Localization on Hopf surfaces
aa r X i v : . [ h e p - t h ] J u l KCL-MTH-14-09July 29, 2014
Localization on Hopf surfaces
Benjamin Assel, Davide Cassani and Dario Martelli
Department of Mathematics, King’s College London,The Strand, London WC2R 2LS, United Kingdom
Abstract
We discuss localization of the path integral for supersymmetric gauge theories withan R-symmetry on Hermitian four-manifolds. After presenting the localization locusequations for the general case, we focus on backgrounds with S × S topology, ad-mitting two supercharges of opposite R-charge. These are Hopf surfaces, with twocomplex structure moduli p, q . We compute the localized partition function on suchHopf surfaces, allowing for a very large class of Hermitian metrics, and prove that this isproportional to the supersymmetric index with fugacities p, q . Using zeta function reg-ularisation, we determine the exact proportionality factor, finding that it depends onlyon p, q , and on the anomaly coefficients a , c of the field theory. This may be interpretedas a supersymmetric Casimir energy, and provides the leading order contribution tothe partition function in a large N expansion. ontents N limit . . . . . . . . . . . . . . . 38 S × S v with arbitrary b , b
47D Non-direct product metric 50E Proof that ( z , z ) ∈ C − (0 ,
52F Reduction of the 4d supersymmetry equations to 3d 54 Regularization of one-loop determinants 58
The complete information of a quantum field theory is contained in the generating functionalof correlation functions; however, in an interacting theory this is very hard to computeexactly. In favourable situations the technique of supersymmetric localization [1] allows oneto perform exact non-perturbative computations of special types of generating functionalsand other observables. In particular, in certain supersymmetric field theories defined oncompact Riemannian manifolds, it is possible to evaluate a class of BPS observables byreducing the functional integrals over all the field configurations to Gaussian integrals arounda supersymmetric locus. In this paper we will present a detailed calculation of the partitionfunction of N = 1 supersymmetric field theories, defined on a four-dimensional complexmanifold.A systematic procedure for constructing supersymmetric field theories in a fixed back-ground geometry has been put forward in [2]. In four dimensions, one way to obtain super-symmetric theories is by taking a suitable limit of new minimal supergravity [3, 4, 5], thatcontains two auxiliary vector fields, one of which is the gauge field for a local chiral symme-try. In such rigid limit, these, together with the metric, provide background fields coupledto a supersymmetric gauge theory with an R-symmetry, comprising ordinary vector andchiral multiplets. Explicit expressions for supersymmetric Lagrangians and supersymmetrytransformations can be obtained from [3, 4, 5] and will be presented below.Supersymmetric theories may be defined only on backgrounds admitting solutions tocertain Killing spinor equations (see (2.1), (2.2) below), which in Euclidean signature areequivalent to the requirement that the four-dimensional manifold is complex and the metricHermitian [6, 7]. In this paper we will construct Lagrangians that are total supersym-metry variations, and therefore can be utilised to implement the localization technique in N = 1 field theories defined on arbitrary Hermitian manifolds. We will then employ theseto compute in closed form the partition function of general supersymmetric gauge theories,in the case that the manifold admits at least two supercharges of opposite R-charge, and hasthe topology of S × S . These manifolds are then Hopf surfaces , with complex structurecharacterised by two parameters p, q , that we will denote as H p,q ≃ S × S .The main result of this paper is the derivation of a formula for the partition function Z of an N = 1 supersymmetric field theory with an R-symmetry, defined on a Hopf surface H p,q , endowed with a very general Hermitian metric. Namely, we will show that Z [ H p,q ] = e −F ( p,q ) I ( p, q ) , (1.1)2here I ( p, q ) is the supersymmetric index with p, q fugacities and F ( p, q ) is a function ofthe complex structure parameters given by F ( p, q ) = 4 π (cid:18) | b | + | b | − | b | + | b || b || b | (cid:19) ( a − c ) + 4 π
27 ( | b | + | b | ) | b || b | (3 c − a ) , (1.2)where p = e − π | b | , q = e − π | b | , and a , c are the R-symmetry traces, appearing in the Weyland R-symmetry anomalies of superconformal field theories [8, 9]. As we will explain, the realparameters b , b characterise an almost contact structure in the three-dimensional theoryobtained from dimensional reduction on S , allowing us to make contact with the resultsof [10], where the localized partition function of three-dimensional N = 2 supersymmetricgauge theories was computed. The supersymmetric index was introduced in [11, 12, 13] inthe context of superconformal field theories, and has been used in [14, 15, 16, 17] (and manyothers) to test non-perturbative dualities.The authors of [18] have shown that very generally the path integral of a supersymmetricfield theory defined on a Hermitian manifold can depend only on complex structure defor-mations of the background. Based on this result, they have conjectured that the partitionfunction defined on a Hopf surface H p,q is proportional to the supersymmetric index I ( p, q ),up to possible local counterterms. Our explicit computation confirms the validity of thisconjecture, although we expect that the ratio e −F between these two quantities genericallycannot be expressed in terms of local counterterms. This provides an interesting quan-tity characterising a four-dimensional supersymmetric field theory, that we will refer to as supersymmetric Casimir energy .Some progress towards obtaining the partition function (1.1) using localization was madein [19], where the one-loop determinant of an N = 1 chiral multiplet on a Hopf surface wascomputed. In particular, in this reference the authors considered a specific Hermitian metriccompatible with | p | = | q | . Localization computations of supersymmetric gauge theories on S × S with a conformally flat metric have appeared in [20, 21].One of our motivations for computing the partition function from first principles arosefrom holography [22]. In situations where there exist simple AdS gravity duals, the gravityside predicts that the logarithm of the partition function, at leading order in a large N expansion, should be proportional to N . In one dimension lower, the analogous problem iswell understood: the N / scaling of the on-shell action on the gravity side can be matchedto the large N limit of the localized free energy [23]; it has been shown in [24] that thisagreement can be extended to a broad class of N = 2 gauge theories, whose partitionfunction was computed in [10]. In four dimensions the supersymmetric index scales like N at large N [12, 25], implying that the N scaling of the logarithm of the partition functionmust arise as an extra contribution. We find that this contribution is contained in (1.2). For simplicity we will restrict attention to the case where the parameters p, q are real. c = a at leading order in N ), we obtain a prediction for the holographically renormalised action offive-dimensional gauged supergravity, evaluated on a solution dual to a supersymmetric fieldtheory defined on a Hopf suface H p,q = ∂M . In particular, we expect that for a solution M ≃ S × R , the renormalised on-shell action will be given by S
5d sugra [ M ] = π G ( | b | + | b | ) | b || b | , (1.3)up to finite local counterterms.The rest of this paper is organized as follows. Section 2 contains a discussion of thebackground geometry of four-manifolds allowing for at least one supercharge, and sets thestage for implementing localization in general four-dimensional N = 1 gauge theories with anR-symmetry. In section 3 we discuss the specific background geometry for Hopf surfaces with S × S topology and U (1) isometries. In section 4 we perform the localization computationon the Hopf surfaces. In section 5 we compare our result for the exact partition function withthe supersymmetric index. We emphasize the presence of the extra pre-factor and define thesupersymmetric Casimir energy. We also comment on the implications of our results forgravity duals. We conclude in section 6 by outlining some perspectives for future work.We also included several appendices. Appendix A contains our conventions. Appendix Bprovides a proof that the partition function is independent of the conformal factor of themetric. Appendix C describes familiar examples of the background geometries considered insection 3. Appendix D elaborates on possible generalizations of our results by consideringnon-direct product metrics, associated to complex values of the complex structure moduli.Appendix E includes computations used in section 3. Appendix F contains details of thereduction of four-dimensional backgrounds to three dimensions. Appendix G contains thedetails of the regularization of one-loop determinants. We begin our analysis by reviewing and elaborating results about the new minimal formu-lation of rigid supersymmetry on curved space. Our considerations in this section will beentirely local, while global properties will be discussed in section 3.
As shown in [2], in the presence of an R-symmetry the supersymmetry transformations andthe Lagrangian of a field theory defined on a curved manifold can be derived by couplingthe theory to the new minimal formulation of off-shell supergravity [3, 4, 5] and freezing the4elds in the gravity multiplet to background values, in such a way that the gravitino variationvanishes. The bosonic fields in the gravity multiplet are the metric and two auxiliary vectorfields A µ , V µ ; after the rigid limit, these play the role of background fields. In Euclideansignature, A µ and V µ are allowed to take complex values, whereas for simplicity the metricwill be constrained to be real.The real part of A µ is associated to u (1) R R-symmetry transformations, and transforms(locally) as a gauge field, while the imaginary part must be a well-defined one-form. Beingthe Hodge dual of a closed three-form, V = ∗ d B is assumed to be a globally defined one-form, constrained by ∇ µ V µ = 0. In Euclidean signature, the condition that the gravitinovariation vanishes corresponds to two independent first-order differential equations( ∇ µ − iA µ ) ζ + iV µ ζ + iV ν σ µν ζ = 0 , (2.1)( ∇ µ + iA µ ) e ζ − iV µ e ζ − iV ν e σ µν e ζ = 0 , (2.2)where ζ and e ζ are two-component complex spinors of opposite chirality, and with oppositecharge under the background gauge field A , associated with the R-symmetry. Solutionsto these equations are either identically zero or nowhere vanishing. Throughout the paper,spinors with no tilde transform in the ( , ) representation of the Spin (4) = SU (2) + × SU (2) − Lorentz group, while spinors with a tilde transform in the ( , ). See appendix A for furtherdetails on our notation and conventions.It was shown in [6, 7] that a necessary and sufficient condition for a Riemannian four-manifold to have a solution ζ to (2.1) is that it admits an integrable complex structure J µν .Lowering an index with the Hermitian metric, the corresponding fundamental two-form canbe constructed as a spinor bilinear, J µν = 2 i | ζ | ζ † σ µν ζ . (2.3)One can also introduce a complex two-form bilinear as P µν = ζ σ µν ζ , which is anti-holomorphicwith respect to the complex structure J µν . Together these define a U (2) structure on thefour-manifold. The solution of (2.1) can be expressed in terms of a nowhere vanishing com-plex function s as ζ α = p s (cid:18) (cid:19) , and the background fields are determined by V µ = − ∇ ρ J ρµ + U µ , (2.4) A µ = A cµ −
14 ( δ νµ − iJ µν ) ∇ ρ J ρν + 32 U µ , (2.5)where A cµ is defined as A cµ = 14 J µν ∂ ν log √ g − i ∂ µ log s , (2.6)5ith g the determinant of the metric in complex coordinates. The solution contains anarbitrariness parametrised by the vector field U µ , which is constrained to be holomorphic,namely J µν U ν = iU µ , and to obey ∇ µ U µ = 0. Note that the combination A cs µ ≡ A µ − V µ is independent of the choice of U µ . Of course a solution e ζ to (2.2) is also equivalent to theexistence of an integrable complex structure defined by e J µν = 2 i | e ζ | e ζ † e σ µν e ζ , (2.7)and leads to expressions for the background fields A µ and V µ analogous to the ones above,with a few sign changes; see [7] for the explicit formulae.When there exist both a non-zero solution ζ to (2.1) and a non-zero solution e ζ to (2.2),namely in the presence of two supercharges of opposite R-charge, the four-dimensional man-ifold is endowed with a pair of commuting complex structures J µν , e J µν , inducing oppositeorientations, and subject to certain compatibility conditions [7]. This means that the man-ifold admits a specific ambihermitian structure [26]. Solutions with two supercharges ofopposite R-charge may be more efficiently characterised by a complex vector field K µ , con-structed as a spinor bilinear as K µ = ζ σ µ e ζ . (2.8)In particular, one can show that K µ is holomorphic with respect to both complex struc-tures and satisfies the algebraic property K µ K µ = 0 as well as the differential condition ∇ ( µ K ν ) = 0, therefore it comprises two real Killing vectors. If K µ commutes with its com-plex conjugate, K ν ∇ ν K µ − K ν ∇ ν K µ = 0, then the vector field U µ above is restricted to takethe form U µ = κK µ , where κ is a complex function such that K µ ∂ µ κ = 0, but otherwisearbitrary [7]. Moreover, introducing adapted complex coordinates w, z (holomorphic withrespect to J µν ) such that the complex Killing vector is K = ∂ w , the metric takes the formd s = Ω [(d w + h d z )(d ¯ w + ¯ h d¯ z ) + c d z d¯ z ] , (2.9)where Ω( z, ¯ z ) and c ( z, ¯ z ) are real, positive functions, while h ( z, ¯ z ) is a complex function. Itis useful to introduce the complex frame e = Ω c d z , e = Ω(d w + h d z ) . (2.10) We denote this as A cs as it is the background field arising when the theory is coupled to conformalsupergravity. Note that the similar term “bihermitian” refers to the different case where the two commuting complexstructures induce the same orientation on the manifold. It is shown in [7] that if [
K, K ] = 0, then the manifold is locally isometric to R × S , with the standardround metric on S . Here e and e are exchanged with respect to those appearing in [7]. This implies that the e ζ given in(2.14) below has swapped components with respect to the one in [7].
6e choose the orientation by fixing the volume form as vol = − e ∧ ¯ e ¯1 ∧ e ∧ ¯ e ¯2 . Then, asa one-form, K reads K = 12 Ω (d ¯ w + ¯ h d¯ z ) = 12 Ω ¯ e ¯2 , (2.11)and the real two-forms associated with the commuting complex structures are J = 2 i Ω K ∧ K − i c d z ∧ d¯ z = − i (cid:16) e ∧ ¯ e ¯1 + e ∧ ¯ e ¯2 (cid:17) , e J = 2 i Ω K ∧ K + i c d z ∧ d¯ z = i (cid:16) e ∧ ¯ e ¯1 − e ∧ ¯ e ¯2 (cid:17) . (2.12)With our choice of orientation J is self-dual while e J is anti-self-dual. Following [7], we willrequire also that K µ ∂ µ κ = K µ ∂ µ | s | = K µ ∂ µ | s | = 0 , (2.13)so that both K and K preserve A and V in addition to the metric. With these restrictions,the functions κ and | s | do not depend on w, ¯ w , but can still have an arbitrary dependenceon z and ¯ z . In the frame (2.10), the spinors ζ and e ζ solving (2.1) and (2.2) read ζ α = r s (cid:18) (cid:19) , e ζ ˙ α = Ω √ s (cid:18) (cid:19) . (2.14)Let us present more explicit formulae for A and V . Noting that ∇ ρ J ρµ d x µ = ∗ d ∗ J = ∗ d J and using the expression for J in (2.12), simple manipulations show that (2.4) and (2.5) canbe written as V = d c log Ω + 2Ω c Im ( ∂ ¯ z h K ) + κK , (2.15) A = 12 d c log (cid:0) Ω c (cid:1) − i (cid:0) Ω − s (cid:1) + (cid:18) κ − i Ω c ∂ ¯ z h (cid:19) K , (2.16)where we used √ g = Ω c .For later applications it is important to observe that we can use the freedom in choosing κ and s to arrange for A to be real . Indeed, requiring Im A = 0 in (2.16) and separating thedifferent components, we obtain the conditions | s | = Ω , κ = 2 i c ∂ ¯ z h , (2.17) Our convention for the Hodge star is ∗ θ a ...a k = − k )! ǫ a ...a k a k +1 ...a θ a k +1 ...a , where θ a denotes a realframe. This is related to the complex frame as e = θ + iθ , e = θ + iθ ; so the volume form introducedabove is vol = θ ∧ θ ∧ θ ∧ θ . For any function f we define d c f = J µν ∂ ν f d x µ = − i ( ∂ − ¯ ∂ ) f . | s | . With these choices of κ and | s | ,the gauge field A takes the simple form A = 12 d c log(Ω c ) + 12 d ω , (2.18)where ω denotes the phase of s , i.e. s = | s | e iω . Note that ω has not been fixed so far, whileit will be determined by our global analysis in section 3. The one-form V in general remainscomplex V = d c log Ω − i c ∂ ¯ z h K + i Ω c ∂ z ¯ h K . (2.19)Recalling that Ω and c are real and depend only on the z , ¯ z coordinates, we can also writemore explicitly A = Im (cid:2) ∂ z log(Ω c ) d z (cid:3) + 12 d ω ,V = 2 Im[ ∂ z log Ω d z ] − i c ∂ ¯ z h K + i Ω c ∂ z ¯ h K . (2.20)Finally, the spinors (2.14) take the form ζ α = r Ω2 e i ω (cid:18) (cid:19) , e ζ ˙ α = r Ω2 e − i ω (cid:18) (cid:19) . (2.21) In this section we present the supersymmetry variations and relevant Lagrangians of thetheories that we consider in this paper. In Euclidean signature, defining N = 1 supersym-metry requires to double the number of degrees of freedom in each multiplet. This can berealized formally by thinking about a given field and its Hermitian conjugate as transformingindependently under supersymmetry. To define the path integral over the fields of a mul-tiplet, one then has to make a choice of reality conditions, reducing the number of degreesof freedom in a multiplet to the usual one. In the following, we will first consider a vectormultiplet and then a chiral multiplet. The N = 1 vector multiplet contains a gauge field A µ , a pair of two-component complexspinors λ , e λ of opposite chirality and an auxiliary field D , all transforming in the adjointrepresentation of the gauge group G . As already noted, a priori in Euclidean signature thefermionic fields λ , e λ are independent, and the bosonic fields A µ , D are not Hermitian. We8efine a covariant derivative as D µ = ∇ µ − i A µ · − iq R A µ , (2.22)where · denotes the action in the relevant representation, and the R-charges q R of the fields( A µ , λ, e λ, D ) are given respectively by (0 , , − , δ A µ = iζ σ µ e λ + i e ζ e σ µ λ ,δλ = F µν σ µν ζ + iDζ ,δ e λ = F µν e σ µν e ζ − iD e ζ ,δD = − ζ σ µ (cid:0) D µ e λ − i V µ e λ (cid:1) + e ζ e σ µ (cid:0) D µ λ + i V µ λ (cid:1) , (2.23)where F µν ≡ ∂ µ A ν − ∂ ν A µ − i [ A µ , A ν ]. Note that the two independent spinorial parameters ζ , e ζ need to be solutions to the equations (2.1), (2.2), and are commuting variables. Itis understood that when one of the two equations only admits the trivial solution, thecorresponding spinor is set to zero in the supersymmetry transformations. The fermionicfields λ , e λ are anti-commuting, and therefore correspondingly the supersymmetry variation δ is defined as a Grassmann-odd operator. Note also that in the above transformations only theconformal invariant and U µ -independent combination of background fields A cs µ ≡ A µ − V µ appears, in the covariant derivative D cs µ ≡ ∇ µ − i A µ · − iq R A cs µ .The supersymmetry algebra is given by { δ ζ , δ ζ } = { δ e ζ , δ e ζ } = 0 , [ δ ζ , δ K ] = [ δ e ζ , δ K ] = 0 , { δ ζ , δ e ζ } = 2 i δ K , (2.24)where δ ζ (respectively, δ e ζ ) means that e ζ (respectively, ζ ) is set to zero in the supersymmetrytransformations (2.23), and on a field of R-charge q R we have δ K = L K − iK µ A µ · − iq R K µ A µ ,where L K is the Lie derivative along K . If there is only one Killing spinor ζ , then one justhas δ ζ = 0 .A tedious calculation shows that the Lagrangian L vector = Tr (cid:20) F µν F µν − D + i λ σ µ D cs µ e λ + i e λ e σ µ D cs µ λ (cid:21) (2.25)is invariant under the supersymmetry transformations (2.23). Here Tr is the trace in theadjoint representation of the gauge group. We will show momentarily that if both spinors9 , e ζ exist, then this Lagrangian is the sum of two supersymmetry variations; this will beimportant for applying the localization argument.Given that in Euclidean signature the degrees of freedom are doubled, it is conceptuallyclearer to impose reality conditions on the fields only after computing the supersymmetryvariations. Therefore, to define various supersymmetry-exact terms, we introduce an invo-lution ‡ acting as ( A µ , D ) ‡ = ( A µ , − D ) , ζ ‡ = ζ † , (2.26)and as complex conjugation on numbers. Then we define L (+)vector = δ ζ V (+) = − δ ζ (cid:18) | ζ | Tr ( δ ζ λ ) ‡ λ (cid:19) = 14 | ζ | Tr ( δ ζ λ ) ‡ δ ζ λ − | ζ | Tr δ ζ (cid:0) ( δ ζ λ ) ‡ (cid:1) λ ≡ δV (+)bos + δV (+)fer . (2.27)The bosonic term is straightforward to evaluate and reads δV (+)bos = 14 Tr (cid:0) F (+) µν F (+) µν − D (cid:1) , (2.28)where F ( ± ) µν = ( F ± ∗F ) µν . The fermionic term reads δV (+)fer = 14 | ζ | Tr (cid:2) − ( ζ † σ µν λ ) δ ζ F µν + i ( ζ † λ ) δ ζ D (cid:3) (2.29)and with some manipulations can be rewritten as δV (+)fer = Tr h i λ σ µ (cid:16) D µ e λ − i V µ e λ (cid:17)i . (2.30)To obtain this we used the following expression for the supersymmetry variation of the gaugefield strength F µν δ F µν = 2 i ζ σ [ ν D µ ] e λ + V [ µ ζ σ ν ] e λ + ǫ µνκλ V κ ( ζ σ λ e λ )+ 2 i e ζ e σ [ ν D µ ] λ − V [ µ e ζ e σ ν ] λ + ǫ µνκλ V κ ( e ζ e σ λ λ ) . (2.31)We have thus shown that L (+)vector = Tr (cid:18) F (+) µν F (+) µν − D + i λ σ µ D cs µ e λ (cid:19) . (2.32) We will not need to define the action of ‡ on λ and e λ .
10f there exists a second Killing spinor e ζ , then the previous computations can be repeatedwith trivial modifications. Namely, we can define L ( − )vector = δ e ζ V ( − ) = − δ e ζ (cid:18) | e ζ | Tr ( δ e ζ e λ ) ‡ e λ (cid:19) = 14 | e ζ | Tr ( δ e ζ e λ ) ‡ δ e ζ e λ − | e ζ | Tr δ e ζ (cid:16) ( δ e ζ e λ ) ‡ (cid:17) e λ ≡ δV ( − )bos + δV ( − )fer , (2.33)with L ( − )vector = Tr (cid:18) F ( − ) µν F ( − ) µν − D + i e λ e σ µ D cs µ λ (cid:19) . (2.34)The sum of the two terms is L (+)vector + L ( − )vector = Tr h F µν F µν − D + i λ σ µ D cs µ e λ + i e λ e σ µ D cs µ λ i = L vector . (2.35)Therefore, we have shown that the vector multiplet Lagrangian L vector in (2.25) is the sumof a δ ζ -exact term and a δ e ζ -exact term. Note that to derive this result we have not imposedany reality condition, and correspondingly at this stage the bosonic part of the Lagrangianis not positive semi-definite.In order to apply the localization arguments, it will be important that L (+)vector and L ( − )vector are separately invariant under both supersymmetries associated with ζ and e ζ , so that δ ζ L ( − )vector = δ ζ δ e ζ V ( − ) = tot der ,δ e ζ L (+)vector = δ e ζ δ ζ V (+) = tot der , (2.36)where “tot der” denotes a total derivative. Recalling that δ ζ = δ e ζ = 0, these are equivalentto the fact that the vector multiplet Lagrangian is invariant under both supersymmetryvariations, namely δ ζ L vector = δ e ζ L vector = tot der . The N = 1 chiral multiplet contains two complex scalars φ, e φ , a pair of two-componentcomplex spinors ψ , e ψ of opposite chirality, and two complex auxiliary fields F, e F . As forthe fields of the vector multiplet, in Euclidean signature the fermionic fields ψ , e ψ and thecomplex scalars φ, e φ , and F, e F are all independent. The fields ( φ, ψ, F ) transform in arepresentation R , while ( e φ, e ψ, e F ) transform in the conjugate representation R ∗ . The R-charges q R entering in (2.22) for the fields ( φ, ψ, F, e φ, e ψ, e F ) are given by ( r, r − , r − , − r, − r +1 , − r + 2) respectively, with r arbitrary. The supersymmetry transformations of the fields11n the multiplet can be read off from [4, 2, 27] and are δφ = √ ζ ψ ,δψ = √ F ζ + i √ σ µ e ζ ) D µ φ ,δF = i √ e ζ e σ µ (cid:16) D µ ψ − i V µ ψ (cid:17) − i ( e ζ e λ ) φ ,δ e φ = √ e ζ e ψ ,δ e ψ = √ e F e ζ + i √ e σ µ ζ ) D µ e φ ,δ e F = i √ ζ σ µ (cid:16) D µ e ψ + i V µ e ψ (cid:17) + 2 i e φ ( ζ λ ) . (2.37)These preserve the Lagrangian L chiral = D µ e φD µ φ + V µ (cid:0) iD µ e φ φ − i e φD µ φ (cid:1) + r R + 6 V µ V µ ) e φφ + e φDφ − e F F + i e ψ e σ µ D µ ψ + 12 V µ e ψ e σ µ ψ + i √ (cid:0) e φλψ − e ψ e λφ (cid:1) . (2.38)This depends on both background fields A and V , except when the R-charge takes thevalue r = 2 /
3, in which case these only appear in the combination A cs = A − V andthe Lagrangian is conformal invariant. Below we will show that the existence of a singlesupersymmetry parameter ζ is enough to express L chiral as a total supersymmetry variation,up to an irrelevant boundary term.In general, one can consider several chiral multiplets with different R-charges r I , withLagrangian given by the sum of the (2.38) for each multiplet, and also add to this a super-potential term L W , as in flat space. The explicit expression in component notation is givenin [2]. The superpotential W can be an arbitrary holomorphic function of the fields φ I , andin order not to break the R-symmetry of the theory it must be homogeneous of degree twoin the R-charges. This follows from the fact that the fermions ψ I have R-charges r I − ∂ W∂φ I ∂φ J ψ I ψ J ∈ L W , (2.39)whose R-charge is r [ W ] − r I − r J + ( r I −
1) + ( r J − In this paper we assume that W is a polynomial in the fields φ I . I one obtains e F I = ∂W∂φ I , F I = ∂ f W∂ e φ I . (2.40)In order to write the supersymmetry-exact terms we extend the action of the involution ‡ used for the vector multiplet to the bosonic fields of the chiral multiplet as( φ, F, e φ, e F ) ‡ = ( e φ, − e F , φ, − F ) . (2.41)While we will not need to define how ‡ acts on ψ , e ψ and on V µ , we will need its action on A µ . There are two natural definitions we can take, which in general are not equivalent. If wedefine A ‡ µ = A µ , then the computation below shows that the Lagrangian L chiral is δ ζ -exact (upto a boundary term) without any restriction on A µ . However, notice that this Lagrangianis not invariant under changes of U µ , and its bosonic part is not positive semi-definite evenafter imposing reality conditions on the dynamical fields. If instead we define A ‡ µ = A † µ , thenthe localizing term that we will choose in the next section does not depend on U µ and itsbosonic part is positive semi-definite after choosing suitable reality conditions. However, forcomplex A µ , this does not reconstruct the Lagrangian (2.38). In the following we will assumethat A µ is real, so that the two definitions are equivalent; as showed at the end of section 2.1,this is certainly possible in the presence of two supercharges of opposite R-charge. Later wewill make some comments about relaxing this choice.We consider δV chiral = δ ζ V + δ ζ V + δ ζ V + δ ζ V U = δ ζ (cid:18) | ζ | h ( δ ζ ψ ) ‡ ψ − e ψ ( δ ζ e ψ ) ‡ i(cid:19) + δ ζ V + δ ζ V U = 12 | ζ | h ( δ ζ ψ ) ‡ δ ζ ψ + δ ζ (cid:0) ( δ ζ ψ ) ‡ (cid:1) ψ + ( δ ζ e ψ ) ‡ δ ζ e ψ + e ψ δ ζ (cid:0) ( δ ζ e ψ ) ‡ (cid:1) + 2 iδ ζ (cid:0) e φ ζ † λ φ (cid:1) − √ δ ζ (cid:0) U µ ζ † σ µ e ψ φ (cid:1)i ≡ δV bos 1 + δV fer 1 + δV bos 2 + δV fer 2 + δV bos 3 + δV fer 3 + δV bos U + δV fer U . (2.42) A priori f W is an arbitrary function of e φ I , but reality conditions will relate this to W . δV bos 1 = − e F F ,δV bos 2 = ( g µν − iJ µν ) D µ e φD ν φ = D µ e φD µ φ − i ( V µ − U µ ) e φD µ φ + r ( R + 6 V µ V µ ) e φφ + J µν e φ F µν φ − i ∇ µ (cid:0) J µν e φD ν φ (cid:1) ,δV bos 3 = − J µν e φ F µν φ + e φDφ ,δV bos U = 2 i U µ D µ e φ φ , (2.43)where to go from the first to the second line in the second term we have used the identity(A.12), and in the last line we used the holomorphicity of U µ , namely J µν U ν = iU µ . As forthe fermionic terms, after some computations involving the Fierz identities in (A.8) we find δV fer 1 = − i D µ e ψ e σ µ ψ − J µν D µ e ψ e σ ν ψ + 12 | ζ | V µ ( ζ σ µ e ψ )( ζ † ψ ) − i √ | ζ | e φ ( ζ λ )( ζ † ψ ) ,δV fer 2 = i e ψ e σ µ D µ ψ − J µν e ψ e σ ν D µ ψ + V µ e ψ e σ µ ψ − | ζ | V µ ( ζ † σ µ e ψ )( ζ ψ ) − i √ e ψ e λ φ ,δV fer 3 = i √ | ζ | e φ ( ζ † λ )( ζ ψ ) δV fer U = − U µ e ψ e σ µ ψ , (2.44)where in the last equality we used holomorphicity of U µ , in the form U µ e σ µ ζ = 0 . The totalfermionic part can be written as δV fer 1 + δV fer 2 + δV fer 3 + δV fer U = i e ψ e σ µ D µ ψ + 12 V µ e ψ e σ µ ψ + i √ e φλψ − e ψ e λφ ) − i D µ (cid:16) ( δ µν − iJ µν ) e ψ e σ ν ψ (cid:17) . (2.45)Adding everything up, we obtain δV chiral = L chiral + ∇ µ Y µ , (2.46)where L chiral is the Lagrangian (2.38) and the total derivative term is Y µ = − iJ µν e φD ν φ − i ( V µ − U µ ) e φφ − i δ µν − iJ µν ) e ψ e σ ν ψ . (2.47)In a similar way, one can see that L chiral is also exact under the variation generated by e ζ .14 .3 Supersymmetric locus equations Let us now discuss how to use the results above to compute the path integral of supersym-metric field theories, using the localization method. The standard localization argumentsrequire to deform the path integral defined by a supersymmetric action by adding a term thatis a supersymmetry variation, and whose bosonic part is positive semi-definite. In this waythe complete path integral is given by the one-loop determinant around the locus where thisbosonic part vanishes. We will address the vector multiplet and chiral multiplet separately.
If the manifold admits one Killing spinor ζ , then we can deform the vector multiplet La-grangian (2.25) by adding to it the δ ζ -exact term (2.27) with an arbitrary parameter t ,namely S = Z d x √ g (cid:0) L vector + t δ ζ V (+) (cid:1) . (2.48)We see that imposing the reality conditions A † µ = A µ , D † = − D implies that the bosonicpart (2.28) of the deformation term is positive semi-definite. The localization locus is givenby δV (+)bos = 0, yielding the conditions F (+) µν = 0 , D = 0 . (2.49)Of course this is also equivalent to δλ = 0, whose independent components give J µν F µν = P µν F µν = 0 = D . The conclusion is that when there exists only one supercharge associatedwith ζ , the localization locus is given by anti-instanton configurations. In the case of a super-charge associated with e ζ , the same argument works by considering the term δ e ζ V ( − ) in (2.33),with the conclusion being that the localization locus is given by instanton configurations.If the manifold admits both ζ and e ζ , then we can deform the vector multiplet Lagrangian(2.25) by adding both the δ ζ -exact and δ e ζ -exact terms, namely S = Z d x √ g (cid:16) L vector + t + δ ζ V (+) + t − δ e ζ V ( − ) (cid:17) . (2.50)To see that the path integral is independent of the parameter t + one notes that δ ζ L vector = δ ζ δ e ζ V ( − ) = tot der. Similarly, the path integral is also independent of the parameter t − . Inthe end one can take t + = t − = t and omit the first term, without affecting the conclusions.The localization locus then is given by δV (+)bos = δV ( − )bos = 0, which is equivalent to the We note that actually the weaker reality condition F (+) † µν = F (+) µν is sufficient to guarantee positivity ofthe deformation term. The condition A † µ = A µ implies that also the original Lagrangian (2.25) has positivebosonic part, but this is not necessary for the localization argument. F µν = 0 , D = 0 , (2.51)so that both the self-dual and the anti-self-dual parts of the gauge field strength vanish. Wewill discuss the solutions to these equations in section 4.1, after specializing the topology ofthe four-dimensional manifold.Notice that the conclusions above are manifestly independent of the choice of the holo-morphic vector field U µ , as well as of the reality properties of the background fields A µ , V µ . If the manifold admits one Killing spinor ζ , then we can deform the chiral multiplet La-grangian (2.38), possibly supplemented by a superpotential, by adding to it the δ ζ -exactterm δ ζ ( V + V ) defined in section 2.2.2. Namely, we consider S = Z d x √ g [ L chiral + L W + t δ ζ ( V + V )] , (2.52)where t is an arbitrary parameter. We must then choose reality conditions such that δV bos 1 and δV bos 2 are positive semi-definite. The former requirement is satisfied imposing e F = − F † . In order to ensure that 2 | ζ | δV bos 2 = ( δ ζ e ψ ) ‡ δ ζ e ψ is positive we require e φ = φ † (hencethe involution ‡ acts as the Hermitian conjugation † ). Note that δV bos 2 does not dependon the background field V µ , thefore there are no reality constraints to impose on the latter.On the other hand, it does depend on the background field A µ , hence its choice may apriori affect positivity. When A is real, the localization locus is defined by the conditions δV bos 1 = δV bos 2 = 0, so that in particular δ ζ ψ = δ ζ e ψ = 0. These are equivalent to F = 0 , J µν D ν e φ = iD µ e φ . (2.53)The second equation means that D µ e φ is a holomorphic vector, or equivalently that e φ is aholomorphic section on a suitable line bundle. These configurations are still very complicatedand in this paper we will not analyse them further. Before moving to the case of twosupercharges, let us briefly comment on the role of U µ . Since this is a holomorphic vector,it drops out from the supersymmetry transformations (2.37), and therefore, if we define A ‡ µ = A † µ , it also drops out from the localizing term and hence from the locus equations(2.53). In this case the positivity property of δV bos 2 is not affected by the choice of U µ .Let us now discuss the case when the manifold admits both ζ and e ζ . In this case, thesame deformation term in (2.52) can be written also as δ e ζ -exact term δ e ζ ( e V + e V ), with tildedand untilded objects appropriately swapped. Assuming the same reality conditions, and in The reason why we are not using simply t L chiral , which is also δ ζ -exact, is that its bosonic part containsthe terms δV bos 3 and δV U , which are not positive after imposing the reality conditions. A µ real (with again no reality condition on V µ ), the localization locusbecomes δ ζ ψ = δ ζ e ψ = δ e ζ ψ = δ e ζ e ψ = 0 . Contracting with appropriate spinors this can berecast into the equations F = 0 , J µν D ν e φ = iD µ e φ , e J µν D ν φ = iD µ φ . (2.54)The last two equations imply K µ D µ φ = K µ D µ e φ = 0. Notice that the locus equations J µν D ν e φ = iD µ e φ and e J µν D ν φ = iD µ φ are derived from two deformation terms that areequal up to a total derivative (exactly equal when integrated over the compact four-manifold).This means that although the two equations may be different locally, they admit the sameglobal solutions.As in the case of the vector multiplet, the solutions to the locus equations (2.54) dependon the global structure of the four-manifold considered. In section 4.1 we will solve (2.54)in the case of M = S × M , where M is topologically a three-sphere, allowing for a verygeneral class of metrics.Before moving to the analysis of the localization on Hopf surfaces, it is interesting to notethat, for manifolds amitting two Killing spinors of opposite R-charge, one can prove thatthe localization locus and one-loop determinants do not depend on the conformal factor Ωof the metric. This argument is presented in appendix B. It is in agreement with [19], thatshowed that the partition function is independent of small metric deformations that do notaffect the complex structures. We will see in section 4.3 how indeed the dependence on Ωdrops from the computation. In this section we focus on a particular class of geometries admitting two spinors of oppositeR-charge, requiring that the four-dimensional manifold has the topology of S × S . Thiswill play an important role in the calculation of the localized partition function in section4. Furthermore, in order to make contact with the results of [10], we will assume that thereexists a third Killing vector commuting with K , and that the metric is a direct product. A Hopf surface is essentially a four-dimensional complex manifold with the topology of S × S , and it may be defined as a compact complex surface whose universal covering is C − (0 , primary Hopfsurface, which is defined as having fundamental group isomorphic to Z [28, 29]. In the For example, on K¨ahler manifolds, the canonical choice is to take A real and V = 0. C − (0 , z , z bya cyclic group ( z , z ) ∼ ( pz + λz m , qz ) , (3.1)where ∼ denotes identification of coordinates, m ∈ N , and p, q, λ are complex parameters,such that 0 < | p | ≤ | q | < p − q m ) λ = 0. See e.g. [30]. It was shown in [28, 29] thatall primary Hopf surfaces are diffeomorphic to S × S . Moreover, it is shown in [18] thatHopf surfaces with λ = 0 admit only one Killing spinor ζ , and we will not consider themfurther. We will only consider Hopf surfaces with λ = 0, showing that these admit a verygeneral class of metrics, compatible with both complex structures J and e J , and hence bothsolutions ζ and e ζ .From the geometric point of view, the question that usually arises is whether on a man-ifold there exists a particular type of metric. In the case of Hopf surfaces, a class of metricthat appears to be of interest is that of locally conformally K¨ahler (LCK) metrics [30]. Thismeans that there exists, at least locally, a conformal rescaling of the metric, to a K¨ahlerone. A simple way to state this property is that the Lee form associated to the complexstructure is closed: d θ = 0. Indeed Ref. [30] constructed a large class of LCK metrics on aHopf surface. However, from the point of view of rigid supersymmetry, there is no naturalcondition on the curvature of a metric, and indeed the LCK property is too restrictive. Fromthe expressions (2.4), (2.5) we see that this property is equivalent to the requirement thatthe curvature of the conformally invariant background field A cs is purely real:Im[d A cs ] = 0 ⇔ LCK . (3.2)Although the Hermitian metric discussed in [22] (see e.g. equation (5.38) of this reference)is indeed LCK, as can be seen from the expression of A cs in (5.10), in general this propertyis not satisfied by Hermitian metrics admitting two Killing spinors of opposite R-charge.Notice also that the metrics written in equation (4.7) of [18] arise from the particularchoice of complex coordinates on C − (0 ,
0) made in this reference. Below we will presenta different construction, where we will start with a smooth metric on S × S , containingarbitrary functional degrees of freedom. This will make transparent the fact that the con-stants p, q parameterise the complex structure of the Hopf surface, while the metric is largelyindependent of these. These are referred to as of “class 0” in [30], while those with λ = 0 are referred to as of “class 1”. .2 Global properties We will discuss the geometries of interest starting from a four-dimensional metric that is byconstruction a non-singular complete metric on S × S . Requiring that this is compatiblewith an integrable complex structure ensures that it is a metric on a Hopf surface [28, 29].The existence of two Killing spinors ζ , e ζ is guaranteed imposing that the metric admits acomplex Killing vector K commuting with its complex conjugate and satisfying K µ K µ = 0.The global analysis of the geometry is facilitated if we assume that there exists an addi-tional real Killing vector commuting with K , so that generically the isometry group of themetric is U (1) , with a U (1) acting on S and a U (1) × U (1) acting on a transverse metricon S . The three-dimensional part is therefore toric , and in particular admits an almostcontact structure and a dual Reeb vector field whose orbits in general do not close. Inappendix D we analyse the most general metric with U (1) isometry, while in the rest of thepresent section we will consider the following metric of direct product form d s = Ω d τ + d s ( M ) ≡ Ω d τ + f d ρ + m IJ d ϕ I d ϕ J I, J = 1 , . (3.3)Here τ ∼ τ +2 π is a coordinate on S , while for M ≃ S we take coordinates ρ, ϕ , ϕ adaptedto the description of S as a T ≃ U (1) fibration over an interval. In these coordinates theKilling vectors generating the U (1) × U (1) isometry are ∂/∂ϕ and ∂/∂ϕ . Without lossof generality we take canonical 2 π periodicities for ϕ , ϕ , and assume 0 ≤ ρ ≤
1, withthe extrema of the interval corresponding to the north and south poles of the three-sphere.For ρ ∈ [0 , ρ ) > f = f ( ρ ) > m IJ = m IJ ( ρ ) is positive-definite. Moreover, in order for the metric to be non-singular,some conditions need to be satisfied at the poles of S , which we will spell out below.Near to an end-point, one of the one-cycles of the torus remains finite, while the otherone-cycle must shrink, in a way such that the associated angular coordinate locally describes,together with ρ , a copy of R . Let us assume that ∂/∂ϕ (respectively, ∂/∂ϕ ) generates theone-cycle that shrinks at ρ → ρ → ρ → f → f , m → m (0) , m = ( f ρ ) + O ( ρ ) , m = O ( ρ ) , (3.4) It would be straightforward to analyse the case where the isometry group of the four-dimensional metricis U (1) . Since a U (1) factor acts on S , the other U (1) is generated by a Reeb vector field on M ≃ S ofregular type. This case is however less interesting. Note that this Riemannian metric is related to a supersymmetric Lorentzian metric with time coordinate t = iτ [31]. This implies that the partition function we will compute in section 4 can also be thought ofas arising from the Euclidean (and compactified) time path integral of a theory defined on R t × M . Thispartially motivates our choice of restricting to a direct product metric. Other motivations are discussed inappendix D. f > m (0) > ρ → f → f , m = f (1 − ρ ) + O ((1 − ρ ) ) , m → m (1) , m = O [(1 − ρ ) ] , (3.5)where f > m (1) > m IJ must degenerate at the poles,since either one of the vectors ∂/∂ϕ I has vanishing norm there. Indeed, as ρ → m IJ ) goes to zero precisely as m (0)( f ρ ) , while when ρ → m (1) f (1 − ρ ) .It is now simple to construct supersymmetric backgrounds preserving two superchargesof opposite R-charge, with metric given by (3.3). As reviewed in section 2.1, a solution ζ and a solution e ζ to equations (2.1), (2.2) exist if the metric admits a complex Killing vector K commuting with its complex conjugate, [ K, K ] = 0, and squaring to zero, K µ K µ = 0. Wechoose K = 12 (cid:20) b ∂∂ϕ + b ∂∂ϕ − i ∂∂τ (cid:21) , (3.6)where b and b are two real parameters, so that the orbits of Re K generically do not close.Notice that Re K is a Reeb vector on M , whose dual one-form defines an almost contactstructure. This clearly satisfies [ K, K ] = 0, while the condition K µ K µ = 0 is equivalent toΩ = b I m IJ b J for ρ ∈ [0 , . (3.7)Note that this can be regarded as a constraint on the g ττ component of the metric (3.3), hencethe three-dimensional part of (3.3) is a non-singular metric on M ≃ S , independent of thetwo parameters b , b [10, 24]. In appendix D we discuss how this condition is generalisedin the case of a non-direct product metric, showing that this is related to complexifying theparameters b , b .The background fields A and V can be determined using the formulae in section 2.1,which require first casting the metric in the canonical complex coordinates w, z . We will dothis in two steps. Firstly, we will show that the metric can be written asd s = Ω (cid:2) d τ + (d ψ + a ) + c d z d¯ z (cid:3) , (3.8)where ψ is an angular coordinate such that ∂∂ψ = b ∂∂ϕ + b ∂∂ϕ , (3.9)and z is a complex coordinate defined in terms of ρ, ϕ , ϕ . Moreover, c = c ( z, ¯ z ) is a real,positive function of z , while a = a z ( z, ¯ z )d z + ¯ a ¯ z ( z, ¯ z )d¯ z is a real one-form. Notice that thethree-dimensional part of the metric (3.8) is precisely of the form implied by new minimalsupersymmetry in three dimensions [27], and used in the analysis of [10]. Secondly, we will20ntroduce another complex coordinate, w , thus arriving at the form (2.9).A convenient choice of Killing vector on M independent of (3.6) is ∂∂χ = b ∂∂ϕ − b ∂∂ϕ , (3.10)with the corresponding change of coordinates given by ϕ = b ( ψ + χ ) , ϕ = b ( ψ − χ ) . (3.11)In terms of the ψ, χ coordinates, the M part of the metric (3.3) becomesd s ( M ) = Ω (cid:2) (d ψ + a ) + Ω − f d ρ + c d χ (cid:3) , (3.12)where Ω is given in (3.7), the function c reads c = 2 | b b | Ω p det( m IJ ) , (3.13)and the one-form a = a χ d χ is given by a χ = 1Ω (cid:0) b m − b m (cid:1) . (3.14)Next, we define the complex coordinate z as z = u ( ρ ) + i χ , where the real function u ( ρ ) isa solution to u ′ = f Ω c , (3.15)with prime denoting derivative with respect to ρ . This differential equation can be solvedfor ρ ∈ (0 , z , together with ψ , covers S everywhere except atthe poles, which are found at Re z → ±∞ ( cf. the expansions in (3.25) below). We then seethat the metric takes the desired form (3.8). In these coordinates, the vector K becomes K = 12 (cid:18) ∂∂ψ − i ∂∂τ (cid:19) , (3.16)while as a one-forms it reads K = 12 Ω (d ψ + a − i d τ ) . (3.17)Note that although the metric components in (3.12) depend explicitly on b , b , this is justan artefact of the choice of coordinates. In particular, global properties of the metric maybe analysed only in the coordinates ρ, ϕ , ϕ , and not in the coordinates ψ, z , as neither ψ The only requirement is that the change of coordinates should be invertible. χ = Im z are period coordinates in general.Let us now cast the metric (3.8) in the form (2.9), introducing a complex coordinate w in addition to z . We take w = ψ + i τ + P ( z, ¯ z ) , (3.18)where P ( z, ¯ z ) is a complex function. With this definition, we have K = ∂/∂w , and the twometrics match if we impose ∂ z P = a z and h = ∂ z ( P − P ) , (3.19)where the first equation can be solved for P , while the second equation determines h . Wecan now discuss the background fields V and A given, for example, in (2.20), with the latterchosen real for convenience. Noting that (3.19) implies ∂ ¯ z h = ∂ ¯ z a z − ∂ z ¯ a ¯ z = − i ∗ (d a ) , (3.20)where ∗ denotes the Hodge star of the 2d metric d z d¯ z , with volume form vol = i d z ∧ d¯ z ,we see that the choice of κ in (2.17), ensuring that A is real, reads κ = ∗ (d a )3Ω c , (3.21)so that κ is real and completely determined by the metric on M . Then the formula for V in (2.20) can be written as V = 2 Im[ ∂ z log Ω d z ] − c ∗ (d a )(d ψ + a ) − i c ∗ (d a ) d τ . (3.22)In the coordinates ρ, ϕ , ϕ , this becomes V = 12 f (cid:20) c Ω ′ − Ω6 c (cid:0) a χ (cid:1) ′ (cid:21) (cid:18) d ϕ b − d ϕ b (cid:19) − Ω6 cf ( a χ ) ′ (cid:18) d ϕ b + d ϕ b + i d τ (cid:19) , (3.23)where the functions Ω( ρ ), c ( ρ ) and a χ ( ρ ) are those in (3.7), (3.13), (3.14). Similarly, theexpression for the real gauge field A in (2.20) becomes A = 14Ω f (Ω c ) ′ (cid:18) d ϕ b − d ϕ b (cid:19) + 12 d ω . (3.24)Having obtained V and A in the ρ, ϕ , ϕ coordinates, we can now discuss their globalproperties, in particular their regularity at the poles of S . Recalling our assumptions on f m IJ , it is easy to see that for ρ close to zero the functions Ω, c and a χ behave asΩ = b [ m (0) + m ′ (0) ρ ] + O ( ρ ) , c = 2 f p m (0) | b || b | ρ + O ( ρ ) , a χ = 1 + O ( ρ ) , (3.25)with analogous expressions holding for ρ →
1. Hence, at leading order in ρ →
0, we see that V behaves as V = k (cid:18) b d ϕ + i τ (cid:19) + O ( ρ ) , (3.26)where k is a constant. This is regular, as neither the one-cycle dual to d ϕ nor the onedual to d τ shrink to zero size at ρ = 0. Regularity of V at ρ = 1 is seen in a similar way.On the other hand, regularity of A is not automatic; by imposing this we determine ω ,namely the phase of s . At leading order in ρ → A = | b | (cid:18) d ϕ b − d ϕ b (cid:19) + 12 d ω + O ( ρ ) , (3.27)while at leading order in (1 − ρ ) → A = − | b | (cid:18) d ϕ b − d ϕ b (cid:19) + 12 d ω + O (1 − ρ ) . (3.28)In order to ensure that A does not have a component along the S that shrinks at eitherpoles, we must take ω = sgn( b ) ϕ + sgn( b ) ϕ . (3.29)To summarise, starting with an arbitrary non-singular metric d s ( M ) on S , we haveconstructed a non-singular (direct-product) metric on S × S , compatible with two com-muting complex structures, and thus admitting two supercharges with opposite R-charge ζ , e ζ . The choice (3.29) guarantees that the background fields A , V are non-singular. Inappendix C we illustrate the formulae above in an explicit example based on the Bergerthree-sphere. The pair (d s , J ) determines a Hopf surface, which must arise as a quotient of C − (0 ,
0) asin (3.1). We now show this explicitly, by relating the complex coordinates w, z to complexcoordinates z , z on C − (0 , p, q in terms of the parameters b , b introduced above. This will provide a relation betweenthe complex structure in four dimensions, and the almost contact structure in the three- This reads k = − b | b | f (cid:2) m ′′ (0) − ( m ′ (0)) − ( f b /b ) (cid:3) . S .Using (3.15), and taking P ( z, ¯ z ) = iQ ( ρ ) with Q ( ρ ) a real function, the first equation in(3.19) becomes Q ′ = f a χ Ω c , (3.30)and we claim that an appropriate choice of complex coordinates on C − (0 ,
0) is given by z = e −| b | ( iw + z ) ,z = e −| b | ( iw − z ) . (3.31)Since these are related to w, z by a holomorphic change of coordinates, they are automaticallycompatible with the complex structure induced by supersymmetry. In terms of the globallydefined coordinates on S × S we have z = e | b | τ e | b | ( Q − u ) e − i sgn( b ) ϕ ,z = e | b | τ e | b | ( Q + u ) e − i sgn( b ) ϕ . (3.32)If ( z , z ) are indeed coordinates on C − (0 , τ ∼ τ + 2 π leads to ( z , z ) ∼ (e π | b | z , e π | b | z ) , (3.33)corresponding to a Hopf surface with parameters p = e − π | b | and q = e − π | b | . Note thatthe choice of p, q is independent of the metric on M , and only affects the four-dimensionalmetric through Ω .It remains to show that z , z are complex coordinates on C − (0 ,
0) when τ is decom-pactified, so that τ ∈ R . From (3.32) it is clear that the phases − sgn( b j ) ϕ j correspond tothe angular directions in polar coordinates for the two copies of C in C = C ⊕ C . Thereforewe have to show that | z | , | z | are appropriate radial directions, and that the point (0 ,
0) isexcluded. The proof is given in appendix E, while below we present a simple example wherethe function Q derived from (3.30) can be obtained explicitly.Consider the Berger sphere M = S v with metricd s ( S v ) = d θ + sin θ d ϕ + v (d ς + cos θ d ϕ ) , (3.34)discussed in detail in appendix C. In the special case b = − b = v > θ = πρ , Ω =1, f = π , c = v sin θ , a χ = cos θ . The equations (3.15) and (3.30) become ∂ θ u = v (sin θ ) − Strictly speaking, it is p = e − π | b | , q = e − π | b | if | b | ≤ | b | and p = e − π | b | , q = e − π | b | if | b | ≤ | b | . ∂ θ Q = v cotan θ and are solved by u ( θ ) = v log tan θ , Q ( θ ) = v log sin θ , (3.35)yielding the coordinates z = √ τ v cos θ − iϕ ,z = √ τ v sin θ − iϕ , (3.36)in agreement with [22]. It is straightforward to see that these indeed cover C − (0 ,
0) when τ ∈ R . In this section we will compute the partition function of a four-dimensional N = 1 super-symmetric gauge theory defined on a background geometry admitting two supercharges ofopposite R-charge, comprising a Hopf surface with arbitrary (real) parameters p, q , and avery general Hermitian metric with U (1) isometry. We will consider gauge theories with avector multiplet transforming in the adjoint representation of a gauge group G , and chiralmultiplets transforming in arbitrary representations of G . The vector multiplet supersymmetric locus given by (2.51) implies that A µ is a flat connec-tion . After having specified an S × S topology, the flat connections are characterized bythe holonomy of constant gauge fields around S . In particular, up to gauge transformations,the localized fields of the vector multiplet are A µ = ( A i , A τ ) = (0 , A ) , D = 0 , (4.1)where A is constant. Notice that this result holds without any further assumption on themetric, therefore it is true also if the metric is not a direct product or/and it has only a U (1) isometry.Let us fix the vector multiplet fields at their locus values (4.1) and proceed to analysethe supersymmetric locus of a chiral multiplet with R-charge r , determined by the equations(2.54). Following the discussion of section 2.3.2, we will choose A µ real and impose the25eality conditions e φ = φ † and e F = − F † on the bosonic fields. Then the locus equations read F = 0 , ( J µν + e J µν ) D ν φ = 0 , ( J µν − e J µν ) D ν φ = − iD µ φ . (4.2)Contracting the second equation with K µ and K µ leads to K µ D µ φ = K µ D µ φ = 0. Usingthe expressions for J , e J and K given in section 2.1, the equations for φ become D τ φ = ∂ τ φ − i A φ = 0 ,D ψ φ = ∂ ψ φ − irA ψ φ = 0 , (4.3) D ¯ z φ = ∂ ¯ z φ − irA ¯ z φ = 0 , where we have used the fact that A τ = 0. The first equation implies that φ is proportionalto e i A τ , which is not globally defined on S , except when A = 0 modulo large gaugetransformations. Therefore in this case we immediately conclude that φ = 0. When A = 0 the analysis is slightly more subtle. The first equation implies that φ is independentof τ , and using (2.18) the two remaining equations are solved by φ = C ( z ) (cid:0) Ω c (cid:1) − r e ir (sgn( b ) ϕ +sgn( b ) ϕ ) , (4.4)with C ( z ) a (locally) holomorphic function of z . In order to obtain a globally defined solution,we must impose periodicity around the two S parametrized by ϕ and ϕ . Recalling that z = u ( ρ ) + i (cid:16) ϕ b − ϕ b (cid:17) , periodicity under the shift ϕ → ϕ + 2 π sgn( b ) yields C (cid:0) z + πi | b | (cid:1) e πir = C ( z ) , (4.5)and similarly periodicity under ϕ → ϕ + 2 π sgn( b ) gives C (cid:0) z − πi | b | (cid:1) e πir = C ( z ) , (4.6)so that in particular C ( z ) is a periodic function in the imaginary direction C (cid:0) z + iπ | b | + | b || b b | (cid:1) = C ( z ). Since | φ | = | C ( z ) | (Ω c ) − r , with Ω c vanishing only at the poles ρ = 0 , ρ = 1 (seeappendix E), we see that in order to have a non-singular solution φ for r > C ( z ) mustvanish at ρ = 0 , ρ = 1, that is lim Re z →±∞ C ( z ) = 0. Extending C ( z ) to the complex ( u, χ )plane, we see that it is a bounded entire function, and therefore Liouville’s theorem implies We discuss these large gauge transformations below. This is true, independently of whether χ is a periodic or a non-compact coordinate.
26t is a constant. The limits at the poles imply C = 0, thus showing that for r >
0, thelocalization locus is φ = 0.If r ≤ C ( z ) = P n ∈ Z C n e −| b | ( r +2 n ) z , where C n are constants. Inserting this into (4.6), we see that for each n ∈ Z , either e πi | b | ( r +2 n )+ πir = 1 or C n = 0. So there can be non-trivial solutions if and onlyif the R-charge r takes the very special form r = − | b | n + 2 | b | m | b | + | b | ≤ , n, m ∈ Z . (4.7)Thus simply assuming that r is not one of the special values (4.7), the chiral multipletlocalization locus is given by F = φ = 0 . (4.8)The full supersymmetric locus is thus completely characterized by the constant Lie al-gebra element A . Correspondingly, the path integral splits into a matrix integral over A ,and a Gaussian integral over all the fluctuations about the saddle point locus (4.1), (4.8).Following a similar discussion in [32], we will now explain how to use the residual gaugefreedom to extract the correct integration measure of the matrix model . First of all, one can use constant gauge transformations to diagonalize A and reduce theintegration to the Cartan subalgebra of the gauge group G , introducing a Vandermondedeterminant ∆ [ A ] = Y α ∈ ∆ + ( α A ) , (4.9)where ∆ + denotes the set of positive roots and α A ≡ α ( A ). In a Cartan basis { H k } wehave A = P r G k =1 a k H k , where r G is the rank of the gauge group G . Then for a root α = { α k } ,we have α A = P k a k α k . One also has to divide by the order of the Weyl group |W| in orderto take care of gauge transformations that permute the elements of the Cartan basis.Furthermore, the path integral must be invariant under large gauge transformations alongthe S , that shift A → A + P k d k H k , where d k ∈ Z . Thus we can restrict the range ofintegration of the constants { a k } to be over the maximal torus T r G of G , parametrised by z = { z k } = { e πia k } ∈ T r G . (4.10) We assume that the gauge field is normalized so that all the matter fields have integer charges. Z = 1 |W| Z T rG d z πiz ∆ [ A ] Z classic [ A ] e Z vector1-loop [ A ] Y J Z chiral ( J )1-loop [ A ] , (4.11)where the integration measure d A has been replaced byd A ≡ r G Y k =1 d a k → d z πiz ≡ r G Y k =1 d z k πiz k . (4.12)Here Z classic [ A ] is the classical contribution from the vector and chiral multiplets. How-ever, for the theories that we consider, with Lagrangians (2.25), (2.38) (plus superpoten-tial couplings), we have Z classic = e − S classic = 1 . The remaining factors e Z vector1-loop [ A ] and Z chiral ( J )1-loop [ A ] are the one-loop determinants of the vector multiplet and chiral multipletsfluctuations around the configurations (4.1) and (4.8).Denoting by A τ and A i the components of the gauge field A µ along S and M , respec-tively, we will impose the following gauge-fixing conditions ∇ τ a = 0 , ∇ i A i = 0 , (4.13)where a ≡ M ) R M A τ . Let us discuss the first condition, while we will deal with thesecond condition later [33, 34]. The Faddeev–Popov determinant det ′ (cid:0) ∇ τ D (0) τ (cid:1) associated to ∇ τ a = 0 can be written in terms of ghost fields γ, ¯ γ , yielding an integral over the followinggauge-fixing term S gauge − fixing a = Z d τ Tr (cid:2) ¯ γ (cid:0) ∇ τ D (0) τ (cid:1) γ + ξ ∇ τ a (cid:3) , (4.14)where D (0) τ = ∇ τ − i [ A , · ] and a prime on the determinant means that it does not containthe zero mode along S . The second term is simply a rewriting of the delta function δ ( ∇ τ a )enforcing the gauge-fixing condition, with ξ a Lagrange multiplier. The gauge fixing action(4.14) can be included in the deformation term by replacing δV → δ ′ V ′ , with δ ′ = δ + δ B ,where δ B is the BRST transformation, and V ′ = V + Tr ¯ γ ∇ τ a [35]. We refer to [1] for amore rigorous treatment of the ghosts.Writing a = A + ∇ τ ϕ and doing the path integral over ϕ introduces a Jacobian factor(det ′ ∇ τ ) − / , which combined with the Faddeev–Popov determinant yields e Z vector1-loop [ A ] = ∆ [ A ] Z vector1-loop [ A ] , (4.15)28here ∆ [ A ] ≡ det ′ D (0) τ = Y α ∈ g Y n =0 ( in − iα A ) , (4.16)and α ∈ g labels both non-zero roots and Cartan generators. A straightforward computationyields ∆ [ A ] = (2 π ) r G Y α ∈ ∆ + ( πα A ) α A , (4.17)where we used the formula sin( πz ) = πz Q ∞ n =1 (cid:0) − z n (cid:1) , and employed zeta function regular-isation to regularise the infinite products. Finally, the matrix model becomes Z = 1 |W| Z T rG d z πiz ∆ [ A ] Z vector1-loop [ A ] Y J Z chiral ( J )1-loop [ A ] , (4.18)with ∆ [ A ] = ∆ [ A ] ∆ [ A ] = (2 π ) r G Y α ∈ ∆ + ( πα A ) . (4.19) Our strategy to compute the one-loop determinants on S × M for the vector and chiralmultiplets is to take advantage of the three-dimensional results of [10]. First we expandthe fields into Kaluza–Klein (KK) modes along the S parametrized by τ . Denoting by Φ ageneric field (bosonic or fermionic), we takeΦ( x i , τ ) = X n ∈ Z Φ n ( x i ) e − inτ . (4.20)The four-dimensional one-loop determinant may be replaced by the product over one-loopdeterminants for the KK modes on M Z [Φ] = Y n ∈ Z Z [Φ n ] . (4.21)The one-loop determinants on M were computed in [10] and our aim is to use the resultstherein for Z [Φ n ]. For this to be possible we need to show that the Gaussian action forfluctuations around the localization locus, resulting from the deformation terms δV , matches Previous studies of relations between the index of four-dimensional gauge theories and the partitionfunction in three dimensions include [36, 37, 38, 39, 40]. A µ , V µ )and the three-dimensional background fields ( ˇ A i , ˇ V i , ˇ h ) are given in (F.23) (we use a ˇ symbolto denote three-dimensional quantities). With our choice of real A µ , the three-dimensionalfields ˇ A i , ˇ V i , ˇ h are also real, as it is assumed in [10]. We denote as B τ and B i the fluctuations of the gauge field A µ along S and M , respec-tively, σ = Ω − B τ and consider the KK fields fluctuations ( B n j , σ n , λ n , e λ n , D n ) around thelocalization locus (4.1), where it is understood that ( e λ n ) α = i ( σ ) α ˙ α e λ ˙ αn . The supersymmetrytransformations (2.23) (with e ζ = 0) read for these KK fields δ B n j = iζ γ j e λ n , δσ n = ζ e λ n ,δλ n = − i ε ijk F n ij γ k ζ − i (cid:0) ∂ j σ n − i ˇ V j σ n + i Ω [ A , B n j ] + i Ω n B n j (cid:1) γ j ζ + (cid:0) ˇ D n − ˇ hσ n (cid:1) ζ ,δ e λ n = 0 ,δ ˇ D n = − iζ γ j (cid:0) ˇ ∇ j − i ˇ A j + i V j (cid:1)e λ n + 12 ˇ V j ζ γ j e λ n + i Ω ζ [ A , e λ n ] + i Ω n ζ e λ n + ˇ h ζ e λ n , (4.22)where we defined ˇ D n = iD n + (ˇ h − ˇ V ψ ) σ n and used the convention γ j = − iσ e σ j for the three-dimensional gamma matrices (see appendix F for more details about the 3d conventions). These transformations correspond to the supersymmetry transformations of the three-dimensional N = 2 vector multiplet fluctuations (cid:0) A j , σ, λ, λ † , D (cid:1) of [10] with respect to In deriving the KK supersymmetry transformations, we have made use of the relation (F.24). We alsopoint out the fact that the three-dimensional free parameter ˇ κ of (F.23) drops from the supersymmetrytransformations and does not affect the whole computation. η = √ ζ , with the map (cid:16) B n j , σ n , √ λ n , √ e λ n , ˇ D n (cid:17) = (cid:0) A j , − σ, λ † , λ, − D (cid:1) ,n + [ A , · ] = [ σ , · ] . (4.23)The evaluation of the one-loop determinant is done by decomposing all KK fields, denotedgenerically Φ n , into the Cartan basis of the gauge algebraΦ n = r G X k =1 Φ n k H k + X α ∈ roots Φ n α E α , (4.24)where H j generate the Cartan subalgebra and E α are the ladder operators. The map (4.23)descends to the α -component multiplets, with n + α A = α ( σ ) . (4.25)The multiplets along the Cartan directions can be associated with “vanishing roots” α = 0.To be able to map the four-dimensional deformation terms to the three-dimensional ones,we note that on S × M the deformation terms (2.27) and (2.33), expanded at quadraticorder around the localization locus, are equal: δ ζ V (+) = δ e ζ V ( − ) . For the fermionic part thisis obvious, while for the bosonic part this follows from the identityTr Z S × M F ∧ F = Tr Z S × M d ( B ∧ d B − i A ∧ B ∧ B ) = 0 . (4.26)Hence we have δV = − | ζ | δ ζ (cid:0) Tr ( δ ζ λ ) ‡ λ (cid:1) . In section 2.3.1, we saw that the reality con-ditions which, along with a real A µ , ensure positivity of the bosonic deformation terms are A † µ = A µ , D † = − D . For the fermions we choose iσ e λ = λ † . For the KK modes thesetranslate into A n µ = A †− n µ , D n = − D †− n and e λ n = λ †− n .Then, using the map (4.23) to three-dimensional fields, the Gaussian action for the n -th The authors of [10] performed localization using a spinor ǫ of positive charge under ˇ A µ and wrote explic-itly the supersymmetry transformations for ǫ . In our derivation, the relations between four-dimensional andthree-dimensional background fields imply that the four-dimensional supersymmetry parameter ζ is mappedto a three-dimensional supersymmetry parameter η of negative charge under ˇ A µ , see appendix F for details.Thus the supersymmetry transformations (4.22) are mapped to the three-dimensional supersymmetry trans-formations with respect to a negative charge spinor. These are not detailed in [10], but they can be derivedfrom the ǫ transformations by changing (in our notations) ǫ α → η α , ( ˇ A j , ˇ V j , A j ) → − ( ˇ A j , ˇ V j , A j ) and e λ ↔ λ (also e Φ ↔ Φ for all fields for the chiral multiplet). They are also given in [27]. The fact that we have anegative charge spinor η in three dimensions does not prevent us from using the results of [10], since thelocalization computation is unchanged if η is used instead of ǫ .
31K mode and α component fluctuations can be expressed as δV ( n,α )(4d) = − | ζ | Tr δ ζ (cid:0) ( δ ζ λ ( n,α ) ) ‡ λ ( n,α ) (cid:1) = − | η | Tr δ η (cid:0) ( δ η λ ( α ) ) ‡ λ ( α ) (cid:1) (3d) = δV (3d) [ σ ( n,α )0 ] , (4.27)where the action of ‡ on the KK modes is Φ ‡ ( n,α ) = Φ ( − n, − α ) , and the constant scalar for theresulting three-dimensional deformation term is σ ( n,α )0 = n + α A . This three-dimensionaldeformation term is the same as the one considered in [10]. The reality conditions on thethree-dimensional fields in α components obtained from this map are Φ (3)( α ) = Φ (3)( − α ) † forbosons and e λ (3)( α ) = λ (3)( − α ) † for fermions, and match the reality conditions of [10]. Moreover,the three-dimensional gauge fixing condition ∇ j B j = 0 chosen above becomes ∇ j A (3) j = 0,reproducing the gauge fixing condition of [10]. We can then use the result of [10] for thethree-dimensional one-loop determinant for each ( n, α )-component multiplet. Note that thecontribution from the Faddeev–Popov determinant of the three-dimensional gauge fixing(namely the second in (4.13)) is included in the result of [10]. We obtain the expectedrelation Z vector1-loop [ A ] = Y α ∈ g Y n ∈ Z Z vector1-loop (3d) (cid:2) σ ( n,α )0 (cid:3) , (4.28)with σ ( n,α )0 = n + α A and here α ∈ g labels both roots and Cartan components.From [10], we extract Z vector1-loop (3d) [ σ ( α )0 ] = 1 iα ( σ ) Y n ,n ≥ n b + n b + iα ( σ ) − ( n + 1) b − ( n + 1) b + iα ( σ ) , (4.29)holding for b , b >
0. A careful re-examination of the three-dimensional one-loop compu-tation in [10] shows that for arbitrary real b , b , the one-loop determinant is given by theformula above with b , b replaced by | b | , | b | .Renaming n → n , our one-loop determinant is expressed by the infinite product: Z vector1-loop = Z Cartan Y α ∈ roots Y n ∈ Z i ( n + α A ) Y n ,n ≥ n b + n b + i ( n + α A ) − ( n + 1) b − ( n + 1) b + i ( n + α A ) ! = Z Cartan ∆ − Y α ∈ roots Y n ∈ Z Y n ,n ≥ n b + n b + i ( n + α A ) − ( n + 1) b − ( n + 1) b + i ( n + α A ) ! . (4.30)We see that the first factor cancels with the matrix model measure ∆ [ A ], while the secondfactor needs to be regularized. We perform this regularization in appendix G, using multipleGamma functions. These manipulations yield the Jacobi theta function θ ( z, p ) and the We thank J. Sparks for discussions about this point. z ; p ), defined for z, p ∈ C and | p | < θ ( z, p ) = Y n ≥ (1 − zp n ) (cid:0) − z − p n +1 (cid:1) , ( z ; p ) = Y n ≥ (1 − zp n ) . (4.31)The result is the following expression for the one-loop determinant Z vector1-loop = e iπ Ψ (0)vec e iπ Ψ (1)vec ( p ; p ) r G ( q ; q ) r G ∆ − Y α ∈ ∆ + θ (cid:0) e πiα A , p (cid:1) θ (cid:0) e − πiα A , q (cid:1) , (4.32)with Ψ (0)vec = i (cid:18) b + b − b + b b b (cid:19) | G | , Ψ (1)vec = − i b + b b b X α ∈ ∆ + α A , (4.33)where p = e − πb , q = e − πb , | G | is the dimension of G , and we have split the prefactorinto a part Ψ (0)vec independent of α A and a part Ψ (1)vec depending on α A . This result lookspuzzling, because the factor e iπ Ψ (1)vec spoils the invariance under the shifts α A → α A + d for d ∈ Z , associated to large gauge transformations A → A + P k d k H k , d k ∈ Z . In otherwords, e iπ Ψ (1)vec is not a function of z α = e πiα A as it must be. For the final matrix modelto be consistent, all such “anomalous” terms breaking the symmetry under large gaugetransformations must cancel. We will see in section 5.1 that this is indeed what happens ifthe theory satisfies relevant physical constraints. The evaluation of the one-loop determinant for the chiral multiplet proceeds in a similarfashion. The KK fields ( φ n , ψ n , F n , e φ n , e ψ n , e F n ) all vanish on the localization locus (4.8),hence we can keep the same notations for their fluctuations around zero. The supersymmetrytransformations (2.37), with respect to the spinor ζ = √ η , and with the vector multipletlocalized to (4.1), read for these KK fields: δφ n = ηψ n , δ e φ n = 0 , δψ n = F n η , δF n = 0 , (4.34) δ e ψ n = − i (cid:16) ˇ D j e φ n + r ∂ j log Ω (cid:17) γ j η − i Ω ( n + A ) e φ n η − r ˇ h e φ n η ,δ e F n = iηγ j (cid:16) ˇ D j − i ˇ V j + r ∂ j log Ω (cid:17) e ψ n − i Ω ( n + A ) η e ψ n − (cid:16) r − (cid:17) ˇ hη e ψ n , (4.35)33ith ˇ D j = ˇ ∇ j + iq R (cid:0) ˇ A j − ˇ V j (cid:1) acting on a field fluctuation of R-charge q R , and where( e ψ n ) α ≡ i ( σ ) α ˙ α e ψ ˙ αn . The match with the three-dimensional multiplet of [10] is given by( φ n , − ψ n , − iF n , e φ n , e ψ n , − i e F n ) = Ω − r/ ( φ, ψ, F, φ † , ψ † , F ) ,n + A = σ . (4.36)The reality conditions ensuring the positivity of the four-dimensional deformation term are φ † n = e φ − n and F † n = − e F − n for bosons, while for the fermions we choose ψ † n = − e ψ − n .It follows that the Gaussian action around the locus solution for the n -th KK mode is δV ( n )4d = δ ζ (cid:16)h ( δ ζ ψ n ) ‡ ψ n − e ψ n ( δ ζ e ψ n ) ‡ i(cid:17) = δ η (cid:16)h ( δ η ψ ) † ψ + ψ † ( δ η ψ † ) † i(cid:17) = δV [ σ ( n )0 ] , (4.37)with σ ( n )0 = n + A and where we have dropped overall factors of Ω that can be cancelled byirrelevant redefinition of the deformation terms. Again we recover the three-dimensionaldeformation term used in [10]. The reality conditions on three-dimensional fields followingfrom our map are (cid:0) Φ (3) † (cid:1) † = Φ (3) for bosons and (cid:0) ψ (3) † (cid:1) † = ψ (3) for fermions, matching[10], so that we are able to use their three-dimensional one-loop determinant for each KKmultiplet.Decomposing the fields along the weight basis of their representation R , Φ n = X ρ weight Φ n, ρ , (4.38)the 4d-3d map holds for the fields Φ ( n,ρ ) with σ ( n,ρ )0 = n + ρ A , where ρ A ≡ ρ ( A ) = P r G k =1 ρ k a k .We obtain the expected result Z chiral1-loop [ A ] = Y ρ ∈ weights Y n ∈ Z Z chiral1-loop (3d) (cid:2) σ ( n,ρ )0 (cid:3) , (4.39)where ρ ∈ weights denotes a sum over the weights of the chiral multiplet representation R .From [10], we extract the result (for b , b > Z chiral1-loop (3d) (cid:2) σ ( ρ )0 (cid:3) = Y n ,n ≥ n b + n b + iρ ( σ ) − r − ( b + b ) n b + n b − iρ ( σ ) + r ( b + b ) . (4.40)For arbitrary real b , b , the one-loop determinant is given by the formula above with | b | , | b | instead of b , b . See also appendix B, where an alternate way to see that Ω does not affect the result is given. Note that the fields with a tilde transform in the complex conjugate representation R ∗ , whose weightsare opposite to the weights of R . n → n , the one-loop determinant is Z chiral1-loop = Y ρ ∈ weights Y n ∈ Z Y n ,n ≥ ρ A + i r − ( b + b ) + n − in b − in b − ρ A − ir ( b + b ) − n − in b − in b , (4.41)Again the regularization of the infinite product is detailed in appendix G. This involves theelliptic gamma function, defined for z, p, q ∈ C and | p | < | q | < e ( z, p, q ) = Y n ,n ≥ − z − p n +1 q n +1 − zp n q n . (4.42)The result is Z chiral1-loop = e iπ Ψ (0)chi e iπ Ψ (1)chi Y ρ ∈ ∆ R Γ e (cid:0) e πiρ A ( pq ) r , p, q (cid:1) , (4.43)withΨ (0)chi = i b + b b b (cid:2) ( r − ( b + b ) − ( r − (cid:0) b + b + 2 (cid:1)(cid:3) |R| , (4.44)Ψ (1)chi = X ρ ∈ ∆ R − ρ A b b − i ( r − b + b b b ρ A + [3( r − ( b + b ) − − b − b ] ρ A b b , where p = e − πb , q = e − πb , ∆ R is the set of weights of the representation R , and |R| is its dimension. As in the case of the vector multiplet, we have split the prefactor into apart Ψ (0)chi independent of A , and an “anomalous” part Ψ (1)chi carrying the inconsistent A dependence. To obtain a consistent result, we will require in the final matrix model thatthese “anomalous” terms vanish. In this section we present our final result for the exact partition function and compare itwith the supersymmetric index. We find that the two quantities match, up to a prefactorthat defines a Casimir energy for a supersymmetric gauge theory on a curved background.
For the matrix model to be well-defined as an integral over the maximal torus T r G , we havepointed out that the sum of the anomalous parts must cancelΨ (1)vec ( A ) + X J Ψ (1)chi , ( J ) ( A ) = 0 , (5.1)35here P J is a sum over the chiral multiplets of the theory. From (4.33), (4.44), assumingarbitrary values of b , b , this gives rise to four constraints on the gauge group and mattercontent of the theory: (i) X J Tr R J (cid:0) A (cid:1) = 0 , (ii) Tr Adj (cid:0) A (cid:1) + X J ( r J −
1) Tr R J (cid:0) A (cid:1) = 0 , (iii) X J ( r J − Tr R J ( A ) = 0 , (iv) X J Tr R J ( A ) = 0 , (5.2)where Adj denotes the adjoint representation of the gauge group G . Using the Cartandecomposition A = P r G k =1 a k H k , with a k ∈ R , and requiring (5.2) for all a k leads to(i) X J Tr R J (cid:0) H ( k H k H k ) (cid:1) = 0 , (ii) Tr Adj (cid:0) H ( k H k ) (cid:1) + X J ( r J −
1) Tr R J (cid:0) H ( k H k ) (cid:1) = 0 , (iii) X J ( r J − Tr R J ( H k ) = 0 , (iv) X J Tr R J ( H k ) = 0 , (5.3)where k = 1 , . . . , r G for all k -indices. These conditions can all be interpreted in terms ofvanishing of triangle Feynman diagrams contributing to various anomalies. Condition (i) isimplied by the requirement of the vanishing of the non-Abelian gauge anomaly; condition (ii)is implied by the vanishing of the ABJ anomaly, responsible for non-conservation of the R-symmetry current in an instanton background; condition (iii) holds requiring the vanishingof the mixed gauge-R symmetry anomaly G × U (1) R ; condition (iv) is equivalent to thevanishing of the mixed gauge-gravitational anomaly. All these anomalies arise from chiralfermions with R-charge r J − R J representation. The contribution from the gauginosappears only in condition (ii), while it drops out from the other ones, because the adjointrepresentation is real.All the conditions are necessary for the preservation of the dynamical gauge symmetry The translation into group theory language is the following: in a representation R with weights { ρ j } ,the matrix representing A in a weight basis is A R = diag[ P k a k ρ jk , ≤ j ≤ |R| ] = diag[ ρ j A , ≤ j ≤ |R| ].More generally ( A R ) n = diag[ ( ρ j A ) n , ≤ j ≤ |R| ] and the trace in the representation R is Tr R ( A n ) =Tr(( A R ) n ) = P |R| j =1 ( ρ j A ) n = P ρ ∈ ∆ R ( ρ A ) n . See [17] for a discussion of anomalies in relation to the supersymmetric index.
36t the quantum level, in a generic background. Notice that the conditions (iii) and (iv) holdautomatically when the gauge group G has no U (1) factors. Moreover, the absence of theABJ anomaly (condition (ii)), is equivalent to the vanishing of the NSVZ exact gauge betafunctions of the theory [41, 42]. In particular, this is satisfied by all theories that flow to aSCFT in the infra-red (IR). However, one can also consider theories exhibiting confinementin the IR, obtained for instance by suitable superpotential deformations [16]. Pure N = 1super Yang–Mills (SYM) is an example of a theory for which the partition function (andhence the supersymmetric index) is ill-defined.Gathering the results of the vector and chiral multiplets (4.33), (4.44), the partitionfunction on S × M is expressed by the exact formula Z [ H p,q ] = e −F ( p,q ) ( p ; p ) r G ( q ; q ) r G |W| Z T rG d z πiz Y α ∈ ∆ + θ ( z α , p ) θ (cid:0) z − α , q (cid:1) Y J Y ρ ∈ ∆ J Γ e (cid:0) z ρ ( pq ) rJ , p, q (cid:1) , (5.4)where z ± α = e ± πiα A , z ρ = e πiρ A , J labels various chiral multiplets of R-charge r J trans-forming in representation R J , ∆ J is the set of weights of R J , and F ( p, q ) = π (cid:18) | b | + | b | − | b | + | b || b b | (cid:19) | G | + X J ( r J − |R J | ! + π
24 ( | b | + | b | ) | b b | X J (cid:0) ( r J − − ( r J − (cid:1) |R J | = 4 π (cid:18) | b | + | b | − | b | + | b || b b | (cid:19) ( a − c ) + 4 π
27 ( | b | + | b | ) | b b | (3 c − a ) , (5.5)where in the second line we have used the following definitions a = 332 (cid:0) R − tr R (cid:1) = 332 h | G | + X J (cid:16) r J − − ( r J − (cid:17) |R J | i , c = 132 (cid:0) R − R (cid:1) = 132 h | G | + X J (cid:16) r J − − r J − (cid:17) |R J | i , (5.6)with R the R-symmetry charge and “tr” runs over the fermionic fields of the multiplets ofthe theory. When the theory flows to a fixed point, a and c are the central charges of theSCFT [8, 43, 42].Comparing with the supersymmetric index I ( p, q ) with fugacities p, q given for instancein [40], we obtain the relation advertised in the introduction Z [ H p,q ] = e −F ( p,q ) I ( p, q ) . (5.7)37he partition function depends on the geometry of S × M only through the complexstructure parameters p = e − π | b | , q = e − π | b | , as predicted by [18]. More precisely, theauthors of [18] have conjectured that the ratio Z [ H p,q ] / I ( p, q ) = e −F ( p,q ) can be set to one bya choice of local counterterms. However, by computing the partition function explicitly in azeta function regularisation scheme, we have found that this ratio depends on the geometryonly through the complex structure parameters, and thus generically it cannot be given interms of integrals of densities local in the background fields. This is clear since generallysuch densities would depend on (functional) degrees of freedom in the metric.Notice that for supersymmetric field theories defined on Hopf surfaces the integrated Weylanomaly vanishes [9] and therefore the corresponding “logarithmic” term in the partitionfunction, arising from conformal transformations of the functional measure [44], is absent.Thus (5.7) is the complete answer for the partition function.In the reminder of this section we will discuss further the interpretation of F ( p, q ). Firstly,we will show that this plays a role in the reduction of the partition function to the partitionfunction of a three-dimensional theory on M , upon taking the limit of small S . Following[40], the reduction along S is performed by setting b = β ˇ b , b = β ˇ b , A = β σ andtaking the limit β → b , ˇ b , σ fixed. In this limit the integration over T r G for e πi A becomes an integration over the Cartan sub-algebra R r G for σ . The limits of thevarious factors in the matrix model are discussed in [40], where it is shown that this reducesto the matrix model of the dimensionally reduced theory on M . However, it was noticedthat a divergent overall factor appears in the reduction of the index I ( p, q ), given byexp " − π β | ˇ b | + | ˇ b || ˇ b || ˇ b | | G | + X J ( r J − |R J | ! = exp h π β | ˇ b | + | ˇ b || ˇ b || ˇ b | ( c − a ) i (5.8)and this was dropped to recover the exact three-dimensional partition function. Our resultsimply that to complete the reduction one should take into account the contribution from theprefactor e −F ( p,q ) . The linear part in β vanishes when β → β precisely cancels (5.8). We conclude that the full four-dimensional partition function reduces to the exact three-dimensional partition function,computed using the regularization in appendix G, reduces to the exact three-dimensionalpartition function of the dimensionally reduced theory. N limit We now discuss how the term linear in β appearing in F may be interpreted as a Casimirenergy, and then comment on the large N limit. In general, the vacuum energy of a field38heory defined on S × M may be defined from the path integral as E Casimir = − lim β →∞ dd β log Z [ β ; M ] , (5.9)where one takes the limit of infinite radius of S , keeping all other parameters fixed. Usingthis definition, our partition function computed with supersymmetric boundary conditionsfor the fermions gives: E susy (ˇ b , ˇ b ) = 4 π (cid:0) | ˇ b | + | ˇ b | (cid:1) ( a − c ) + 4 π
27 ( | ˇ b | + | ˇ b | ) | ˇ b || ˇ b | (3 c − a ) , (5.10)that we refer to as supersymmetric Casimir energy . This arises from the β → ∞ limit of(5.5), and we used the fact that lim β →∞ dd β I = 0 . We see that E susy depends on the complexstructure parameters of the geometry, and on both the central charges a and c , characterisingthe field theory. Since the parameter β enters both in the g ττ component of the metric andin V τ , one can see that E susy receives contributions both from the energy-momentum tensorand from the currents in the R-multiplet. When p = q , with | b | = | b | ≡ β π , this reduces to E susy = 427 ( a + 3 c ) , (5.11)which agrees with the expression for the “index Casimir energy” given in appendix B of[45]. The latter was defined as tr[( − F H ], where H is the Hamiltonian commuting withthe supercharges, and a particular supersymmetric regularisation was adopted. Extendingto general p, q a prescription given therein for p = q , we find that our E susy can be expressedin terms of the letter indices [12, 14, 25] f chiral ( p, q ) = ( pq ) r − ( pq ) − r (1 − p )(1 − q ) , f vector ( p, q ) = 2 pq − p − q (1 − p )(1 − q ) , (5.12)with p = e − πβ ˇ b , q = e − πβ ˇ b , as E susy (ˇ b , ˇ b ) = −
12 lim β → dd β X all fields (cid:16) f chiral ( p, q ) + f vector ( p, q ) (cid:17) − πβ | ˇ b | + | ˇ b || ˇ b || ˇ b | ( a − c ) , (5.13)where the finite part reproduces E susy and the O ( β − ) term is proportional to a − c .In order to compare our E susy (ˇ b , ˇ b ) with other Casimir energies in the literature weshould restrict to the sub-space p = q , and assume that the metric is the round one on Up to a factor of 2 / arbitrary metric on M ≃ S , asanticipated in [22]. × S . In this case, it was shown in [46, 47], that in a conformal field theory (not necessarilysupersymmetric) the Casimir energy, defined as E = Z S h T i vol( S ) , (5.14)is proportional to the trace anomaly coefficient a , namely E = 34 a in a CFT . (5.15)Note that this result is valid for an arbitrary CFT, where a and c are not necessarily related.For an N = 1 SCFT defined on the round S × S , when both can be computed, E and E susy are two different measures of the vacuum energy of a theory.Notice that in the particular case of N = 4 SYM theory on S × S with G = SU ( N ),the Casimir energy, can be computed in the free field limit [48] and agrees with E , while itdiffers from E susy by a numerical factor, namely E free = 3( N − E , E susy = 4( N − N = 4 SYM . (5.16)Although E susy is valid for any value of the coupling constant (and for any N ) and inparticular at weak coupling in the N = 4 SYM theory, a priori it does not have to coincidewith E free or E . It would be interesting to understand precisely the relationships betweenthese Casimir energies.Finally, let us discuss the implications of our results for field theories that admit a gravitydual. For concreteness, we will now assume that the gauge theory is a quiver, with gaugegroup G = SU ( N ) k and chiral fields transforming in bi-fundamental representations ( N , N ).We also assume that there is a non-trivial superpotential, and that the theory flows to aninteracting fixed point in the IR, with a = c + O (1) = O ( N ), in the limit N → ∞ .These theories are expected to admit a gravity dual solution in type IIB supergravity withgeometry M × Y , where Y is a Sasaki–Einstein manifold [49] and M is a deformation ofAdS , supported by N units of five-form flux. Moreover, it should be possible to constructsuch solutions within the consistent truncation to minimal gauged supergravity and thenuplift these to ten dimensions, as illustrated in [22]. In these cases, at leading order in alarge N expansion, the prefactor (5.5) in the partition function simplifies to F ( p, q ) = 4 π
27 ( | b | + | b | ) | b || b | a , (5.17)and using the AdS/CFT relation exp( − S gravity [ M ]) = Z QFT [ ∂M ], we obtain the following40rediction for the five-dimensional holographically renormalised on-shell action: S
5d sugra [ M ] = π G ( | b | + | b | ) | b || b | . (5.18)Here we used the relation a = c = πℓ G (at leading order in N ), with G denoting the Newtonconstant of the five-dimensional supergravity, and we have set the AdS radius ℓ = 1.In the solution of [22] this formula was found valid, up to some local counterterms. Inparticular, in that solution p = q = e − β , albeit the boundary metric comprises a biaxiallysquashed three-sphere (see appendix C) and hence it is not conformally flat. In the case ofa solution of the form AdS × Y , the expression (5.18) reduces to S
5d sugra [AdS ] = πβ G andagain this should be contrasted with the computation in [48], giving S
5d sugra [AdS ] = πβ G .When Y = S the latter agrees with the large N limit of E free = E CFT above, while theformer gives a different value. We expect that this difference can be traced to the use ofdifferent holographic regularisation procedures. However, this interesting problem deservesto be studied in a future occasion.Finally, it is tantalizing to compare (5.18) with analogous formulae for the on-shell actionsin the case of four-dimensional and six-dimensional gauged supegravities, S
4d sugra [ M ] = π G ( | b | + | b | ) | b || b | , S
6d sugra [ M ] = π G ( | b | + | b | + | b | ) | b || b || b | , (5.19)put forward in [24] and [50], respectively. Here we simply note that these are expressions forthe holographically renormalised on-shell action of supersymmetric solutions dual to fieldtheories defined on backgrounds with topology of S and S , respectively, referring to [24]and [50] for more details. In this paper we have computed the partition function of N = 1 supersymmetric gaugetheories — comprising a vector multiplet for a general gauge group, chiral multiplets withgeneric R-charges and possibly a superpotential — defined on a primary Hopf surface H p,q .We have found that this depends on the background only through the complex structuremoduli p, q of the Hopf surface, and is proportional to the supersymmetric index I ( p, q )with fugacities p, q . We have carried out the computation reducing the path integral to amatrix integral over the holonomy of the gauge field around S , and evaluating explicitlythe one-loop determinant using the method developed in [10]. The second formula was verified in several explicit examples in [50], and conjectured to hold for generalsolutions with the topology of the six-ball. In [50] it is presented in terms of positive coefficients b , b , b ,parameterising a contact structure on the five-sphere. −F by performing a careful regularisation ofthe infinite products, employing generalised zeta function techniques. This factor defines a supersymmetric Casimir energy , depending on the anomaly coefficients a , c and containingthe leading contribution of log Z in the large N limit. We believe that this term cannot beexpressed as a supersymmetric local counterterm and therefore it should be independent ofthe details of the regularisation scheme. We plan to investigate this further, for example byclassifying the possible supersymmetric counterterms.Perhaps a related question is that of clarifying the dependence of the partition functionon the function κ , parametrising the freedom in choosing the background fields A µ , V µ [18].Throughout this paper we have worked with the specific choice of κ in (2.17), dictated byrequiring that A µ is real . The general arguments presented in [18] imply that the partitionfunction should not depend on κ , at least when the path integral is well defined. However,for a generic choice of κ the Lagrangian (2.38) does not have positive-definite bosonic part,so that the localization arguments become more formal. It would be nice to analyse thedependence on κ more explicitly.There are several directions for future work. It would be interesting to apply our methodto compute other BPS observables, such as a supersymmetric Wilson loop. It should also bepossible to prove factorisation of the index [51, 21] using a generalisation of the argumentsin section 5.2 of [10]. As a simple generalisation of our analysis, it should be possible toconsider non-direct-product metrics, thus allowing for general complex parameters p, q (seeappendix D). A more challenging extension is that of performing a localization computa-tion on Hermitian manifolds with different topologies, requiring only the existence of onesupercharge.One of the motivations for this work was to clarify the results of [22], by obtaining aprecise prediction for the holographically renormalised on-shell action in five-dimensionalgauged supergravity, which we presented in (5.18). It would be interesting to reproduce thisformula directly from the dual gravitational perspective. We have noted that in dimensionsfour, five, and six, the relevant on-shell actions appear to follow a precise pattern, and weexpect that explaining this will improve our general understanding of the gauge/gravityduality. Acknowledgments
We thank J. Sparks for discussions and communications on Ref. [50]. B. A. and D. M.are supported by the ERC Starting Grant N. 304806, “The Gauge/Gravity Duality andGeometry in String Theory”. D. M. also acknowledges partial support from the STFC grantST/J002798/1. D. C. is supported by the STFC grant ST/J002798/1.42
Conventions and identities
In this appendix we spell out our conventions and give some identities, useful for the com-putations in the main text.Our spinor conventions are as in [7]. A two-component notation is used: left-handedspinors carry an undotted index, as ζ α , α = 1 ,
2, while right-handed spinors are denoted by atilde and carry a dotted index, as e ζ ˙ α . These transform in the ( , ) and ( , ) representationsof Spin (4) = SU (2) + × SU (2) − , respectively. The Hermitian conjugate spinors have indexstructure ( ζ † ) α = ( ζ α ) ∗ , ( e ζ † ) ˙ α = ( e ζ ˙ α ) ∗ , (A.1)and the spinor norms are given by | ζ | = ζ † α ζ α and | e ζ | = e ζ † ˙ α e ζ ˙ α .The Clifford algebra is generated by 2 × σ aα ˙ α = ( ~σ, − i ) , e σ a ˙ αα = ( − ~σ, − i ) , (A.2)where a = 1 , . . . , ~σ = ( σ , σ , σ ) are the Pauli matrices. Thegenerators of SU (2) + and SU (2) − are given by σ ab = 14 ( σ a e σ b − σ b e σ a ) , e σ ab = 14 ( e σ a σ b − e σ b σ a ) , (A.3)and satisfy 12 ǫ abcd σ cd = σ ab , ǫ abcd e σ cd = − e σ ab , (A.4)with ǫ = 1, namely they are self-dual and anti-self-dual, respectively. The sigma matriceshave the following hermiticity properties( σ a ) † = − e σ a , ( σ ab ) † = − σ ab , ( e σ ab ) † = − e σ ab , (A.5)and satisfy the relations σ a e σ b + σ b e σ a = − δ ab , e σ a σ b + e σ b σ a = − δ ab ,σ a e σ b σ c = − δ ab σ c + δ ac σ b − δ bc σ a + ǫ abcd σ d , e σ a σ b e σ c = − δ ab e σ c + δ ac e σ b − δ bc e σ a − ǫ abcd e σ d ,σ ab σ cd = ( − ǫ abcd − δ ad σ bc + 2 δ ac σ bd − δ bc σ ad + 2 δ bd σ ac − δ ac δ bd + δ ad δ bc ) , e σ ab e σ cd = (+ ǫ abcd − δ ad e σ bc + 2 δ ac e σ bd − δ bc e σ ad + 2 δ bd e σ ac − δ ac δ bd + δ ad δ bc ) . (A.6)Our supersymmetry parameters ζ , e ζ are commuting spinors, with the supersymmetryvariation δ ζ , δ e ζ being Grassmann-odd operators; on the other hand, the dynamical spinorfields are assumed anti-commuting. The spinor indices are raised or lowered acting from43he left with the antisymmetric symbol ε αβ = − ε αβ = ε ˙ α ˙ β = − ε ˙ α ˙ β , chosen such that ε = +1. When constructing a spinor bilinear, the indices are contracted as ζ χ = ζ α χ α and e ζ e χ = e ζ ˙ α e χ ˙ α . Then one has the following relations for commuting spinors ζ χ = − χζ , e ζ e χ = − e χ e ζ ,ζ σ a e χ = e χ e σ a ζ , ζ σ ab χ = χσ ab ζ , ( σ a e ζ ) χ = − e ζ e σ a χ , ( σ ab ζ ) χ = − ζ σ ab χ , ( ζ χ ) † = χ † ζ † , ( e ζ e χ ) † = e χ † e ζ † , ( ζ σ a e χ ) † = − e χ † e σ a ζ † , ( ζ σ ab χ ) † = − χ † σ ab ζ † , (A.7)as well as the Fierz identities( ζ χ )( e ζ e χ ) = − ( ζ σ a e χ )( χσ a e ζ ) , ( χ χ )( χ χ ) = − ( χ χ )( χ χ ) − ( χ χ )( χ χ ) . (A.8)When the spinors are anti-commuting one has to include an extra minus sign whenever therelation involves swapping two of them.The spinor covariant derivative is given by ∇ µ ζ = ∂ µ ζ − ω µab σ ab ζ , ∇ µ e ζ = ∂ µ e ζ − ω µab e σ ab e ζ , (A.9)where ω µab is the spin connection, defined from the vielbein e aµ and its inverse e µa as ω µab = 2 e ν [ a ∂ [ µ e b ] ν ] − e ν [ a e b ] ρ e cµ ∂ ν e dρ . (A.10)From the spin connection we can construct the Riemann tensor via R µνab = ∂ µ ω νab − ∂ ν ω µab + ω µac ω νcb − ω νac ω µcb . (A.11)The integrability condition of the supersymmetry equation (2.1) implies the followingrelations ( R + 6 V µ V µ ) ζ = 4 i ( ∂ µ A ν − ∂ ν A µ ) σ µν ζ ,R + 6 V µ V µ = 2 J µν ( ∂ µ A ν − ∂ ν A µ ) . (A.12)The first is derived using [ ∇ µ , ∇ ν ] ζ = − R µνab σ ab ζ , contracted with σ µν , and implies the Our spin connection and Riemann tensor differ by a sign from those of [7] (so our Ricci scalar is positiveon a round sphere).
B Weyl transformations
In this appendix we discuss how the supersymmetry transformations and Lagrangians areaffected by a conformal rescaling of the geometry and of the dynamical fields, in the casewhen there exist two supercharges of opposite R-charge. This will explicitly show that theconformal factor Ω can be rescaled away from the localizing terms, and therefore does notaffect the result of the computation of the one-loop determinants.We consider a Weyl rescaling of the general metric (2.9), g µν = Λ ˆ g µν , (B.1)corresponding to redefining the conformal factor Ω asΩ = Λ ˆΩ , (B.2)where here and below a hat denotes the transformed quantities. We assume that Λ is areal, positive function depending on z, ¯ z only, so that rescaled background still admits twosupercharges of opposite R-charge. If Λ is chosen equal to Ω, then the conformal factor ofthe new metric is simply ˆΩ = 1 . The vielbein and the spin connection transform as e aµ = Λ ˆ e aµ , ω µab = ˆ ω µab + ˆ e cµ ( δ ca ˆ e ν b − δ cb ˆ e νa ) ∂ ν log Λ , (B.3)while the two-form J µν transforms in the same way as the metric, J µν = Λ ˆ J µν , and thecomplex structure J µν remains invariant. As a vector, K is invariant, while as a one-formit transforms as K µ = Λ ˆ K µ . Starting from (2.15), (2.16), we can now deduce how thebackground fields A and V transform. We will also assign a weight to | s | and κ , | s | = Λ | ˆ s | , κ = Λ − ˆ κ , (B.4)so that both the imaginary part of A and the one of V remain invariant. Note that theseconditions are consistent with those ensuring that A is real, given in (2.17). Then from (2.15)and (2.16) we obtain V µ = ˆ V µ + (d c log Λ) µ , A µ = ˆ A µ + 32 (d c log Λ) µ , (B.5) The transformation of | s | is necessary to make sure that the spinors transform correctly and that theimaginary part of A does not transform. The transformation of κ is imposed for simplicity: as explainedin section 2 any choice of U µ = κK µ drops from the supersymmetry variations and the localizing terms, aslong as one defines A ‡ µ = A † µ . c log Λ) µ = J µν ∂ ν log Λ = − e J µν ∂ ν log Λ . Finally, from (2.14) we see that the spinorstransform as ζ = Λ / ˆ ζ , e ζ = Λ / ˆ e ζ . (B.6)We now consider the variations of the fields in the supersymmetry multiplets, showingthat these are covariant if the Weyl transformation is accompanied by suitable rescaling ofthe fields. Let us start with the gauge multiplet, where we assign the standard conformalweights A µ = ˆ A µ , λ = Λ − / ˆ λ , e λ = Λ − / ˆ e λ , D = Λ − ˆ D . (B.7)It is easy to see that the supersymmetry variations (2.23) transform covariantly as δ A µ = ˆ δ ˆ A µ , δλ = Λ − / ˆ δ ˆ λ , δ e λ = Λ − / ˆ δ ˆ e λ , δD = Λ − ˆ δ ˆ D , (B.8)where the variation ˆ δ uses ˆ ζ , ˆ e ζ , and is done on the transformed background defined by ˆ g µν ,ˆ V and ˆ A . The only non-trivial check is for the relation involving D : this follows using thefact that the A cs µ = A µ − V µ is invariant under the Weyl transformation, and the followingidentity e ζ e σ µ ∇ µ λ ≡ Λ − / ˆ e ζ ˆ e σ µ (cid:0) ˆ ∇ µ − ˆ σ µν ∂ ν log Λ (cid:1)(cid:0) Λ − / ˆ λ (cid:1) = Λ − ˆ e ζ ˆ e σ µ ˆ ∇ µ ˆ λ , (B.9)where we used e σ a σ ab = − e σ b .It is also easy to see that the localizing terms, as well as the Lagrangian (2.25) for thevector multiplet scale as Λ − , so that the action is invariant, namely Z d x √ g L vector = Z d x p ˆ g b L vector . (B.10)We then pass to the chiral multiplet, whose supersymmetry variations were given in (2.37).For the scalar φ we take φ = Λ − w ˆ φ , and choose the conformal weight w such as w = 3 r/ ψ , e ψ is w + 1 /
2, while the one of F , e F is w + 1. Again, one canshow that the supersymmetry variations are covariant under the rescaling, namely δφ = Λ − w ˆ δ ˆ φ , δψ = Λ − w − / ˆ δ ˆ ψ , δF = Λ − w − ˆ δ ˆ F , (B.11)with exactly the same relations for e φ , e ψ and e F . While this is straightforward for the variation46f φ , it is less obvious for the others. For instance, in the variation of e ψ in (2.37) we have e σ µ ζ D µ e φ = Λ − / ˆ e σ µ ˆ ζ (cid:16) ˆ D µ + 32 i rJ µν ∂ ν log Λ (cid:17)(cid:16) Λ − w ˆ e φ (cid:17) = Λ − w − / ˆ e σ µ ˆ ζ (cid:20) ˆ D µ ˆ e φ − (cid:16) w δ µν − riJ µν (cid:17) ∂ ν log Λ ˆ e φ (cid:21) . (B.12)Since we set w = r , the second term vanishes because the vector X µ = ( δ µν − iJ µν ) ∂ ν log Λis holomorphic, and therefore satisfies X µ e σ µ ζ = 0 . We can now discuss how the localizingterm δ ζ ( V + V ) for the chiral multiplet transforms. Given that this is constructed as acombination of supersymmetry variations, it is also covariant under the Weyl transformation.Specifically, it transforms as δ ζ ( V + V ) = Λ − w − ˆ δ ˆ ζ ( b V + b V ) . (B.13)Now consider taking Λ = Ω, so that ˆΩ = 1. If as a localizing term we consider thefollowing modified integral weighted by the suitable power of Ω Z d x √ g Ω w − δ ζ ( V + V ) , (B.14)then we see that this precisely equal to the original localizing term, in a background withΩ = 1, namely Z d x √ g Ω w − δ ζ ( V + V ) = Z d x p ˆ g ˆ δ ˆ ζ ( b V + b V ) . (B.15)In this way the background dependence on Ω in the localizing term can be reabsorbed by aredefinition of the dynamical fields.In conclusion, we have shown that the localizing terms on the left hand side of (B.10)and (B.15) are equivalent upon rescaling the dynamical fields to the same localizing termsdefined on a background having Ω = 1. This is in agreement with the results of [18]. C S × S v with arbitrary b , b S v , namely the biaxially squashed three-sphere with SU (2) × U (1) isometry and squashing parameter v . For any value of v , this yields a familyof four-dimensional supersymmetric backgrounds S × S v , depending on the two parameters b and b which define the Killing vector (3.6). The results of the present paper show thatthe partition function depends on b , b , and not on v . A similar construction of three-dimensional backgrounds, obtained from a dual holographic perspective, has been presented47n [24].We take a four-dimensional metricd s = Ω d τ + d s ( S v ) , (C.1)where the metric on the Berger sphere in standard form isd s ( S v ) = d θ + sin θ d ϕ + v (d ς + cos θ d ϕ ) , (C.2)with θ ∈ [0 , π ], ϕ ∈ [0 , π ], ς ∈ [0 , π ], and v > ϕ = ϕ + ϕ , ς = ϕ − ϕ . (C.3)Identifying θ = πρ , so that f = π , the matrix m IJ reads m = 4 cos θ (cid:18) sin θ v cos θ (cid:19) , m = (1 − v ) sin θ ,m = 4 sin θ (cid:18) v sin θ θ (cid:19) . (C.4)Given the choice of Killing vector K in (3.6), the supersymmetry condition K µ K µ = 0 yieldsΩ = b sin θ + v ( b − + b + cos θ ) , (C.5)where b ± = b ± b . The background fields A and V are obtained from eqs. (3.23), (3.24) byfirst evaluating the functions c and a χ appearing in the form (3.8) of the metric. We find c = 4 v | b b | Ω sin θ ,a χ = 1Ω (cid:2) b + b − sin θ + v ( b − + b + cos θ )( b + + b − cos θ ) (cid:3) , (C.6)with the map to the ψ, χ coordinates being ϕ = b + ψ + b − χ , ς = b − ψ + b + χ . (C.7)One can also determine the complex coordinate z = u ( θ ) + i b b ( b + ς − b − ϕ ) entering in (3.8)by integrating (3.15), which takes the formd u d θ = Ω( θ )4 v | b b | sin θ , (C.8)48nd can be solved in closed form. Then from (3.24) we obtain A = v sgn( b b )2 Ω (cid:2) b cos θ sin θ + v ( b − + b + cos θ )( b − cos θ + b + cos(2 θ )) (cid:3) ( b + d ς − b − d ϕ )+ 12 d ω, (C.9)with ω = 12 [sgn( b )( ϕ + ς ) + sgn( b )( ϕ − ς )] , (C.10)while (3.23) gives V = v | b b | Ω n − v ( b + + b − cos θ )( b − + b + cos θ ) + v b + (cid:2) b − + 7 b − b (C.11)+ b + (cid:0) (22 b − + 4 b ) cos θ + 16 b − b + cos(2 θ ) + 2( b − + 2 b ) cos(3 θ ) + b − b + cos(4 θ ) (cid:1) (cid:3) + 2 b (cid:2) b − b − ) cos θ + b − b + (3 + cos(2 θ )) (cid:3) sin θ o ( b + d ς − b − d ϕ )+ v sgn( b b )3 Ω (cid:2) b + (cid:0) b − cos θ + b + (1 + cos θ ) (cid:1) − v ( b − + b + cos θ ) (cid:3)(cid:18) b + d ϕ − b − d ς b b + i τ (cid:19) . These expressions simplify in the following two special cases.
Case b = − b , with v arbitrary If we choose b = − b = b/ >
0, we obtainΩ = b v , c = sin θv , a χ = cos θ . (C.12)The complex coordinate z is given by z = b (cid:0) log tan θ + iϕ (cid:1) . The background fields A and V reduce to the SU (2) × U (1) × U (1) invariant expressions A = 12 (d ς + cos θ d ϕ ) ,V = v (cid:18) d ς + cos θ d ϕ + i b d τ (cid:19) , (C.13)with the conformally invariant combination being A cs = A − V = 12 (1 − v ) (d ς + cos θ d ϕ ) − i b v d τ . (C.14)The gravity dual of superconformal field theories on S × S v with this SU (2) × U (1) × U (1)invariant choice of background one-forms has been studied in [22].49 ase v = 1 , with b and b arbitrary Let us keep b and b arbitrary, and set v = 1, so that the metric (C.2) becomes the one ofthe round three-sphere. Then Ω , c and a χ simplify toΩ = 4 (cid:18) b cos θ b sin θ (cid:19) , c = 4 | b b | sin θ Ω ,a χ = 4Ω (cid:18) b cos θ − b sin θ (cid:19) , (C.15)and the background fields read A = sgn( b b )4 Ω (cid:2) (cid:0) b + b (cid:1) cos θ + (cid:0) b − b (cid:1) (1 + 3 cos(2 θ )) (cid:3) ( b + d ς − b − d ϕ ) + 12 d ω , (C.16) A cs = A − V = − sgn( b b )2 Ω ( b + d ϕ − b − d ς ) − i | b b | Ω d τ + 12 d ω . (C.17)As a final remark, we observe that the class of three-sphere metrics (3.3) also comprisesthe elliptically squashed three-sphere with U (1) isometry. This may be obtained redefiningthe coordinate ρ into a coordinate ϑ ∈ [0 , π/
2] such that f d ρ = [ γ sin ϑ + γ cos ϑ ] / d ϑ ,and taking m = γ cos ϑ , m = γ sin ϑ , m = 0; here, γ and γ are real parameters,with the squashing being controlled by γ /γ . The particular choice γ = 1 /b and γ = 1 /b leads to simpler expressions (for instance eq. (3.7) gives Ω = 1 and the background fields alsosimplify), however we stress that this choice is not necessary; again, the partition functiondepends on b , b and not on γ , γ . D Non-direct product metric
In this paper we consider supersymmetric backgrounds having S × S topology and admit-ting two supercharges of opposite R-charge. In the main text we focused on direct productmetrics with U (1) isometry, together with a complex Killing vector K depending on tworeal parameters b , b , cf. eqs. (3.3) and (3.6), respectively. We discussed how these dataare sufficient to characterize the supersymmetric background. In this appendix, we relax thedirect product condition and make a preliminary analysis of the more general case in which S is fibered over S , still preserving a U (1) isometry. As we show below, this generaliza-tion allows to consider complex values of the moduli b and b parametrising the complexstructure on the Hopf surface and appearing in the supersymmetric partition function.The most general metric with U (1) invariance on the topological product S × S can50e written asd s = Ω (cid:0) d τ + c I d ϕ I + ˜ c d ρ (cid:1) + f d ρ + m IJ (cid:0) d ϕ I + n I d ρ (cid:1) (cid:0) d ϕ J + n J d ρ (cid:1) , (D.1)where all the metric functions depend solely on the ρ coordinate. An immediate semplifica-tion occurs by noting that one can set n I = ˜ c = 0 by a suitable redefinition of the angularcoordinates ϕ I and τ ; hence with no loss of generality we can restrict to the simpler metricd s = Ω (cid:0) d τ + c I d ϕ I (cid:1) + f d ρ + m IJ d ϕ I d ϕ J . (D.2)Further, the Killing vector K in (3.6) can be generalised by analytically continuing theparameters b and b to complex values K = 12 (cid:20) b ∂∂ϕ + b ∂∂ϕ − i ∂∂τ (cid:21) , (D.3)where b I = b I + ik I , with b I and k I real. Since [ K, K ] = 0 is still satisfied, for the backgroundto be supersymmetric we just need to solve the condition K µ K µ = 0 . This constrains themetric as Ω (cid:0) i c I b I (cid:1) = b I m IJ b J . (D.4)Separating the real and imaginary parts, we obtain b I c I = Ω − Im p b I m IJ b J ,k I c I = 1 − Ω − Re p b I m IJ b J . (D.5)In the generic case where the 2 × (cid:18) b I k I (cid:19) = (cid:18) b b k k (cid:19) is invertible, these equations canbe solved for the c I . In the main text we considered instead the non-generic case k I = 0, withthe second equation solved by Ω = b I m IJ b J , and the first satisfied by setting c I = 0, namelyassuming a direct product metric on S × M . Note that in the generic case one cannot set c I = 0 . In both cases, the metric on M remains arbitrary, in particular independent of the b I .Let us discuss regularity of the metric in the generic case. In addition to the conditionsstated in section 3.2, ensuring regularity of the metric on M , we need that the one-formdescribing the S fibration be well-defined on M . This amounts to requiring that c → ρ → ϕ shrinks to zero size), and that c → ρ → ϕ shrinks). Let us study the behavior at ρ →
0, the case ρ → ρ → c → Ω(0) − p m (0) ( b b + k k ) − b b k − b k ,c → − Ω(0) − p m (0) | b | + b b k − b k . (D.6)The regularity condition c (0) = 0 fixes Ω(0) = p m (0) | b | b , which then gives c → k | b | .Apart for the behavior at the poles, in this generic case Ω( ρ ) is arbitrary.In order to complete the global analysis, and check regularity of the background fields A and V as well, we should proceed as done in the main text for the direct product case: definecomplex coordinates w, z and then use the formulae in section 2.1. Although straightforward,we will not pursue this in the present paper. E Proof that ( z , z ) ∈ C − (0 , Below we complete the proof that the coordinates (3.32), namely z = e −| b | ( iw + z ) = e | b | τ e | b | ( Q − u ) e − i sgn( b ) ϕ ,z = e −| b | ( iw − z ) = e | b | τ e | b | ( Q + u ) e − i sgn( b ) ϕ , (E.1)where the function Q ( ρ ), u ( ρ ) obey Q ′ = f a χ Ω c , u ′ = f Ω c , (E.2)span C − (0 , c = 2 | b b | Ω p det( m IJ ) ,a χ = 1Ω (cid:0) b m − b m (cid:1) , Ω = b I m IJ b J , (E.3)with f arbitrary, and obey certain boundary conditions near to the end-points of the interval[0 , | z | = e | b | δ for finite δ ∈ R and solving for τ = δ − Q − u , we obtain | z | = e | b | δ e − | b | u , | z | = e | b | δ , (E.4)52nd similarly fixing | z | = e | b | δ for finite δ ∈ R and solving for τ = δ − Q + u we obtain | z | = e | b | δ , | z | = e | b | δ e | b | u . (E.5)The expansion near to ρ → ρ → u ′ ( ρ ) = 12 | b | ρ + O ( ρ ) ,u ′ ( ρ ) = 12 | b | (1 − ρ ) + O ((1 − ρ ) ) , (E.6)leading to u ( ρ ) = 12 | b | log ρ + O ( ρ ) ,u ( ρ ) = − | b | log(1 − ρ ) + O ((1 − ρ ) ) . (E.7)Using these, and noticing that u ( ρ ) is a monotonically increasing function of ρ , since u ′ = f Ω c ≥
0, we see that u ( ρ ) is a bijection (0 , → ( −∞ , + ∞ ). Therefore, at fixed non-zero | z | ,the radial coordinate | z | covers R > (once) and at fixed non-zero | z | , the radial coordinate | z | covers R > (once).So far we have seen that for ( τ, ϕ , ϕ , ρ ) ∈ R × [0 , π ) × [0 , π ) × (0 , z , z ) cover C − { ( C , } − { (0 , C ) } . The cases u = ±∞ , corresponding to ρ = 0 and ρ = 1, must be considered separately, since we may not be able to solve for τ ∈ R in thosecases ( τ = ±∞ / ∈ R !). Again solving for Q and u near to ρ → ρ →
1, we obtain Q − u = O ( ρ ) , Q − u = − | b | log(1 − ρ ) + O ((1 − ρ ) ) ,Q + u = 1 | b | log ρ + O ( ρ ) , Q + u = O ((1 − ρ ) ) . (E.8)In the limit ρ = 0 we have | z | = e | b | τ , | z | = 0 , (E.9)while in the limit ρ = 1 we have | z | = 0 , | z | = e | b | τ . (E.10)Then we observe that at | z | = 0, | z | covers R > (once) and at | z | = 0, | z | covers R > (once). This concludes the proof that ( z , z ) covers C − (0 , Reduction of the 4d supersymmetry equations to 3d
In this appendix we revisit the 4d →
3d reduction of the supersymmetry equations (2.1),(2.2) discussed in [27, app. D] (see also [6]), including a more general identification betweenthe background fields as well as a non-trivial dilaton. Then we show that the 4d backgrounddescribed in section 3.2 reduces to the 3d background considered in [10].
General reduction
Similarly to the four-dimensional case, in three dimensions the supersymmetry equationarising from the rigid limit of “new minimal” supergravity contains different signs dependingon whether the spinor parameter has R-charge +1 or −
1. In terms of a spinor ǫ with R-charge+1 and a spinor η with R-charge −
1, one has [27] (cid:0) ˇ ∇ i − i ˇ A i (cid:1) ǫ + i h γ i ǫ + i ˇ V i ǫ + 12 ǫ ijk ˇ V j γ k ǫ = 0 , (F.1) (cid:0) ˇ ∇ i + i ˇ A i (cid:1) η + i h γ i η − i ˇ V i η − ǫ ijk ˇ V j γ k η = 0 , (F.2)where i, j, k are 3d curved indices, and we append a ˇ on 3d quantities that may be confusedwith 4d ones. The 3d spinor covariant derivative is defined asˇ ∇ i ǫ = (cid:16) ∂ i + i ω i ˇ a ˇ b ǫ ˇ a ˇ b ˇ c γ ˇ c (cid:17) ǫ , (F.3)(same for η ), where ˇ ω i ˇ a ˇ b is the 3d spin connection, and ˇ a, ˇ b, ˇ c are 3d flat indices. Moreover, ˇ A i is the 3d background gauge field coupling to the R-current, while ˇ V i and ˇ h are a backgroundone-form and a background scalar, respectively. Our 3d gamma matrices are defined as( γ ˇ a ) αβ = σ ˇ a Pauli . These are related to the 4d sigma matrices as σ ˇ aα ˙ α = i ( γ ˇ a ) αβ σ β ˙ β , e σ ˇ a ˙ αα = − i e σ αβ ( γ ˇ a ) βα , (F.4)which imply σ ˇ a = − i γ ˇ a , σ ˇ a ˇ b = − i ǫ ˇ a ˇ b ˇ c γ ˇ c , e σ ˇ a = − i e σ γ ˇ a σ , e σ ˇ a ˇ b = + i ǫ ˇ a ˇ b ˇ c e σ γ ˇ c σ . (F.5)In this way, a 4d left-handed spinor ζ α directly reduces to a 3d spinor, while a 4d right-handedspinor e ζ ˙ α is mapped to a 3d spinor via iσ α ˙ α e ζ ˙ α .54et us consider a 4d metric of the formd s = ˇ g ij ( x )d x i d x j + e x ) (cid:0) d τ + c i ( x )d x i (cid:1) , (F.6)where we are splitting the 4d coordinates as x µ = ( x i , τ ), and ˇ g ij , c i , Φ are a 3d metric, a 3done-form and a dilaton function, respectively, depending on the 3d coordinates only. The 4dvielbein and its inverse can be written as e aµ = (cid:18) ˇ e ˇ ai Φ c i e Φ (cid:19) , e µa = (cid:18) ˇ e i ˇ a − c j ˇ e j ˇ a e − Φ (cid:19) , (F.7)where ˇ e ˇ ai is a vielbein for ˇ g ij , with inverse ˇ e i ˇ a . The 4d spin connection ω cab splits as ω ˇ c ˇ a ˇ b = ˇ e i ˇ c ˇ ω i ˇ a ˇ b , ω a ˇ b = − e Φ ∂ [ i c j ] ˇ e i ˇ a ˇ e j ˇ b ,ω ˇ c b = e Φ ∂ [ i c j ] ˇ e i ˇ b ˇ e j ˇ c , ω b = ˇ e i ˇ b ∂ i Φ . (F.8)We now reduce the 4d equation for ζ given in (2.1) along the Killing direction ∂/∂τ .Assuming that ζ is independent of τ , we obtain the following 3d equations (cid:20)
14 e Φ v i γ i − i ∂ i Φ γ i − i e − Φ A τ + i e − Φ V τ −
12 ( V i − c i V τ ) γ i (cid:21) ζ = 0 , (F.9) (cid:20) ˇ ∇ i + 14 e Φ ǫ ijk v j γ k − i ( A i − c i A τ ) + i ( V i − c i V τ ) + 12 e − Φ V τ γ i + 12 ǫ ijk ( V j − c j V τ ) γ k (cid:21) ζ = 0 , (F.10)where we introduced v i = − i ǫ ijk ∂ j c k . (F.11)The first equation is solved by requiring that the 4d one-form U µ = ( U i , U τ ) = (cid:16) V i − c i V τ + 2 c i A τ −
12 e Φ v i + i ∂ i Φ , A τ − V τ (cid:17) (F.12)satisfies U µ e σ µ ζ = 0 ⇔ ( U i − c i U τ ) γ i ζ + i e − Φ U τ ζ = 0 , (F.13)which is equivalent to J µν U ν = i U µ , meaning that U µ is of type (0 ,
1) with respect to thecomplex structure J defined by ζ . Then eq. (F.10) can be matched with either one of the3d supersymmetry conditions (F.1), (F.2). As we will need to precisely recover the solutionstudied in [10], we choose to match the equation (F.2) for η , although this leads to a mapbetween 4d and 3d background fields containing some awckward minus signs. Identifying55he spinor parameters as η = ζ , the 3d background fields are given byˇ A i = − ( A i − c i A τ ) −
12 e Φ v i , ˇ V i = − ( V i − c i V τ ) −
12 e Φ v i , ˇ h = − i e − Φ V τ . (F.14)The reduction of the equation (2.2) for a spinor e ζ works similarly. In this case, we needto require that the one-form e U µ = ( e U i , e U τ ) = (cid:16) V i − c i V τ + 2 c i A τ −
12 e Φ v i − i ∂ i Φ , A τ − V τ (cid:17) (F.15)(differing from U µ just by the sign of ∂ i Φ) satisfies e U µ σ µ e ζ = 0 ⇔ (cid:0) e U i − c i e U τ (cid:1) γ i σ e ζ − i e − Φ e U τ σ e ζ = 0 , (F.16)namely is of type (0 ,
1) with respect to the complex structure e J defined by e ζ . Identifying thespinors as ǫ = iσ e ζ , eq. (F.1) is retrieved by taking exactly the same 3d background fieldsas in (F.14).From (F.14), we see that if we want both the 4d A and the 3d ˇ A to be real, then thepurely imaginary v has to vanish. In this case, it is possible to set c to zero by redefiningthe τ coordinate, so that the 4d metric takes a direct product form.We observe that the 3d background fields are not uniquely determined though, as the3d equations are invariant under certain shifts [27]. This remains true even if the analogousshift freedom in 4d has been fixed. For our purposes, it will be enough to discuss this forreal 3d background fields ˇ A i , ˇ V i , ˇ h . In this case, given a solution ǫ to (F.1), one also has asolution to (F.2) by taking the charge conjugate, η = ǫ c . This implies the existence of a realKilling vector ˇ K i = ǫ † γ i ǫ . Then the equations (F.1), (F.2) are invariant under shifting thebackground fields asˇ A → ˇ A + 32 ˇ κ ˇ K p ˇ K i ˇ K i , ˇ V → ˇ V + ˇ κ ˇ K p ˇ K i ˇ K i , ˇ h → ˇ h + ˇ κ , (F.17)where ˇ κ is a real function. The identifications (F.14) between 4d and 3d background fieldsfor a general ˇ κ becomeˇ A + 32 ˇ κ ˇ K p ˇ K i ˇ K i = − A i d x i , ˇ V + ˇ κ ˇ K p ˇ K i ˇ K i = − V i d x i , ˇ h + ˇ κ = − i e − Φ V τ , (F.18)where we have assumed that the 4d metric is in a direct product form, i.e. c i = v i = 0, asthis is the case that will be relevant below. 56 eduction of our background We now apply the formulae above and show that the S × M background given in section 3.2reduces to the 3d background studied in [10]. Here, neither the fact that the 3d metric admits U (1) isometry, nor the global constraints discussed in section 3.2 will play any role. Thesolution in [10] has real background fields and supercharges related by charge conjugation.The metric takes the general formd s = Ω (cid:2) (d ψ + a ) + c d z d¯ z (cid:3) , (F.19)where Ω = Ω( z, ¯ z ), c = c ( z, ¯ z ), a = a z ( z, ¯ z )d z + ¯ a ¯ z ( z, ¯ z )d¯ z , and for the spinors we take ǫ = √ s there (cid:18) (cid:19) , η = − iσ ǫ ∗ = p s ∗ there (cid:18) (cid:19) , (F.20)where | s there | = Ω. Then the 3d Killing vector is ˇ K = ∂/∂ψ , which as a one-forms readsˇ K = Ω (d ψ + a ). Finally, the background fields given in [10] read A there = − Im (cid:2) ∂ z log (cid:0) Ω c (cid:1) d z (cid:3) + 12 dArg( s there ) + ∗ (d a ) c (d ψ + a ) ,V there = − ∂ z log Ω d z ] + ∗ (d a ) c (d ψ + a ) ,h there = ∗ (d a )2Ω c . (F.21)These expressions are obtained expanding eqs. (2.11)–(2.16) therein and translating to ournotation (in particular c there = Ω c ).Reducing our 4d metric (3.8) along ∂/∂τ clearly matches (F.19). In order to match thespinors in (F.20) with our spinors (2.21), we need to identify ζ α = 1 √ η α , iσ α ˙ α e ζ ˙ α = 1 √ ǫ α ⇒ ω = − Arg( s there ) . (F.22)Using the formulae derived above, we can also check that the background fields reduce asneeded. Since the 4d metric is a direct product, we set c i = v i = 0; in addition, we takeΦ = log Ω. Starting from our expressions (2.18), (3.22) for the 4d fields A and V , it is easy57o check that the conditions on U and e U are indeed satisfied. Then (F.18) givesˇ A + 32 ˇ κ Ω(d ψ + a ) = − A i d x i = − Im (cid:2) ∂ z log (cid:0) Ω c (cid:1) d z (cid:3) −
12 d ω , ˇ V + ˇ κ Ω(d ψ + a ) = − V i d x i = − ∂ z log Ω d z ] + 13 c ∗ (d a )(d ψ + a ) , ˇ h + ˇ κ = − i Ω V τ = − ∗ (d a )6Ω c . (F.23)Comparing (F.21) and (F.23), we see that ˇ A , ˇ V and ˇ h agree with A there , V there and h there ifwe pick ˇ κ = − c ∗ (d a ) . However, in the main text it will be not necessary to fix ˇ κ , as itactually drops from the 3d supersymmetry transformations.The condition on U translates into the relation (cid:0) ˇ V i − i ∂ i log Ω (cid:1) γ i η + 1Ω ˇ V ψ η = 0 . (F.24)This is useful in the 4d →
3d reduction of the supersymmetry variations in section 4.3.
G Regularization of one-loop determinants
In this appendix we proceed with the regularization of one-loop determinants for the vectormultiplet and for the chiral multiplet.For the vector multiplet the one-loop determinant is given by the infinite product (4.30) Z vector1-loop = Z Cartan ∆ − Y α ∈ roots Y n ∈ Z Y n ,n ≥ n b + n b + i ( n + α A ) − ( n + 1) b − ( n + 1) b + i ( n + α A ) ! = Z Cartan ∆ − Y α ∈ roots F [ α A , ib , ib ] , (G.1)with b > , b > and ζ . The first step is to rewrite the infinite product above, labelled by aroot α , with triple gamma functions: F [ w α , τ, σ ] ≡ Y n ∈ Z Y n ,n ≥ w α + n − n τ − n σw α + τ + σ + n + n τ + n σ (G.2)= Y n ,n ,n ≥ w α + n − n τ − n σw α + τ + σ + n + n τ + n σ Y n ,n ,n ≥ − w α + n + n τ + n σ − w α − τ − σ + n − n τ − n σ = Γ ( w α + τ + σ | , τ, σ ) Γ (1 − w α − τ − σ | , − τ, − σ )Γ (1 + w α | , − τ, − σ ) Γ ( − w α | , τ, σ )58here w α = α A , and we renamed the parameters b , b into τ = ib and σ = ib for the easeof comparison with references [52, 53]. Then using formula (6.4) in [52] we get: F [ w α , τ, σ ] = e iπ { ζ (0 , − w α | ,τ,σ ) − ζ (0 ,w α + τ + σ | ,τ,σ ) } Y n ,n ≥ − e πi ( − w α + n τ + n σ ) − e πi ( w α +( n +1) τ +( n +1) σ ) = e iπ { ζ (0 , − w α | ,τ,σ ) − ζ (0 ,w α + τ + σ | ,τ,σ ) } e Γ e ( − w α , τ, σ ) , (G.3)where e Γ e is the elliptic gamma function defined for x, τ, σ ∈ C and Im( τ ) , Im( σ ) > e Γ e ( x, τ, σ ) = Y n ,n ≥ − e πi ( − x +( n +1) τ +( n +1) σ ) − e πi ( x + n τ + n σ ) . (G.4)In the product over roots α we can combine the factors for the roots α and − α and use someformulae in [53]: F [ w α , τ, σ ] F [ − w α , τ, σ ] = e iπ Ψ( w α ,τ,σ ) e Γ e ( − w α , τ, σ ) e Γ e ( w α , τ, σ ) = e iπ Ψ( w α ,τ,σ ) θ ( w α , τ ) θ ( − w α , σ ) , (G.5)where Ψ( w α , τ, σ ) = ζ (0 , − w α | , τ, σ ) − ζ (0 , w α + τ + σ | , τ, σ )+ ζ (0 , w α | , τ, σ ) − ζ (0 , − w α + τ + σ | , τ, σ ) (G.6)and θ is the Jacobi theta function, defined for w α , τ ∈ C , Im( τ ) > θ ( w α , τ ) = Y n ≥ (cid:0) − e πi ( nτ + w α ) (cid:1) (cid:0) − e πi (( n +1) τ − w α ) (cid:1) . (G.7)Formula (5.24) in [52] gives:Ψ( w α , τ, σ ) = w α (cid:18) τ + 1 σ (cid:19) + 16 (cid:18) τ + σ + 1 τ + 1 σ (cid:19) . (G.8)In total we have Z vector1-loop = Z Cartan ∆ − Y α ∈ ∆ + e iπ Ψ α θ ( α A , ib ) θ ( − α A , ib ) . (G.9)The contribution of a Cartan component corresponds to the contribution of a root α = 0.To evaluate it we can simply take the square root of the contribution of a positive root α α to zero, Z Cartan = (cid:16) lim α → p Z α (cid:17) r G , Z α = e iπ Ψ α θ ( α A , ib ) θ ( − α A , ib )4 sin( πα A ) , (G.10)where r G is the rank of G ( i.e. the number of Cartan generators). This yields Z Cartan = e iπ Ψ(0 ,τ,σ ) r G (e − πb ; e − πb ) r G (e − πb ; e − πb ) r G , (G.11)with the Pochhammer symbol defined for x, q ∈ C , | q | <
1, by ( x ; q ) = Q n ≥ (1 − xq n ). Withthe change of notation θ ( x, y ) = θ (e πix , e πiy ), we have Z vector1-loop = e iπ Ψ (0)vec e iπ Ψ (1)vec ( p ; p ) r G ( q ; q ) r G ∆ − Y α ∈ ∆ + θ (cid:0) e πiα A , p (cid:1) θ (cid:0) e − πiα A , q (cid:1) , Ψ (0)vec = i (cid:18) b + b − b + b b b (cid:19) | G | , Ψ (1)vec = − i b + b b b X α ∈ ∆ + α A , (G.12)with p = e − πb , q = e − πb and | G | is the dimension of G , and we have split the prefactorinto a part Ψ (1)vec depending on α A and a part Ψ (0)vec independent of α A .The regularization of the chiral multiplet one-loop determinant proceeds similarly Z chiral1-loop = Y ρ ∈ weights Y n ∈ Z Y n ,n ≥ ρ A + i r − ( b + b ) + n − in b − in b − ρ A − ir ( b + b ) − n − in b − in b = Y ρ ∈ weights Γ ( u ρ | , τ, σ ) Γ (1 − u ρ | , − τ, − σ )Γ (1 + u ρ − τ − σ | , − τ, − σ ) Γ ( − u ρ + τ + σ | , τ, σ ) , (G.13)where we have regularized the infinite product using triple Gamma functions , and we havedefined ρ A ≡ ρ ( A ), u ρ = ρ A + r ( τ + σ ), and again τ = ib , σ = ib .Using formula (6.4) of [52] leads to Z chiral1-loop = Y ρ ∈ weights e iπ Ψ( u ρ ,τ,σ ) Y n ,n ≥ − e πi ( − u ρ +( n +1) τ +( n +1) σ ) − e πi ( u ρ + n τ + n σ ) ! = Y ρ ∈ weights e iπ Ψ( u ρ ,τ,σ ) e Γ e ( u ρ , τ, σ ) , (G.14) We consider the square root because the α -factor contains both the contribution of the roots α and − α . The product Q n ∈ Z has been split into Q n > × Q n ≤ in the numerator and Q n ≥ × Q n < in thedenominator. u ρ , τ, σ ) = ζ (0 , τ + σ − u ρ | , τ, σ ) − ζ (0 , u ρ | , τ, σ ) = ( u ′ ρ ) τ σ + 2 − τ − σ τ σ u ′ ρ , (G.15)and u ′ ρ = u ρ − τ + σ = ρ A + r − ( τ + σ ). The full chiral multiplet one-loop determinant is Z chiral1-loop = e iπ Ψ (0)chi e iπ Ψ (1)chi Y ρ ∈ ∆ R Γ e (cid:0) e πiρ A ( pq ) r , p, q (cid:1) , Ψ (0)chi = i b + b b b (cid:2) ( r − ( b + b ) − ( r − (cid:0) b + b + 2 (cid:1)(cid:3) |R| , Ψ (1)chi = X ρ ∈ ∆ R − ρ A b b − i ( r − b + b b b ρ A + [3( r − ( b + b ) − − b − b ] ρ A b b , (G.16)where p = e − πb , q = e − πb , |R| = dim( R ), ∆ R is the set of weights of R and we haveredefined the e Γ e function as e Γ e ( x, τ, σ ) = Γ e (e πix , e πiτ , e πiσ ). References [1] V. Pestun,
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