSMs) of K¨ahler manifolds at a r X i v : . [ h e p - t h ] S e p Daniel S. Park intermediate IR scales, but also explains how to understand their IR fixed points. a The former fact implies that we are able to compute physical observables of non-linear sigma models, which in turn have geometric interpretations, via gauge theorycomputations. Furthermore, the fact that 2d N “ p , q gauge theories can be usedto engineer N “ p , q superconformal field theories (SCFTs) in the IR also entailsthat gauge theories can be used to compute observables of type IIA or IIB stringcompactifications.The utility of supersymmetric field theories is that there are often exactly com-putable quantities. One such quantity is the partition function of a supersymmetrictheory on a manifold given that the theory has enough supersymmetry. This is be-cause with a certain amount of supersymmetry, the full path integral yielding thepartition function can reduce to a integral over a finite dimensional space. This iscalled localization , whose first application to quantum gauge theory appeared in theseminal work Ref. 5, which studies 4d N “ SU p q super-Yang-Mills theory. Notethat when the manifold is curved, the Lagrangian of a theory on that manifold isnot completely determined by its flat space Lagrangian—there is an ambiguity re-garding terms that vanish in the flat-space limit. If one chooses the right curvatureterms, there exists a supersymmetry, the supercharge operator of which we denote Q , that is preserved.In the case of Ref. 5, the supercharge in question satisfies Q “
0. Let us writethe partition function Z M of the theory on a manifold M schematically as Z M “ ż r d Φ s exp p´ S r Φ sq , (1)where Φ denotes all the configuration of fields in the theory. In this review, weexclusively work in the case where M is Euclidean. The expectation value of a Q -exact operator vanishes, i.e., ż r d Φ s exp p´ S r Φ sq t Q , O s “ r Q , S s “ . (3)Thus Z M can be computed also by the path integral Z M “ ż r d Φ s exp p´ S r Φ s ´ t t Q , V uq (4)for any t . The idea of localization is to choose a Q -exact action S loc “ t Q , V u thatis bounded below by zero. Upon taking t to infinity, the integral (4) localizes ontothe locus in field space where S loc “
0, which we call the localization locus . The a While Ref. 1 focused on abelian gauge theories, the results were generalized to non-abelian gaugetheories in subsequent work. ecent developments in 2d N “ p , q supersymmetric gauge theories localization locus may have several different components, which we label by c . Thefinal integral is then given by Z M “ ÿ c ż X c r dx s exp p´ S r Φ x sq Z r Φ x s (5)where X c is the component of the localization locus labeled by c , and x are co-ordinates of the localization locus, whose number one would aim to make finite. S r Φ x s is the classical action evaluated at the field configuration Φ x parametrizedby x . Z is the one-loop determinant of all the fields around that particular fieldconfiguration.The supercharge studied in Ref. 5 is topological, in that the stress-energy ten-sor of the theory is Q -exact, and thus the partition function of the theory on themanifold is independent of the metric. There are two distinct topological supersym-metric backgrounds for 2d theories with N “ p , q supersymmetry, coined the A -model and the B -model. While these backgrounds have been mostly employedto compute physical observables in non-linear sigma models, localization computa-tions using the A -model supercharge have been carried out to compute gauge theorycorrelators in Ref. 8.The localization scenario described above can be slightly generalized to incorpo-rate Q such that Q “ J , where J is some symmetry of field space. Oftentimes, J is associated to an isometry of the space-time manifold M . Given that r Q , S s “ V that satisfies r J, V s “
It would be remiss to go on withoutemphasizing that the Ω-background
12, 13 used to compute the exact effective K¨ahlerpotential of N “ (curved) supersymmetric backgrounds , stud-ied and developed in Refs. 14, 15 and 16, can be understood as expectation valuesof fields (including the auxiliary fields) of a supergravity multiplet on manifold M . b While the expectation values of the fields do not necessarily satisfy equations ofmotion since gravity is not dynamical in this setting, they are required to preservesome amount of supersymmetry. After the supersymmetry equations are solved,the gravity multiplet may be coupled to a supersymmetric field theory to yielda supersymmetric Lagrangian which in turn may be used to compute the parti-tion function of the theory. We note that in this approach, a field theory musthave off-shell supersymmetry for it to couple to the supersymmetric background. Asystematic classification of supersymmetric backgrounds in two dimensions comingfrom the N “ p , q new-minimal supergravity multiplet has been carried out inRef. 19. c b Also see Refs. 17, 18. c See Refs. 20, 21 for subsequent developments.
Daniel S. Park
The current approach to computing exact partition functions is thus to firstpick a supergravity multiplet, classify supersymmetric backgrounds with the givenmultiplet, use localization to compute the partition function and interpret the result.While pointing out every major development in this direction of research is outsidethe scope of this review, we have collected a small list of work, starting from thework of Pestun computing the exact partition function of N “ that have appeared leading up to computations of the samespirit being carried out in two dimensions, as well as some early higher-dimensionalcomputations that are directly related to 2d localization. Localization computations for 2d N “ p , q gauge theories took off after Refs.23, 24 computed the exact partition function of the gauge theories on the “roundsphere.” Exact partition functions (and/or correlators) on the flat torus, thehemisphere, RP , the Ω-deformed sphere
32, 33 and the A -twisted higher-genusRiemann surface were computed subsequently. While the result of the localizationcomputation for certain backgrounds, such as the A -twist or the flat torus, alreadyhad well-established expectations, results of some—such as the round sphere par-tition function for theories with conformal fixed points
20, 51, 52 —turned out to havesurprising physical interpretations. Meanwhile, the meaning of some quantities—such as correlation functions on the Ω-deformed sphere—still remain to be under-stood.What makes these backgrounds useful is that the path integral can be localizedonto relatively unsophisticated saddles, whose moduli space is finite-dimensional.Save for the flat torus, the path integral can be localized onto a Coulomb-branchpath integral, where the saddles of the path integral are parametrized by zero modesof the sigma-fields—the scalar components in the vector multiplet—in the Cartansubalgebra of the gauge algebra. At these saddles, the chiral fields are forced tovanish, and can be completely integrated out. This should be contrasted to thelocalization computation of Ref. 8, where the saddles used are vortex configurationswhere both the chiral fields and the gauge fields take on non-trivial expectationvalues.The goal of this review is to present the results of supersymmetric localizationof 2d N “ p , q supersymmetric gauge theories on supersymmetric backgroundsand to discuss their physical and geometric interpretation. We place emphasis onexplaining how to actually carry out the computations, and various subtleties thatcan arise when attempting to evaluate the partition function.We choose to focus on three backgrounds: the “round sphere” or equivalently, thetwo-sphere with no R -flux, the torus, and the equivariant A -twist on the sphere (Ω-deformed sphere). The gauge theories we consider are those whose gauge multipletslie within the basic vector multiplet, and whose matter consist of chiral multiplets. d The partition function of a gauge theory on the A -twisted higher-genus Riemann surface can beobtained by dimensionally reducing a three-dimensional computation, which also appeared in Ref.50. ecent developments in 2d N “ p , q supersymmetric gauge theories Thus the scope of the review is restricted in two directions—the type of gaugetheories we consider, e and the type of backgrounds we focus on.This review is structured as follows. We begin by going over basics of N “ p , q gauge theories we study throughout the review in section 2. We present the field con-tent and the data that specify the gauge theories and explain important aspects ofglobal symmetries. We also introduce and review three (classes of) model theories—the CP N f ´ model, the quintic GLSM and U p N q gauge theories with fundamentaland anti-fundamental matter—whose partition functions we compute on varioussupersymmetric backgrounds throughout the course of the review. In section 3, wereview the basic framework for understanding the supersymmetric backgrounds.Supersymmetric backgrounds can be understood as vacuum expectation values ofbosonic components of a supergravity multiplet coupled to the gauge theories. Wereview the N “ p , q new minimal supergravity multiplet and write explicit expres-sions for the Lagrangian of a gauge theory coupled to a supersymmetric backgroundobtained by giving expectation values to components of this multiplet.In the following three sections, sections 4, 5 and 6, we present localization for-mulae for the round sphere, the torus, and the sphere with the equivariant A -twist,respectively. For each of these backgrounds, we describe the localization locus aswell as background expectation values for the vector multiplets coupled to the flavorcurrent multiplets that can be turned on. We present the localization formula forgauge theories, and use it to compute the partition function or correlators of modelgauge theories introduced in section 2. We also explain the interpretation of thepartition functions and correlators, when they are known.In section 8 we summarize some applications of the localization computations.On the side of physical applications, we focus on how localization has been usedto verify various dualities
4, 23, 57, 58 proposed for 2d N “ p , q gauge theories. Wealso review how supersymmetric partition functions and correlators are related togeometric invariants of complex K¨ahler manifolds.We must acknowledge that we have, inevitably, focused on certain topics morethan others. It should be emphasized that this is not for the lack of importance ofthe topics less covered, but simply due to the inability of the author to due justiceto all subjects. We give a brief summary of supersymmetric backgrounds that wehave not been able to discuss in detail in section 7. We list some more applicationsof supersymmetric localization in subsection 8.3.
2. 2d N “ p , q gauge theories We begin by describing the gauge theories that we wish to study. While there area plethora of multiplets that can be utilized to construct field theory Lagrangians, e Exact partition functions of field theories with multiplets other than the standard vector and chiralmultiplets have been computed in the literature. Partition functions for theories with twisted chiralmatter and/or twisted vector multiplets have been computed in Refs. 52–54. Refs. 55, 56 studytheories with semi-chiral multiplets.
Daniel S. Park we stick to gauge theories whose gauge fields lie in vector multiplets, and whosematter fields form chiral multiplets.While we relegate expressions for the components of the various multiplets andhow the various parameters show up in the Lagrangian of the theory to the nextsection, let us summarize the data that specify the theories that we study. Theingredients that go into specifying the theory are as follows: ‚ The gauge group of the theory G with gauge algebra g . The gauge fieldssit in the g valued vector multiplet, which we denote V .(a) We denote the rank of G , rk p G q .(b) The Cartan subalgebra h of g has rk p G q generators, which we indexby the labels a, b, ¨ ¨ ¨ .(c) The components of V , in Wess-Zumino gauge, are given by p a µ , σ, r σ, λ ˘ , r λ ˘ , D q .(d) Twisted chiral and twisted anti-chiral multiplets may be constructedby taking supercovariant derivatives of the vector fields in flat space,whose bottom components are given by σ and r σ , respectively:Σ “ iD ´ r D ` V , r Σ “ iD ` r D ´ V . (7)We note that in space-times with Euclidean signature, these two su-perfields are independent, as opposed to the case when the space-timesignature is Lorentzian.(e) Throughout the review, we refer to the entries of the Lie-algebra valuedbosonic component σ of the gauge multiplet as sigma fields . ‚ Charged chiral and anti-chiral multiplets Φ i , r Φ i in the representations R i , R i of G .(a) We denote the U p q R -charge of the chiral multiplet r i . The charges r i must be quantized for certain supersymmetric backgrounds.(b) Given the flavor symmetry group G f , we can turn on twisted masses m F for each factor of the Cartan subalgebra À F u p q f,F of the flavorsymmetry group. We denote the flavor charge of the chiral fields under u p q f,F , q iF .(c) On manifolds with a non-trivial fundamental group, holonomies forthe R -symmetry or flavor symmetries may be turned on along non-contractible loops.(d) The components of Φ i and r Φ i are given by p φ i , ψ i, ˘ , F i q and p r φ i , r ψ i, ˘ , r F i q , respectively. f ‚ The superpotential W p Φ i q , which is a function of the chiral fields Φ i . W must be invariant under gauge and flavor symmetry, and must have charge-two under the U p q R symmetry. f The placement of the flavor indices i with respect to the field symbols vary throughout the reviewas to make the equations more readable. We hope that this does not cause the reader any confusion. ecent developments in 2d N “ p , q supersymmetric gauge theories ‚ The gauge algebra c of the center C of the gauge group is either empty, oris a direct sum of abelian components, which we index by I “ , ¨ ¨ ¨ , n : c “ n à I “ u p q I . (8)Here, n ď rk p G q , the equality being saturated when the gauge group isabelian. We introduce the linear twisted superpotential x W “ ÿ I t I σ I , (9)where σ I is the complex scalar component of the I th abelian vector multi-plet. We may write this superpotential in a basis-invariant form as x W p σ q “ t p σ q “ ÿ I t I tr I σ , (10)where t is a complex vector in the dual of the Cartan subalgebra c ˚ C Ă h ˚ C ,and the parenthesis denotes the canonical pairing between elements of h ˚ and h . The parameters t I are complexified Fayet-Iliopoulos (FI) parameters,whose real part encodes the theta-angle of u p q I . We often write: t I “ iξ I ` θ I π . (11)Before we move on, let us be a bit pedantic and review the difference betweenan R -symmetry and a flavor symmetry. R -symmetry is a global symmetry underwhich supercharges carry charge—it follows that components lying in the samemultiplet have different representations under the R -symmetry. Flavor symmetry,in the context of supersymmetric theories, are global symmetires that commutewith supersymmetry—it is not part of the supersymmetry algebra. Thus the flavorcharges of all fields in a given chiral multiplet are equivalent.Classically, in the absence of twisted masses, the R -symmetry group is givenby U p q R ˆ U p q A , given that the superpotential W of the theory has U p q R -charge 2, as we have assumed. We denote the first and second factor the vec-tor and axial R -symmetry group, respectively. The U p q R -charges of the compo-nents p φ i , ψ i, ´ , ψ i, ` , F i q and p r φ i , r ψ i, ´ , r ψ i, ` , r F i q of the chiral and anti-chiral multi-plets are given by p r i , r i ´ , r i ´ , r i ´ q and p´ r i , ´ r i ` , ´ r i ` , ´ r i ` q ,while the U p q A charges are given by p , , ´ , q and p , ´ , , q . The flavorcharges, on the other hand, are given by p q iF , q iF , q iF q and p´ q iF , ´ q iF , ´ q iF q , re-spectively. Meanwhile, the vector and axial R -symmetry charges of the vector mul-tiplet components p a µ , σ, r σ, λ ´ , λ ` , r λ ´ , r λ ` , D q are given by p , , , , , ´ , ´ , q and p , , ´ , , ´ , ´ , , q , respectively. Note that the twisted masses can be un-derstood as giving a vacuum expectation value to the scalar components of back-ground vector fields coupling to the flavor symmetries. Thus the twisted massesbreak U p q A symmetry explicitly. Daniel S. Park
While a classical U p q R symmetry group is still preserved in the quantum theory,a classical U p q A symmetry can be broken by anomalies. Thus when we refer to R -charge or R -symmetry without further explanation, it should be understood that weare referring to the vector R -symmetry. Whether the axial R -symmetry of a theoryexists in the quantum theory is crucial for determining the properties for its IR fixedpoint. If the symmetry group U p q A is unbroken in the quantum theory, the gaugetheory is expected to flow to a superconformal fixed point, as both left and rightmoving R -symmetries, which are part of the superconformal algebra, stay intact. Asnoted above, the axial R -symmetry may be broken by twisted masses classically, orin the absence of twisted masses, broken by mixed anomalies. The mixed anomaliesare computed by fermion loop diagrams with two vertices—the axial R -symmetrycurrent on one vertex, and the flavor or gauge current on the other. While the mixedanomaly between the R -symmetry current and the flavor current can be avoidedwhen the background vector field coupling to the flavor current is not turned on,the mixed anomaly with the gauge current is unavoidable. Nevertheless, a discretesubgroup of U p q A can be shown to survive quantum mechanically. We describethese subgroups in relevant examples, which we now present.Throughout the review, we use three model gauge theories to illustrate how toapply the various localization formulae. The three theories are the following:(i) The CP N f ´ model.
1, 60 (a) Gauge group: U p q .(b) Charged matter: N f chirals Φ i with gauge charge 1 and vanishing U p q R -charge.(c) x W “ tσ . W “ U p q A symmetry broken to Z N f .(ii) The quintic GLSM. (a) Gauge group: U p q .(b) Charged matter: 5 chirals Φ i with gauge charge 1 and U p q R -charge0, and 1 chiral P with gauge charge ´ U p q R -charge 2.(c) x W “ tσ . W “ P G p Φ i q , G is a homogenous polynomial of degree 5.(d) U p q A symmetry is unbroken.(iii) U p N q theory with N f fundamental and N a antifundamental matter.(a) Gauge group: U p N q (b) Charged matter: N f fundamental chiral fields Q F , labeled by F Pr N f s : “ t , ¨ ¨ ¨ , N f u , and N a antifundamental chiral fields q Q A , labeledby A P r N a s . We set the U p q R -charges r F and q r A to be arbitrary fornow. g g Only for this theory are the chiral fields labeled by two different indices F and A . As noted before,the index i is used to label all chiral fields, and the index F is used to label elements of the Cartansubalgebra of the flavor symmetry group for all other cases. ecent developments in 2d N “ p , q supersymmetric gauge theories (c) Without loss of generality, we assume N f ě N a . We also assume N f ě N , so that there exists a supersymmetric ground state.(d) x W “ t tr σ . W is a generic gauge invariant polynomial of U p q R -charge 2, which must be a function of the mesons q Q A Q F .(e) When all twisted masses are turned off, U p q A symmetry unbroken for N f “ N a ; broken to Z p N f ´ N a q for N f ‰ N a .We turn on generic twisted masses for the matter in the U p N q theory unless statedotherwise. The flavor symmetry of the theory, in the absence of twisted masses andassuming all R -charges are equal, is given by S r U p N f q ˆ U p N a qs , whose Cartansubgroup is given by U p q N f ` N a ´ . We choose the conventions where the funda-mental fields of U p N q transform as an anti-fundamental of the U p N f q component,and the antif-fundamental fields of U p N q transform as fundamental fields of U p N a q .The rank of the Cartan subgroup being p N f ` N a ´ q , we may turn on as manytwisted masses. We choose the more convenient route of turning on N f ` N a twistedmasses, which we denote s F and q s A , and identify them under the equivalence rela-tion p s F , q s A q ” p s F ` s, q s A ` s q .
23, 61, 62
We also can turn on flavor holonomies in asimilar manner when we put the gauge theory on T .Before going on further, let us comment on the RG flow of these theories. Wefirst discuss theory (i) in some detail, and briefly comment on theories (ii) and (iii).Much of what we discuss can be found in Refs. 1, 2, 3, 63 and 64. We follow theexposition of Ref. 61 when discussing theory (iii).Theory (i) flows to a CP N f ´ sigma-model in the intermediate IR regime. Letus explain what we mean by the intermediate IR regime. The gauge coupling e ofa two-dimensional gauge theory has the dimension of mass. Thus the gauge theorybecomes strongly coupled at energy scales Λ withΛ ă e . (12)Meanwhile, there is a second coupling in the game when the gauge group has abeliancomponents, namely the FI parameters. The FI parameters are classically marginal,but can flow—the beta function for the FI parameters are one-loop exact and isgiven by β “ ´ b πi , b “ ÿ i ÿ ρ P Λ R i ρ , (13)where we take the view that β , as t , is a vector in h ˚ C —in fact, iβ must be a vector in i c ˚ Ă h ˚ C . It proves to be useful to define the vector b as defined for future purposes.For theory (i), it follows that β “ ´ N f {p πi q . Thus the effective FI coupling at scaleΛ is given by q p Λ q “ ˆ Λ UV Λ ˙ N f q UV , (14)where we have defined q “ e πit , (15) Daniel S. Park for the FI parameter t . We thus find that q becomes large whenΛ ă Λ UV | q UV | { N f . (16)Now we assume that q UV is small enough—i.e., the imaginary part of t is positiveand very large—so that | q UV | ! ˆ e Λ UV ˙ N f . (17)We define the intermediate IR scale to be when Λ is between the two scales wherethe gauge coupling becomes strongly coupled, and when q becomes small:Λ UV | q UV | { N f ! Λ ! e . (18)In this regime, the effective theory is given by a two-dimensional sigma model intoa CP N f ´ manifold, whose complexified K¨ahler parameter is given by t p Λ q “ πi ln q p Λ q . (19)The K¨ahler parameter is a single complex number, as the second homology groupof CP N f ´ has rank-one. q being small in the regime implies that the imaginarypart ξ of the K¨ahler parameter is large and positive, or “very positive”. Only whenthe imaginary part of t is very positive is the sigma model controllable, since thenon-perturbative effects are suppressed by the large volume of the target space.In the far IR, when Λ ! Λ UV | q UV | { N f , (20)the sigma-model is no longer reliable—the theory becomes massive. To be moreconcrete, there are N f ground state vacua, and all the fluctuations around a givenground state are massive with masses much larger than Λ. This can be argued byshowing that the extrema of the quantum twisted superpotential x W p σ q are givenby the N f roots of σ N f “ q (21)and that the ground states are reliably represented by these roots. By carefullyreinstating all the scales in place, it can be shown that all fluctuations around thesevacua have masses of order | q UV | { N f Λ UV " Λ . (22)Theory (ii) is quite interesting in two aspects. First, the FI parameter t is exactlymarginal and the theory flows to an interacting superconformal theory in the IR. Thevariable t parametrizes the twisted chiral conformal manifold. Second, dependingon the value of t , the effective IR theory that describes the theory at length scaleswhere the effective gauge coupling becomes large, takes on different guises. When | q | !
1, i.e., when the imaginary part of t is very positive, the effective theory isdescribed by a sigma model into the famous quintic Calabi-Yau (CY) threefold. ecent developments in 2d N “ p , q supersymmetric gauge theories This is often called the large-volume limit of the SCFT. We should note that thetarget space of an NLSM that is superconformal has a vanishing canonical class,i.e., is Calabi-Yau. Meanwhile, when the imaginary part of t is very negative, theeffective theory is a Landau-Ginzburg orbifold theory. It is quite surprising that theextended conformal manifolds of these two seemingly very different theories are thesame, and that in fact the two theories have an interpretation as different “phases”of the same superconformal theory. Note that when discussing theory (ii), we donot mention an intermediate IR scale, as the only energy scale in play is the scalee , where the effective gauge coupling becomes large. As before, in the “geometricphase” where the theory flows to an NLSM into the quintic threefold, t is identifiedwith the complexified K¨ahler parameter of the manifold. Meanwhile, the coefficientsof the superpotential, which are also exactly marginal deformations of the SCFT,parametrize the complex structure moduli space of the manifold.Theory (iii) has many moving parts. Let us first discuss the theory where alltwisted masses are turned off, and the R -charges are set to zero. The beta functionfor the FI parameter is given by β “ p N a ´ N f q{p πi q , and the theory flows toa superconformal theory in the IR only when N f “ N a . When N a “
0, and theimaginary part of t is taken to be very positive, the theory flows to an NLSM ofthe Grassmannian Gr p N, N f q , which is the space of complex N -planes in C N f , inthe intermediate IR regime. Each anti-fundamental matter can be interpreted as acopy of the tautological bundle S of the Grassmannian. Thus in general, when theimaginary part of t is taken to be sufficiently large, the theory flows to an NLSMof the bundle S ‘ N a Ñ Gr p N, N f q in the IR. For N f “ N a , the theory flows to theNLSM of the manifold S ‘ N f Ñ Gr p N, N f q for both limits | q | ! | q | "
1. Thetwo manifolds, however, are not equivalent, and are related by a flop. In particular,the complexified K¨ahler parameter of the manifold for | q | ! t ,while the K¨ahler parameter of the manifold obtained for | q | " ´ t .The elements of the cohomology of the manifold X : “ S ‘ N a Ñ Gr p N, N f q are represented by gauge-invariant polynomials of the sigma fields, as explainedin Ref. 2. h The operators built out of the sigma fields are elements of the quan-tum cohomology ring of the manifold, and their correlation functions encode theGromov-Witten invariants
6, 66, 67 of these manifolds. Now the flavor symmetry of thegauge theory translates into isometries of the NLSM one obtains in the IR. Giventhe existence of the U p q N a ` N f ´ isometry of manifold X , one could ask about theequivariant version
68, 69 of the quantum cohomology and Gromov-Witten invari-ants of X . These can be computed by turning on the twisted masses s F and q s A for the isometries. Below the energy scale of the twisted masses, the theory quickly h While all the elements of the cohomology of S ‘ N a Ñ Gr p N, N f q are represented this way fortheory (iii) with all R -charges set to zero, this is not the case in general. For example, in thegeometric phase of theory (ii), there are 204 elements of the third cohomology of the quinticthreefold that cannot be represented using the sigma fields.2 Daniel S. Park flows to a theory of ` N f N ˘ isolated massive vacua.
63, 64
The correlators of this theory,however, encode the equivariant quantum cohomology of the original manifold M with equivariant parameters s F and q s A .
3. Supersymmetric backgrounds
In this section, we discuss general aspects of supersymmetric backgrounds that wecan use to localize gauge theory on. We also write down the Lagrangians of gaugetheories in the supersymmetric backgrounds. We follow the approach of Ref. 19,where the supersymmetric backgrounds are obtained by turning on expectationvalues for fields in the N “ p , q new minimal supergravity multiplet. We notethat all the formulae for the supersymmetric backgrounds and Lagrangians in thissection are borrowed from Ref. 19. i As explained in the introduction, the supersymmetric backgrounds can be un-derstood as vacuum expectation values (VEVs) of components of the N “ p , q newminimal supergravity multiplet. We utilize the bosonic components of this multipletdenoted by g µν , A µ , C µ , r C µ (23)in Ref. 19. Obviously, g µν is the background metric, while A µ is the vector field thatcouples to the R -current. C µ and r C µ are vector fields that couple to the currentassociated to the complex central charge of the supersymmetry algebra. As in Ref.19, it is useful to introduce the field strengths H “ ´ i(cid:15) µν B µ C ν , r H “ ´ i(cid:15) µν B µ r C ν . (24)Following Ref. 19, we use “ R ” to denote the opposite of the scalar curvature in thissection. Given that these components are turned on, the Lagrangian of the theoryis given by L gauge ` L chiral ` L W ` L x W . (25)Each term is written explicitly as follows. i While we focus on gauge theories with chiral matter in this work, it is straightforward to coupleany N “ p , q field theory whose flat space Lagrangian can be written in superspace, such asa non-linear sigma model, to these supersymmetric backgrounds. Curvature couplings of 2d N “ p , q NLSMs have also been studied in Ref. 71. ecent developments in 2d N “ p , q supersymmetric gauge theories The term L gauge is the gauge-kinetic term: j L gauge “ « D r σD ¯1 σ ` D ¯1 r σD σ ` r σ, r σ s ` i r λ ` D λ ` ´ i r λ ´ D ¯1 λ ´ ` i r λ ´ r σ, λ ` s ´ i r λ ` r r σ, λ ´ s` ˆ if ` r H σ ´ H r σ ˙ ´ ˆ D ` r H σ ` H r σ ˙ ff . (26)We note that we use the frame indices 1 and ¯1, where the metric is written ds “ g z ¯ z dzd ¯ z “ e e ¯1 . (27)Note that by definition, f is imaginary. The covariant derivative D µ defined to be D µ “ ∇ µ ´ irA µ ` s r C µ ´ r sC µ (28)where ∇ µ is the covariant derivative including the metric and gauge connections.Here, r is the R -charge of the field, while s and r s are the complex central chargesof the field. We note that s “ r s “ L chiral “ ÿ i L i (29)where L i “ D r φ i D ¯1 φ i ` D ¯1 r φ i D φ i ´ r F i F i ` i r ψ i ` D ψ i ` ´ i r ψ i ´ D ¯1 ψ i ´ ` r φ i Dφ i ´ ˆ r i R ´ H r s i ´ r H s i ˙ r φ i φ i ` r φ i ´r s i ´ r σ ´ r i r H ¯ ´ s i ´ σ ´ r i r H ¯ φ i ` r φ i r σ, r σ s φ i ` i r ψ i ` ´r s i ´ r σ ´ r i r H ¯ ψ i ´ ´ i r ψ i ´ ´ s i ´ σ ´ r i H ¯ ψ i ` ` i ? p r ψ i ` r λ ´ ´ r ψ i ´ r λ ` q φ i ` i ? r φ i p λ ` ψ i ´ ´ λ ´ ψ i ` q . (30)Here, all the fields in the vector multiplet can be understood as being matrices in the R i representation of g , and that the indices of the fields are contracted accordingly.When we wish to turn on real twisted masses only, we may take s i “ r s i “ ÿ F s F q iF , (31)where s F are twisted masses for the flavor symmetry u p q f,F , and q iF is the flavorcharge of Φ i . j When the gauge algebra can be decomposed into a direct sum of sub-algebras, we may assignto each component a separate gauge coupling. The gauge coupling, however, does not play aprominent role in the backgrounds we study, as we see later on.4
Daniel S. Park
It turns out to be more useful to allow more elaborate flavor backgrounds byturning on supersymmetric expectation values for background vector multipletscoupled to the flavor symmetry. This is particularly useful if one wishes to ultimatelypromote a subgroup of the flavor symmetry group into a gauge symmetry. To do so,we need to set all the central charges s and r s to zero, and couple the chiral fields tothe background flavor vector multiplet V F . Now turning on a constant real twistedmass s F for a flavor symmetry is equivalent to turning on the supersymmetricexpectation values s F “ r s F , a F,µ “ ´ i s F r C µ ` i r s F C µ , D F “ ´ s F r H ´ r s F H (32)for the fields in the vector multiplet V F . There are more general supersymmetricconfigurations of V F that may be turned on depending on the supersymmetricbackground the theory is coupled to. We explore such configurations further in thesubsequent sections.The superpotential terms of the Lagrangian are determined by functions W and Ă W of Φ i : L W “ F i B i W p φ i q ` ψ i ´ ψ j ` B i B j W p φ i q ` r F i r B i Ă W p r φ i q ` r ψ i ´ r ψ j ` r B i r B j Ă W p r φ i q . (33)for the superpotential W p Φ i q of the theory. While W and Ă W do not have to relatedin Euclidean signature, we restrict ourselves in the case where Ă W is related to W by Ă W p φ i q “ W p φ i q , (34)where the bar denotes complex conjugation.We mostly consider twisted superpotentials that are linear in the field strengthmultiplet of the abelian factors of the gauge group. The twisted superpotential termsare then given by: L x W “ ´ ÿ I ξ I tr I D ` i ÿ I θ π tr I p if q , (35)where f is the field strength of the gauge field. For future reference, let us notethat for generic twisted superpotential functions x W and Ăx W of twisted superfieldsΩ i with components p ω i , η i ˘ , G i q , the twisted superpotential terms are given by L x W “ G i B i x W p ω i q ` η i ´ r η j ` B i B j x W p ω i q ´ i r H x W p ω i q` r G i r B i Ăx W p r ω i q ´ r η i ´ η j ` r B i r B j Ăx W p r ω i q ` i H Ăx W p r ω i q . (36)As before, we restrict ourselves to cases when the function Ăx W is defined with respectto x W such that Ăx W p ω i q “ x W p ω i q . (37) ecent developments in 2d N “ p , q supersymmetric gauge theories We note that the components of the twisted chiral field-strength multiplet con-structed from the vector multiplet has components p σ, ? λ ´ , ´? r λ ` , iD ´ f ` i r H σ q . (38)The supersymmetric backgrounds are found by asking which expectation valuesof (23) preserve some supersymmetry. The supersymmetry transformations of fieldsin the supergravity multiplet can be found in Ref. 19, as well as a systematic classifi-cation of supersymmetric backgrounds. Now in all the supersymmetric backgroundswe discuss in this review, the Lagrangians L gauge , L chiral and L W are exact, in thatthey can be written as t Q , V u for a preserved supercharge Q . k In fact, we use " Q , V gauge ` V chiral * “ L gauge ` L chiral (39)as the localizing Lagrangian of equation (4) to localize the path integral to theCoulomb branch, i.e., we take e , g Ñ
0. Thus one might naively expect that thepartition function is only dependent on parameters of the twisted superpotential Ă W . This is almost true—the one-loop determinants of the theory depend on thequadratic fluctuations around the localization locus, and these can depend on addi-tional parameters of the theory, such as the R -charge, twisted masses or backgroundfluxes of flavor symmetries, as we see in specific examples later on. This is consis-tent with the fact that such parameters appear in the supersymmetry algebra ofthe localizing supercharge being used.
4. The round sphere
In this section, we review the round sphere partition function (or equivalently, thepartition function on the S with no R -flux) first computed in Refs. 23 and 24. Thebackground supergravity fields for the round sphere partition function are givenby: ds “ p ` | z | q dzd ¯ z , A µ “ , H “ i , r H “ i , (40)where we have set the radius of the sphere to 1. We note that the gauge field A µ that couples to the R -symmetry current is set to vanish in this background.The partition function can be made to localize onto the Coulomb branch locus where all the components of the chiral fields vanish and the fields σ , r σ , D and f take on the constant values σ “ p σ ´ i m , r σ “ p σ ` i m , D “ ´ i p σ , if “ ´ m . (41) k While most of the supersymmetric backgrounds for N “ p , q studied can be understood asvacuum expectation values of the new minimal supergravity multiplet, the torus partition functionwe study in section 5 is an exception, and should be understood as a background of a N “ p , q supergravity multiplet. There, a background gauge field coupling to the left-moving R -current,which does not exist in the N “ p , q supergravity multiplet studied here, is turned on.6 Daniel S. Park
Here p σ and m are taken to be elements of the Cartan subalgebra i h . We often writethe components of p σ and m explicitly, i.e., p σ “ p σ a T a , m “ m a T a , (42)where T a are generators of the Cartan subalgebra. As we have set the radius of thesphere to be the unit of length, m is identified with the magnetic flux through thesphere, and must be GNO quantized. Thus the path integral of the theory on theround sphere localizes onto a sum over the magnetic flux m , and a finite dimensionalintegral over the continuous real parameters p σ a .As discussed previously, we may turn on complex twisted masses in these back-grounds by turning on components of the background flavor vector multiplet V F .These complex masses are turned on by giving a vacuum expectation value to thescalar components s F and r s F of V F . By supersymmetry, the vector field a F,µ andauxiliary field D F of the multiplet must also take the following expectation values: s F “ Re p s F q ´ i m F , r s F “ Re p s F q ` i m F , D “ ´ i Re p s F q , if F, “ ´ m F . (43)We have denoted the imaginary part of s F , m F to emphasize that m F is a back-ground magnetic flux for the flavor symmetry. This flux must be quantized so that q iF m F P Z for all chiral fields Φ i . As before, we denote the complex twisted massesfor the multiplets Φ i and r Φ i , s i “ q iF s F , r s i “ q iF r s F . (44)The round sphere partition function is given by l Z S “ | W | ÿ m ż ˜ź a d p σ a π ¸ Z cl m p p σ q Z (cid:96) V , m p p σ q ź i Z (cid:96)i, m p p σ q , (45)where the sum m runs over all GNO quantized fluxes. | W | denotes the order ofthe Weyl group of the gauge algebra. Z cl m p p σ q is the classical action evaluated at thesaddles: m Z cl m p p σ q “ exp ˜ ´ πi ÿ I ξ I tr I p p σ q ` i ÿ I θ I tr I p m q ¸ “ e ´ π ř I p t I tr I σ ´ ¯ t I tr I r σ q “ e ´ πt p σ q` π ¯ t p r σ q . (46)As we see as we go on, the expressions become more elegant, once the variables σ and r σ in equation (41) are used. The other factors of the integrand come from the l Operators and defects may be inserted in the path integral, although we do not explore thispossibility in this section. The insertion of vortex defects in this background has been studied inRef. 73. m We note that it is not entirely accurate to call this piece the classical piece, in that the renormal-ized real FI parameter ξ I at the scale of the radius of the sphere should be plugged into equation(46). ecent developments in 2d N “ p , q supersymmetric gauge theories one-loop determinant of the vector and chiral multiplets. Z (cid:96) V , m p p σ q comes from thevector multiplets: Z (cid:96) V , m p p σ q “ ź α ą ˆ ´ α p p σ q ´ α p m q ˙ “ ź α ą ´ ´ α p σ q α p r σ q ¯ (47)where the product runs over the positive roots α ą g . n Theone-loop factor coming from integrating out Φ i and r Φ i is given by Z (cid:96)i, m p p σ q “ ź ρ P Λ R i Γ ´ r i ´ iρ p p σ q ´ ρ p m q ´ is i ¯ Γ ´ ´ r i ` iρ p p σ q ´ ρ p m q ` i r s i ¯ “ ź ρ P Λ R i Γ ` r i ´ iρ p σ q ´ is i ˘ Γ ` ´ r i ` iρ p r σ q ` i r s i ˘ , (48)where Λ R i Ă i h ˚ denote the set of weights of representation R i .The integral (45) is a real integral, in that the contour of integration for p σ a isalong the real line, when all the twisted masses s i “ r s i are real. In many interestingcases, the integrand of (45) may have poles along the real axis. The correct way todeal with those cases is to first shift the R -charge of the chiral fields whose one-loopdeterminant is responsible for the poles by a small positive amount δr . o After thisdeformation, the integral (45) will not have any poles along the real axis, and maybe evaluated. The desired partition function may then be obtained by taking thelimit δr Ñ p To be concrete, let us write the explicit matrix integrals for the round spherepartition function for theories (i), (ii) and (iii) introduced in section 2. For theory(i), the partition function is given by ÿ m P Z ż d p σe ´ πiξ p σ ` iθ m ˜ Γ ` ´ i p σ ´ m ˘ Γ ` ` i p σ ´ m ˘ ¸ N f . (49)For theory (ii), it is given by ÿ m P Z ż d p σe ´ πiξ p σ ` iθ m ˜ Γ ` ´ i p σ ´ m ˘ Γ ` ` i p σ ´ m ˘ ¸ Γ ` ` i p σ ` m ˘ Γ ` ´ i p σ ` m ˘ . (50)For theory (iii), we write the partition function in a particular way that turnsout to be quite useful for multiple purposes. To do so, we introduce some notation, n There is an additional minus sign on each of the factors of the product in equation (47) comparedto Refs. 23, 24, which has been correctly accounted for in Ref. 30. o Depending on the properties one wants to preserve, this may not be possible. To be more concrete,one might want certain superpotential terms to be present, and thus may want to impose linearconstraints on the supercharges of the various chiral fields in the theory. In this case, one needs toshift the R -charges in a manner consistent with the linear constraints, such that all gauge invariantpolynomials of the chiral fields have positive R -charge. p While the partition function Z S often has a well defined δr Ñ δr . Daniel S. Park following Ref. 61. Note that the Cartan subalgebra of u p N q is given the diagonalentries and thus the saddles are given by the diagonal matrices p σ “ diag p p σ a q , m “ diag p m a q . (51)with m a P Z . We define the “packaged” variables σ a, ` “ iσ a “ i p σ a ` m a , σ a, ´ “ i r σ a “ i p σ a ´ m a F, ˘ “ i Re p s F q ˘ m F ` r F , q Σ A, ˘ “ i Re p q s A q ˘ q m A ´ q r A , (52)and the traces Σ ˘ “ ÿ a σ a, ˘ . (53)We introduce the following differences to condense our notation. We distinguish thevarious entries of the differences by their indices as follows:Σ ab ˘ “ σ a ˘ ´ σ b ˘ , Σ aF ˘ “ σ a ˘ ´ Σ F ˘ , Σ aA ˘ “ σ a ˘ ´ q Σ A ˘ , Σ F F ˘ “ Σ F ˘ ´ Σ F ˘ , Σ FA ˘ “ Σ F ˘ ´ q Σ A ˘ , Σ A A ˘ “ q Σ A ˘ ´ q Σ A ˘ . (54)The partition function for theory (iii) may now be succinctly written as e iϕ N N ! ÿ m P Z N ż ˜ź a dσ a π ¸ q Σ ` ` q Σ ´ ´ ź a ă b Σ ab ` Σ ab ´ ź F Γ p´ Σ aF ` q Γ p ` Σ aF ´ q ź A Γ p Σ aA ` q Γ p ´ Σ aA ´ q (55)with the phase ϕ N “ N p N ´ q π {
2. We have previously introduced the exponentialof the complexified FI parameter q “ e πit . q ˘ are related to q by q ` “ e ´ iπ p N ` q q , q ´ “ e iπ p N ` q ¯ q . (56)The contour of integration of the integral (55) needs to be commented on, as wehave turned on flavor fluxes. By carefully reviewing the prescription for the contour,one should be able to convince oneself that the contour should be taken such thatall the poles coming from the one-loop determinant of the fundamental fields shouldbe positioned below the p σ a -contour, while those coming from the determinant ofthe anti-fundamental fields should be positioned above the contour. In other words,the contour should be taken such that it divides the two classes of poles.When the beta function of an FI parameter ξ is negative or zero, the asymptoticsof the integrand becomes such that we may deform the contour of integration tothe lower-half plane for each p σ a coupled to ξ linearly, once we take ξ to be verypositive. The final integral then can be written as a sum of residues of poles ofthe integrand over poles lying below the contour of integration. When the betafunction is positive or zero, we can similarly deform the contour of integration tothe upper-half p σ a plane once we take ξ to be very negative.For the U p N q theories at hand, we can deform the contour of integration to thelower-half plane and pick up the poles at σ a ` “ Σ F a ` ` n ` , σ a ´ “ Σ F a ´ ` n ´ , (57) ecent developments in 2d N “ p , q supersymmetric gauge theories for n ˘ P Z ě . The partition function then decomposes into the form ÿ (cid:126)F P C p N,N f q Z (cid:126)F Z (cid:126)F ` Z (cid:126)F ´ . (58)Here, C p N, N f q is the set of N -tuples of integers F a such that1 ď F ă F ă ¨ ¨ ¨ ă F N ď N f . (59)The component labeled by (cid:126)F comes from picking up precisely the poles (57) forcomponents of (cid:126)F .The decomposition (58) has a beautiful interpretation, as the result of the lo-calization computation on a different set of saddles, which is often called the Higgsbranch locus . A vector (cid:126)F P C p N, N f q labels a Higgs sector of the theory, wherethe fundamental fields F , ¨ ¨ ¨ , F N take vacuum expectation values. In order for thefields to do so, the sigma fields must take vacuum expectation values such that σ ” diag p s F a q , r σ ” diag p r s F a q , (60)where the symbol “ ” ” in this equation is used to indicate equivalence up to conju-gation. The Higgs branch saddles are such that in the bulk of the sphere the fun-damental fields F , ¨ ¨ ¨ , F N take constant VEVs and the sigma fields take constantvalues (60). One then needs to sum over the point-like vortices and anti-vorticeslocalized at the two poles of the sphere, in which the fundamental fields F , ¨ ¨ ¨ , F N are turned on. The factor Z (cid:126)F is the one-loop determinant of the fields around theconstant field configuration, while Z (cid:126)F ` and Z (cid:126)F ´ denote the vortex and anti-vortexpartition functions , coming from summing over all the vortex configurations lo-calized at the two poles of the sphere. Quite amazingly, the sphere partition functionallows one to compute the vortex partition function reliably without addressing thevortex moduli space.While we do not write the explicit expression for Z (cid:126)F , which can be found in manyplaces in the literature,
23, 24, 61, 62 let us note that the vortex partition functions aregiven by Z (cid:126)F ` “ Z (cid:126)F v p Σ F ` ; q Σ A ` ; q q , Z (cid:126)F ´ “ Z (cid:126)F v p Σ F ´ ; q Σ A ´ ; p´ q N a ´ N f ¯ q q , (61)where we have used the shorthand notation Z (cid:126)F v “ Z (cid:126)F , O “ for the function Z (cid:126)F , O v p Σ F ; q Σ A ; q q “ ÿ n ě q n ÿ |p n a q|“ n O p Σ F a ` n a q ¨ N ź a “ ś N a A “ p Σ F a A q n a ś Nb “ p´ Σ F a F b ´ n a q n b ś N b “ p´ Σ F a F cb ´ n a q n a (62)defined for any symmetric polynomial O of | (cid:126)F | “ N variables. We have denoted p n a q to be N -tuples of non-negative integers, and |p n a q| : “ ř a n a . (cid:126)F c is a N “ p N f ´ N q -tuple in C p N f , N f ´ N q , whose elements are given by the complement of (cid:126)F withrespect to r N f s . p a q n is the Pochammer symbol, given by Γ p a ` n q{ Γ p a q . Daniel S. Park
When the theory flows to a N “ p , q superconformal fixed point in the IR,the sphere partition function Z S has a beautiful interpretation first conjectured inRef. 51, and proven in Refs. 20, 52. The FI parameters t I of the gauge theory aremarginal deformations of the superconformal theory, and thus span the conformalmanifold of twisted chiral couplings of the IR theory. This conformal manifold has aK¨ahler metric, which can be identified with the Zamolodchikov metric. Thesphere partition function can be related to the K¨ahler potential of this metric by Z S p t I , ¯ t I q “ e ´ K p t I , ¯ t I q . (63)Note that the K¨ahler potential K is determined up to shifts K p t I , ¯ t I q Ñ K p t I , ¯ t I q ` f p t I q ` f p t I q , (64)where f is a holomorphic function of the parameters t I . These come from localcounterterms in the field theory,
19, 20 which vanish in flat space. Thus, it is appro-priate to think of the partition function Z S as a section of a bundle over the twistedchiral conformal manifold, rather than a function.When the superconformal theory in the IR is an NLSM with a target manifold X , the FI parameters t I can be identified with the K¨ahler parameters of X , andthe conformal manifold with its extended K¨ahler moduli space. Thus, in this case, Z S turns out to encode sophisticated geometric invariants, more about which wediscuss in section 8.We end the discussion of the round sphere partition function by noting that while Z S has an elegant interpretation when the gauge theory flows to a superconformalfixed point in the IR, its meaning is not entirely clear when this is not the case. Itwould be interesting to understand the significance of Z S when the IR theory isnot superconformal.
5. The torus
We now study the torus partition functions, computed in Refs. 25, 26 and 27. Wefirst review their results, which assumes that the gauge group is connected. We thenslightly extend these results to the case when the gauge group contains a discretefactor in section 5.1.The supersymmetric background for the torus is quite simple—the metric isflat, and a flat connection for the gauge field A Lµ , which couples to the left-moving R -current of the theory, may be turned on. We can think of the torus as beingobtained by quotienting the complex plane with complex coordinate w with respectto two independent shifts such that w „ w ` „ w ` τ , (65)where τ is the complex structure of the torus. The flat connection for the left-moving R -symmetry, which we denote U p q L , is parametrized by the complex holonomy z “ ¿ t A L ´ τ ¿ s A L , (66) ecent developments in 2d N “ p , q supersymmetric gauge theories where t and s denote the temporal, and spatial cycles on the torus. In order to turnon a holonomy z for the U p q L -charge, all twisted masses for the chiral multipletsmust be turned off, as their introduction completely breaks the left-moving R -symmetry classically. Meanwhile, flat connections of background flavor symmetriesmay be turned on, parametrized by q u F “ ¿ t a F ´ τ ¿ s a F . (67)The parameters z and u F both lie on a torus with complex structure τ , as theyhave the periodicities (65). Following Refs. 25–27, we define the exponentiated pa-rameters: q q “ e πiτ , y “ e πiz , q x F “ e πi q u F . (68)We note that when the left-moving R -symmetry of the theory is broken into adiscrete subgroup Γ of U p q L by anomalies, the partition function is not well definedunless y is restricted to be an element of Γ.The path integral for the torus partition function localizes onto the space offlat connections, parametrized by the holonomies of the connection around the twocycles of the torus. As with the background flavor and R -symmetry connections,this can be packaged into a complex parameter for each element of the Cartansubalgebra of the gauge group, which we denote u a , following Refs. 25–27. After anintegration by parts, the path integral can be shown to be Z T “ | W | ż C ź a du a πi Z (cid:96) V p u q ź i Z (cid:96)i p u q , (69)with a middle-dimensional integration cycle C in the rk p G q -complex dimensionaltorus, which we denote r T . The torus r T can be identified with the smooth cover ofthe space of flat connections on T . As before, | W | is the order of the Weyl groupof the gauge group. We note that there is no sum over distinct topological sectorswhen the gauge group is connected. The one-loop determinants coming from thevector multiplet, and the chiral multiplet i are given by Z (cid:96) V “ ˆ πη p q q θ p q, y ´ q ˙ rk p G q ź α θ p q, x α q θ p q, y ´ x α q , Z (cid:96)i “ ź ρ P Λ R i θ p q, y r i { ´ x ρ q x q i q θ p q, y r i { x ρ q x q i q , (70)where r i is the U p q R -charge of the chiral multiplet. As before, α runs over the rootsof the gauge algebra, while Λ R i denotes the weights of the representation R i of G .We have introduced the notation such that x ρ “ e πiρ p u q , q x q i “ e πiq iF q u F . (71) q In this section and this section only do we use the variable q to denote the exponentiated complexstructure of the torus. This is not to be confused with the exponentiated FI parameters, whichthe torus partition function does not depend on.2 Daniel S. Park
The eta function and theta function have the expansions η p q q “ q ź n “ p ´ q n q , θ p q, y q “ ´ iq y ź k “ p ´ q k qp ´ yq k qp ´ y ´ q k ´ q (72)when the arguments are small. Note that these functions are multivalued with re-spect to the variables q and y , and should really be viewed as functions of τ and z for y “ e πiz . Following Refs. 26, 27, we write the function as θ p τ | z q when we wantto emphasize this fact. θ p τ | u q does not have poles with respect to u , while it haszeros at u “ n ` mτ . We find that θ p τ | u ´ z q θ p τ | u q „ y m θ p q, y ´ q πη p q q p u ´ n ´ mτ q (73)around this point. It is also worth noting that θ p τ | z ` n ` mτ q “ p´ q n ` m e ´ πimz ´ iπm τ θ p τ | z q . (74)Now the most non-trivial part of the equation (69) lies in the determinationof the contour of integration C . Note that the integrand of the integral (69) isholomorphic at a generic point on the complex rk p G q -dimensional torus. Thus, theintegral (69) boils down to a sum of residues of the integrand. The singularity ofthe factors of the integrand lie along the hyperplanes: H I “ ! u : Q I p u q ` r I z ` q I F q u F ” p mod Z ` τ Z q ) (75)where I labels all the charged components of the chiral multiplets in the theory. Tobe more concrete, the label I belongs to the set I P t p ρ, i q : ρ P Λ R i u . (76)Recall that we use the index i to label the chiral multiplets throughout this review.Then we find that p Q I , r I , q I F q “ p ρ, r i , q iF q . (77)Now let us define the set of codimension-rk p G q singularities of the integrand of(69), r T ˚ sing . For any u ˚ P r T ˚ sing , there are s ě rk p G q hyperplanes H I , H I , ¨ ¨ ¨ , H I s intersecting at u ˚ . We use Q p u ˚ q to denote the charges associated to those hyper-planes: Q p u ˚ q “ t Q I , ¨ ¨ ¨ , Q I s u . (78)The torus partition function (69) can then be written as Z T “ ÿ u ˚ P r T ˚ sing JK-Res u “ u ˚ r Q p u ˚ q , η s ω T p u q , (79)where we have defined the differential form ω T p u q : “ Z (cid:96) V p u q ź i Z (cid:96)i p u q du ^ ¨ ¨ ¨ ^ du rk p G q . (80) ecent developments in 2d N “ p , q supersymmetric gauge theories In order to evaluate this formula we must review the definition of the Jeffrey-Kirwanresidue (JK residue).
A vector η P i h ˚ must be introduced to compute theseresidues. While the individual residues depend on the choice of η , the final sum,which yields the partition function, is independent of this vector.After shifting the position of the pole to the origin, the definition of the JKresidue boils down to defining the residue of a rational rk p G q -form Ω that canhave poles along the hyperplanes Q I p u q “ Q I P Q p u ˚ q . The space of suchdifferential forms R Q p u ˚ q can be understood as a graded vector space over thecomplex numbers. There is a subspace S Q p u ˚ q of this vector space, spanned by ´ | Q p u ˚ q| rk p G q ¯ rk p G q -forms: ω S “ ź Q I P S Q I p u q du ^ ¨ ¨ ¨ ^ du rk p G q (81)where S is an arbitrary subset of Q p u ˚ q with rk p G q distinct elements. For ω S , wedefine the JK residue to beJK-Res u “ r Q p u ˚ q , η s ω S “ | det p S q| when η P Cone p S q η R Cone p S q , (82)where Cone p S q is the cone in i h ˚ spanned by the elements of S , and det p S q is thedeterminant of the rk p G q ˆ rk p G q matrix of charge vectors Q I P S . This defines theJK residue for any element of the vector space S Q p u ˚ q . There is a natural projection π from R Q p u ˚ q to S Q p u ˚ q , which is the analogue of extracting the z ´ term ofthe Laurent series of a rational function f p z q at z “
0. Then, the JK residue ofany Ω P R Q p u ˚ q is defined to be the JK residue of the projection π p Ω q of Ω into S Q p u ˚ q .
33, 82, 83
In order for ω T p u q to be a single-valued differential form on the torus r T , it mustbe invariant under the shifts u a Ñ u a ` u a Ñ u a ` τ for each index a . Onecan check that while ω T p u q is automatically invariant under the shifts u a Ñ u a ` z for the left-moving R -symmetry turned on. The failure of ω T to be invariant under this shiftreflects the U p q L -anomaly of the theory.Let us touch on a rather technical point, before moving on further. When thecharges of the hyperplanes meeting at a singularity are contained within a half-spaceof i h ˚ , the hyperplane arrangement is said to be projective. When the arrangementof the intersecting hyperplanes at some u ˚ is not projective, the JK residue isnot well defined. To resolve such a situation, one must judicially shift parameterssuch as flavor charges and R -charges to split-up the non-projective singularity in r Refs. 26, 27 re-introduced Jeffrey-Kirwan residues into the physics literature, which has beenshowing up in localization computations in diverse dimensions and backgrounds since.
A generalization of the JK residue, coined the Jeffrey-Kirwan-Grothendieck residue, appears inlocalization computations of half-twisted correlators of N “ p , q GLSMs. Daniel S. Park to projective ones, compute the partition function, and take the limit where theparameters are set to their initial values.Let us now write down some expressions for the torus partition function of themodel theories (i), (ii) and (iii), all of which can be found in the original works.
26, 27
For theory (i), we have ω T p u q “ πη p q q θ p q, y ´ q ˆ θ p q, y ´ x q θ p q, x q ˙ N f du . (83)It is simple to verify that ω T p u ` τ q “ y N f ω T p u q , (84)and ω T is not single-valued on r T unless y is an element of the multiplicative group Z N f . In order to employ the formula (79), we must turn on y such that y ‰
1, sincewhen y “ ω T diverges, and is not well-defined. Obviously, the torus partitionfunction computed by the formula (79) vanishes for y N f “ y ‰ η “ ´ Z T “ N f for y “ s This is consistent with the fact that the CP N f ´ theory has N f massive vacuarepresented by the N f solutions of σ N f “ e πit , (85)where t is the complexified FI parameter of the theory. For this theory, the toruspartition function computes a trace of the operator y J , J being the charge of U p q L ,with respect to these ground states. The sigma field σ has unit charge under theaction of the left-moving R -symmetry. The k -th vacuum | k y corresponds to theexpectation value | k y : σ “ e πik { N f e πit { N f (86)of the sigma field. We see that a generic element of the left-moving R -symmetrygroup maps a vacuum to a non-vacuum. A discrete subgroup of the R -symmetry,however, is preserved, since the set of vacua are preserved by a Z N f subgroup. Thegenerator ω “ e πi { N f of this subgroup maps | k y to | k ` y . Thus the vacua may beorganized in representations of Z N f : | ˜ k y “ N f ´ ÿ k “ e ´ πi ˜ kk { N f | k y . (87)It is simple to see that ω | ˜ k y “ e πi ˜ k { N f | ˜ k y (88) s Alternatively, since we have turned off the Wilson line for the left-moving R -symmetry, we mayintroduce twisted masses and use the localization formula for the A -twist on the torus to find that Z T “ N f . ecent developments in 2d N “ p , q supersymmetric gauge theories where ω on the left-hand-side of the equation is understood to be an operator, ratherthan a number. Then the trace over the ground states can be computed explicitly:tr ω k “ N f ´ ÿ ˜ k “ e πik ˜ k { N f “ N f when k ” p mod N f q ω T p u q “ πη p q q θ p q, y ´ q ¨ θ p q, x ´ q θ p q, yx ´ q ¨ θ p q, y ´ x q θ p q, x q du . (90)The theory is superconformal, and thus ω T is well-defined on r T for any value of y .We can take η “ ´ x “ e πi p z ` k ` lτ q{ for 0 ď k, l ď Z T “ ÿ k,l “ y ´ l ˆ θ p q, e πi p´ z ` k ` l q{ q θ p q, e πi p z ` k ` l q{ q ˙ . (91) ω T p u q for theory (iii) is given by ω T p u q “ N ! ˆ πη p q q θ p q, y ´ q ˙ N ˜ź a ‰ b θ p q, x a x ´ b q θ p q, y ´ x a x ´ b q ¸ź a ˜ź F θ p q, y r F { ´ x a ˘ x ´ F q θ p q, y r F { x a ˘ x ´ F q ź A θ p q, y q r A { ´ x ´ a q x A q θ p q, y q r A { x ´ a q x A q ¸ du ^ ¨ ¨ ¨ ^ du N (92)where we have turned on holonomies ˘ x F “ e πi ˘ u F and q x A “ e πi q u A for the flavorsymmetry. We refer the reader to the original reference Ref. 27 for the evaluationof this partition function.The torus partition function, as computed, has a natural interpretation as theelliptic genus —a weighted sum over the Ramond-Ramond states of the two-dimensional theory Z T “ Tr RR p´ q F ¯ q H R q H L y J ź F x K F F , (93)where H R and H L are the right- and left-moving Hamiltonian operators while J and K F are the charge operators for the left-moving R -symmetry and the maximaltorus of the flavor symmetry, respectively. F is the fermion number. Note that whenthe left-moving R -symmetry is broken to a finite subgroup Γ of U p q L , y J must berestricted to be an element of Γ. When y and x F are all taken to 1, we arrive atthe celebrated Witten index, which counts the (graded) number of ground statesof the theory.When the gauge theory flows to an NLSM in the IR, the elliptic genus hasan interpretation as the index of a Dirac operator in the loop space of the targetmanifold. This is a geometric invariant that can be computed by integrating a Daniel S. Park certain elliptic density over the manifold. Now for an NLSM that has a (compact)Calabi-Yau threefold as a target space, the elliptic genus is completely determinedby the Euler number χ E of the threefold:
91, 92 Tr RR p´ q F ¯ q H R q H L y J “ χ E φ , { p q, y q “ χ E p y { ` y ´ { q ` O p q q . (94)Here, φ , { is a weak Jacobi form, written out explicitly, for example, in Ref. 92. Wecan check that the computation (91) for the quintic GLSM reproduces the Eulernumber χ E “ ´
200 of the quintic threefold by utilizing the expansion (72) andtaking the limit q Ñ Z quintic T | q Ñ “ ´ p y ´ { ` y { q . (95) Discrete gauge symmetry
While we have discussed theories with a connected gauge group up to this point,there are many interesting theories whose gauge group has multiple components.In this section, we discuss the simplest case, when the gauge group factors into acontinuous, and a discrete factor: G “ G cont ˆ Γ . (96)In this case, we must sum over all non-trivial principal Γ bundles over the manifold M the gauge theory lives on. This can be readily computed by Hom p π p M q , Γ q{ Γ (97)where the quotient is taken with respect to the adjoint action of Γ . We thus seethat the sphere does not have any non-trivial principal Γ -bundles, while the torusdoes. The elements of (97) for the torus can be explicitly written out to be t p g, h q : gh “ hg u{ „ (98)where p g , h q „ p g , h q when p g , h q “ p g ´ g g, g ´ h g q for some g P Γ .Now the chiral fields Φ i in the theory transform as representations of Γ . Thusfor each g P Γ , there exists a matrix Λ p g q ij such that the action of g is given by g p Φ q i “ Λ p g q ij Φ j . (99)Let us introduce the twisted partition function Z T p g, h q , which is the torus partitionfunction with twisted boundary conditions such thatΦ i p w ` q “ Λ p g q ij Φ j p w q , Φ i p w ` τ q “ Λ p h q ij Φ j p w q . (100)Then the partition function of the theory Z T must be given as a weighted sumover the twisted partition functions Z T p g, h q for all commuting pairs of g and h .To find the correct weights, we can view the gauge theory as a G cont gauge theoryorbifolded by the global symmetry Γ . Following Ref. 94, we then arrive at Z T “ | Γ | ÿ gh “ hg Z T p g, h q . (101) ecent developments in 2d N “ p , q supersymmetric gauge theories We note that one can do something more interesting, while we do not explorethis possibility further here. For Γ whose second group cohomology H p Γ , U p qq isnon-trivial, we can consider a theory with discrete torsion which can be identifiedwith a choice of an element γ of this cohomology group. Once the torsion is turnedon, the torus partition function is given by Z T “ | Γ | ÿ gh “ hg (cid:15) γ p g, h q Z T p g, h q (102)for some non-trivial phases (cid:15) γ p g, h q . From the gauge theory point of view, this isequivalent to coupling the G cont gauge theory with global symmetry Γ to a non-trivial topological field theory. Now let us consider two simple examples of gauge theories that flow to a Calabi-Yau threefold in the IR, both Z orbifolds of the quintic threefold theory, at specialpoints in the complex structure moduli space where the theory has a Z symmetry.Both theories have G “ U p q ˆ Z (103)as their gauge group, with six chiral multiplets as matter: Φ i with U p q charge 1and P with U p q charge ´
5, only differing in their charges under Z . The charges q i of the chiral multiplets under Z for the two theories, which we denote (ii)-Aand (ii)-B, are listed in table 1. Now the superpotential of these theories must befurther restricted to be invariant under the given Z symmetry—this is equivalentto moving to a point in complex-structure moduli space where the threefold has theappropriate Z isometry. Table 1. Charges of chiral fields of theory(ii)-A and (ii)-B under Z .Φ Φ Φ Φ Φ P (ii)-A 0 1 2 3 4 0(ii)-B 0 0 0 1 4 0 Now in the geometric phase of the IR theory, the action of the Z group in the IRbecomes an orbifolding action of the target manifold. The Euler numbers of the CYmanifolds obtained by orbifolding the quintic by the actions of table 1 can be foundin Ref. 97, among other places. We note that for (ii)-A, the action is free, i.e., doesnot have any fixed points, and the resulting theory does not have a twisted sector.This means that all the states of the orbifold theory can be obtained by projectingthe states of the quintic theory down to Z -invariant subspace. This results in theEuler number becoming a fifth of the Euler number of the quintic, i.e., χ E “ ´ χ E “ ´ Daniel S. Park
These Euler numbers can be nicely computed by the torus partition function,using the formula (101). Γ being abelian, the torus partition function can be writtenas Z T “ ÿ r,s “ Z T p e πir { , e πis { q“ ÿ r,s “ ÿ u ˚ P r T ˚ sing JK-Res u “ u ˚ r Q p u ˚ q , η s ω r,sT p u q . (104)We can readily compute this partition function for theories (ii)-A and (ii)-B.For (ii)-A, ω r,sT is given by ω r,sT “ πη p q q θ p q, y ´ q θ p q, x ´ q θ p q, yx ´ q ź m “ θ p q, y ´ xe πim p s ´ rτ q{ q θ p q, xe πim p s ´ rτ q{ q du (105)We now show that Z T p e πir { , e πis { q “ p r, s q ‰ p , q . For such p r, s q , let us define ζ “ e πi p s ´ rτ q{ . Now taking η “ x “ ζ ´ m for m “ , ¨ ¨ ¨ ,
4. The JK residue at the pole x “ ζ ´ m , however, is given by θ p q, ζ ´ m q θ p q, yζ ´ m q ź ď m p‰ k qď θ p q, y ´ ζ m ´ k q θ p q, ζ m ´ k q “ , (107)since ζ “ q ´ r , θ p q, q rm q “ θ p q, ζ m ´ k q are all non-zero when p r, s q ‰ p , q . Hence we find that Z (ii)-A T “ Z quintic T , (108)consistent with the claim that the twisted sectors of theory (ii)-A should be empty.Thus it is easy to see that Z (ii)-A T | q Ñ “ Z (ii)-A T | q Ñ “ ´ p y ´ { ` y { q (109)reproducing the Euler number χ E “ ´
40 for the orbifold.Theory (ii)-B, on the other hand, has states in the twisted sector. ω r,sT for thistheory is given by ω r,sT “ πη p q q θ p q, y ´ q ¨ θ p q, x ´ q θ p q, yx ´ q ¨ ˆ θ p q, y ´ x q θ p q, x q ˙ ¨ θ p q, y ´ xe πi p s ´ rτ q{ q θ p q, xe πi p s ´ rτ q{ q ¨ θ p q, y ´ xe πi p s ´ rτ q{ q θ p q, xe πi p s ´ rτ q{ q du . (110)We may take η “ ´ x “ e πi p k ` lτ q{ of the second factor of equation (110) for each of the 25 twisted partition functions.Taking the q Ñ Z (ii)-B T | q Ñ “ ´ p y ´ { ` y { q , (111) ecent developments in 2d N “ p , q supersymmetric gauge theories reproducing the Euler number χ E “ ´
6. The Ω-deformed sphere
Let us move on to describing the equivariant A -twisted sphere, or the Ω-deformedsphere. This background was studied mainly in Ref. 33, but could be obtainedby dimensionally reducing a supersymmetric background on S ˆ S along the S direction. We follow the exposition of Ref. 33.The supersymmetric background is given by the expectation values ds “ g z ¯ z p| z | q dzd ¯ z , A µ “ ω µ , H “ (cid:15) Ω (cid:15) µν B µ V ν , r H “ , (112)where ω µ is the spin connection of the metric and V µ is defined to be the Killingvector V µ “ iz B z ´ i ¯ z B ¯ z (113)of the U p q isometry. The metric can be any smooth metric with the isometerygenerated by V µ . The localizing supercharge on this background squares to thegenerator for the action of the isometry, (cid:15) Ω being the equivariant parameter. Thusthis background is the two-sphere analogue of the omega deformation
12, 13 in fourdimensions. Note that the background U p q gauge field coupling to the R -charge hasunit magnetic flux through the sphere. Thus, in order to couple a theory consistentlyto the background, all the fields of the theory must have integer R -charge.We take the localizing action to be the standard gauge and chiral kinetic terms. t The saddle of the action is simple in the zero-flux sector—it is given by setting σ “ r σ to a constant real value with vanishing field strength 2 if . Meanwhile,explicit expressions for the supersymmetric field configuration of the saddle points insectors with non-zero gauge flux have not been obtained. Nevertheless, assuming theexistence of such saddles, just enough information to compute the partition functioncan obtained by utilizing supersymmetry and index theorems. In particular, it canbe shown that the bosonic zero modes, which can be identified as the coordinatesof the moduli space of saddle points, are given by p σ a “ p σ a q S ` p σ a q N , (114)where the subscript denotes the value of the field at the south or north pole of thesphere. By supersymmetry, it can be shown that the saddles parametrized by p σ a ina given flux sector satisfies p σ a q N “ p σ a ´ (cid:15) Ω m a , p σ a q S “ p σ a ` (cid:15) Ω m a , (115) t As in the case of the round sphere, one may choose a localizing action that localizes to a Higgsbranch locus. Daniel S. Park where the m a are the fluxes of the Cartan elements of the gauge group:12 π ż e e ¯1 p´ if q “ m a T a . (116)Recall that we may turn supersymmetric vacuum expectation values of the vec-tor multiplets that couple to the flavor symmetries of the theory. We may thus turnon supersymmetric field configurations of the sigma fields and the gauge fields inthe flavor vector multiplets parametrized by s F and m F so that p s F q N “ s F ´ (cid:15) Ω m F , p s F q S “ s F ` (cid:15) Ω m F , π ż p da F q “ m F , (117)where p s F q S,N are used to denote the vacuum expectation values of the sigma fieldof the flavor vector multiplet at the south and north poles. We note that unlikein the case of the round sphere, s F may be any complex number. Setting m F “ ÿ m ż d p σ d rp σ d p λ d rp λ d p D Z m p p σ, rp σ, p λ, rp λ, p D q (118)where p p σ, rp σ, p λ, rp λ, p D q form a zero-mode multiplet, and the sum over magnetic fluxes m is taken. Due to the fact that Z m is invariant under the relevant supersymmetries,it can be shown that the integrals, for each m , reduces to a holomorphic integral for p σ a over a middle dimensional contour in C rk p G q .The partition function of a gauge theory coupled to this background genericallyvanishes. One can nevertheless insert operators at the poles of the sphere to com-pute expectation values or correlators of operators. While more general operatorspreserving the supersymmetry can be constructed, we consider the correlation func-tions of gauge invariant operators constructed using the sigma fields of the vectormultiplet. For example, for the U p N q theory, the ring of these operators is generatedby tr σ k , k “ , ¨ ¨ ¨ , N . (119)We note that any such operator O can be written as a polynomial of the eigenvalues σ a of σ that is invariant under the Weyl group of the gauge group. We often write O p σ a q to denote the polynomial corresponding to the operator O .The result of the path integral, with operators O and O inserted at the northand south poles, is given by the weighted sum of the JK residues x O ˇˇ N O ˇˇ S y (cid:15) Ω “ p´ q N ˚ | W | (cid:15) Ω ´ d grav ¨ ÿ m e πit s p m q (cid:15) Ω b p m q ÿ p σ ˚ P Ă M m sing JK-Res p σ “ p σ ˚ r Q p p σ ˚ q , ξ UVeff s I m p O , O q , (120) ecent developments in 2d N “ p , q supersymmetric gauge theories of the differential form I m p O , O q “ (cid:15) Ωrk p G q ¨ O p p σ N q ¨ O p p σ S q d p σ ^ ¨ ¨ ¨ ^ d p σ rk p G q ¨ ź α ą ˆ α p p σ N q α p p σ S q (cid:15) Ω2 ˙ ź i ź ρ P Λ R i Γ ´ ρ p p σ N q` s i, N (cid:15) Ω ` r i ¯ Γ ´ ρ p p σ S q` s i, S (cid:15) Ω ´ r i ` ¯ . (121)Here we have used the packaged variables p σ N “ p σ ´ (cid:15) Ω m , p σ S “ p σ ` (cid:15) Ω m ,s F, N “ s F ´ (cid:15) Ω m F , s F, S “ s F ` (cid:15) Ω m F . (122)and s i, N { S “ q iF s F, N { S inspired by the saddles (115) and (117). As before, the sumof m is taken over the GNO quantized magnetic fluxes, while the product over α istaken over all positive roots. d grav , defined by d grav “ ´ dim p G q ´ ÿ i p r i ´ q dim R i , (123)coincides with the complex dimension of the target space when the gauge theoryflows to an NLSM in the IR. N ˚ is an integer, whose determination we do not getinto here, while t s P c ˚ C is the complexified FI parameter shifted by a multiple of1 {
2, which amounts to the shift of the theta-angle by a multiple of π : t s ” t ` ÿ α ą α p mod h ˚ Z q . (124)Some explanation is due regarding the evaluation of equation (120). Ă M m sing de-notes the codimension-rk p G q singularities of the integrand I m on C rk p G q . The sin-gularities lie where s ě rk p G q hyperplanes H I ,k m “ ! p σ : Q I p p σ N q ` q I F s F, N ` r I (cid:15) Ω “ ´ k(cid:15) Ω ) , (125)for integers k with 0 ď k ď Q I p m q ` q I F m F ´ r I (126)intersect. The indices I and the variables defining the hyperplane equations aredefined in equations (76) and (77). For each such singular point p σ ˚ , we define, asbefore, the set of associated charges Q p p σ ˚ q “ t Q I , ¨ ¨ ¨ , Q I s u . (127)Now the JK residues at the codimension-rk p G q poles may be evaluated by the choiceof a JK vector. An important difference between this sphere partition function andthe torus partition function is that this choice matters—it must be chosen to takethe value ξ UVeff “ ξ ` π b log R , R " , (128) Daniel S. Park where b is defined in equation (13). When the IR fixed point is conformal with b “
0, the meaning of ξ UVeff is clear. To explain equation (128) for b ‰
0, we mustremind ourselves that the choice of the JK vector, at the end of the day, is choosingan rk p G q -dimensional chamber C ξ,b among the chambers separated by cones ofdimension ă rk p G q spanned by the charges Q I . Equation (128) instructs that C ξ,b should be chosen such that D R ą ξ ` π b log R P C ξ,b @ R ą R . (129)As in the torus partition function, the formula (120) is not well-defined in thepresence of non-projective singularities. The usual prescription of dealing with suchcases—deforming the theory by some twisted masses to resolve the singularities andtaking the limit where the masses vanish—applies here as well. Meanwhile, it maybe the case that ξ UVeff P i c ˚ lies squarely on a lower-dimensional cone spanned bythe charges. In this case, one should slightly deform the JK vector, possibly to liein i h ˚ z i c ˚ , and evaluate the formula.Let us now write out correlators x σ n ˇˇ N y (cid:15) Ω for theory (i). The integration measurefor the correlator is given by I m “ d ˆ p σ(cid:15) Ω ˙ ¨ ˆp σ ´ (cid:15) Ω m ˙ n ¨ $’’’&’’’%ś m p “ ´ p σ(cid:15) Ω ´ m ` p ¯ ´ N f m ě m “ ´ ś ´ m ´ p “ ´ p σ(cid:15) Ω ` m ` p ¯ N f m ě ´ m , which is now just an integer. We see the integration measuredoes not have any poles when m ă b “ N f ą
0. Since all the charges of thematter are given by Q I “ ą
0, this means we need to sum over all the poles ofthe integrand I m . After introducing the variable x “ p p σ { (cid:15) Ω ´ m { q , we arrive at theformula: x σ n ˇˇ N y (cid:15) Ω “ (cid:15) Ω1 ´ N f ÿ m “ q m (cid:15) Ω N f m m ÿ (cid:96) “ Res x “´ (cid:96) (cid:15) Ω n x n ś m p “ p x ` p q N f “ ´ (cid:15) Ω1 ´ N f ÿ m “ q m (cid:15) Ω N f m Res x “8 (cid:15) Ω n x n ś m p “ p x ` p q N f . (131)We can explicitly evaluate the residues to obtain x σ n ˇˇ N y (cid:15) Ω “ $’’&’’% n ď N f ´ n “ N f ´ n “ N f . (132) ecent developments in 2d N “ p , q supersymmetric gauge theories For larger n , we may use the following identity x σ N f f p σ q ˇˇ N y (cid:15) Ω “ ´ (cid:15) Ω1 ´ N f ÿ m q m (cid:15) Ω N f Res x “8 (cid:15) Ω N f x N f f p (cid:15) Ω x q ś m p “ p x ` p q N f “ ´ q(cid:15) Ω1 ´ N f ÿ m q p m ´ q (cid:15) Ω N f p m ´ q Res x “8 f p (cid:15) Ω x ´ (cid:15) Ω q ś m ´ p “ p x ` p q N f “ q x f p σ ´ (cid:15) Ω q ˇˇ N y (cid:15) Ω , (133)to compute the expectation values, where we shifted the variable x Ñ x ´
6, 98, 99 of CP N f ´ , as is discussed further in section 8.2.The correlators for theory (ii), the quintic GLSM can be similarly written, wherewe set ourselves in the geometric phase of the theory ξ ą x σ n ˇˇ N y (cid:15) Ω “ (cid:15) Ω n ´ ÿ m “ Res x “8 ś m j “ p´ x ´ j q ś m p “ p x ` p q x n . (134)We now write down correlators for theory (iii). The formulae being quite long,we set (cid:15) Ω “
1, and use the packaged variables extensively. We define σ a, N “ p σ a ´ m a , σ a, S “ p σ a ` m a , Σ F, N “ s F ´ ˆ m F ´ r F ˙ , Σ F, S “ s F ` ˆ m F ´ r F ˙ , q Σ A, N “ q s A ´ ˆ q m A ` r A ˙ , q Σ A, S “ q s A ` ˆ q m A ` r A ˙ , (135)and the traces Σ N { S “ ÿ a σ a, N { S . (136)As in the case of the round sphere partition function, we also introduce the followingdifferences: Σ ab, N { S “ σ a, N { S ´ σ b, N { S , Σ aF, N { S “ σ a, N { S ´ Σ F, N { S , Σ aA, N { S “ σ a, N { S ´ q Σ A, N { S , Σ F F , N { S “ Σ F , N { S ´ Σ F , N { S , Σ FA, N { S “ Σ F, N { S ´ q Σ A, N { S , Σ A A , N { S “ q Σ A , N { S ´ q Σ A , N { S . (137)As explained before, the operators we concern ourselves with can be expressed asWeyl-invariant polynomials of the eigenvalues of the sigma fields. In the case of the U p N q theory, these are none other than the symmetric polynomials of N variables.The correlators then can be computed to give x O ˇˇ N O ˇˇ S y (cid:15) Ω “ p´ q N ˚ N ! ÿ m P Z N e πit s p Σ S ´ Σ N q ÿ p σ ˚ P Ă M m sing JK-Res p σ “ p σ ˚ r Q p p σ ˚ q , ξ UVeff s I m , (138) Daniel S. Park where the integration measure I m is given by I m “ N ľ a “ d p σ a ¨ O p Σ a, N q ¨ O p Σ a, S q¨ ź a ă b ` Σ ab, N Σ ab, S ˘ N ź a “ ˜ N f ź F “ Γ p Σ aF, N q Γ p Σ aF, S ` q N a ź A “ Γ p´ Σ aA, N q Γ p´ Σ aA, S ` q ¸ . (139)Taking the FI parameter ξ to be positive, we find that the poles picked up bythe contour integral are located atΣ a, N “ S F a , N ´ n a, N , Σ a, S “ S F a , S ` n a, S (140)for some (cid:126)F P C p N, N f q for non-negative integers n a, N { S . At the end of the day, theintegral factorizes, much like the round sphere partition function, into the form ÿ (cid:126)F P C p N,N f q Z (cid:126)F Z (cid:126)F , O N Z (cid:126)F , O S . (141)The functions Z (cid:126)F , O N { S are related to the vortex partition functions defined in equation(62) by Z (cid:126)F , O N “ Z (cid:126)F , O v p´ Σ F, N ; ´ q Σ A, N , p´ q N f ` N q q , Z (cid:126)F , O S “ Z (cid:126)F , O v p Σ F, S ; q Σ A, S , p´ q N f ` N a q q . (142)We note that analogous results in higher dimensions have been obtained in Refs.100, 101.The expectation values (120) reproduce the expectation values of operators inthe A -twisted theory when (cid:15) Ω is taken to zero:lim (cid:15) Ω Ñ x O ˇˇ N O ˇˇ S y (cid:15) Ω “ x O O y A . (143)This is evident from the supersymmetric background (112). This computation canbe viewed as the Coulomb-branch counterpart of the Higgs-branch computation ofthe A -twisted correlators carried out in Ref. 8. More discussion on these correlatorsfrom the geometric point of view is presented in section 8.2.The interpretation of the correlators on the Ω-deformed sphere remains myste-rious from the field theoretic point of view. The operation of composing operatorsat the poles becomes non-associative in the presence of (cid:15) Ω , which is evident, forexample, in equation (133) for correlators in the CP N f ´ model. While one maywonder if this has to do with the fact that operators constructed out of the sigmafields preserve supersymmetry only when they are placed at the poles, no clearphysical picture of the supersymmetric operators has been given yet. u It would bedesirable to gain an understanding of these correlators based on a solid frameworkcomparable to that of the A -twisted theory. u Some hints on the nature of these correlators exist in the literature, for example, in the discussionabout gravitational descendant invariants in chapters 26-30 of Ref. 102. ecent developments in 2d N “ p , q supersymmetric gauge theories
7. More backgrounds
Before moving on to applications of localization computations on supersymmetricbackgrounds, let us give a brief summary of backgrounds and partition functionsthat we have not been able to review in detail.We begin with the hemisphere partition function computed in Refs. 28–30. Sincethe hemisphere has a boundary, additional data living at the boundary must beintroduced in addition to the supersymmetric background specified in the bulk. Theappropriate data turns out to be a Z -graded hermitian Chan-Paton vector space,and a certain polynomial function related to the superpotential of the theory. Upon localizing the supersymmetric gauge theories introduced in section 2, thehemisphere partition function turns out to be a function of the FI parameters, thetwisted masses, and the Chan-Paton data. In fact, the Chan-Paton data specifiesa B -brane B , while the partition function itself is conjectured to computethe central charge of that brane. This central charge can be understood as anoverlap between the canonical Ramond-Ramond (RR) ground state and the RRstate corresponding to the brane: Z D p B q “ RR x B | y RR . (144)When the gauge theory flows to a Calabi-Yau manifold, the B -branes can be thoughtof as D-branes wrapping holomorphic cycles of the manifold in the large-volumelimit. Upon choosing a suitable basis of branes, both the round sphere and Ω-deformed sphere partition functions can be written as a weighted sum over a productof hemisphere partition functions,
29, 30 i.e., the hemisphere partition functions canbe thought of as building blocks for sphere partition functions. Hemisphere partitionfunctions have also been used to confirm the role of the gamma class incomputing the central charge of B -branes. v Meanwhile, the RP partition function, when the IR theory of the gauge theoryis a sigma model into a Calabi-Yau manifold, can be interpreted as the central chargeof orientifold planes in the large-volume limit: Z RP p C q “ RR x C | y RR . (145)Since RP is an unoriented manifold, there is no sum over fluxes when computingthe partition function. The fundamental group of RP , however, is nontrivial—itis Z . Thus the RP partition function must be computed by summing over the Z valued holonomies of the gauge fields. Upon choosing the appropriate weightbetween the distinct topological sectors, one can compute the crosscap amplitude,or the central charges of space-time filling orientifolds for Calabi-Yau manifolds inthe large-volume limit. When the gauge theory has a Z -valued flavor symmetry, aholonomy with respect to such a symmetry may be turned on along the Z element ofthe fundamental group of the RP . When the theory flows to an NLSM in the IR, theflavor symmetry implies the existence of a Z isometry of the target manifold. The v See also Refs. 31, 113, 114.6
Daniel S. Park partition function with the flavor holonomy activated then turns out to computethe central charge of lower dimensional orientifold planes wrapping submanifoldslocated along the fixed points of the corresponding Z isometry.The A -twisted partition function and correlators of a gauge theory on a closed,orientable Riemann surface Σ g of genus g ě w Thegenus- g partition function of the A -twisted theory can be understood as a g -pointfunction of the handle-operator which has been computed for gauge theories, forexample, in Ref. 115. The localization computation correctly reproduces this result.
8. Applications
Up to now, we have described various supersymmetric backgrounds that may beutilized to compute exact correlation functions of N “ p , q gauge theories. Whilethe fact that we are able to compute expectation values of gauge theory observablesexactly is satisfying in and of itself, it has further reaching physical and mathe-matical applications. While we mainly focus on applications of of supersymmetriclocalization to the study of 2d dualities (section 8.1), and to quantum cohomol-ogy (section 8.2), we have collected other important applications and point to therelevant literature in section 8.3. Dualities
Duality refers to either the equivalence of two different Lagrangian theories underthe map of their parameters, or the equivalence of their subsectors. In this section,we concern ourselves with infra-red dualities, which implies the equivalence of the IRfixed points, or even the IR effective theories of two distinct Lagrangian theories. Theexact partition function or correlators of supersymmetric gauge theories can be usedto confirm such dualities. While there are many dualities of two-dimensional N “p , q gauge theories that have been proposed and studied,
4, 23, 57, 58, 61, 62, 116, 117 wechoose to focus on Hori-Vafa duality and cluster dualities of quiver theories withunitary gauge group factors in this section. We also briefly touch upon dualities oftheories with adjoint matter at the end of the subsection.Hori-Vafa duality refers to the equivalence between gauge theories with chiralmatter and Landau-Ginzburg (orbifold) models of twisted chiral fields. More pre-cisely, it refers to the duality between a gauge theory, specified by the data givenin section 2, with a theory of twisted chiral fields with the following data: ‚ Σ, with bottom component σ , is a twisted chiral field valued in the Cartansubalgebra h C of the gauge algebra g of the original theory. We use thenotation tr I σ to denote the projection of σ to the element I of c C P h C . w The 2d localization formula of Ref. 34 can be obtained by dimensionally reducing the partitionfunction of a three-dimensional gauge theory on Σ g ˆ S . The 3d computation of Ref. 34 alsoappears in Ref. 50. ecent developments in 2d N “ p , q supersymmetric gauge theories ‚ For each chiral field Φ i in the original theory, there is a corresponding setof twisted chiral fields Y i,ρ , with bottom component y i,ρ , labeled by theweights ρ of R i . Y i,ρ are periodic, i.e., Y i,ρ „ Y i,ρ ` πi . ‚ The twisted superpotential is given by x W p σ, y q “ ÿ I t I tr I σ ´ i π ÿ a σ a ÿ i ÿ ρ P Λ R i ρ a y i,ρ ´ i π ÿ i ÿ ρ P Λ R i e ´ y i,ρ (146) ‚ The Weyl group W of G is a discrete global symmetry of the Landau-Ginzburg theory of Σ and Y . When W is non-trivial, it is gauged.Here we have ignored possible twisted masses and R -charges of the chiral fields inthe original theory, but they are straightforward to incorporate. Hori-Vafa dualityis the statement that correlators of the original gauge theory is reproduced bythe correlators of this Landau-Ginzburg theory with an additional insertion of theoperator ź α ą | α p σ q| , (147)to the path integral, where α runs over the positive roots of g . Note that whilethe algebra g is used in defining this Landau-Ginzburg theory, it is not a gaugesymmetry of the theory.Now the round sphere partition function of the Landau-Ginzburg theory can becomputed to be compared with that of the gauge theory. The partition function oftwisted chiral fields, which we schematically denote by Y for the moment, localizeson saddles where the twisted chiral fields take constant values: Z S “ ż dY dY e ´ π x W p Y q` π x W p Y q , (148)where our conventions slightly differ from the original reference. We can then com-pute the sphere partition function of the Landau-Ginzburg theory that should matchthat of the gauge theory—it is given by Z LGS “ | W | ż dσdσ ź α ą | α p σ q| e ´ πt p σ q` π ¯ t p ¯ σ q ¨ ź i ź ρ P Λ R i ˆż dy R ż π ´ π dy I e iρ p σ R q y R ` iρ p σ I q y I e ie ´ y R sin y I ˙ , (149)where the factor of 1 {| W | in the front of the equation is due to the orbifoldingaction. We have introduced the subscripts R and I to denote the real and imaginarypart of the variables involved. Now note that unless ρ p σ I q are half-integers, the dy I integrals vanish. Thus we find that the imaginary part of the sigma fields must beGNO quantized: σ “ p σ ´ i m , ρ p m q P Z . (150) Daniel S. Park
The integral can now be written as a sum over GNO quantized fluxes: Z LGS “ | W | ÿ m ż dσdσ ź α ą | α p σ q| e ´ πiξ p p σ q` iθ p m q ¨ ź i ź ρ P Λ R i ˆż dy R ż π ´ π dy I e iρ p p σ q y R ´ iρ p m q y I e ie ´ y R sin y I ˙ . (151)The y integrals can be carried out explicitly to reproduce the sphere partition (45)exactly. While Hori-Vafa duality was proven for abelian gauge theories in the originalwork, the non-abelian case was best described as a conjecture except in a limitingnumber of examples. The sphere partition function computation provides strongevidence for it being true for gauge theories in general.Let us now discuss cluster duality, elements of which have appeared in Refs. 4,23,58,119. Cluster duality, some crucial components of were also discovered in Ref.62, is based on the Seiberg-like duality of U p N q theories with N f fundamentalsand N a anti-fundamentals, i.e., theory (iii).
4, 23
The claim is that theory (iii) is dualto theory (iv):(iv) U p N f ´ N q theory with N f antifundamental and N a fundamental matter.(a) Gauge group: U p N q with N “ N f ´ N .(b) Charged matter: N f antifundamental chiral fields Q F , labeled by F and N a antifundamental chiral fields q Q A , labeld by A . The U p q R charges are given by 1 ´ r F and 1 ´ q r A .(c) There is a single chiral meson M , that transforms as a fundamental inthe U p N f q subgroup and as an antifundamental in the U p N a q subgroupof the flavor symmetry. When generic twisted masses and R -chargesare assigned, the flavor symmetry group breaks up into U p q N f ` N a ´ and the meson breaks up into N f ˆ N a massive chiral fields.(d) The superpotential is given by W “ W p M AF q ` ÿ F,A M AF Q F q Q A , (152)where W is the superpotential of theory (iii), with the gauge invariantmesons q Q A Q F of theory (iii) replaced by the singlets M AF .(e) The twisted superpotential of the theory is given by x W “ t tr σ ` « ˆ t ` N ˙ ÿ F s F ` N ÿ A q s A ` πi δ N f ,N a ln p ` z q ˜ ´ ÿ F s F ` ÿ A q s A ¸ ff , (153)for t “ ´ t ` N a {
2, where it is useful to recall that the twisted masseslie within a background vector multiplet. We remind the reader that wealways assume that N f ě N a . We have ignored various contact terms ecent developments in 2d N “ p , q supersymmetric gauge theories that do not depend on the (dynamical/background) vector multiplets.Here we have defined a convenient parameter z “ e iπ p N f ´ N q e πit (154)to make the equation simpler.Much of the data regarding the duality can be succinctly captured by a quiverdiagram. A quiver diagram is made up of circular and square nodes with inscribedpositive integers and directed edges which connect a pair of nodes. The nodes encodethe gauge and flavor symmetry group—the circular nodes with inscribed numbers N p stand for the U p N p q gauge group factors, while the squares stand for flavorsubgroups. Meanwhile, each edge corresponds to bifundamental matter, that is inthe fundamental representation of the group at the tail, and an antifundamentalrepresentation of the group at the head. To each gauge node, we also associatea complex number z that encodes the FI parameter of the corresponding unitarygauge group factor. The type of gauge theories whose gauge/matter content can beencoded into such a diagram is called a quiver gauge theory . The duality betweentheories (iii) and (iv) can then be expressed by figure 1.The bracketed terms of equation (153), which are dependent on the twistedmasses, have a surprising effect when the background flavor vector fields are pro-moted to dynamical gauge fields, i.e., when a subgroup of the flavor group is pro-moted to a gauge group. This happens when one takes a quiver gauge theory withunitary gauge group factors and dualizes the theory with respect to a gauge node,which we denote p for now. The bracketed terms in equation (153) amount to shift-ing the FI parameters of the neighboring nodes of node p by a function of e πit p ,where t p is the complex FI parameter of U p N p q . Needless to say, from the rules wehave learned from the Seiberg-like duality of the U p N q theory, the dual quiver theoryshould have a different rank, and a different quiver. Quite surprisingly, the rules formutating the quiver and modifying the various FI couplings by this duality map, ex-plained in all their glory in Ref. 61, has been studied in detail in mathematics—theyare precisely the mutation rules studied in cluster algebras, originally formulatedby Fomin and Zelevinsky. The bracketed terms in equation (153) were found in Refs. 61, 62 by examiningthe sphere partition function of theory (iii). Recall from section 4 that the sphere NN f N a z 1/zN'N f N a Fig. 1. The quiver diagram of the dual theories (iii) (left) and (iv) (right).0
Daniel S. Park partition function of theory (iii) can be expressed as ÿ (cid:126)F P C p N,N f q Z (cid:126)F Z (cid:126)F ` Z (cid:126)F ´ , Z (cid:126)F ` “ Z (cid:126)F v p Σ F ` ; q Σ A ` ; e πit q , (155)where the vortex partition function Z (cid:126)F v “ Z (cid:126)F , O “ is defined via equation (62).Now we can also write the sphere partition function of theory (iv) but without theadditional terms proportional to the twisted masses in equation (153) by ÿ (cid:126)F c P C p N,N f ´ N q Z (cid:126)F c Z (cid:126)F c ` Z (cid:126)F c ´ , Z (cid:126)F c ` “ Z (cid:126)F v p ´ Σ F ` ; ´ ´ q Σ A ` ; p´ q N a e πit q , (156)where the ordered tuple (cid:126)F c , as a set, is the complement of (cid:126)F with respect to r N f s : t F cI u “ r N f szt F I u . (157)While the perturbative pieces satisfy the relation Z (cid:126)F ” e πi ´´ t ` N ¯ ř F Σ F ` ` N ř A q Σ A ` ¯ e ´ πi ´´ ¯ t ` N ¯ ř F Σ F ´ ` N ř A q Σ A ´ ¯ Z (cid:126)F c (158)up to an overall common factor independent of the twisted masses involved, thevortex partition functions Z (cid:126)F v and Z (cid:126)F c v satisfy the relation: Z (cid:126)F ` “ Z (cid:126)F c ` ˆ $’’&’’% N f ě N a ` e ´ z N f “ N a ` p ` z q ´ ř F Σ F ` ` ř A q Σ A ` `p N f ´ N q N f “ N a . (159)The relative factors in (158) and (159) are precisely accounted for by the bracketedterms of equation (153) in the dual of the U p N q theory.The cluster dualities are IR dualities in a stronger sense in that the effectivetheory of the intermediate IR regimes of the dual theories, as well as their fixedpoints, are equivalent. In particular, when a unitary quiver gauge theory flows toan NLSM at an intermediate IR scale, the duality manifests itself as an equivalencebetween distinct constructions of the same target space. For example, the duality ofthe U p N q theory has an interpretation as the canonical isomorphism of the Grass-mannian.
2, 3, 122
When the theory is conformal, the duality rules indicate how thecoordinates of the quantum-corrected K¨ahler moduli space of a Calabi-Yau mani-fold are mapped under such equivalences. Cluster dualities of quiver theories alsohave been approached from the point of view of the gauge/Yang-Baxter equationcorrespondence using the torus partition function in Refs. 124, 125.Dualities of U p N q gauge theories with adjoint matter have also been exploredusing round sphere partition functions. Duality of N “ p , q ˚ theories, which are N “ p , q theories broken by a twisted mass, have been studied in Refs. 61, 62.Meanwhile, Kutasov-Schwimmer-like dualities of theories with adjoint matter canalso be verified by similar methods. These dualities have a beautiful applicationin the context of the famed Alday-Gaiotto-Tachikawa (AGT) correspondence, as ecent developments in 2d N “ p , q supersymmetric gauge theories the sphere partition function of certain N “ p , q theories can be identified withcorrelation functions of certain conformal field theories on a Riemann surface. Webriefly discuss this point in section 8.3. Geometric applications
In may instances, the N “ p , q gauge theories of study flow to non-linear sigmamodels of a K¨ahler manifold, which we denote X throughout this section, in theIR. The various partition functions and correlators compute geometric quantitiesof the target space X of the IR theory. The most basic example of this is the toruspartition function. As reviewed in section 5, the Euler character of the target spacegeometry of the IR theory is encoded in the torus partition function.As noted earlier, when the gauge theory flows to an NLSM of a Calabi-Yau man-fold, the round sphere partition function computes the quantum K¨ahler potentialof the twisted chiral conformal manifold of the theory:
20, 51, 52 Z S “ e ´ K p t I , ¯ t I q . (160)This has rather profound geometric implications. In particular, when the Calabi-Yau manifold happens to be complex-three-dimensional, all the genus-zero Gromov-Witten invariants,
6, 66, 67 which “count” pseudo-holomorphic curves of given degree,can be extracted from this partition function. Recall that the exponentiated FIparameters q I “ e πit I (161)parametrize the K¨ahler moduli space of the target manifold X of the IR theory.These coordinates are often called “algebraic coordinates” of the K¨ahler modulispace. The coordinates q I are natural from the point of view of the gauge theory,as they are straightforwardly related to physical UV couplings, in which the variouspartition functions are readily expressed.Meanwhile, there is a separate set of coordinates on this moduli space, denotedthe “flat coordinates”, that are more appropriate to extracting the Gromov-Witten invariants of the manifold. These flat coordinates x I are related to q I bythe “mirror map” of the form x I “ log q I πi ` x I ` f I p q q (162)where f I is a holomorphic function of the algebraic coordinates. The constants x I and the functions f I can be extracted from the fact that the quantum K¨ahler Daniel S. Park potential, or the sphere partition function of the theory, is given by the form e ´ K p x I , ¯ x I q “ ´ i ÿ I,J,K κ IJK p x I ´ ¯ x I qp x J ´ ¯ x J qp x K ´ ¯ x K q ` ζ p q π χ p X q` i p πi q ÿ η N η ` Li p e πix ¨ η q ` Li p e ´ πi ¯ x ¨ η q ˘ ´ i p πi q ÿ η,I N η ` Li p e πix ¨ η q ` Li p e ´ πi ¯ x ¨ η q ˘ η I p x I ´ ¯ x I q (164)where η runs over the elements of H p X, Z q , χ p X q denotes the Euler character of X , and Li k p z q “ ÿ n “ z n n k . (165)The numbers N η are the integral genus-zero Gromov-Witten invariants labeled bythe homology class η . Thus, once the map (162) is established, it can be inverted towrite the quantum K¨ahler potential in the flat coordinates, from which the Gromov-Witten invariants can be extracted. Some applications of the round sphere partitionfunction in this context can be found in Refs. 131–134.As noted in section 6, the A -twisted correlation functions of operators in thetwisted chiral ring can be computed by taking the (cid:15) Ω Ñ A -twisted sphere. The operators studied in section 6, i.e., thegauge-invariant polynomials of sigma fields, can be identified as elements of the“vertical” cohomology dim X à n “ H n,n p X q (166)of the target manifold X , which forms a subset of the A -twisted operators of thenon-linear sigma model. The vector space of gauge-invariant polynomials of thesigma fields have a natural grading, which is the degree of the polynomials withrespect to the elements of σ . This grading can be identified with the grading n ofthe vertical cohomology of equation (166).The A -twisted correlators satisfy quantum cohomology ring relations, which isa deformation of the classical cohomology ring. To be concrete, let us consider thecase when X “ CP N f ´ . The classical cohomology of the theory is a ring generated x In order to arrive at the given formula, an appropriate frame must be chosen, or equivalently, aproduct of a holomorphic and antiholomorphic function of the q I coordinates must be multipliedto the sphere partition function: e ´ K “ F p q I q F p q I q Z S . (163)The choice of the appropriate function F p q I q requires the knowledge of the Euler character χ p X q ,which can be obtained by computing the torus partition function. ecent developments in 2d N “ p , q supersymmetric gauge theories by the hyperplane class, represented by the sigma field σ . Now σ , being a p , q form, must satisfy σ N f “ , (167) CP N f ´ being complex p N f ´ q dimensional. Thus the cohomology ring of CP N f ´ is given by Z r σ s{p σ N f q . In the quantum theory, however, this ring is deformed to Z r σ s{p σ N f ´ q q with q “ e πit .
6, 98, 99
This is realized in the A -twisted correlationfunctions: x σ N f ¨ f p σ qy A “ q x f p σ qy A , (168)which can be obtained from equation (133) by taking (cid:15) Ω Ñ
0. Here, f p σ q is anarbitrary polynomial of σ . The localization formulae for A -twisted correlation func-tions for Calabi-Yau GLSMs also reproduce classic results obtained by employingmirror symmetry or by direct counting of holomorphic curves. For example, forthe quintic GLSM, equation (134) reproduces the famous result x σ k y A “ ` q when k “
30 otherwise, (169)when (cid:15) Ω is taken to vanish.Meanwhile, the (cid:15) Ω Ñ A -twisted correlators of Calabi-Yau NLSMs that have not been computed before. Forexample, new correlation functions of operators when X is the Gulliksen-Neg˚ard(GN) manifold, which is a submanifold of P ˆ Gr p , q , have been obtained thisway. Let us explain this example in a little bit more detail. The GLSM for theGN manifold is a U p q ˆ U p q theory. Thus the sigma field can be written as σ “ σ ‘ σ with σ P u p q and σ P u p q , and there exist two algebraic K¨ahlercoordinates obtained by exponentiating the FI parameters, which we denote z and w . The vertical cohomology of P ˆ Gr p , q is generated by the elementstr σ , tr σ , tr σ , (170)where the subscript on the traces label the algebra with respect to which the traceis being taken. Note that the existence of the inherently non-abelian operator tr σ is linked to the fact that Gr p , q is not toric. While the vertical cohomology of P ˆ Gr p , q is bigger, the basis elements of the vertical cohomology of X Ă P ˆ Gr p , q are given by six elements:1 P H , p X q , p tr σ q ˇˇ X P H , p X q , tr σ ˇˇ X , tr σ ˇˇ X P H , p X q , p tr σ q ˇˇ X , p tr σ q ˇˇ X P H , p X q . (171)Here the notation “ ˇˇ X ” has been used to denote that the given cohomology classhas been pulled back to, or restricted to, X . Now since the operator tr σ flows toa four-form in the IR theory, it must be thattr σ ˇˇ X “ a p tr σ q ˇˇ X ` b p tr σ q ˇˇ X (172) Daniel S. Park for some coefficients a and b , which are dependent on the two algebraic coordinates z and w on the K¨ahler moduli space of X . These coefficients can be computedexactly by solving the linear equations a x tr σ ˇˇ X p tr σ q ˇˇ X y A ` b x tr σ ˇˇ X p tr σ q ˇˇ X y A “ x tr σ ˇˇ X tr σ ˇˇ X y A ,a x tr σ ˇˇ X p tr σ q ˇˇ X y A ` b x tr σ ˇˇ X p tr σ q ˇˇ X y A “ x tr σ ˇˇ X tr σ ˇˇ X y A . (173)All the correlations functions listed in this equation have been computed by takingthe (cid:15) Ω Ñ A -twisted sphere in Ref. 33,which thus leads to the values of a and b . To the author’s knowledge, this resulthas not been obtained before Ref. 33, as the computation of correlators involvingnon-abelian operators have only been performed for a limiting number of examplesbefore the recent advances in 2d N “ p , q localization techniques.Turning (cid:15) Ω on has an interesting effect. As discussed earlier, the most conspicu-ous is that the quantum cohomology ring undergoes a non-associative deformation—the existence of (cid:15) Ω renders the action of composition of fields, while still commu-tative, non-associative. A simple example is that the quantum cohomology relationof equation (168) is deformed into x σ N f f p σ q ˇˇ N y (cid:15) Ω “ q x f p σ ´ (cid:15) Ω q ˇˇ N y (cid:15) Ω , (174)as derived in section 6. The correlators for Calabi-Yau GLSMs also become moreinteresting once (cid:15) Ω is turned on. The correlators of the sigma fields for the quinticGLSM can be evaluated using equation (134): x σ k ˇˇ N y “ p k “ , , q , x σ ˇˇ N y “ ` q , x σ ˇˇ N y “ (cid:15) Ω ¨ q p ` q q , x σ ˇˇ N y “ (cid:15) Ω2 q p´ ` ¨ q qp ` q q , ... (175)The meaning of these correlators are not entirely clear from the geometric point ofview, although it seems sensible to conjecture that they are computing equivariantclasses of the moduli space of holomorphic maps from a two-punctured sphere to X . This moduli space has a natural C ˚ action, and (cid:15) Ω may be identified with theequivariant parameter with respect to this action. More discussions along theselines can be found in Ref. 139.
More applications
We conclude with listing and providing references for some important topics we didnot cover in the previous subsections.One place that N “ p , q gauge theories appear is as worldvolume theoriesof surface operators in four-dimensional theories with N “ y A related context in which 2d N “ p , q gauge theories appear in the study 4d N “ ecent developments in 2d N “ p , q supersymmetric gauge theories When the N “ G , an interestingclass of surface operators can be described by a gauge theory whose flavor currentis coupled to the four-dimensional dynamical gauge fields. The localizationtechniques discussed in this review have been used to compute the partition functionof these coupled 2d-4d systems on various backgrounds. The supersymmetric indexof the N “ S partition functions of N “ S Ă S have been computed in Refs. 62, 150. Bythe AGT correspondence, these partition functions have an interpretation ascorrelation functions of certain 2d conformal field theories on Riemann surfaces. Thevarious two-dimensional duality relations presented in section 8.1 can be interpretedas symmetries of these correlation functions. The gauge-Bethe correspondence
63, 64 is the correspondence between physicalobservables of certain 2d N “ p , q gauge theories and integrable systems. Theround sphere partition function and the A -twisted correlators of these gaugetheories have been studied and interpreted in this context.Supersymmetric partition functions have also been used to study gauge theoriesthat flow to NLSMs of unconventional geometries in the IR. Localization compu-tations have been carried out in Refs. 55, 56 for gauge theories with semi-chiralmultiplets, which flow to geometries with torsion in the IR. The spectrum ofstring states for ALE and ALF spaces have been studied using the torus partitionfunction of gauge theories with both chiral and twisted chiral matter in Ref. 54.The Gromov-Witten invariants of non-commutative resolutions of singular spaceshave been computed in Ref. 155 using the round sphere partition function.
Acknowledgments
I thank Francesco Benini, Cyril Closset, Stefano Cremonesi, Jaewon Song and PengZhao for collaborating on work presented in this review—most of what I know aboutthe subject has been gained through the experience of working with them. I shouldalso thank Allan Adams, Marcos Crichigno, Tudor Dimofte, Ethan Dyer, AbhijitGadde, Davide Gaiotto, Sergei Gukov, Kentaro Hori, Bei Jia, Peter Koroteev, VijayKumar, Josh Lapan, Jaehoon Lee, Sungjay Lee, Bruno Le Floch, Noppadol Meka-reeya, Dave Morrison, Nikita Nekrasov, Wolfger Peelaers, Martin Roˇcek, MauricioRomo, Eric Sharpe and Yuji Tachikawa for educating discussions on related topicsover the years. I would like to thank Francesco Benini, Noppadol Mekareeya, DaveMorrison and Jaewon Song again for helpful comments on the draft, and especiallyCyril Closset and Stefano Cremonesi for being kind and patient with me while Ihave riddled them with questions throughout the course of this work. I also thankthe Korea Institute for Advanced Study for hospitality while this work was beingcarried out. This work is supported by DOE grant DOE-SC0010008. theories can be found in Refs. 141–144. There, the 2d gauge theories are identified as effectivetheories of vortex strings in the Higgs branch of the 4d theories.6
Daniel S. Park
References
1. E. Witten,
Nucl. Phys.
B403 , 159 (1993), arXiv:hep-th/9301042 [hep-th] , doi:10.1016/0550-3213(93)90033-L.2. E. Witten (1993), arXiv:hep-th/9312104 [hep-th] .3. R. Donagi and E. Sharpe,
J. Geom. Phys. , 1662 (2008), arXiv:0704.1761[hep-th] , doi:10.1016/j.geomphys.2008.07.010.4. K. Hori and D. Tong, JHEP , 079 (2007), arXiv:hep-th/0609032 [hep-th] , doi:10.1088/1126-6708/2007/05/079.5. E. Witten, Commun. Math. Phys. , 353 (1988), doi:10.1007/BF01223371.6. E. Witten,
Commun. Math. Phys. , 411 (1988), doi:10.1007/BF01466725.7. E. Witten (1991), arXiv:hep-th/9112056 [hep-th] .8. D. R. Morrison and M. R. Plesser,
Nucl. Phys.
B440 , 279 (1995), arXiv:hep-th/9412236 [hep-th] , doi:10.1016/0550-3213(95)00061-V.9. E. Witten,
J. Geom. Phys. , 303 (1992), arXiv:hep-th/9204083 [hep-th] , doi:10.1016/0393-0440(92)90034-X.10. G. W. Moore, N. Nekrasov and S. Shatashvili, Commun. Math. Phys. , 97 (2000), arXiv:hep-th/9712241 [hep-th] , doi:10.1007/PL00005525.11. G. W. Moore, N. Nekrasov and S. Shatashvili,
Commun. Math. Phys. , 77 (2000), arXiv:hep-th/9803265 [hep-th] , doi:10.1007/s002200050016.12. N. A. Nekrasov,
Adv. Theor. Math. Phys. , 831 (2003), arXiv:hep-th/0206161[hep-th] , doi:10.4310/ATMP.2003.v7.n5.a4.13. N. Nekrasov and A. Okounkov, Prog. Math. , 525 (2006), arXiv:hep-th/0306238[hep-th] , doi:10.1007/0-8176-4467-9 15.14. G. Festuccia and N. Seiberg,
JHEP , 114 (2011), arXiv:1105.0689 [hep-th] ,doi:10.1007/JHEP06(2011)114.15. T. T. Dumitrescu, G. Festuccia and N. Seiberg, JHEP , 141 (2012), arXiv:1205.1115 [hep-th] , doi:10.1007/JHEP08(2012)141.16. T. T. Dumitrescu and G. Festuccia, JHEP , 072 (2013), arXiv:1209.5408[hep-th] , doi:10.1007/JHEP01(2013)072.17. A. Adams, H. Jockers, V. Kumar and J. M. Lapan, JHEP , 042 (2011), arXiv:1104.3155 [hep-th] , doi:10.1007/JHEP12(2011)042.18. C. Klare, A. Tomasiello and A. Zaffaroni, JHEP , 061 (2012), arXiv:1205.1062[hep-th] , doi:10.1007/JHEP08(2012)061.19. C. Closset and S. Cremonesi, JHEP , 075 (2014), arXiv:1404.2636 [hep-th] ,doi:10.1007/JHEP07(2014)075.20. J. Gomis, P.-S. Hsin, Z. Komargodski, A. Schwimmer, N. Seiberg and S. Theisen, JHEP , 022 (2016), arXiv:1509.08511 [hep-th] , doi:10.1007/JHEP03(2016)022.21. J. Bae, C. Imbimbo, S.-J. Rey and D. Rosa, JHEP , 169 (2016), arXiv:1510.00006[hep-th] , doi:10.1007/JHEP03(2016)169.22. V. Pestun, Commun. Math. Phys. , 71 (2012), arXiv:0712.2824 [hep-th] , doi:10.1007/s00220-012-1485-0.23. F. Benini and S. Cremonesi,
Commun. Math. Phys. , 1483 (2015), arXiv:1206.2356 [hep-th] , doi:10.1007/s00220-014-2112-z.24. N. Doroud, J. Gomis, B. Le Floch and S. Lee,
JHEP , 093 (2013), arXiv:1206.2606[hep-th] , doi:10.1007/JHEP05(2013)093.25. A. Gadde and S. Gukov, JHEP , 080 (2014), arXiv:1305.0266 [hep-th] , doi:10.1007/JHEP03(2014)080.26. F. Benini, R. Eager, K. Hori and Y. Tachikawa, Lett. Math. Phys. , 465 (2014), arXiv:1305.0533 [hep-th] , doi:10.1007/s11005-013-0673-y.27. F. Benini, R. Eager, K. Hori and Y. Tachikawa,
Commun. Math. Phys. , 1241 ecent developments in 2d N “ p , q supersymmetric gauge theories (2015), arXiv:1308.4896 [hep-th] , doi:10.1007/s00220-014-2210-y.28. S. Sugishita and S. Terashima, JHEP , 021 (2013), arXiv:1308.1973 [hep-th] ,doi:10.1007/JHEP11(2013)021.29. D. Honda and T. Okuda, JHEP , 140 (2015), arXiv:1308.2217 [hep-th] , doi:10.1007/JHEP09(2015)140.30. K. Hori and M. Romo (2013), arXiv:1308.2438 [hep-th] .31. H. Kim, S. Lee and P. Yi, JHEP , 103 (2014), arXiv:1310.4505 [hep-th] , doi:10.1007/JHEP02(2014)103.32. F. Benini and A. Zaffaroni, JHEP , 127 (2015), arXiv:1504.03698 [hep-th] , doi:10.1007/JHEP07(2015)127.33. C. Closset, S. Cremonesi and D. S. Park, JHEP , 076 (2015), arXiv:1504.06308[hep-th] , doi:10.1007/JHEP06(2015)076.34. F. Benini and A. Zaffaroni (2016), arXiv:1605.06120 [hep-th] .35. A. Kapustin, B. Willett and I. Yaakov, JHEP , 089 (2010), arXiv:0909.4559[hep-th] , doi:10.1007/JHEP03(2010)089.36. D. L. Jafferis, JHEP , 159 (2012), arXiv:1012.3210 [hep-th] , doi:10.1007/JHEP05(2012)159.37. N. Hama, K. Hosomichi and S. Lee, JHEP , 127 (2011), arXiv:1012.3512[hep-th] , doi:10.1007/JHEP03(2011)127.38. K. Hosomichi, R.-K. Seong and S. Terashima, Nucl. Phys.
B865 , 376 (2012), arXiv:1203.0371 [hep-th] , doi:10.1016/j.nuclphysb.2012.08.007.39. S. Kim,
Nucl. Phys.
B821 , 241 (2009), arXiv:0903.4172 [hep-th] , doi:10.1016/j.nuclphysb.2012.07.015,10.1016/j.nuclphysb.2009.06.025, [Erratum: Nucl.Phys.B864,884(2012)].40. V. Pestun,
JHEP , 067 (2012), arXiv:0906.0638 [hep-th] , doi:10.1007/JHEP12(2012)067.41. Y. Imamura and S. Yokoyama, JHEP , 007 (2011), arXiv:1101.0557 [hep-th] ,doi:10.1007/JHEP04(2011)007.42. N. Hama, K. Hosomichi and S. Lee, JHEP , 014 (2011), arXiv:1102.4716[hep-th] , doi:10.1007/JHEP05(2011)014.43. C. Krattenthaler, V. P. Spiridonov and G. S. Vartanov, JHEP , 008 (2011), arXiv:1103.4075 [hep-th] , doi:10.1007/JHEP06(2011)008.44. J. Gomis, T. Okuda and V. Pestun, JHEP , 141 (2012), arXiv:1105.2568[hep-th] , doi:10.1007/JHEP05(2012)141.45. A. Kapustin and B. Willett (2011), arXiv:1106.2484 [hep-th] .46. F. Benini, T. Nishioka and M. Yamazaki, Phys. Rev.
D86 , 065015 (2012), arXiv:1109.0283 [hep-th] , doi:10.1103/PhysRevD.86.065015.47. Y. Ito, T. Okuda and M. Taki,
JHEP , 010 (2012), arXiv:1111.4221 [hep-th] ,doi:10.1007/JHEP03(2016)085,10.1007/JHEP04(2012)010, [Erratum:JHEP03,085(2016)].48. L. F. Alday, M. Fluder and J. Sparks, JHEP , 057 (2012), arXiv:1204.1280[hep-th] , doi:10.1007/JHEP10(2012)057.49. C. Closset and I. Shamir, JHEP , 040 (2014), arXiv:1311.2430 [hep-th] , doi:10.1007/JHEP03(2014)040.50. C. Closset and H. Kim (2016), arXiv:1605.06531 [hep-th] .51. H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison and M. Romo, Com-mun. Math. Phys. , 1139 (2014), arXiv:1208.6244 [hep-th] , doi:10.1007/s00220-013-1874-z.52. J. Gomis and S. Lee,
JHEP , 019 (2013), arXiv:1210.6022 [hep-th] , doi:10.1007/JHEP04(2013)019. Daniel S. Park
53. N. Doroud and J. Gomis,
JHEP , 99 (2013), arXiv:1309.2305 [hep-th] , doi:10.1007/JHEP12(2013)099.54. J. A. Harvey, S. Lee and S. Murthy, JHEP , 110 (2015), arXiv:1406.6342[hep-th] , doi:10.1007/JHEP02(2015)110.55. J. Nian and X. Zhang, JHEP , 047 (2015), arXiv:1411.4694 [hep-th] , doi:10.1007/JHEP11(2015)047.56. F. Benini, P. M. Crichigno, D. Jain and J. Nian, JHEP , 060 (2016), arXiv:1505.06207 [hep-th] , doi:10.1007/JHEP01(2016)060.57. K. Hori and C. Vafa (2000), arXiv:hep-th/0002222 [hep-th] .58. K. Hori, JHEP , 121 (2013), arXiv:1104.2853 [hep-th] , doi:10.1007/JHEP10(2013)121.59. S. R. Coleman, Annals Phys. , 239 (1976), doi:10.1016/0003-4916(76)90280-3.60. A. D’Adda, P. Di Vecchia and M. Luscher,
Nucl. Phys.
B152 , 125 (1979), doi:10.1016/0550-3213(79)90083-X.61. F. Benini, D. S. Park and P. Zhao,
Commun. Math. Phys. , 47 (2015), arXiv:1406.2699 [hep-th] , doi:10.1007/s00220-015-2452-3.62. J. Gomis and B. Le Floch,
JHEP , 183 (2016), arXiv:1407.1852 [hep-th] , doi:10.1007/JHEP04(2016)183.63. N. A. Nekrasov and S. L. Shatashvili, Nucl. Phys. Proc. Suppl. , 91 (2009), arXiv:0901.4744 [hep-th] , doi:10.1016/j.nuclphysbps.2009.07.047.64. N. A. Nekrasov and S. L. Shatashvili,
Prog. Theor. Phys. Suppl. , 105 (2009), arXiv:0901.4748 [hep-th] , doi:10.1143/PTPS.177.105.65. C. Vafa, Topological mirrors and quantum rings, in
In *Yau, S.T. (ed.): Mirrorsymmetry I* 97-120 , (1991). arXiv:hep-th/9111017 [hep-th] .66. M. Gromov,
Invent. Math. , 307 (1985), doi:10.1007/BF01388806.67. M. Dine, N. Seiberg, X. G. Wen and E. Witten, Nucl. Phys.
B289 , 319 (1987),doi:10.1016/0550-3213(87)90383-X.68. M. F. Atiyah and R. Bott,
Topology , 1 (1984), doi:10.1016/0040-9383(84)90021-1.69. E. Witten, J. Geom. Phys. , 303 (1992), arXiv:hep-th/9204083 [hep-th] , doi:10.1016/0393-0440(92)90034-X.70. A. B. Givental, Internat. Math. Res. Notices , 613 (1996), doi:10.1155/S1073792896000414.71. B. Jia and E. Sharpe,
JHEP , 031 (2013), arXiv:1306.2398 [hep-th] , doi:10.1007/JHEP09(2013)031.72. P. Goddard, J. Nuyts and D. I. Olive, Nucl. Phys.
B125 , 1 (1977), doi:10.1016/0550-3213(77)90221-8.73. K. Hosomichi,
JHEP , 155 (2016), arXiv:1507.07650 [hep-th] , doi:10.1007/JHEP02(2016)155.74. D. S. Park and J. Song, JHEP , 142 (2013), arXiv:1211.0019 [hep-th] , doi:10.1007/JHEP01(2013)142.75. Y. Honma and M. Manabe, JHEP , 102 (2013), arXiv:1302.3760 [hep-th] , doi:10.1007/JHEP05(2013)102.76. S. Shadchin, JHEP , 052 (2007), arXiv:hep-th/0611278 [hep-th] , doi:10.1088/1126-6708/2007/08/052.77. N. Seiberg, Nucl. Phys.
B303 , 286 (1988), doi:10.1016/0550-3213(88)90183-6.78. V. Periwal and A. Strominger,
Phys. Lett.
B235 , 261 (1990), doi:10.1016/0370-2693(90)91961-A.79. S. Cecotti and C. Vafa,
Nucl. Phys.
B367 , 359 (1991), doi:10.1016/0550-3213(91)90021-O.80. A. B. Zamolodchikov,
JETP Lett. , 730 (1986), [Pisma Zh. Eksp. Teor. ecent developments in 2d N “ p , q supersymmetric gauge theories Fiz.43,565(1986)].81. L. C. Jeffrey and F. C. Kirwan,
Topology , 291 (1995), doi:10.1016/0040-9383(94)00028-J.82. M. Brion and M. Vergne, Ann. Sci. ´Ecole Norm. Sup. (4) , 715 (1999), doi:10.1016/S0012-9593(01)80005-7.83. A. Szenes and M. Vergne, Invent. Math. , 453 (2004), doi:10.1007/s00222-004-0375-2.84. K. Hori, H. Kim and P. Yi,
JHEP , 124 (2015), arXiv:1407.2567 [hep-th] , doi:10.1007/JHEP01(2015)124.85. C. Closset, W. Gu, B. Jia and E. Sharpe, JHEP , 070 (2016), arXiv:1512.08058[hep-th] , doi:10.1007/JHEP03(2016)070.86. K. Pilch, A. N. Schellekens and N. P. Warner, Nucl. Phys.
B287 , 362 (1987), doi:10.1016/0550-3213(87)90109-X.87. E. Witten,
Commun. Math. Phys. , 525 (1987), doi:10.1007/BF01208956.88. O. Alvarez, T. P. Killingback, M. L. Mangano and P. Windey,
Commun. Math. Phys. , 1 (1987), doi:10.1007/BF01239011.89. E. Witten,
Nucl. Phys.
B202 , 253 (1982), doi:10.1016/0550-3213(82)90071-2.90. E. Witten, The index of the Dirac operator in loop space, in
Elliptic curves andmodular forms in algebraic topology (Princeton, NJ, 1986) , Lecture Notes in Math.Vol. 1326 (Springer, Berlin, 1988) pp. 161–181.91. T. Kawai, Y. Yamada and S.-K. Yang,
Nucl. Phys.
B414 , 191 (1994), arXiv:hep-th/9306096 [hep-th] , doi:10.1016/0550-3213(94)90428-6.92. C. A. Keller and H. Ooguri,
Commun. Math. Phys. , 107 (2013), arXiv:1209.4649 [hep-th] , doi:10.1007/s00220-013-1797-8.93. D. Husem¨oller, M. Joachim, B. Jurˇco and M. Schottenloher,
Basic bundle theoryand K -cohomology invariants , Lecture Notes in Physics, Vol. 726 (Springer, Berlin,2008). With contributions by Siegfried Echterhoff, Stefan Fredenhagen and BernhardKr¨otz.94. L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Nucl. Phys.
B261 , 678 (1985),doi:10.1016/0550-3213(85)90593-0.95. C. Vafa,
Nucl. Phys.
B273 , 592 (1986), doi:10.1016/0550-3213(86)90379-2.96. R. Dijkgraaf and E. Witten,
Commun. Math. Phys. , 393 (1990), doi:10.1007/BF02096988.97. P. S. Aspinwall, C. A. Lutken and G. G. Ross,
Phys. Lett.
B241 , 373 (1990), doi:10.1016/0370-2693(90)91659-Y.98. E. Witten,
Nucl. Phys.
B340 , 281 (1990), doi:10.1016/0550-3213(90)90449-N.99. K. A. Intriligator,
Mod. Phys. Lett. A6 , 3543 (1991), arXiv:hep-th/9108005[hep-th] , doi:10.1142/S0217732391004097.100. A. Cabo-Bizet (2016), arXiv:1606.06341 [hep-th] .101. F. Nieri and S. Pasquetti, JHEP , 155 (2015), arXiv:1507.00261 [hep-th] , doi:10.1007/JHEP11(2015)155.102. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil andE. Zaslow, Mirror symmetry , Clay mathematics monographs, Vol. 1 (AMS, Provi-dence, USA, 2003).103. A. Kapustin and Y. Li,
JHEP , 005 (2003), arXiv:hep-th/0210296 [hep-th] ,doi:10.1088/1126-6708/2003/12/005.104. I. Brunner, M. Herbst, W. Lerche and B. Scheuner, JHEP , 043 (2006), arXiv:hep-th/0305133 [hep-th] , doi:10.1088/1126-6708/2006/11/043.105. K. Hori and J. Walcher, Comptes Rendus Physique , 1061 (2004), arXiv:hep-th/0409204 [hep-th] , doi:10.1016/j.crhy.2004.09.016. Daniel S. Park arXiv:0803.2045 [hep-th] .107. H. Ooguri, Y. Oz and Z. Yin,
Nucl. Phys.
B477 , 407 (1996), arXiv:hep-th/9606112[hep-th] , doi:10.1016/0550-3213(96)00379-3.108. K. Hori, A. Iqbal and C. Vafa (2000), arXiv:hep-th/0005247 [hep-th] .109. A. Libgober,
Math. Res. Lett. , 141 (1999), doi:10.4310/MRL.1999.v6.n2.a2.110. H. Iritani, ArXiv e-prints (December 2007), arXiv:0712.2204 [math.AG] .111. H. Iritani,
Adv. Math. , 1016 (2009), doi:10.1016/j.aim.2009.05.016.112. L. Katzarkov, M. Kontsevich and T. Pantev, Hodge theoretic aspects of mirror sym-metry, in
From Hodge theory to integrability and TQFT tt*-geometry , , Proc. Sympos.Pure Math. Vol. 78 (Amer. Math. Soc., Providence, RI, 2008) pp. 87–174.113. J. Halverson, H. Jockers, J. M. Lapan and D. R. Morrison,
Commun. Math. Phys. , 1563 (2015), arXiv:1308.2157 [hep-th] , doi:10.1007/s00220-014-2157-z.114. S. Galkin, V. Golyshev and H. Iritani (2014), arXiv:1404.6407 [math.AG] .115. N. A. Nekrasov and S. L. Shatashvili,
JHEP , 100 (2015), arXiv:1405.6046[hep-th] , doi:10.1007/JHEP01(2015)100.116. P. Putrov, J. Song and W. Yan, JHEP , 185 (2016), arXiv:1505.07110 [hep-th] ,doi:10.1007/JHEP03(2016)185.117. A. Gadde, S. S. Razamat and B. Willett, JHEP , 163 (2015), arXiv:1506.08795[hep-th] , doi:10.1007/JHEP11(2015)163.118. A. Bertram, I. Ciocan-Fontanine and B.-s. Kim (2003), arXiv:math/0304403[math-ag] .119. A. Hanany and K. Hori, Nucl. Phys.
B513 , 119 (1998), arXiv:hep-th/9707192[hep-th] , doi:10.1016/S0550-3213(97)00754-2.120. N. Seiberg,
Nucl. Phys.
B435 , 129 (1995), arXiv:hep-th/9411149 [hep-th] , doi:10.1016/0550-3213(94)00023-8.121. S. Fomin andA. Zelevinsky,
J. Amer. Math. Soc. , 497 (2002), arXiv:math/0104151 [math] ,doi:10.1090/S0894-0347-01-00385-X.122. E. Sharpe, A few recent developments in 2d (2,2) and (0,2) theories, in StringMath 2014 Edmonton, Alberta, Canada, June 9-13, 2014 , (2015). arXiv:1501.01628[hep-th] .123. M. Yamazaki,
J. Statist. Phys. , 895 (2014), arXiv:1307.1128 [hep-th] , doi:10.1007/s10955-013-0884-8.124. J. Yagi,
JHEP , 065 (2015), arXiv:1504.04055 [hep-th] , doi:10.1007/JHEP10(2015)065.125. M. Yamazaki and W. Yan, J. Phys.
A48 , 394001 (2015), arXiv:1504.05540[hep-th] , doi:10.1088/1751-8113/48/39/394001.126. D. Kutasov andA. Schwimmer,
Phys. Lett.
B354 , 315 (1995), arXiv:hep-th/9505004 [hep-th] ,doi:10.1016/0370-2693(95)00676-C.127. L. F. Alday, D. Gaiotto and Y. Tachikawa,
Lett. Math. Phys. , 167 (2010), arXiv:0906.3219 [hep-th] , doi:10.1007/s11005-010-0369-5.128. D. R. Morrison, J. Amer. Math. Soc. , 223 (1993), doi:10.2307/2152798.129. P. Deligne, Local behavior of Hodge structures at infinity, in Mirror symmetry, II ,, AMS/IP Stud. Adv. Math. Vol. 1 (Amer. Math. Soc., Providence, RI, 1997) pp.683–699.130. D. R. Morrison,
Ast´erisque , 243 (1993), Journ´ees de G´eom´etrie Alg´ebrique d’Orsay(Orsay, 1992).131. G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko,
JHEP , 038 (2014), arXiv:1306.0432 [hep-th] , doi:10.1007/JHEP01(2014)038. ecent developments in 2d N “ p , q supersymmetric gauge theories Commun. Math. Phys. , 717(2015), arXiv:1307.5997 [hep-th] , doi:10.1007/s00220-014-2193-8.133. S. Nawata,
Adv. Theor. Math. Phys. , 1277 (2015), arXiv:1408.4132 [hep-th] ,doi:10.4310/ATMP.2015.v19.n6.a4.134. A. Gerhardus and H. Jockers (2016), arXiv:1604.05325 [hep-th] .135. P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, Nucl. Phys.
B359 , 21(1991), doi:10.1016/0550-3213(91)90292-6.136. T. H. Gulliksen and O. G. Neg˚ard,
C.R.Acad.Sci.Paris S´er.A-B (1972).137. H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison and M. Romo,
JHEP , 166(2012), arXiv:1205.3192 [hep-th] , doi:10.1007/JHEP11(2012)166.138. A. Givental, The mirror formula for quintic threefolds, in Northern California Sym-plectic Geometry Seminar , , Amer. Math. Soc. Transl. Ser. 2 Vol. 196 (Amer. Math.Soc., Providence, RI, 1999) pp. 49–62.139. K. Ueda and Y. Yoshida (2016), arXiv:1602.02487 [hep-th] .140. S. Gukov and E. Witten (2006), arXiv:hep-th/0612073 [hep-th] .141. A. Hanany and D. Tong,
JHEP , 037 (2003), arXiv:hep-th/0306150 [hep-th] ,doi:10.1088/1126-6708/2003/07/037.142. R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Nucl. Phys.
B673 , 187(2003), arXiv:hep-th/0307287 [hep-th] , doi:10.1016/j.nuclphysb.2003.09.029.143. N. Dorey, S. Lee and T. J. Hollowood,
JHEP , 077 (2011), arXiv:1103.5726[hep-th] , doi:10.1007/JHEP10(2011)077.144. H.-Y. Chen, N. Dorey, T. J. Hollowood and S. Lee, JHEP , 040 (2011), arXiv:1104.3021 [hep-th] , doi:10.1007/JHEP09(2011)040.145. D. Gaiotto, JHEP , 090 (2012), arXiv:0911.1316 [hep-th] , doi:10.1007/JHEP11(2012)090.146. T. Dimofte, S. Gukov and L. Hollands, Lett. Math. Phys. , 225 (2011), arXiv:1006.0977 [hep-th] , doi:10.1007/s11005-011-0531-8.147. D. Gaiotto, S. Gukov and N. Seiberg, JHEP , 070 (2013), arXiv:1307.2578[hep-th] , doi:10.1007/JHEP09(2013)070.148. L. F. Alday, M. Bullimore, M. Fluder and L. Hollands, JHEP , 018 (2013), arXiv:1303.4460 [hep-th] , doi:10.1007/JHEP10(2013)018.149. M. Bullimore, M. Fluder, L. Hollands and P. Richmond, JHEP , 62 (2014), arXiv:1401.3379 [hep-th] , doi:10.1007/JHEP10(2014)062.150. J. Lamy-Poirier (2014), arXiv:1412.0530 [hep-th] .151. L. F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, JHEP , 113(2010), arXiv:0909.0945 [hep-th] , doi:10.1007/JHEP01(2010)113.152. G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko (2015), arXiv:1505.07116[hep-th] .153. H.-J. Chung and Y. Yoshida (2016), arXiv:1605.07165 [hep-th] .154. P. M. Crichigno and M. Roˇcek, JHEP , 207 (2015), arXiv:1506.00335 [hep-th] ,doi:10.1007/JHEP09(2015)207.155. E. Sharpe, J. Geom. Phys. , 256 (2013), arXiv:1212.5322 [hep-th]arXiv:1212.5322 [hep-th]