Topological anomalies for Seifert 3-manifolds

We study globally supersymmetric 3d gauge theories on curved manifolds by describing the coupling of 3d topological gauge theories, with both Yang-Mills and Chern-Simons terms in the action, to background topological gravity. In our approach the Seifert condition for manifolds supporting global supersymmetry is elegantly deduced from the topological gravity BRST transformations. A cohomological characterization of the geometrical moduli which affect the partition function is obtained. In the Seifert context Chern-Simons topological (framing) anomaly is BRST trivial. We compute explicitly the corresponding local Wess-Zumino functional. As an application, we obtain the dependence on the Seifert moduli of the partition function of 3d supersymmetric gauge theory on the squashed sphere by solving the anomalous topological Ward identities, in a regularization independent way and without the need of evaluating any functional determinant.


Introduction
In the last few years there has been considerable progress in the analytical evaluation of partition functions and observables of supersymmetric gauge theories in different dimensions on certain compact manifolds equipped with suitable metrics. The common theme of these computations is localization. Localization is a long-known property of supersymmetric and topological theories, by virtue of which semi-classical approximation becomes, in certain cases, exact [1]. In more recent times this property has been exploited with considerable success in the work by Pestun [2] and in many following papers. In Pestun's approach no twisting of supersymmetry is performed. One rather seeks for manifolds and metrics supporting (generalized) covariantly constant spinors which ensure that certain supersymmetry global charges are unbroken. The global supersymmetry charges, even if spinorial in character, function essentially as topological BRST charges. Under favourable conditions one can choose a Lagrangian for which the semi-classical computation in the supersymmetric background is exact.
In three dimensions, a host of results is available. Explicit exact computations have been performed for 3-spheres, both with round and "squashed" metrics, and for Lens spaces. The best understood case is the one for which the complex conjugate of the (generalized) covariantly constant spinor is also covariantly constant. This is referred to as the "real" case in [3]. In all these cases the existence of (generalized) convariantly constant spinors implies in turn the existence of a Seifert structure on the 3-manifold. This refers to 3-manifolds with an almost contact metric structure and associated Reeb Killing vector field.
As a matter of fact Seifert 3-manifolds had already made their appearance earlier, in the study of non-supersymmetric pure Chern-Simons (CS) gauge theories. It was discovered first "experimentally" [4] and then explained using various approaches by different authors [5], [6] that the semiclassical approximation for CS theories becomes exact precisely for Seifert 3-manifolds. Later, starting from [7], this result was rederived by considering the supersymmetric extension of CS: indeed this model is equivalent, after integrating out non-dynamical auxiliary fields, to the bosonic theory. In this way, computability of CS on Seifert manifolds was brought within the more general paradigm of convariantly constant spinors and localization.
In some cases, it is possible to perform localization computations not just for a single isolated Seifert structure, but for families of Seifert metrics depending on some continuous parameters. A significant example is provided by the the squashed metric on the 3-sphere [8]. 1 In those instances the partition function (and the observables) turns out to depend non-trivially on (some of) those parameters. Take for example the case of (supersymmetric) Chern-Simons theory on the squashed spheres, with the metric ds 2 =ḡ µν (x; b) dx µ ⊗ dx ν = (sin 2 θ + b 4 cos 2 θ) dθ 2 + cos 2 θ dφ 2 1 + b 4 sin 2 θ dφ 2 2 . (1.1) At first sight, the fact that the partition function is a non-trivial function of the squashing parameter b 2 is not, per se, surprising. Indeed, even if CS theory is topological at classical level, topological invariance is anomalous at quantum level [10]. This means that the quantum CS action does depend on the background metric in a way which is controlled by the anomaly functional: where ψ µν = δg µν is the variation of the metric and c is a computable anomaly coefficient. However, if one plugs the squashed metric (1.1) and the variation ψ µν = b ∂ b g µν into the anomaly form (1.2), one finds that the anomaly vanishes identically In this work we will solve this conundrum: we will see that the vanishing of the topological anomaly for the squashed spheres is compatible with the non-trivial dependence of the partition function on the squashing parameter. As a matter of fact, we will show that the topological anomaly captures the precise dependence of the partition function on b. We will extend this results to generic three-dimensional supersymmetric theories, with both Yang-Mills and Chern-Simons terms in the action (YM+CS), involving vector multiplets only. The resolution of our puzzle will require understanding the appropriate renormalization prescription for quantum effective actions on Seifert manifolds. The time-honored method to identify the renormalization prescriptions associated to certain symmetries is to introduce backgrounds fields which act as sources for the currents associated to those symmetries. This approach has been forcefully advocated more recently in the specific context of supersymmetric gauge theories in [11] and in several following papers.
We also will introduce backgrounds, but our treatment will differ from the one which has become common in the literature on localization of the last few years. Instead of coupling the supersymmetric gauge theory to supergravity, we will first consider its topological version and then couple it to topological gravity.
This will have the advantage of obtaining the Seifert condition for global supersymmetry in the most straightforward way by avoiding all the complications of spinors. Moreover and most importantly the topological gravity formulation will make immediate to identify the subsets of the topological transformations which preserve the Seifert structure. In the Seifert context Chern-Simons topological (framing) anomaly is BRST trivial [5]. In this paper we compute explicitly the corresponding local Wess-Zumino (WZ) functional. To our knowledge this has not been done before. We will use the WZ local action to derive the dependence on the Seifert moduli of the quantum action directly from the anomalous Ward identity associated to the topological anomaly. Our computation will be manifestly regularization and gauge independent: We will do it without the need of computing explicitly any functional determinant.

Coupling 3d Chern-Simons to topological gravity
The classical Chern-Simons action [12] is is a 1-form gauge field on a closed 3-manifold M 3 . T a , with a = 1, . . . , dim G, are generators of the Lie algebra of the simple, connected and simply connected gauge group G. Gauge invariance leads to the nilpotent BRST transformation rules 2 where c = c a T a is the ghost field carrying ghost number +1 and D c ≡ d c + [A, c] + is the covariant differential. The classical action (2.4) is both invariant under diffeomorphisms and independent of the 3-dimensional background metric g µν . In order to study the fate in the quantum theory of this global topological symmetry one must couple the theory to suitable backgrounds. This has been done in [13] where it is explained that the backgrounds appropriate for the topological symmetry in question are those of equivariant topological gravity [14] s g µν = ψ µν − L ξ g µν , s ψ µν = L γ g µν − L ξ ψ µν , where ξ µ is the ghost of reparametrizations of ghost number +1, ψ µν is the topological gravitino of ghost number +1 and γ µ is the ghost-for-ghost of ghost number +2. The coupling to background topological gravity induces both deformations in the BRST transformations of the matter fields and extra terms in the action. It also requires introducing new matter fieldsÃ andc which sit in the same BRST multiplet as c and A.Ã andc are Lie algebra-valued 2 and 3-forms of ghost number −1 and −2 respectively. All the matter fields fit niceley into a single super-field, or polyform, A, whose total fermionic number, given by the form degree plus ghost number, is +1. 3 The action of the BRST transformation on the supermultiplet (2.9) before coupling to topological gravity writes in the compact form where is the "rigid" coboundary operator of total fermion number +1. It is immediate to check that (2.10) reduces to (2.7) when restricted to the fields c and A.
To describe the coupling to topological gravity it is convenient to consider the operator rather than the nilpotent BRST operator s. On the functionals of the backgrounds independent of ξ µ , S satisfies and it is therefore nilpotent on the space of equivariant functionals of the topological gravity multiplet.
After these preliminaries, one discovers that the coboundary operator appropriate to describe the BRST transformation rules of the system after coupling to topological gravity is simply where δ writes in the same form as the rigid ones When written for the component fields, these transformations become The action also gets modified after coupling to topological gravity: one can check that is invariant by transforming both the fields according to (2.16) and the backgrounds according to (2.7). The fieldsÃ andc have a natural interpretation, in the Batalin-Vilkovisky formalism, as the anti-fields of A and c. In this traditional BV framework, which is the one adopted in [15], [13], the action Γ should be supplemented with the canonical piece for all the fields and backgrounds: The full BV action generates the BRST transformations of both fields and anti-fields via the familiar BV formulas. However in this paper we will argue that an alternative -although exotic -point of view is available in our context. In the approach that we proposeÃ andc are independent auxiliary fields whose function is to close the BRST transformations off-shell: at the same time, our action we will be just Γ CS+t.g. , not the full Γ BV .
This approach is consistent since the BRST transformations close on the fields (c, A,Ã,c) and leave Γ CS+t.g. invariant. Of course, in the formulation in which (Ã,c) are auxiliary fields, Γ CS+t.g. maintains the original non-abelian gauge symmetry, which, eventually, will have to be fixed: both the gauge non-abelian symmetry and the the global vector supersymmetry associated to the topological invariance of the matter theory are captured by the BRST symmetry in (2.16). In other words, the anti-fields of the BV formulation can be reinterpreted as the auxiliary fields which are necessary to close the global supersymmetry algebra of the supersymmetric CS, after twisting that supersymmetry to obtain a topological model.
As a matter of fact, the 3d action (2.17) has more local gauge invariance than just the standard non-abelian gauge invariance: it is invariant also under the fermionic local symmetrỹ where χ is a fermionic scalar gauge parameter in the adjoint of the gauge group. Thus, the commuting fieldc can be viewed as the ghost associated to this additional local symmetry. This extra gauge invariance is fixed by replacing Γ CS+t.g. with where b is a 0-form anti-ghost of ghost number -2, Λ is a Lagrange multiplier of ghost number -1 and * is the Hodge dual with respect to the background metric g µν . Summarizing: The action Γ ′ CS+t.g. has the background topological supersymmetry captured by (2.7) and (2.16) and no other local gauge-invariance beyond the standard non-abelian gauge invariance:Ã andc can now be consistently thought of as auxiliary, non propagating, fields rather than anti-fields.
3 Coupling 3d topological YM with CS term to topological gravity The 3d topological YM theory is characterized by the BRST transformations Ψ is a fermionic 1-form of ghost number 1 and σ a bosonic 0-form of ghost number 2. Both of them are in the adjoint of the gauge group. It is convenient to introduce a super-field or polyform of total fermionic number (ghost number + form degree) equal to 2: The transformations (3.23) write in a nice compact form in terms of the "rigid" coboundary operator where It is important to observe that the super-field or polyform containing the gauge connection which is appropriate for the YM theory is not the same as the connection polyform A (2.9) of CS. Let us again denote by s the nilpotent BRST operator after coupling to topological gravity. As seen in the previous section, it is convenient to introduce the operator where ξ is the ghost associated to reparametrizations. The coboundary operator for topological YM coupled to topological gravity is again given by a formula identical to (2.14) These transformations write in components: Finally, the action of pure topological YM consists only of a s-trivial term: As just remarked, the superfield (3.26) appropriate for YM theory is quite different from the corresponding polyform relevant for Chern-Simons theory. However, we will now show that it is possibile to recast the topological YM BRST transformations purely in terms of the Chern-Simons superfield A. To this end, let us pick a contact structure k on the 3-manifold, M 3 which is dual to the background vector field γ µ : 1-forms ω on M 3 are naturally decomposed along the horizontal and vertical directions as follows Let us therefore decompose the fermionic 1-form Ψ of the topological YM multiplet according to whereÃ is a 2-form of ghost number -1. Note that the decomposition (3.37) is not unique and it has the gauge invarianceÃ When written in terms of ζ andÃ the YM topological transformations (3.30) rewrite as where we introduced the ghost-for-ghostc to take into account the gauge-invariance (3.38). When written in this form, the emergence of the CS superfield becomes apparent. Indeed the transformations (3.39) can be expressed entirely in terms of A: where Φ is the polyform of total ghost number +2 The relation (3.43) shows that the YM BRST operator δ Y M has the same algebraic content as the CS BRST operator δ. As a matter of fact one sees from (3.39) that δ Y M differs from the CS δ only because it also includes, on top of the CS transformations, the shift symmetry together with σ, the BRST partner of ζ. This shift symmetry was originally introduced in [5], to explain localization of CS theory on Seifert manifolds. Since ζ, σ make a trivial BRST doublet, the physical content of δ Y M and δ is the same. Indeed, from (3.44) one derives the identity .
In particular from the S CS -invariant action (2.17), one obtains the S Y M invariant actioñ which is equivalent to the CS action coupled to topological gravity backgrounds. This action is essentially the same as the Beasley-Witten's action [5]. In the symplectic formalism of [5] the term quadratic inÃ is interpreted as the symplectic 2-form on the space of connections living on the base of the Seifert fibration. In our approach this term emerges naturally from the coupling to the topological backgrounds. Summarizing, it is possible to include a CS-term in the action for the topological YM gauge theory coupled to topological gravity: this is given in (3.48). The total action of the topological YM+CS system coupled to the topological gravity backgrounds has therefore the form

The supersymmetric point
The quantum partition function of the topological YM + CS system in presence of topological gravity backgrounds 4 is an equivariant functional of the topological gravity multiplet. This means that it is both independent of ξ µ and invariant under reparametrizations. At classical level it satisfies the Ward identity which can be -and actually is -broken by quantum anomalies. We postpone the discussion regarding quantum topological anomalies to Section 6.
We can now look for bosonic backgrounds which are left invariant by S: The second equation above says that the superghost backgroundγ µ (x) is a Killing vector of the 3-dimensional metricḡ µν (x). These geometrical data define a so-called Seifert structure on a 3-dimensional manifold [5]. We see therefore that supersymmetric YM+CS theories admitting a rigid topological supersymmetry are precisely those defined on 3-dimensional Seifert manifolds. Hence one anticipates that supersymmetric YM+CS theories on Seifert manifolds enjoy localization properties [2]. This fact, originally discovered in a "phenomenological" way in [4], has been subsequently explained using various approaches by different authors [5], [7]. In our approach this follows straighforwardly from the BRST trasformations of topological gravity. The topological YM+CS action on a fixed Seifert manifold is therefore invariant under the following BRST transformations which encode both gaugeinvariance and global topological supersymmetry

The relation between topological and physical supersymmetry
The rigid topological theory that we obtained by coupling topological YM+CS to topological gravity computes certain (semi)-topological observables of the "physical" globally supersymmetric YM+CS theory living on the same manifold. In particular the topological partition function, which is the object that we consider in this paper, is the same as the superpartition function of the "physical" theory, i.e. the partition function with supersymmetric boundary conditions on both bosons and fermions. Indeed, as we mentioned in the introduction, the almost totality of the computations of those (semi)-topological observables performed in recent years, were developed directly in the context of the "physical" theory with spinorial supercharges. We argued in the introduction that the topological gravity viewpoint provides some benefits, both conceptual and practical. For starters, we just saw in the previous section that the Seifert condition emerges from topological gravity directly -without the necessity to go through covariantly constant spinors [7], add extra symmetries [5], or pick up ingenious gauges [6]. But, above all, the coupling to topological gravity will allow us, in the next sections, to compute the moduli dependence of the partition function of supersymmetric YM+CS theory (involving only vector multiplets) by solving the anomalous topological Ward identities, in a completely regularization and gauge independent way. In this section we will describe more precisely the relation between the topological YM+CS obtained via the coupling to topological gravity and "physical", spinorial, supersymmetric theory. We will also elucidate how the action that emerges from topological gravity encompasses the topological actions which were introduced in either [5] or [18].
Supersymmetric CS (SCS) theory on curved space has been studied starting from [7], who considered the special example of S 3 . The supersymmetric extension of the CS action in flat space [16] writes where the scalars D, σ and the Dirac spinor λ are in the adjoint of the gauge group. Since D, σ and λ are auxiliary non-dynamical fields, supersymmetric CS theory (5.55) is physically equivalent to pure CS theory. The action (5.55) is invariant under the global supersymmetry transformations which have the structure The classical action of ordinary, non supersymmetric, CS action has the peculiarity of being invariant under local diffeomorphisms without the need of introducing a space-time metric. This means that one can study quantum CS theory on any fixed curved manifold: topological invariance of CS theory must be thought of as a global symmetry, in the sense that one does not need to integrate over space-time metrics to make sense of the quantum theory. This global symmetry is actually broken by anomalies: but, precisely because one is dealing with a global symmetry, this does not spoil the consistency of the quantum theory.
One might imagine that, by analogy, SCS theory might be made invariant under local supersymmetry transformations without the need of explicitly introducing supergravity backgrounds. If this were so, SCS could be formulated consistently on any manifolds. Let us discuss why this is not the case.
The standard recipe for putting a generic supersymmetric theory on a curved manifold is to first couple it to supergravity, by promoting the global supersymmetry transformations (5.57) to local ones. For the SCS theory, this would mean in principle to couple (5.55) to the Noether supercurrents by changing the action where ψ µ is the gravitino field and the dots denote the higher order terms of the Noether procedure. The coupled action Γ curved SCS is invariant -at linearized level -under local supersymmetry transformations (5.57) of the fields if the gravitino background also transforms as However SCS theory is "almost" topological. This is reflected by the fact that the supercurrents (5.58) vanish on shell where α is an arbitrary parameter. Since, when ǫ is space-dependent, the supersymmetry variation of the flat space action writes in terms of the supercurrents as follows one sees there is an alternative way to make (the diffeomorphism invariant extension of) Γ SCS locally supersymmetric. This alternative method does not require introducing the gravitino: thanks to on-shell vanishing of the supercurrents one can simply modify the supersymmetry variations of λ, λ † and Dδ ǫ λ = δ ǫ λ + i α γ µ D µ ǫ σ , for space-time dependent ǫ's. The trouble with this "alternative" way to make the supersymmetry local is that, for α arbitrary and for generic manifolds, the local supersymmetry algebra does not close. By analyzing the supersymmetry commutation relations one discovers [19] that closure of the algebra requires both that the condition is met and that the space-time dependence of ǫ be restricted by the differential equation The lesson of this discussion is that, even for the "almost topological" supersymmetric CS theories one cannot neglect the coupling of the (classically vanishing) supercurrents to the supergravity backgrounds. Indeed it has since been understood [17], [3] that the conditions (5.65) and (5.66) are to be interpreted, in a model independent way, as the equations for the vanishing of the supersymmetry variation of the gravitino background The nice feature of Eqs. (5.67), which are readily seen to imply (5.66), is their universality: they do not depend on the specific theory one is considering and they characterize manifolds on which field theories with global supersymmetry may be constructed. The specific form of supersymmetry does instead depend on both the solution of (5.67) and the form of the coupling of the supergravity multiplet to the theory at hand. It can be shown that in the real case when a solution ǫ of δ ǫ ψ µ = 0 defines by conjugation a solution of δǭ ψ † µ = 0, the vectorγ is a (real) Killing vector of the underlying 3-manifold This explains in particular why CS theories on 3-manifolds admitting a U(1) action -known as Seifert 3-manifolds -enjoy the localization property which was originally discovered in [4] in an experimental way. We have seen that in our topological approach, the Seifert condition (5.69) emerged directly from the topological gravity BRST transformation laws, with no reference to (generalized) covariantly constant spinors.
We can consider also the supersymmetric YM action in the SCS theory: This latter action is not only invariant under the global supersymmetry transformations (5.57), (5.63), (5.67), but also supersymmetric exact Therefore, cohomologically, the SCS+SYM system is equivalent to SCS theory. The supersymmetric SCS+SYM on a fixed Seifert manifold can be twisted to give a model with a topological rigid symmetry. This was done in [18]. The physical supersymmetric vector multiplet (5.57) includes a Dirac fermion λ which has 4 real components. After the twist of [18], three of those form the topological gaugino Ψ. This, together with the scalar σ and the gauge connection A form the multiplet of topological YM. The twisted supersymmetry transformations of this supermultiplet turn out to have the form (3.30), in which the topological gravity field γ µ is replaced by the Reeb vector fieldγ µ . The remaining fermion gives rise to a scalar α of ghost number +1 which form, together with the auxiliary scalar field D of ghost number +2, an additional BRST trivial doublet, The twist of the physical SCS action (5.55) discussed in [18] gives instead the Note that the choice of X(A, σ) introduces a spurious dependence of the BRST operator on the metric compatible with the vector fieldγ µ which defines the Seifert structure. This dependence should of course drop out in physical observables, but this is not explicit in the framework of [18]. The reason of course is that twisting a physical supersymmetric action corresponds to make a specific choice for the gauge-fixing term of the topological action. This, although sometimes convenient to perform explicit computations, leads to gauge-dependent BRST transformations, somehow obscuring the geometric content of the topological symmetry. One appealing feature of our treatment is that it makes manifest that the theory only depends on the Seifert structure.

Topological Anomaly for Seifert manifolds
The classical Ward identity (4.51) can be broken by quantum anomalies The topological anomaly describes therefore the response of the quantum action density Γ[g µν ] of the YM+CS topological system under a generic variation of the metric δg µν ≡ ψ µν 1 [g µν , δg µν ] . (6.78) The topological anomaly 3-form A 1 is a local cohomology class of ghost number +1 of the BRST operator of topological gravity, which must satisfy the Wess-Zumino consistency condition (6.79) Topological anomalies were classified in [13]. In 3 dimensions we have a single representative of ghost number +1 c is the anomaly coefficient. From the structure of the anomaly, it is clear that the parity invariant topological YM part of the action cannot contribute to c. A non-trivial c can only come from the CS part Γ CS+t.g. of the action. Since this theory is equivalent to bosonic CS, we conclude that c is nothing but the coefficient of the framing anomaly of pure bosonic CS. For SU(N) gauge theories this has been computed in [10] 6 : 3-dimensional diffeomorphisms are not anomalous. Hence, there exists a renormalization prescription which gives rise to an effective (non-local) action Γ[g µν ] which transforms as a 3-form under 3-dimensional generic diffeomorphisms. To express this condition, it is useful to introduce the Bardeen-Zumino BRST operator S diff [20] associated to 3-dimensional diffeomorphisms: where ξ = ξ µ ∂ µ is the reparametrization ghost in 3-dimensions, and L ξ denotes the action of the Lie derivative along ξ on the metric g µν . The equation precisely expresses the fact that the quantum action density Γ[g µν ] transforms as 3-form under reparametrizations. After these preliminaries, let us now make our main observation: When considering YM+CS topological theories on Seifert manifolds, one relaxes the request (6.83) of full 3-dimensional reparametrization invariance. One is satisfied with invariance under reparametrizations which preserve the Seifert structure: these are reparametrizations whose ghost fields ξ µ commute with the Reeb vectorγ µ . Let us denote by the Bardeen-Zumino BRST operator associated to diffeomorphisms preservingγ µ . One also restricts the topological gravity background fields to those left invariant under Lγ Lγ g µν = Lγ ψ µν = Lγ γ µ = 0 . (6.85) To parametrize solutions of (6.85) it is useful to introduce systems of coordinates adapted to the Seifert structure associated toγ µ : (ds) 2 M = e σ k ⊗ k + g ij dx i ⊗ dx j = = e σ dy ⊗ dy + 2 e σ a i dx i ⊗ dy + (g ij + e σ a i a j ) dx i ⊗ dx j , (6.86) where k is the contact 1-form dual to the Reeb vector field iγ(k) = 1 , (6.88) σ, g ij and a i are fields on the two-dimensional surface Σ 2 , associated to the Seifert fibration π : M → Σ 2 . The invariant ψ µν are analogously parametrized by fermions ζ, ψ ij and ψ i living on Σ 2 , defined as follows: ψ µν = e σ ζ e σ ψ i + e σ ζ a i e σ ψ i + e σ ζ a i ψ ij + e σ (ψ i a j + ψ j a i + ζ a i a j ) .
Finally, the invariant ξ µ and γ µ can be written in terms of fields living on Σ 2 as Therefore, in the Seifert case, the effective action Γ Seif [g ij , σ, a i ] is a functional of the fields σ, g ij and a i , and the appropriate renormalization prescription writes The action density Γ[g µν ] which satisfies the (strong) prescription (6.83) defines, of course, once written in Seifert adapted coordinates, also an action density Γ Seif [g ij , σ, a i ] satisfiying the (weaker) Seifert renormalization condition (6.91): This effective action satisfies the topological anomaly equation The invariant gravitational background fields split into three multiplets: one is the 2-dimensional topological gravity multiplet (ξ i , γ i , g ij , ψ ij ). Then there is an abelian topological gauge multiplet (ξ 0 , γ 0 , a i ψ i ): their BRST properties are not just the "flat" ones, but they are modified by the coupling to 2-dimensional gravity. Finally there is also an uncharged scalar topological multiplet (σ, ζ): this too is coupled to 2-dimensional topological gravity. Writing A one finds the following expression for A: which is the scalar field dual to the U(1) field strength f (2) ≡ da. The important fact, now, is that A where the Wess-Zumino action Γ Seif W Z is the following local functional 7 is a legitimate Wess-Zumino action since it is both local and invariant under reparametrizations which preserve the Seifert structure It should be kept in mind, however, that Γ Seif W Z [g ij , σ, a i ] -unlike the non-local Γ[g µν ] in eq. (6.92) -is not invariant under the full 3-dimensional S diff . 7 The topological anomaly A is not a globally defined 3-form, the anomaly is indeed non-trivial in the 3dimensional sense. In [5] it is pointed out that on a Seifert manifold M there exists a natural trivialization of the double tangent T M ⊕ T M which leads to a specific definition of the Chern-Simons invariant Γ GCS [g]. This provides a geometric explanation of why the topological anomaly is BRST trivial in the Seifert context. To the best of our knowledge however an explicit local Wess-Zumino functional of the adapted Seifert metric as Γ Seif W Z [g ij , σ, a i ] has not been presented earlier. Note that Γ GCS [g] evaluated for an adapted Seifert metric and Γ Seif W Z [g ij , σ, a i ] differ by a not-globally defined total derivative, whose precise form is described in Appendix B.
Hence one can define the effective actioñ Summarizing, we have shown that it is always possible to define through Eq. (6.101) a quantum action densityΓ Seif [g ij , σ, a i ] which depends on the moduli parametrizing the Seifert structures (which we will characterize in Section 7) but not on the specific adapted metric which one picks to quantize the theory.

Moduli
In this Section we will identify cohomologically the parameters on which the quantum partition function of 3d supersymmetric gauge theory depends.
In the general "non-real" case [3], the space of infinitesimal deformations of a given supersymmetric background can be identified [21] with the appropriate cohomology of a certain differential operator∂ which, loosely speaking, is the 3d generalization of the 2-dimensional Dolbeault op-erator∂. The cohomology of∂ has not been well studied yet, and, to our knowledge, it is not explicitly known for general 3-manifolds. On the other hand, in the more special "real" context to which our analysis is confined one can give a very explicit and general characterization of the space of infinitesimal deformations of (topological) supersymmetric backgrounds.
We should also emphasize that the analysis of topological quantum anomalies contained in the previous Section makes our identification of the parameters which affect the partition function valid both at the non-linear and the quantum level.
We have seen that supersymmetric topological backgrounds correspond to solutions of the Killing equations Given a solution {ḡ µν ,γ µ } of (7.104) we want explore nearby supersymmetric backgrounds {ḡ µν + δg µν , γ µ + δγ µ }. The deformations {δg µν , δγ µ } must satisfy the linear equation Lγ δg µν + L δγḡµν = 0 . where We are interested in characterizing physical deformations, i.e. solutions of this equation modulo gauge equivalences. Gauge-invariance includes infinitesimal diffeomorphisms (δγ µ , δg µν ) ∼ (δγ µ , δg µν ) + (L ξ δγ µ , L ξ δg µν ) , (7.109) where ξ µ is a vector field on M 3 . But Lγ-invariant topological deformations of the metric should also be treated as a gauge invariances (δγ µ , δg µν ) ∼ (δγ µ , δg µν + ψ µν ) , (7.110) for any Lγ-invariant ψ µν We can therefore define a linear operator Q −1 which captures both gauge equivalences (7.109) and (7.110) where We have One can consider therefore the short exact sequence The associated cohomology space The cokernel of ϕ is, on the other hand, characterized by the Lγ-invariant ψ µν 's which are orthogonal to Img ϕ where v µ ≡ḡ µλDν ψ λν . (7.122) The vector v µ is Lγ-invariant, sinceḡ µν and ψ µν are: Hence v µ ∈ Cγ. But since, according to (7.121) v µ is orthogonal to whole Cγ, it vanishes v µ =ḡ µλDν ψ λν = 0 . where the abelian gauge connection a and its field strength f (2) are given by a = 1 2 cos α dβ + d ǫ , The curvature R 2 of the 2-dimensional metric (g 2 ) ij on the S 2 base of the Seifert fibration is where we introduced the ratio b 2 ≡l l . (8.140) The scalar field which is dual to the abelian field strength is therefore We learnt in the previous Section that, givenḡ µν (x; l,l) andγ = 1 l ∂ φ 1 + 1 l ∂ φ 2 , the deformations of the Seifert structure are associated to the isometries which commute withγ modulō γ. For generic (l,l) the isometries which commute withγ are ∂ φ 1 and ∂ φ 2 . Hence we see that b 2 parametrizes precisely the inequivalent deformations of the Seifert structure around a generic point b = 1. The point b = 1 corresponds to the "round" sphere, which has an enhanced symmetry SU(2) L × SU(2) R . Around this point more general deformations are possible, since the isometries which commute with, let us say, J R 3 form the full SU(2) L . Let us compute the topological anomaly for the squashed sphere metricḡ µν (x; l,l). Since A 1 [g µν , ψ µν ] depends only on the conformal class of the metric, we can take, without loss of generality l = 1 ,l = b 2 (8.142) and putḡ µν (x; 1, b 2 ) ≡ḡ µν (x; b). Then where the b-derivative is taken by keeping the Hopf coordinates constant. It is easy to verify that the topological anomaly for these backgrounds vanishes for all b's: This implies that the effective action Γ[g µν ] evalutated for the squashed sphere metricḡ µν ( where X(x; b) ≡ (y, α, β) are the coordinates adapted to the Seifert structure parametrized by b 2 . However, we explained in Section 6 that Γ[g µν ] is not the action renormalized with the correct Seifert prescrition (6.91). The actionΓ Seif [g ij , σ, a i ] renormalized according to the Seifert prescription is given by (6.101). When this latter action is evaluated onḡ µν (x; b), one obtains We have just shown that, due to (8.144), the first (non-local) term in the r.h.s. of the equation above is b-independent. But the second (local) one is not. Indeed, by plugging (8.139) inside(6.99) one computes which therefore encodes the whole dependence of the Seifert partition function on b 2 : where Z(1) is the partition function on the "round" sphere, corresponding to b = 1. Let us compare the anomaly equation (8.148) with the YM+CS partition function on the squashed sphere computed directly by means of localization in [19], taking for simplicity the case of SU(2) gauge group 10 Here the exponential in the integrand comes from the value of the action on the saddle point, while the hyperbolic sine factors are the results of the 1-loop determinants. Expressing the sinh factors in terms of exponentials, the integration reduces to the sum of Gaussian integrals:

Conclusions
The current paradigm for localization relies on spinorial global supercharges. Since the fate and properties of quantum global symmetries are best studied by introducing background fields coupled to currents, the same paradigm has lead to studying the coupling of "physical" supersymmetric theories to off-shell supergravity. In particular the search for globally supersymmetric models has been reduced to the study of generalized covariantly constant spinors. In this paper we proposed an alternative route. Localization is naturally understood in terms of topological scalar supercharges -i.e. in terms of topological theories and BRST symmetries. In this framework it is the coupling of topological field theories to topological gravity, not supergravity, which is relevant. For this reason we worked out the coupling of both CS and topological Yang Mills theory -i.e. of a generic vector twisted supermultiplet -to topological gravity. The BRST structure of the Chern-Simons supermultiplet looks very different from that of the topological YM theory, when the latter is presented in its familiar formulation valid in arbitrary dimension. We exhibited, however, a new formulation of the BRST transformations of topological YM in 3d purely in terms of the CS supermultiplet. This allowed us to derive a unique (anomalous) Ward identity which characterizes the coupling of a 3d generic twisted vector supermultiplet to topological gravity.
One first advantage of the topological gravity viewpoint is that the structure of topological gravity is the same in all dimensions, a fact which makes the characterization of supersymmetric bosonic backgrounds straightforward. For example, in the context of 3d gauge theories which is the one of this paper, we have seen that the Seifert condition emerges quite immediately without the need to go through generalized covariantly constant spinors or similar indirect routes peculiar to other approaches. Moreover we have found that, in the 3d context, the off-shell coupling of topological (YM+CS) gauge theories to topological gravity is easily achieved by suitably covariantizing the "rigid" coboundary BRST operator with a universal term which is the form-contraction with the super-ghost field of topological gravity. We also discovered that the anti-fields of the BV formulations of CS theory are nothing but the auxiliary fields which are required to close off-shell the topological supersymmetry algebra.
But the real payoff of the topological approach was that it made straightforward to identify the subset of local topological transformations which preserves the Seifert backgrounds. These turned out to be 2d topological gravity transformations coupled to topological abelian gauge transformations. This allowed us to give a cohomological characterization of the Seifert background moduli. Moreover we were able to explicitly solve the anomalous Ward identity associated to topological transformations of the gravitational background. The solution involved a Wess-Zumino local action, invariant under the reparametrizations which preserve the Seifert structure.
The triviality of the topological (framing) 3d anomaly when restricted on Seifert backgrounds shows, rigorously and in a completely regularization independent way, that the quantum effective action of the gauge theory on Seifert manifold depends on the Seifert moduli but not on the specific metric adapted to the Seifert structure which one picks to quantize the theory. The Wess-Zumino Seifert action also completely determines the dependence of the partition function on the Seifert moduli. We explicitly showed this in the case of the squashed sphere, for which we recovered the dependence of the partition function on the squashing modulus without computing any functional determinant.
Our discussion in this paper was restricted to (twisted) vector supermultiplets in 3d. It would be interesting to extend our results to (twisted) chiral matter. To do this it would be necessary to work out the coupling of topological chiral matter to topological gravitational backgrounds: something which, to our knowledge, has not be done yet 11 . The dependence on the Seifert moduli of the quantum effective action of chiral theories is considerably more complicated than that of vector supermultiplets. We expect therefore that the coupling of topological chiral theories to topological gravitational backgrounds involves some new ingredients. It would be equally interesting to apply our methods to higher dimensions. Realizing this program might reduce the computation of the dependence on the moduli of quantum effective actions of localizable theories to the solution of appropriate anomalous Ward identities. We hope to come back to these problems in the future.

B Gravitational CS action and the Seifert WZ action
The topological anomaly is, by definition, non-trivial: there is no local functional of the 3dimensional metric, trasforming as a 3-form, whose BRST variation gives the anomaly. As a matter of fact, one has 1 is indeed non-trivial. Let us discuss the relation between the Seifert Wess-Zumino action (6.99) and the gravitational CS action (B.159). Let us introduce the 3-form is the gravitational CS action evaluated for the metric adapted to the Seifert structure (6.86). Therefore, there exists a local 2-form Ω (2) such that where S diff is the BRST operator associated to 3-dimensional diffeomorphisms: with ξ = ξ µ ∂ µ the reparametrization ghost in 3-dimensions. Q   In other words the functional Ω (2) trivializes the 2-dimensional gravitational anomaly: this is possibile since Ω (2) is a functional not only of the 2-dimensional metric g ij but also of the abelian field strength f .