The Twisted Index and Topological Saddles
Mathew Bullimore, Andrea E. V. Ferrari, Heeyeon Kim, Guangyu Xu
aa r X i v : . [ h e p - t h ] J u l Prepared for submission to JHEP
The Twisted Index and Topological Saddles
Mathew Bullimore, a Andrea E. V. Ferrari, a,b
Heeyeon Kim, b Guangyu Xu a a Department of Mathematical Sciences, Durham University,Stockton Road, Durham DH1 3LE, UK b Mathematical Institute, University of Oxford,Woodstock Road, Oxford, OX2 6GG, UK
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The twisted index of 3d N = 2 gauge theories on S × Σ has an algebro-geometricinterpretation as the Witten index of an effective supersymmetric quantum mechanics. In thispaper, we consider the contributions to the supersymmetric quantum mechanics from topologicalsaddle points in supersymmetric localization of abelian gauge theories. Topological saddles areconfigurations where the matter fields vanish and the gauge symmetry is unbroken, which exist fornon-vanishing effective Chern-Simons levels. We compute the contributions to the twisted indexfrom both topological and vortex-like saddles points and show that their combination recovers theJeffrey-Kirwan residue prescription for the twisted index and its wall-crossing.
July 24, 2020 ontents U (1) + 1 Chiral 206.2 U (1) k + 1 Chiral 21 SU (2) A powerful feature of supersymmetric localisation is the ability to introduce exact deformations andscaling limits that lead to different mathematical representations of the same partition function. Inthe case of 3d N = 2 supersymmetric gauge theories, the localisation schemes used in the literaturefall into two broad classes: – 1 – In Coulomb branch localisation, the path integral localises onto configurations where thevectormultiplet scalar is non-zero and the gauge group is broken to a maximal torus. Thisexpresses the partition function as a contour integral of special functions, for example, as inthe first supersymmetric localisation computations on S [1–3]. • In Higgs branch localisation, the path integral localises instead onto configurations solvingvortex-like equations. This expresses the path integral in terms of integrals of characteristicclasses over moduli spaces of vortices, which may often be reduced to isolated fixed points byturning on mass parameters for flavour symmetries [4, 5].In many cases, the individual residues of the contour integral in Coulomb branch localisation re-produce the isolated fixed point contributions in Higgs branch localisation. Further background onlocalisation in three dimensions can be found in the reviews [6, 7].In this paper, we focus on the twisted index of 3d N = 2 supersymmetric gauge theories on S × Σ, where Σ is a closed Riemann surface of genus g [8–10]. This is a rich observable amenableto supersymmetric localisation and has important applications to exact microstate counting forsupersymmetric black holes in AdS [11–14] and the Bethe/gauge correspondence [15–17]. Theaim to expand upon and generalise the geometric interpretation of the twisted index as the Wittenindex of an effective supersymmetric quantum mechanics [18–20].The route to such a geometric interpretation of the twisted index is through a Higgs branchlocalisation scheme and for large classes of theories the twisted index is a generating function ofenumerative invariants associated to moduli spaces of vortices on the Riemann surface Σ. However,Higgs branch localisation of the twisted index of a generic 3d N = 2 gauge theory leads to additionalsaddle points beyond vortex configurations. For a U (1) gauge theory, these additional saddle pointsare characterised by1. All chiral multiplets vanish Φ = 0;2. The vectormultiplet scalar is fixed σ = σ ;3. The gauge symmetry is unbroken.Such “topological” saddle points exist whenever the effective U (1) Chern-Simons level in asymptoticregions of the Coulomb branch is non-zero. We compute the contributions of topological saddlepoints to the twisted index and show that they reproduce the contributions from residues at infinityin Coulomb branch localisation. These contributions are crucial for the consistency of the geometricinterpretation of the twisted index and wall-crossing phenomena studied in [20] in more generalclasses of theories. Let us first summarise the Coulomb branch localisation scheme for the twisted index of 3d N = 2gauge theories on S × Σ, which was introduced for g = 0 in [8] and extended to g > U (1) gauge theory this leads to a representation of the twisted index as a contour integral I = X m ∈ Z q m Z Γ dx πix I ( x, m ) , (1.1)where each summand is the contribution from configurations with m units of flux on Σ . Theintegrand receives contributions from a 1-loop determinant and an integral over gaugino zero modes.The contour is a Jeffrey-Kirwan residue prescription, building on computations of the elliptic genus We assume there are no monopole operators in the superpotential so there is a U (1) topological symmetry – 2 –f 2d N = (2 ,
2) gauge theories [21, 22]. However, a novel feature in three dimensions is theexistence of poles at x ± →
0. The contour prescription for these poles is determined by theeffective Chern-Simons levels k ± eff in asymptotic regions of the Coulomb branch.Here we introduce an exact deformation of the Lagrangian depending on a real parameter τ .This can be understood as a 1d FI parameter from the perspective of supersymmetric quantummechanics on S . In Coulomb branch localisation, it modifies the contour prescription Γ for thepoles at x ± → k ± eff = 0.This has two important consequences: • Away from walls in the parameter space of τ , the contour Γ is always well-defined for eachindividual flux m , before summing over m ∈ Z . This feature is necessary for a hamiltonianinterpretation of the twisted index as counting supersymmetric ground states and in particularfor compatibility with any Higgs branch localisation scheme. • It leads to interesting wall-crossing phenomenon in τ [20], via the same mechanism as super-symmetric quantum mechanics [23].Indeed, in the presence of the 1d FI parameter τ , it is possible to consider an alternativescaling limit in the path integral that leads to a Higgs branch localisation scheme. This provides arepresentation of the twisted index in the form I = X m ∈ Z q m Z ˆA( M m ) ch( F ) , (1.2)where M m denotes the moduli space of supersymmetric saddle points with flux m and ch( F ) isthe contribution of massive fluctuations and Chern-Simons terms. Each summand in (1.2) is thecontribution from an effective supersymmetric quantum mechanics in the topologically distinctsector labelled by the flux m . The moduli space M m in general has contributions from two typesof saddle points:1. Vortex Saddles
These are solutions of abelian vortex equations on Σ depending on τ . In the presence ofgeneric mass parameters for flavour symmetries, their contribution to the moduli space M m has a concrete description as a disjoint union of symmetric products of Σ. We show that theircontribution to (1.2) reproduces residues of the contour integral (1.1) at poles at finite valuesof x .2. Topological Saddles
These are configurations where all chiral multiplets vanish, σ has a fixed expectation value,and the U (1) gauge symmetry is unbroken. Topological saddles only exist if an effectiveChern-Simons level is non-vanishing, k ± eff = 0. Their contribution to the moduli space M m is roughly the Picard variety parametrising holomorphic line bundles on Σ of degree m . Weshow that their contribution to (1.2) reproduces residues of (1.1) at poles at x ± → τ and the moduli space M m may jump as this parameter crosses walls proportional to the flux m . Taking this into account,we show that equation (1.2) exactly reproduces the Coulomb branch residue prescription in thepresence of τ for a broad class of U (1) gauge theories.Finally, the twisted index has a hamiltonian interpretation as a Witten index I = X m ∈ Z q m Tr H m ( − F , (1.3)– 3 –here H m is the space of supersymmetric ground states with flux m on Σ. It is therefore naturalto identify the supersymmetric ground states with some form of cohomology of the moduli spacesof vortex and topological saddle points contributing to M m [18]. However, the twisted indexexhibits cancelations between contributions from residues at finite x and x ± → ∞ . This indicatesthe presence of instanton corrections between perturbative ground states associated to vortex andtopological vacua. We hope to return to this phenomenon in the future. The outline of the paper is as follows. In section 2 we review the Coulomb branch localisation ofthe twisted index of U (1) gauge theories. In section 3 we introduce an alternative Higgs branchlocalisation scheme and discuss general features of vortex and topological saddles. In sections 4 and5 we evaluate the contribution of vortex and topological saddles respectively to the twisted index.In section 6 we evaluate these contributions explicitly in examples with a single chiral multiplet.Finally, in section 7 we perform a preliminary investigation of an SU (2) gauge theory deformed bya 1d FI parameter for the Cartan subalgebra. We consider a 3d N = 2 gauge theory with G = U (1) and N chiral multiplets Φ j of charge Q j and R-charge r j ∈ Z . We will restrict attention to the cases where Q j = ±
1. We introduce asupersymmetric Chern-Simons term at level k for the gauge group and a mixed gauge-R symmetryChern-Simons term k R . The quantization condition requires k + 12 N X j =1 Q j ∈ Z ,k R + 12 N X j =1 Q j ( r j − ∈ Z . (2.1)The latter condition implies that the R-symmetry line bundle that we use for the topological twistingis well-defined. In this paper, we set the superpotential to vanish.There is a global topological symmetry U (1) t with associated real Fayet-Iliopoulos (FI) param-eter ζ ∈ R . In some cases, this is enhanced to a non-abelian symmetry in the infrared. In addition,there is a global flavour symmetry G f with maximal torus T f ∼ = h × Nj =1 U (1) j i /U (1) , (2.2)where U (1) j rotates Φ j with charge +1 and the quotient is by the gauge group. Correspondingly,we introduce real mass parameters m j such that the total mass of Φ j is m j + Q j σ where σ is the realscalar in the vectormultiplet. The real masses are defined up to a constant shift m j → m j + Q j c ,which can be absorbed by σ → σ − c .Integrating out chiral multiplets in the presence of generic real masses generates effective su-persymmetric gauge and mixed gauge R-symmetry Chern-Simons levels k eff ( σ ) = k + 12 N X j =1 Q j sign( m j + Q j σ ) ,k R, eff ( σ ) = k R + 12 N X j =1 Q j ( r j −
1) sign( m j + Q j σ ) , (2.3)– 4 –hich are piece-wise constant in σ . The quantization condition (2.1) ensures that the constantvalues are integers. It is also useful to define k ± eff := lim σ →±∞ k eff = k ± N X j =1 Q j | Q j | , (2.4)which controls the gauge charges of monopole operators with U (1) t topological charge ±
1. Similarly,the asymptotic values of the effective mixed Chern-Simons level controls of the R-charge of the samemonopole operators.
Following [8–10], we consider the twisted index on S × Σ with a closed orientable Riemann surfaceΣ of genus g . The twist is performed using the unbroken R-symmetry, which preserves an N = (0 , S with a pair of supercharges Q, ¯ Q .The twisted index has a hamiltonian interpretation as counting supersymmetric ground stateson S × Σ annihilated by Q , ¯ Q . The space of supersymmetric ground states H transforms as arepresentation space of the global symmetry U (1) t × G f and I = Tr H ( − F q J t N Y j =1 y J j j , (2.5)where J t and J j denote the Cartan generators of U (1) t and U (1) j respectively and q and y j are thefugacities. A basic assumption is that H is locally finitely graded, meaning that the coefficient of agiven monomial in q and y j is a finite integer. The twisted index can also be computed using supersymmetric localisation applied to the path inte-gral construction [8–10]. We now briefly summarise the various Lagrangians used in supersymmetriclocalisation.First, the vectormultiplet Yang-Mills lagrangian L YM and the chiral multiplet lagrangian L Φ are exact with respect to both supercharges Q, ¯ Q . In supersymmetric localisation, their coefficientsare typically sent to infinity so that the saddle point approximation is exact.The FI and mass parameter Lagrangians L FI and L m are not exact and the twisted index de-pends on these parameters. These parameters are naturally complexified by Wilson lines around S for the associated global symmetries. In particular, we can identify the fugacities in the hamiltoniandefinition (2.5) of the twisted index as q = e − πβ ( ζ + iA t ) , y = e − πβ ( m + iA f ) , (2.6)where A t and A f denote the constant background connections for U (1) t and T f respectively. Thetwisted index is a meromorphic function of q and y with poles at loci where non-compact masslessdegrees of freedom appear and the spectrum is no longer gapped.The supersymmetric Chern-Simons lagrangian L CS is not exact and the index depends on thelevel k .Following [20], we introduce an additional exact term L τ = iτ Q + ¯ Q )( λ + ¯ λ ) = − iτ D , (2.7)where D := D − F (2.8)– 5 –nd τ is a real parameter valued in the Lie algebra of the topological symmetry U (1) t . This isinterpreted as a 1d FI parameter since the combination D is the auxiliary field in the N = (0 , ζ , it cannot be complexified. Asthis parameter is real and exact, the twisted index depends in a piecewise constant manner on τ but may jump accross walls. We now briefly review the derivation of the contour integral formula for the twisted index usingsupersymmetric localisation.First, the localisation scheme used in [8–10] starts from the lagragian L = 1 e L YM + 1 g L Φ + L CS + L FI . (2.9)Schematically, localisation is done by sending e → g → I = X m ∈ Z q m πi I C d xx H ( x ) g Z ( x, m ) , (2.10)where the contour C is in the complexified maximal torus of G = U (1) parameterised by x and thesummation is over the magnetic flux m ∈ Z . The integrand is constructed from the supersymmetricChern-Simons and 1-loop contributions Z ( x, m ) = x k m x ( g − k R N Y i =1 " ( x Q i y i ) − x Q i y i Q i m +( g − r i − (2.11)and hessian factor H ( x ) = k + N X j =1 Q j (cid:18)
12 + x Q j y j − x Q j y j (cid:19) , (2.12)which arises from integration over gaugino zero modes.The contour is given explicitly by12 πi I C d xx = X x ∗ JK-Res x = x ∗ ( Q ∗ , η ) d xx , (2.13)where JK-Res x =0 ( Q, η ) d xx := Θ( Qη ) sign( Q ) (2.14)and η = 0 is an auxiliary real parameter. The sum is over poles of the integrand and Q ∗ denotesthe JK charge associated to a pole at x = x ∗ . The poles at solutions of x Q i y i = 1 arise from theelementary chiral multiplet Φ i and their JK charge is simply the gauge charge Q i . The JK chargesassigned to the poles at x = 0 and x = ∞ are x = 0 : Q + = − k +eff ,x = ∞ : Q − = + k − eff , (2.15)which are the gauge charges of monopole operators of U (1) t charge ± η if all JK charges are non-vanishing. However, if a monopole operator is gauge neutral, when Q = 0 or Q ∞ = 0, then the JK– 6 –esidue operation requires further specification. In such cases, [8–10] introduce an additional termin the lagrangian as a regulator with the result that the residue at x = 0 or x = ∞ should not betaken in such cases. This leads to a meaningful result that is independent of η after summing overmagnetic flux m ∈ Z .In reference [20], the 1d FI parameter τ was introduced to ensure a meaningful result for eachindividual flux m ∈ Z . This feature is necessary if we want to unambiguously interpret the coefficientof q m as counting the supersymmetric ground states with U (1) t charge m , as in the Hamiltoniandefinition (2.5). The starting point is now the lagrangian L = 1 t (cid:18) e L YM + L τ (cid:19) + 1 g L Φ + L CS + L FI . (2.16)and the localisation proceeds as in supersymmetric quantum mechanics by taking the limit t → e finite [23]. This leads to an identical contour integral formula but with a different assignmentof JK charges to the poles.For the poles associated to chiral multiplets Φ i the JK charge is again Q i . For the polesassociated to monopole operators, the JK charges are now assigned according to Q + = ( − k +eff if k +eff = 0 m − τ ′ otherwise , (2.17a) Q − = ( + k − eff if k − eff = 0 m − τ ′ otherwise , (2.17b)where τ ′ := e vol(Σ)2 π τ . (2.18)The contribution from each magnetic flux m ∈ Z is now separately independent of the auxiliaryparameter η provided τ ′ = m . However, the twisted index may now jump accross the wall τ ′ = m according to I ( τ ′ = m + ǫ ) − I ( τ ′ = m − ǫ ) = q m h δ k +eff , Res x =0 + δ k − eff , Res x = ∞ i d xx H ( x ) g Z ( x, m ) . (2.19)where ǫ → + .In what follows, we therefore require τ ′ / ∈ Z . This ensures the JK charges are always non-vanishing and the contribution to the twisted index from each flux m ∈ Z is meaningful.Finally, the contour prescription used in [8–10] is recovered by sending τ ′ → + ∞ with η > τ ′ → −∞ with η <
0. That this is independent of the auxiliary parameter η is equivalent to thestatement that sum of (2.19) over m ∈ Z is proportional to a formal delta function at q = 1. The hamiltonian definition of the twisted index (2.5) can be viewed as the Witten index of aneffective N = (0 ,
2) supersymmetric quantum mechanics obtained by twisting on S × Σ. Based onthe general structure of supersymmetric quantum mechanics of this type we can expect a geometricconstruction of the twisted index in the form I = X m ∈ Z q m Z ˆA( M m ) ch( F ) . (3.1)– 7 –n such an expression, M m denotes a moduli space parametrising saddle points of the localised pathintegral with magnetic flux m ∈ Z , while F is schematically a complex of vector bundles arisingfrom the massive fluctuations of chiral multiplets and supersymmetric Chern-Simons terms. Theintegral should be understood equivariantly with respect to the flavour symmetry T f , leading to thedependence on the parameters y i .For such an interpretation to be meaningful, it is necessary for the contribution to the twistedindex from each individual flux m ∈ Z to be unambiguous. This necessitates the introduction of the1d FI parameter τ . The wall-crossing phenomena in τ are then reflected in jumps in the structureof the moduli spaces M m and complexes E m .This general expectation was verified in previous work [19, 20] for a special class of theories(for example those with N = 4 supersymmetry) where for generic τ ′ = m the moduli spaces M m exclusively parametrise vortex-like configurations on Σ where the gauge group is completelybroken. The purpose of this paper is to extend the geometric interpretation to theories where thereare “topological” saddle points where there is an unbroken gauge symmetry and the moduli spaces M m must be described as quotient stacks.There is an important distinction between saddle points where the unbroken gauge symmetryis the whole G = U (1) or a discrete subgroup. The latter involves a relatively mild extension of [20]to deal with moduli spaces with orbifold singularities and our constraint | Q i | = 1 is designed toavoid such cases. We therefore consider theories with topological saddle points where G = U (1) isfully unbroken and the moduli space M m has a component that is the Picard stack parametrisingdegree m holomorphic line bundles on Σ. To arrive at such a geometric interpretation we introduce an alternative supersymmetric localisationfor the twisted index [20], which is similar to the Higgs branch localisation schemes for 2d N = (2 , N = 2 theories [4, 5]. However, in addition to the usual vortex-likesaddle points, here there will be additional topological saddle points where the matter fields vanishand the gauge group is unbroken.The starting point is to consider the same lagrangian from equation (2.16) including the 1d FIparameter τ but first set g = t to obtain L = 1 t (cid:18) e L YM + L τ + L Φ (cid:19) + L CS + L FI . (3.2)The second step is to consider the limit t → e finite. The supersymmetric saddlepoints are then solutions to the following set of equations,1 e ∗ F A + N X j =1 Q j | φ j | − t σ k eff ( σ )2 π − τ = 0 , (3.3a)d A σ = 0 , ¯ ∂ A φ i = 0 , (3.3b)( m i + Q i σ ) φ i = 0 ∀ i = 1 , . . . , N , (3.3c)where F A is the curvature of the gauge connection A , and ∗ is the Hodge star operator on Σ. Inwriting these equations, φ j should be understood as a section of K r j / ⊗ L Q j where K Σ is thecanonical bundle on Σ and L is the holomorphic gauge bundle on Σ.Note that the dependence on the 3d FI parameter ξ has dropped out but the equations dependcritically on the 1d FI parameter τ . We keep the contribution proportional to the effective Chern-Simons term in the limit t → | σ | → ∞ with σ := t σ finite. – 8 – .3 Saddle Points The solutions to equations (3.3) fall into topologically distinct sectors labelled by the flux m := 12 π Z Σ F A ∈ Z . (3.4)A constraint on the existence of saddle points with a given flux m is found by integrating equa-tion (3.3a) over the Riemann surface Σ to give( τ ′ − m ) + e vol(Σ)4 π t σk eff ( σ ) = N X j =1 Q j k φ j k , (3.5)where k φ j k := e π Z Σ ¯ φ j ∧ ∗ φ j (3.6)is a positive definite inner product on sections of K r j Σ ⊗ L Q j and the normalised FI parameter τ ′ isdefined in equation (2.18).Assuming the 1d FI parameter is generic, meaning τ = m , there are two classes of solutionswith a given magnetic flux m ∈ Z . They can be described as follows:1. Vortex Saddles
Vortex saddle points are solutions where σ remains finite in the limit t → e ∗ F A + N X j =1 Q j | φ j | = τ , ¯ ∂ A φ i = 0 , ( m i + Q i σ ) φ i = 0 . (3.7)For generic mass parameters m , . . . , m N , the space of solutions decomposes as a disjoint unionof components where a single φ i is non-vanishing and σ = − m i /Q i . From the constraint (3.5),a component of the moduli space where φ i is non-vanishing exists ifsign( τ ′ − m ) = sign Q i . (3.8)2. Topological Saddles
Topological saddle points are solutions where | σ | → ∞ in the limit t →
0, such that σ := t σ remains finite and has a unique non-vanishing solution. This requires φ j = 0 for all j = 1 , . . . , N and therefore the constraint (3.5) becomes τ ′ − m = − e vol(Σ)4 π σ k ± eff . (3.9)A unique solution with ± σ > k ± eff = 0 andsign( τ ′ − m ) = sign Q ± . (3.10)In addition, if k ± eff = 0 then a non-compact Coulomb branch parametrised by ± σ > τ ′ = m , which is responsible for the wall-crossing phenomena in equation (2.19). These threeclasses are analogous to the trichotomy of flat space supersymmetric vacua in [27].If we align the auxiliary parametersign( τ ′ − m ) = sign η , (3.11)– 9 –omponents of the moduli space of saddles with flux m are in one-to-one correspondence with thepoles selected by the contour prescription in section 2. There is a component of the vortex modulispace with φ i = 0 when the pole at x Q i y i = 1 is selected. Similarly, there is a topological saddlepoint with ± σ > x ± → The moduli space of vortex saddle points consists of solutions to1 e ∗ F A + N X j =1 Q j | φ j | = τ , ¯ ∂ A φ j = 0 , ( m j + Q j σ ) φ j = 0 , (4.1)for all j = 1 , . . . , N , modulo gauge transformations. The moduli space is a disjoint union oftopologically distinct components M m labelled by the magnetic flux m ∈ Z . The entire moduli spaceis realised as an infinite-dimensional K¨ahler quotient and under our assumptions each component M m is a finite-dimensional smooth K¨ahler manifold.For generic mass parameters m j , the moduli space further decomposes as a disjoint union ofcomponents M m ,i where a single chiral multiplet φ i is non-vanishing and σ = − m i /Q i . Eachcomponent parametrises solutions to the abelian vortex equations1 e ∗ F A + Q i | φ i | = τ , ¯ ∂ A φ i = 0 , (4.2)where φ i transforms as a section of K r i / ⊗ L Q i . A small modification of the standard analysisapplies and each component is either a symmetric product of the curve or empty, Q i = +1 : M m ,i = ( Σ d i if τ ′ > m ∅ if τ ′ < m Q i = − M m ,i = ( ∅ if τ ′ > m Σ d i if τ ′ < m (4.3)where d i := Q i m + r i ( g −
1) (4.4)is the degree of K r i / ⊗ L Q i and Σ d := Sym d Σ with the understanding that this is empty for d < d i parametrises the positions of the vortices. The assumption | Q i | = 1 isimportant to get a symmetric product, otherwise the moduli space has orbifold singularities wherea discrete gauge subgroup is unbroken.Note that if the auxiliary parameter η is aligned with τ ′ − m , meaningsign( τ ′ − m ) = sign( η ) , (4.5)then the component M m ,i of the moduli space is non-empty whenever the JK residue prescriptionselects the pole at x Q i y i = 1 from the chiral multiplet Φ i . The task in the remainder of this sectionis to reproduce the residue at this pole from supersymmetric localisation.It is useful to use an algebraic description of moduli spaces of abelian vortices in terms ofholomorphic pairs. Let us assume sign( τ ′ − m ) = sign Q i so that the vortex moduli space M m ,i is– 10 –on-empty. Then the Hitchin-Kobayashi correspondence says that there is an algebraic descriptionparametrising pairs ( L, φ i ) where L is a holomorphic line bundle of degree m and φ i is a non-zerosection of K r i / ⊗ L Q i . The symmetric product Σ d i in equation (4.3) parametrises the zeros of thesection φ i . The contribution to the twisted index from a component M m ,i of the vortex moduli space is I m ,i = Z ˆ A ( M m ,i ) ˆ A ( E ) e ( E ) ch( L k ⊗ L k R R ))= Z ˆ A ( M m ,i ) ch( L k ⊗ L k R R )ch(ˆ ∧ • E ) (4.6)where E is a perfect complex of sheaves encoding the massive fluctuations of chiral multiplets aroundvortex configurations and L , L R are holomorphic line bundles arising from the gauge and mixedgauge R-symmetry Chern-Simons terms respectively. We omit the labels m , i from the bundles forbrevity.This integral should be understood equivariantly with respect to the flavour symmetry T f withparameters y i . It can be evaluated using intersection theory on symmetric products and convertedinto a contour integral following [28, 29]. This extends a computation performed in [19] to a widerclass of theories.For simplicity and to avoid a profusion of signs at intermediate steps in the calculation, we set Q i = +1 with τ ′ > m . A similar computation applies to Q i = − We first consider the contribution from the tangent directions to M m ,i , which is the symmetricproduct Σ d i when τ ′ > m and otherwise empty.Let us briefly summarise some notation for the intersection theory on a symmetric product.There are standard generators ζ a , ¯ ζ a ∈ H (Σ d ) with a = 1 , . . . g and η ∈ H (Σ d ) arising fromcohomology classes on Σ. We then define θ a := ζ a ∧ ¯ ζ a and θ := P ga =1 θ a .From reference [28], the Chern character of the tangent bundle isch( T Σ d i ) = ( g −
1) + (( d i − g + 1) − θ ) e η = ( g −
1) + ( d i − g + 1) e η + g X a =1 e η − θ a . (4.7)From here we can evaluate the ˆ A -genus as followsˆ A (Σ d i ) = exp (cid:18) − d i − g + 12 η + θ (cid:19)(cid:18) η − e − η (cid:19) d i − g +1 g Y a =1 η − θ a − e − η + θ a = exp (cid:18) − d i − g + 12 η + θ (cid:19)(cid:18) η − e − η (cid:19) d i − g +1 g Y a =1 η − θ a − e − η (1 + θ a )= exp (cid:18) − d i − g + 12 η + θ (cid:19)(cid:18) η − e − η (cid:19) d i − g +1 g Y a =1 η − θ a (1 − e − η ) − e − η θ a = exp (cid:18) − d i − g + 12 η + θ (cid:19)(cid:18) η − e − η (cid:19) d i − g +1 g Y a =1 η − θ a − e − η (cid:18) − e − η − e − η θ a (cid:19) − – 11 – exp (cid:18) − d i − g + 12 η + θ (cid:19)(cid:18) η − e − η (cid:19) d i − g +1 g Y a =1 η − e − η (cid:18) − θ a η + e − η − e − η θ a (cid:19) = exp (cid:18) − d i − g + 12 η + θ (cid:19)(cid:18) η − e − η (cid:19) d i − g +1 g Y a =1 exp (cid:20) θ a (cid:18) − η + e − η − e − η (cid:19)(cid:21) = (cid:18) ηe − η/ − e − η (cid:19) d i − g +1 exp (cid:20) θ (cid:18) − η + e − η − e − η (cid:19)(cid:21) , (4.8)where we have made repeated use of θ a = 0 for any a = 1 , . . . , g . We now consider the fluctuations of each of the remaining massive chiral multiplets Φ j with j = i around configurations in M m ,i .At a point ( L, φ i ) on the moduli space M m ,i , each chiral multiplet Φ j with j = i generates 1d N = (0 ,
2) chiral and Fermi multiplet fluctuations valued in the following vector spaces: • Chiral multiplets: E j := H (cid:16) Σ , L Q j ⊗ K r j / (cid:17) , • Fermi multiplets: E j := H (cid:16) Σ , L Q j ⊗ K r j / (cid:17) . If we move around in the moduli space M m ,i the dimensions of these vector spaces may jump butby the Riemann-Roch theorem the difference of their dimensions is constant,dim E j − dim E j = h ( L Q j ⊗ K r j / ) − h ( L Q j ⊗ K r j / )= Q j m + ( g − r j − d j − g + 1 . (4.9)We can therefore formally regard the difference of these vector spaces as the fibre of a holomorphicvector bundle on the moduli space M m ,i of rank d j − g + 1, or at least this will define a reasonableK-theory class for use in the computation of the twisted index.To make this more precise, we recall the construction of the universal bundle on a symmetricproduct. We consider the pair of projection mapsΣ d i × ΣΣ d i Σ π p . (4.10)There is a unique universal line bundle L on Σ d i × Σ with the property that its restriction to a point(
L, φ i ) on the symmetric product is the holomorphic line bundle L on Σ. We also define K := p ∗ K Σ to be the pull-back of the canonical bundle on the curve. With this in hand, the fluctuations of Φ j transform in a perfect complex of sheaves on Σ d defined by the derived push-forward E • j := R • π ∗ ( L Q j ⊗ K r j / ) . (4.11)The stalks of E • j over a point ( L, φ i ) on the symmetric product are the vector spaces E • j .We can extract the Chern roots of E • j following standard computations [29]. The starting pointis the Chern character of the universal bundlech (cid:0) L Q j (cid:1) = e Q j η (cid:0) Q j m η Σ + Q j γ − Q j η Σ θ (cid:1) (4.12)and ch( K r j / ) = 1 + r j ( g − η Σ . (4.13)– 12 –ere we abuse notation and identify the cohomology classes η , θ with their pull-backs by π . In asimilar way, η Σ denotes the class of a point on Σ and its pullback by p . Finally, γ is built from thepull-back of 1-form generators and will not play a role in what follows.An application of the Groethendiek-Riemann-Roch theorem to π givesch( E • j ) = π ∗ n ch( L Q j ⊗ K r j / ) td(Σ d i ) o = e Q j η (cid:0) ( d j − g + 1) − Q j θ (cid:1) . (4.14)On vortex saddle points parametrised by the moduli space M m ,i , the real vectormultiplet scalar isfixed to σ = − m i and the real mass of Φ j fluctuations is m j − Q j m i . We therefore promote thisresult to a T f -equivariant Chern characterch( E • j ) = z j e Q j η (cid:0) ( d j − g + 1) − Q j θ (cid:1) , (4.15)where z j := y j /y Q j i . (4.16)The fluctuations from all the massive chiral multiplets is encoded in E = O j = i E • j . (4.17)The equivariant Chern roots have a similar structure to those in (4.7) and the contribution of thesefluctuations to the twisted index can be evaluated in a similar way, with the resultˆ A ( E ) e ( E ) = Y j = i ( e − Q j η z j ) − e − Q j η z j ! d j − g +1 exp (cid:20) Q j θ (cid:18)
12 + e − Q j η z j − e − Q j η z j (cid:19)(cid:21) . (4.18) The supersymmetric Chern-Simons terms generate holomorphic line bundles on the moduli space M m ,i ∼ = Σ d i according to the general mechanism in [30]. A careful translation into the algebraicframework of this paper leads to the conclusion that the Chern-Simons levels k , k R generate holo-morphic line bundles L k , L k R R with c ( L ) = θ − m η ,c ( L R ) = − ( g − η . (4.19)The contribution to the integrand of equation (4.6) is thereforech( L k ⊗ L k R R ) = e k ( θ − m η ) e − k R ( g − η . (4.20)This result passes a consistency check. It is compatible with the contribution (4.18) from massivefluctuations of chiral multiplets and the fact that integrating out a massive chiral multiplet of charge Q j and R-charge r j with real mass m j → ±∞ shifts k → k ± Q j ,k R → k R ± Q j r j − . (4.21)– 13 – .3 Evaluation of Witten Index Collecting the the contributions from the tangent directions to the moduli space, the fluctuationsof massive chiral multiplets and the supersymmetric Chern-Simons terms, the contribution (4.6) tothe twisted index from the component M m ,i of the vortex moduli space is I m ,i = Z Σ di e k ( θ − m η ) e − k R ( g − η (cid:18) ηe − η/ − e − η (cid:19) d i − g +1 exp (cid:20) θ (cid:18) − η + e − η − e − η (cid:19)(cid:21) × N Y j = i ( e − Q j η z j ) − e − Q j η z j ! d j − g +1 exp (cid:20) Q j θ (cid:18)
12 + e − Q j η z j − e − Q j η z j (cid:19)(cid:21) , (4.22)provided τ ′ > m and vanishes otherwise.The final step is to convert the integration over the symmetric product into a contour integralusing the following useful result [31], Z Σ d A ( η ) e θB ( η ) = I u =0 d uu A ( u ) [1 + u B ( u )] g u d . (4.23)The integral in equation (4.22) has precisely this form with A ( η ) = e − k m η e − k R ( g − η (cid:18) ηe − η/ − e − η (cid:19) d i − g +1 Y j = i " ( e − Q j η z j ) − e − Q j η z j d j − g +1 , (4.24) B ( η ) = k + (cid:18) − η + e − η − e − η (cid:19) + X j = i Q j (cid:18)
12 + e − Q j η z j − e − Q j η z j (cid:19) , (4.25)and therefore we find I m ,i = I u =0 d u e − k m u e − k R ( g − u (cid:18) e − u/ − e − u (cid:19) d i − g +1 N Y = i " ( e − Q j u z j ) − e − Q j u z j d − g +1 × k + 12 (cid:18) e − u − e − u (cid:19) + N X j = i Q j (cid:18) e − Q j u z j − e − Q j u z j (cid:19) g (4.26)= I x = y − i d xx x k m + k R ( g − ( xy i ) − xy i ! d i − g +1 N Y j = i " ( x Q j y j ) − x Q j y j d j − g +1 × k + 12 (cid:18) xy i − xy i (cid:19) + N X j = i Q j (cid:18) x Q j y j − x Q j y j (cid:19) g , (4.27)where the substitution e − u = xy i has been made in the second line. A similar calculation can beperformed in the case Q i = − | Q i | = 1 is that the contribution to the twistedindex from vortex saddle points parameterised by M m ,i is I m ,i = I x = y − /Qii d xx x k m + k R ( g − N Y j =1 " ( x Q i y j ) − x Q j y j d j − g +1 k + N X j =1 Q j (cid:18)
12 + x Q j y j − x Q j y j (cid:19) g (4.28)when sign( τ ′ − m ) = sign Q i and zero otherwise. This exactly reproduces the contribution to thetwisted index from the pole at x Q i y i = 1 when the auxiliary parameter η is chosen such thatsign η = sign( τ ′ − m ). – 14 – Topological Saddles
Topological saddle points are configurations where φ j = 0 for all j = 1 , . . . , N and there is a uniquefinite expectation value for σ := t σ that solves the equation τ ′ − m = − e vol(Σ)4 π σ k ± eff (5.1)in the region ± σ >
0. Topological saddle points exist provided k ± eff = 0 and sign( τ ′ − m ) = sign Q ± .If we choose the auxiliary parameter such that sign η = sign( τ ′ − m ), there are topological saddlepoints with ± σ > x ± →
0. The taskin this section is to reproduce the residue at this pole.The only massless bosonic fluctuations around a topological saddle are those of the gaugeconnection A . Topological saddle points with flux m ∈ Z are therefore parametrised by connections A on a principle U (1) bundle satisfying ∗ F A = 2 π vol(Σ) m , (5.2)modulo gauge transformations on Σ. As for vortex saddle points, the contribution to the twistedindex is expected to be the Witten index of a supersymmetric quantum mechanics whose target isthe moduli space of solutions to these equations. However, gauge transformations act trivially on F A and σ , so the U (1) gauge symmetry is unbroken and the quantum mechanics is gauged.To describe the supersymmetric quantum mechanics concretely, we use the algebraic descrip-tion of solutions to (5.2) as holomorphic line bundles L of degree c ( L ) = m . We then expect asupersymmetric sigma model to the Picard variety Pic m (Σ), parametrised by the complex structure¯ ∂ A which transforms as a chiral multiplet under N = (0 ,
2) supersymmetry.However, any holomorphic line bundle has a C ∗ worth of automorphisms, corresponding tounbroken complexified gauge transformations. It is therefore more appropriate to describe thesupersymmetric quantum mechanics as a sigma model to the Picard stack, M m = Pic m (Σ) . (5.3)We can make this more concrete at the cost of introducing an auxiliary base point p ∈ Σ. Decom-posing complex gauge transformations into those trivial at p and constant gauge transformations,we have M m = M m × [pt / C ∗ ] , (5.4)where M m = Pic m (Σ) ≃ T g . (5.5)In this way, the supersymmetric quantum mechanics is a hybrid of a non-linear sigma model withtarget space T g and a U (1) gauge theory.The supersymmetric quantum mechanics is not, however, a product due to the massive fluc-tuations of the chiral multiplets Φ j . They transform in a perfect complex on M m generated byfluctuations annihilated by ¯ ∂ A . Choosing an auxiliary base point as above, this becomes a C ∗ -equivariant complex on M m . So the fluctuations roughly transform as sections of a holomorphicvector bundle on the target space T g of the sigma model and are also charged under the unbroken U (1) gauge symmetry. – 15 – .2 Contributions to Index The contributions to the twisted index from topological saddle points can be expressed in the sameform as vortex saddle points, Z ˆ A ( M m ) ch( L k ⊗ L k R R )ch(ˆ ∧ • E ) , (5.6)where E is a perfect complex arising from fluctuations of the massive chiral multiplet and L , L R are holomorphic line bundles arising from the gauge and mixed gauge R-symmetry Chern-Simonsterms.To make this more precise, we choose an auxiliary base point on Σ and decompose the modulistack M m = M m × [pt / C ∗ ]. The characteristic classes in equation (5.6) are then to be understoodas C ∗ -equivariant classes on M m . The integral over the moduli stack decomposes into two parts: • A regular integral over the moduli space M m ∼ = Pic m (Σ). This is the usual contribution froman N = (0 ,
2) supersymmetric non-linear sigma model. • A contour integral 12 πi I C d xx , where x is the Chern character of the trivial C ∗ -equivariant holomorphic vector bundle withweight +1. This is the contribution due to the unbroken U (1) gauge symmetry.The purpose of the contour integral is of course to project onto gauge invariant contributions. Thisis not meaningful as it stands because the integrals of C ∗ -equivariant classes in equation (5.6) overthe moduli space M m produce rational functions of x . It is therefore necessary to specify whetherthe integrand should be expanded inside or outside the unit circle, which correspond to the residuesat x = 0 or x = ∞ respectively.Our prescription will be guided by by physical intuition. First, note that the path integralconstruction identifies x = e − πβ ( σ + iA ) where σ is the real vectormultiplet scalar and A is a constantgauge connection around the circle. Topological saddle points with σ > x →
0, while those with σ < x → ∞ . The natural expectationfor the contour C is therefore σ > πi Z x =0 d xx ,σ < πi Z x = ∞ d xx . (5.7)This gains further support from the hamiltonian interpretation of the twisted index as count-ing supersymmetric ground states. The supersymmetric ground states depend on the sign of thereal mass of fluctuations, which is dominated by σ as | σ | → ∞ . For example, the ground statewavefunctions of a 1d chiral multiplet of charge +1 are σ > φ n e − σ | φ | , n ≥ ,σ < φ n e − σ | φ | ¯ ψ , n ≥ σ > x + x + · · · = 11 − x ,σ < − x − − x − + · · · = 11 − x . (5.9)– 16 –o projecting onto uncharged states at the level of the index is equivalent to σ > πi Z x =0 d xx − x = 1 ,σ < πi Z x = ∞ d xx − x = 0 , (5.10)which select the coefficient of x in the expansions around x = 0 and x = ∞ respectively. Thegeneral prescription (5.7) is basically a broad generalisation of this observation.In summary, we have two contributions from potential topological vacua with σ > σ < I = 12 πi Z x =0 d xx Z ˆ A ( M m ) ch( L k ⊗ L k R R )ch(ˆ ∧ • E ) ,I ∞ = 12 πi Z x = ∞ d xx Z ˆ A ( M m ) ch( L k ⊗ L k R R )ch(ˆ ∧ • E ) , (5.11)where we interpret E and L , L R as C ∗ -equivariant objects on the moduli space M m ∼ = T g . In thenext section we evaluate these explicitly and show that they reproduce the appropriate contributionsto the twisted index according to the contour prescription (2.17). Let us first summarise some notation for intersection theory on the Picard variety M m ≃ T g . Thecohomology ring is generated by classes ζ a ∈ H , ( T g , Z ) and ¯ ζ a ∈ H , ( T g , Z ) with a = 1 , . . . g .We define θ a := ζ a ∧ ¯ ζ a and θ := P ga =1 θ a with normalisation Z T g θ g g ! = 1 . (5.12)The tangent bundle is flat and therefore ˆ A ( M m ) = 1. We now consider the massive fluctuations arising from each of the chiral multiplets Φ j . At a pointon the moduli space corresponding to a holomorphic line bundle L , each chiral multiplet generateschiral and Fermi multiplet fluctuations solving¯ ∂ A φ j = 0 , ¯ ∂ A η j = 0 , (5.13)where φ j and η j transform as 0-form and 1-form sections of L Q j ⊗ K r j / respectively. The fluctu-ations of Φ j around this point therefore generate the vector spaces • Chiral multiplets: E j := H (cid:16) Σ , L Q j ⊗ K r j / (cid:17) , • Fermi multiplets: E j := H (cid:16) Σ , L Q j ⊗ K r j / (cid:17) . As L varies in Pic m (Σ) the dimensions of these vector spaces may jump but by the Riemann-Rochtheorem the difference is constant and equal todim E j − dim E j = Q j m + ( g − r j −
1) (5.14)= d j − g + 1 . (5.15)This means the difference behave like the fibre of a holomorphic vector bundle on Pic m (Σ) for thepurpose of K-theoretic computations involved in the twisted index.– 17 –o make this more precise, it is again useful to consider a universal construction. This iscanonical for the moduli stack but for concreteness we pick a base point p ∈ Σ and pass the modulispace M m . There is a corresponding diagram M m × Σ M m Σ π p (5.16)and universal line bundle L such that on restriction to a point on M m corresponding to a holomor-phic line bundle L , L| p ≃ L . The universal line bundle is not unique: due to C ∗ automorphismsthere is the possibility to transform L → L ⊗ π ∗ N . However, this can be fixed by demanding L istrivial on restriction to p . We also define K = p ∗ K Σ .The massive fluctuations of the chiral multiplet Φ i generate a perfect complex E • i of sheavesdefined by derived push-forward E • j := R • π ∗ (cid:16) L Q j ⊗ K r j / (cid:17) . (5.17)The stalks of E • j at L ∈ M m are the vector spaces E • j considered above. The class ch (cid:0) E • j (cid:1) =ch( E j ) − ch( E j ) makes sense in equivariant K-theory and the complex behaves like a vector bundleof rank d j − g + 1 for the purpose of such computations.To compute the contribution to the twisted index, we begin by computing the Chern characterof E • j . This is a small modification of a standard argument presented in [29]. In what follows, weagain abuse notation and identify the class θ with its pull-back via π . Similarly η Σ denotes the classof a point on Σ and its pull-back via p .First, the Chern class of the universal line bundle is c ( L ) = m η Σ + γ , (5.18)where γ = − η Σ θ . We therefore findch (cid:0) L Q i (cid:1) = e Q i c ( L ) = 1 + Q i m η Σ + Q i γ + Q i γ = 1 + Q i m η Σ + Q i γ − Q i η Σ θ . (5.19)Similarly, from c ( K ) = (2 g − η Σ , (5.20)we find ch (cid:16) K r j / (cid:17) = e rj c ( K ) = e r j ( g − η Σ = 1 + r j ( g − η Σ . (5.21)Combining these results ch (cid:16) L Q j ⊗ K rj (cid:17) = 1 + d j η Σ + Q j γ − Q j η Σ θ . (5.22) There is a unique universal line bundle on M m × Σ without such a choice. – 18 –e can now compute the Chern character of E • j using the Groethendiek-Riemann-Roch theorem,ch (cid:0) E • j (cid:1) = π ∗ h ch( L Q j ⊗ K rj ) Td(Pic m Σ × Σ) i (5.23)= π ∗ h ch (cid:16) L Q i ⊗ K ri (cid:17) (1 − ( g − η Σ ) i = π ∗ (cid:2) d j − g + 1) η Σ + Q i γ − Q i η Σ θ (cid:3) = ( d j − g + 1) − Q i θ (5.24)= ( d j − g + 1) + g X a =1 e − Q j θ a , (5.25)where in the final line we have expressed the result in such a way that the Chern roots are manifest.This is promoted to an equivariant Chern characterch (cid:0) E • j (cid:1) = x Q j y j (( d j − g + 1) + g X a =1 e − Q j θ a ) . (5.26)The contribution to the twisted index is now given by the equivariant ˆ A -genus of the complex E • j . This is straightforward to compute from the equivariant Chern roots by a now familiar set ofmanipulations,ˆ A ( E • j ) e ( E • j ) = (cid:0) x Q j y j (cid:1) dj − g +12 e Q jθ (cid:18) − x Q j y j (cid:19) d j − g +1 g Y a =1 − x Q j y j e Q j θ a = (cid:0) x Q j y j (cid:1) dj − g +12 e Q jθ (cid:18) − x Q j y j (cid:19) d j − g +1 g Y a =1 − x Q j y j (1 + Q j θ a )= (cid:0) x Q j y j (cid:1) dj − g +12 e Q jθ (cid:18) − x Q j y j (cid:19) d j − g +1 g Y a =1 (cid:18) x Q j y j − x Q j y j Q j θ a ) (cid:19) = (cid:0) x Q j y j (cid:1) dj − g +12 e Q jθ (cid:18) − x Q j y j (cid:19) d j − g +1 exp (cid:18) x Q j y j − x Q j y j Q j θ (cid:19) = ( x Q j y j ) − x Q j y j ! d j − g +1 exp (cid:18)(cid:18)
12 + x Q j y j − x Q j y j (cid:19) Q j θ (cid:19) , (5.27)where we have made repeated use of θ a = 0. The supersymmetric Chern-Simons again induce holomorphic line bundles over the moduli space M m ∼ = T g . In the algebraic framework the Chern-Simons levels k , k R induce holomorphic linebundle L k , L k R R with c ( L ) = θ ,c ( L R ) = 0 , (5.28)and transform equivariantly with weights m and ( g −
1) respectively. The equivariant Chern char-acters are therefore ch( L k ) = ( x m e θ ) k = x k m e θ , ch( L k R R ) = x k R ( g − . (5.29)This is compatible with the contribution (5.27) from fluctuations of Φ j and the fact that integratingout a massive chiral multiplet of charge Q j and R-charge R j shifts the supersymmetric Chern-Simonslevels as in equation (4.21). – 19 – .3 Evaluation of Witten Index Combining all these contributions, the contribution to the integrand from the integration over themoduli space M m = Pic m (Σ) ∼ = T g is Z Pic m (Σ) x k m e kθ x k R ( g − N Y j =1 ( x Q j y j ) − x Q j y j ! d j − g +1 exp (cid:18)(cid:18)
12 + x Q j y j − x Q j y j (cid:19) Q j θ (cid:19) = x k m + k R ( g − N Y j =1 ( x Q j y j ) − x Q j y j ! d j − g +1 Z Pic m (Σ) exp k + N X j =1 Q j (cid:18)
12 + x Q j y j − x Q j y j (cid:19) θ = x k m + k R ( g − N Y j =1 ( x Q j y j ) − x Q j y j ! d j − g +1 k + N X j =1 Q j (cid:18)
12 + x Q j y j − x Q j y j (cid:19) g . (5.30)The contributions from topological saddle points therefore exactly reproduce the potential residuesat x = 0 and x = ∞ in the JK residue prescription with η aligned with τ ′ − m . We consider a U (1) Chern-Simons theory at level k ∈ + Z ≥ with one chiral multiplet Φ of R-charge r = 1 and charge Q = +1. The flavour symmetry T f is trivial and there are no real massparameters. The effective Chern-Simons level coupling is k eff ( σ ) = k + 12 sign( σ ) (6.1)and so k ± eff = k ± . (6.2)The cases k = and k > are quite different. The former has a neutral monopole operator andis mirror to a free chiral multiplet. This difference is reflected in the structure of the saddle pointsin our computation of the twisted index and therefore we treat the two cases separately. We alsorestrict attention to the twisted index with g > U (1) + Chiral
First consider k = . In this case k eff ( σ ) = (1 + sign( σ )) and therefore k +eff = 1 and k − eff = 0.There is a neutral monopole operator and the theory is mirror to a free chiral multiplet, togetherwith specific background mixed Chern-Simons couplings.The contour integral for the twisted index is I = X m ∈ Z ( − q ) m πi I C d xx x m (1 − x ) m + g . (6.3)where we have shifted q → − q compared to above. In the presence of a 1d FI parameter τ , thecontour is a JK residue prescription with charges Q + = − , Q = 1 , Q − = m − τ ′ . (6.4)Note that the charge Q − associated to the residue at x = ∞ now depends on the 1d FI parameter τ according to equation (2.17) since k − eff = 0. – 20 –or g > x = ∞ vanishes and there is no wall-crossing phenomena. The twistedindex is given by computing the residue at x = 1 ( η >
0) or equivalently minus the residue at x = 0( η < I = ( − g q − g (1 − q ) g − . (6.5)While the twisted index is non-zero only for fluxes 1 − g ≤ m ≤
0, there are in fact supersymmetricground states for all m ≥ − g [18].We now reproduce this result by evaluating the contributions from vortex and topological saddlepoints. The existence of vortex and topological saddle points is constrained by equation (3.5), whichbecomes ( τ ′ − m ) + e vol(Σ)4 π σ k eff ( σ ) = k φ k , (6.6)together with the equation σφ = 0. The existence of solutions depends on the sign of τ ′ − m . • When τ ′ − m >
0, there are vortex saddle points with σ = 0. The moduli space of vortexsolutions with flux m is the symmetric product M m = Σ d where d = m + g −
1. Following thecomputations in section 4, the contribution to the twisted index is Z Σ d ˆA( T Σ d ) Ch( L / ) = Z Σ d (cid:18) ηe − η − e − η (cid:19) m exp (cid:20) θ (cid:18) − η + e − η − e − η (cid:19)(cid:21) = 12 πi Z x =1 d xx x m (1 − x ) m + g . (6.7) • When τ ′ − m <
0, there are topological saddle points with φ = 0 , σ = − π e vol(Σ) ( τ ′ − m ) > . (6.8)The moduli space of topological solutions with flux m is the Picard variety M m = Pic m Σ.Following the computations in section 5, the contribution to the twisted index is Z Σ d ˆA( T Σ d ) Ch = 12 πi Z x =0 d xx Z Pic m Σ (cid:18) x − x (cid:19) m exp (cid:20)(cid:18) − x (cid:19) θ (cid:21) = 12 πi Z x =0 d xx x m (1 − x ) m + g , (6.9)where the residue at x = 0 is taken since σ > σ < τ ′ − m = 0 and so there is the potential forwall-crossing. However, the vanishing of the residue at x = ∞ means that the twisted index isindependent of τ . This reproduces the JK residue prescription with charges (6.11) and sign η =sign( τ ′ − m ). The result is independent of η for each flux m by construction. U (1) k + Chiral
Now consider k > such that k ± eff = k ± >
0. There are no gauge neutral monopole operatorsand the structure of the twisted index differs considerably.The contour integral for the twisted index is now I = X m ∈ Z ( − q ) m πi I C d xx x k m (cid:18) x / − x (cid:19) m (cid:18) k + 12 1 + x − x (cid:19) g , (6.10)where the contour is a JK residue prescription with charges Q + = − k − < , Q = 1 , Q − = k − > . (6.11)– 21 –he index is now manifestly independent of τ and there is no wall-crossing. We therefore enumeratethe residues at x = 1 and x = ∞ ( η >
0) or equivalently minus the residues at x = 0 ( η < g = 2 and k = we find −I = 1 q − , I = 1 q − − q − q − q − q − · · · = 1 − q + √ − q q , I ∞ = − q + 2 q + 5 q + 14 q + · · · = 1 − q − √ − q q . (6.12)Notice that the contributions I and I ∞ are not rational function of q and so cannot individuallyreproduce a reasonable index. In fact they do not count honest supersymmetric ground states butonly perturbative ground states. These are subject to instanton corrections that remove pairs ofperturbative ground states corresponding to cancelations in the sum I + I ∞ = −I .We can reproduce these contributions from an analysis of vortex and topological saddle points.The saddle points are again constrained by( τ ′ − m ) + e vol(Σ)4 π σ k eff ( σ ) = k φ k , (6.13)and depend on the sign of τ ′ − m . • When τ ′ − m >
0, there are both vortex saddle points and topological saddle points with σ <
0. The contributions from these saddle points reproduce the residues at x = 1 and x = ∞ respectively. • When τ ′ − m <
0, there are topological saddle points with σ >
0, whose contributionreproduces the residue at x = 0.There is no Coulomb branch at τ ′ − m = 0 and the twisted index is independent of τ . Thisreproduces precisely the JK residue prescription with charges (6.11) and sign η = sign( τ ′ − m ). Theresult is independent of η for each flux m by construction. SU (2) In this section, we briefly explore the extensions of our results to G = SU (2) Chern-Simons mattertheories, highlighting some novelties and difficulties compared to G = U (1). For simplicity, wefocus on SU (2) Chern-Simons theory at level k coupled to N fundamental chiral multiplets. The contour integral formula reads I = X m ∈ Z πi I Γ d xx H ( x ) g Z ( x, m ) , (7.1)where Z ( x, m ) = 12 x m k (cid:0) − x (cid:1) − g (cid:0) − x − (cid:1) − g N Y i =1 (1 − x − y − i ) m − ( g − r i − (1 − xy − i ) m +( g − r i − , (7.2) The quantization condition requires k + N ∈ Z . In this section, we will assume N ∈ Z and k ∈ Z . – 22 –ith m valued in the co-character lattice of G = SU (2). We also have H = k + N X i =1 xy − i − xy − i ) + 1 + x − y − i − x − y − i ) ! . (7.3)The factor (1 − x ) − g (1 − x − ) − g in equation (7.2) originates from the W -bosons in breakingthe gauge group from SU (2) to U (1). The corresponding poles for g > x ± → Q + = − k +eff ,U (1) = − k ,Q − = k − eff ,U (1) = 2 k , (7.4)and the contour encloses poles whose charges are of the same sign of the auxiliary parameter η .This prescription becomes ambiguous when k = 0. In analogy to the U (1) theories discussed above, we now study the saddle point equation in orderto interpret the index geometrically. The supersymmetric saddle points are determined by theequations 1 e ∗ F A + N X i =1 φ † i φ i − t σ k eff ( σ )2 π = 0 ,d A σ = 0 , ¯ ∂ A φ i = 0 , ( σ + m i ) φ i = 0 , (7.5)where σ = σ a T a . For each i , the chiral multiplet φ i transforms as a section of the vector bundle K r i / ⊗ E i . Suppose that the mass m i is generic. The SU (2) gauge group is unbroken only at σ = 0, where the effective level k eff is given by k eff ,SU (2) = k + 12 X i T ( R i )sign( m i ) . (7.6)For σ = 0, the gauge group is broken to U (1) and the effective levels are given by (7.4).These equations exhibit the same types of solutions as their abelian counterparts: there arevortex and topological saddles. For general k , there is an important subtlety due to the existenceof topological saddles with both σ = 0 and φ = 0. The SU (2) gauge symmetry is unbroken and therelevant moduli space is that of SU (2) flat connections on Σ. They are not topologically disjointfrom vortex and topological saddles where the gauge group is broken to U (1).In order to circumvent this issue, in the discussion below, we deform the moduli space byturning on a small 1d FI parameter τ ∈ R that explicitly breaks SU (2) to U (1). This modifies thetop line of equation (7.5) to e ∗ F A + N X i =1 φ † i φ i − t σ k eff ( σ )2 π = τ . (7.8) For convenience, in the discussion below we will take the normalised τ defined in (2.18) to be | τ ′ | < , (7.7)although in principle any value of τ ′ is allowed. – 23 –or k = 0, unless τ ′ / ∈ Z , the topological saddles with SU (2) unbroken are removed and the twistedindex is well-defined and independent of τ . For k = 0, there exist a non-compact Coulomb branchparametrised by constant σ := t σ and therefore the twisted index may jump as we vary τ ′ acrossinteger values. Vortex saddles are solutions with φ i = 0 for some i . The final equation in (7.5) implies that σ = σ = 0 and σ = ± m i . This equation breaks the gauge group SU (2) → U (1) by itself.Accordingly, the vector bundle decomposes into a sum of line bundles, whose i -th summand is E i = L i ⊕ L − i . (7.9)We denote by φ ,i and φ ,i the sections of L i and L − i respectively. In the limit t → m i , the moduli space of vortex saddles is a union of disjointcomponents M + m ,i and M − m ,i , which are parametrised by solutions ( A, φ ) that satisfy1 e ∗ F A + 12 | φ ,i | = τ , ∂ A φ ,i = 0 (7.10)and 1 e ∗ F A − | φ ,i | = τ , ∂ A φ ,i = 0 (7.11)respectively. Here A is a connection on the line bundle L i of degree m . We then have M + m ,i = ( Σ d + i if m < τ ′ ∅ if m > τ ′ , M − m ,i = ( Σ d − i if m > τ ′ ∅ if m < τ ′ , (7.12)with d ± i := ± m + r i ( g − . (7.13)The contribution to the twisted index from the vortex saddles can be studied in the same wayas in section 4.2. Each component contributes I ± i = Z ˆ A ( M ± m ,i ) ch( L k )ch(ˆ ∧ • E ) . (7.14)Here the index bundle E is the contribution from the fluctuation of the chiral and vector multiplets: E = E • V ⊗ O j = i E • + ,j ⊗ N O j =1 E •− ,j , (7.15)where E •± ,j is the contribution from the chiral multiplets φ ,i and φ ,i : E •± ,j = R • π ∗ ( L ± ⊗ K r j / ) . (7.16)The first term E • V is the contribution from fluctuations of W-bosons. This decomposes into 1d N = (0 ,
2) chiaral and Fermi multiplets: • chiral multiplets ( a z , Λ z ) : E V := H (Σ , L ⊕ L − ) , • Fermi multiplets ( σ, λ ) : E V := H (Σ , L ⊕ L − ) ,– 24 –hich can be written in terms of the universal bundle as E • V = R • π ∗ ( L ⊕ L − ) . (7.17)From this we can compute1ch(ˆ ∧ • E ) = ( e − η − e η ) m − g +1 ( e η − e − η ) − m − g +1 Y j = i (cid:18) ( e − η z j ) / − e − η z j (cid:19) d + j − g +1 N Y j =1 (cid:18) ( e η z j ) / − e η z j (cid:19) d − j − g +1 exp θ X j = i (cid:18)
12 + e − η z j − e − η z j (cid:19) + θ N X j =1 (cid:18)
12 + e η z j − e η z j (cid:19) , (7.18)where we defined z j = y j /y i with y i = e − πm i as before.Finally, the SU (2) Chern-Simons term at level k generates a holomorphic line bundle L k m ,i onthe moduli space M ± m ,i , which contributesch( L k m ,i ) = e k ( θ − m η ) . (7.19)By the same manipulations that follow (4.22), we get an agreement of the integrand with (7.1).The integration picks the poles of the chiral multiplets according to the alignment of η and τ ′ − m ,which for m = 0 and τ ′ chosen as in (7.7) is the same as − m . Topological saddles are characterised by φ i = 0 for all i . With τ turned on, we only have solutionswith σ = 0 and the vector bundle decomposed into a sum of line bundles E = L ⊕ L − . The BPS equations in the limit t → | σ | → ∞ with σ = t σ kept finite. The equations uniquely determine the value of σ . Therefore this componentof the moduli space is parametrised by connections A of the U (1) bundle L satisfying ∗ F A = 2 π vol(Σ) m , (7.20)where deg( L ) = m . The low-energy theory is described by the U (1) Chern-Simons theory with k ± eff ,U (1) = 2 k . (7.21)The geometric description of the twisted index can be given in the same way as in section 5.Algebraically, this topological saddles can be described by the Picard stack M m = Pic m (Σ) , (7.22)which contributes to the twisted index as Z ˆ A ( M m ) ch(Θ k )ch(ˆ ∧ • E ) . (7.23)As discussed in section 5.2, this integral decomposes into an integral over the moduli space M m and a contour integral around x = 0 , ∞ that projects onto C ∗ -invariant contributions. In the limit τ →
0, the moduli space of flat SU (2) connections appears, as we discussed above. – 25 –he factor ch(ˆ ∧ • E ) − is the contribution from fluctuations of chiral and vector multiplets on M m . Following the discussion in section 5.2, we find1ch(ˆ ∧ • E ) = ( x − − x ) m − g +1 ( x − x − ) − m − g +1 N Y i =1 (cid:18) ( xy i ) / − xy i (cid:19) m +( g − r i − (cid:18) ( x − y i ) / − x − y i (cid:19) − m +( g − r i − exp " N X i =1 θ (cid:18)
12 + xy i − xy i (cid:19) + θ (cid:18)
12 + x − y i − x − y i (cid:19) . (7.24)The effective U (1) CS coupling contributesch(Θ k ) = e kθ e k m . (7.25)The result agrees once again with (7.1). Projecting onto C ∗ -invariant configuration is done bytaking residues at σ → ±∞ where x = e − πσ as usual. Poles are then picked up according to theassignment of charges (2.17), which we rewrite for convenience: Q = ( − k if k = 0 m − τ ′ otherwise , (7.26a) Q ∞ = ( +2 k if k = 0 m − τ ′ otherwise . (7.26b)The final answer for the index computation may or may not depend on τ , depending on thepresence of vectormultiplet poles and on whether the level k vanishes. We conclude by studyingsome examples on P at level k = 0, which are immune of these problems. At genus g = 0, all holomorphic vector bundles split as a sum of holomorphic line bundles by theBirkhoff-Groethendieck theorem. Therefore, the topological vacua that preserve full SU (2) gaugegroup are absent and the subtleties described in the paragraph below (7.6) do not arise. This isreflected in the fact that there are no vectormulitplet poles in the integrand of the contour formula.The SU (2) theory can then be viewed as a 1d U (1) theory prior to the insertion of the regulator.For k = 0, we do not expect a dependence of the result on the regulator τ .Let us consider the example of G = SU (2) k theory with N = 2 fundamental chiral multiplets.The holomorphic vector bundle E decomposes into a sum of line bundle E = L ⊕ L − with deg( L ) = m ∈ Z . The D-term equation reduces to1 e ∗ F A + 12 X i =1 (cid:0) | φ ,i | − | φ ,i | (cid:1) − t kσ π = 0 , (7.27)We find that the solutions of the BPS equations exists in the following regions, depending on signof m : • m > σ = m i and topological solutions at σ → ∞ . • m < σ = − m i . and topological solutions at σ → −∞ . • m = 0 : (cid:20) τ > σ = − m i and topological solution at σ → −∞ τ < σ = m i and topological solutions at σ → ∞ . We remind that for convenience, we take the regulator to be small. – 26 –n the other hand, the JK charges for the poles at infinities are assigned as Q = − k , Q ∞ = 2 k. (7.28)Let Q + be the collection of positive U (1) charges at σ = − m i and Q − be the negative U (1) chargesat σ = m i . Let also η >
0. The JK residue integral then picks up the poles that correspondsto charges Q + and Q ∞ for all values of m ∈ Z and τ . By the residue theorem, we find that thisprescription agrees with the geometric interpretation of the indices at each flux sector. Acknowledgments
MB gratefully acknowledges support from the EPSRC Early Career Fellowship EP/T004746/1“Supersymmetric Gauge Theory and Enumerative Geometry”.
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