The topological structure of supergravity: an application to supersymmetric localization

The BRST algebra of supergravity is characterized by two different bilinears of the commuting supersymmetry ghosts: a vector $\gamma^\mu$ and a scalar $\phi$, the latter valued in the Yang-Mills Lie algebra. We observe that under BRST transformations $\gamma$ and $\phi$ transform as the superghosts of, respectively, topological gravity and topological Yang-Mills coupled to topological gravity. This topological structure sitting inside any supergravity leads to universal equivariant cohomological equations for the curvatures 2-forms which hold on supersymmetric bosonic backgrounds. Additional equivariant cohomological equations can be derived for supersymmetric backgrounds of supergravities for which certain gauge invariant scalar bilinears of the commuting ghosts exist. Among those, $N= (2,2)$ in $d=2$, which we discuss in detail in this paper, and $N=2$ in $d=4$.

and the particular generalized Killing spinor equations depend on the global symmetries of the specific SQFT one is interested in: indeed, conserved currents of the SQFT that one would like to probe couple to gauge fields which must sit in supergravity multiplets.
It was observed in [3] and [4] that the generalized Killing spinor equations for certain extended supergravities in 3 and 2 dimensions are equivalent to cohomological equations which are obtained by setting to zero the BRST variations of fermionic fields of topological gravity coupled to certain topological gauge systems. In this paper we will provide a conceptual explanation of the equivalence between the generalized Killing spinor equations of supergravity and the cohomological equations derived from topological gravity.
Our starting point will be the BRST formulation of supergravity. We will revisit this in Section 2. The BRST structure of a given supergravity theory is specified by the number of its local supersymmetries and by its bosonic local gauge symmetries. For each local symmetry one introduces ghost fields -anti-commuting for bosonic local symmetries and commuting for fermionic ones. The bosonic local gauge symmetries always include at least local reparametrizations and local Lorentz transformations: on top of those one can consider additional Yang-Mills local symmetries. These may be associated, for example, to global R-symmetries of the matter SQFT to which supergravity can be coupled.
We will show that the BRST algebra of any supergravity theory takes the form Here S is obtained from the nilpotent BRST operator s by subtracting the transformations associated to the bosonic gauge symmetries; L γ is the Lie derivative along the vector field γ µ of ghost number 2 where ζ i , with i = 1, . . . , N , are the commuting supersymmetry Majorana ghosts 2 and Γ µ , with µ = 1, . . . , d, are the Dirac matrices in curved space-time of dimension d; δ c denotes a gauge transformation with parameter c and i γ (A) is the contraction of the YM gauge field A, belonging in the supergravity multiplet, with the vector field γ µ ; the scalar field φ lives in the adjoint representation of the bosonic YM gauge symmetry group. φ, like γ µ , is a combination of ghost number 2 of the supergravity fields, bilinear in the commuting ghosts ζ i and it is independent of the other ghost fields. The vector ghost bilinear γ µ and the adjoint-valued scalar ghost bilinear φ completely specify the supergravity BRST algebra -and thus the supergravity model. It should be remarked that the vector ghost bilinear γ µ has, for any supergravity, the universal form (1.2); on the other hand the dependence of the adjoint-valued scalar ghost bilinear φ on the the bosonic fields is model dependent. We will provide explicit expressions for φ for the supergravity theories that we discuss in this paper.
The algebra (1.1) encodes the connection between supergravity and topological theories: indeed both the field γ µ and the field φ admit a natural topological interpretation on which we will elaborate in Section 4. We will show that, under supergravity BRST transformations, the ghost number 2 vector γ µ transforms precisely as the superghost of topological gravity while φ transforms as the superghost of a topological Yang-Mills multiplet. Explicitly, the supergravity BRST operator S acts on φ as follows where F is the field strength 2-form associated to bosonic Yang-Mills symmetries and λ is a ghost number 1 combination of supergravity fields which we will call the (composite) "topological gaugino". λ is defined by the supergravity BRST variation of the gauge field The BRST transformations (1.3) are identical to the BRST rules of the topological Yang-Mills multiplet (F, λ, φ) coupled to topological gravity, once one identifies the supergravity bilinear γ µ with the topological gravity superghost.
The topological gravity BRST transformations read indeed where g µν is the metric and ψ µν is the topological gravitino. The consistency of (1.5) with (1.3) hinges on the supergravity transformations for the supersymmetry ghosts and vierbein where ψ i are the gravitinos and e a = e a µ dx µ the vierbein 1-forms. Note that the supergravity BRST rules (1.6) are universal, i.e. independent of the specific supergravity theory. The fact that the supergravity transformations (1.6) imply, in particular, the invariance of the bilinear (1.2) was observed first in [5].
In Section 3 we will review the coupling, in generic dimensions, of topological gravity to topological YM in any dimensions. This was worked out in [6] and [3] in a three-dimensional context. It is well-known that the BRST operator of "rigid" topological YM is geometrically the de Rham differential on the space A of gauge connections, equivariant with respect to gauge transformations. Analogously, the BRST operator of topological gravity in (1.5) is the de Rham differential on the space Met of metrics, equivariant with respect to space-time diffeomorphisms. The BRST operator S in (1.3), which will be discussed in Section 3, is instead the de Rham differential on the space A × Met, equivariant with respect to both diffeomorphisms and gauge transformations. Independently of the application to supergravity that we consider in this work, this operator should have applications to the study of metric dependence of Donaldson invariants.
In the rest of this paper we apply to supersymmetric localization the topological structure that we have discovered sitting inside the supergravity BRST algebra. Supersymmetric bosonic backgrounds are obtained by setting to zero the supergravity BRST variations of all the fermionic supergravity fields. We will refer to the set of such backgrounds as the localization locus. On the localization locus also the supergravity BRST variations of the composite fields λ = S A and ψ µν = S g µν , must vanish. The resulting equations will be analyzed in Section 5. By imposing that the supergravity BRST variation of the topological gravitino vanishes one gets the equation which expresses the request that the vector bilinear γ µ be an isometry of the space-time metric g µν . The fact that, on the localization locus, the vector bilinear γ µ must be a Killing vector of the space-time metric is well-known in the supergravity literature. The vanishing of the supergravity BRST variation of the topological gaugino gives instead the equation This equation is universal, in the sense that it is valid, on the localization locus, for any supergravity in any dimensions. It appears to be a novel equation which has not been yet explored in either supergravity or topological field theory literature.
The differential D γ ≡ d−i γ is the coboundary operator defining the de Rham cohomology of forms on space-time, equivariant with respect to the action associated to the Killing vector γ µ . We will refer to this cohomology as the

The BRST formulation of supergravity
In the BRST framework one introduces ghost fields of ghost number +1 in correspondence to each of the local symmetries. The bosonic local symmetries of supergravity are diffeomorphisms and YM gauge symmetries. We will denote by ξ µ the anti-commuting vector ghost field associated to diffeomorphisms, and by c the anti-commuting ghost associated to the YM gauge symmetry which takes values in the adjoint representation of the YM algebra. The YM gauge symmetries always include local Lorentz transformations. Beyond local Lorentz gauge symmetry, we will also allow for additional YM gauge symmetries. For the application to localization, for example, these additional YM gauge symmetries include the R-symmetries of the supersymmetric quantum field theory whose coupling to supergravity one is considering.
In correspondence with the N local supersymmetries, one introduces commuting supergravity spinorial Majorana ghosts ζ i , with i = 1, . . . N , whose BRST transformation rules have the form In this equation s is the nilpotent BRST operator

12)
ψ i = ψ i µ dx µ are the Majorana gravitinos and γ µ is the following vector bilinear of the commuting ghosts 3 where e µ a are the inverse of the vierbein e a ≡ e a µ dx µ . The vector γ µ has ghost number +2.
Both the Majorana ghosts ζ i and gravitinos ψ i carry a label i = 1, . . . N on which the O(N ) subgroup of the R-symmetry group acts. The full Rsymmetry group can however be as large as U (N ).
The BRST transformations of the vierbein are universal s e a = iψ i Γ a ζ i + diffeos + gauge transfs (2.14) We will denote the action of diffeomorphisms with L ξ , the Lie derivative associated with the vector field ξ µ . The BRST transformations of the diffeomorphism ghost are It was noted in [5] that the BRST transformations (2.11) and (2.14) imply that the vector ghost bilinear γ µ transforms as follows This transformation law coincides precisely with the BRST transformation rule for the superghost of topological gravity [7]. Indeed, the full BRST transformations of topological gravity write where g µν is the metric, ψ µν is the topological gravitino and γ µ the vector superghost. From this it is apparent that also the supergravity BRST transformation rules (2.15) match the topological gravity ones, once the topological gravity superghost is identified with the supergravity ghost bilinear γ µ according to (2.13). The formal coincidence of the topological relations (2.17) with the supergravity ones (2.15), (2.16) is the first hint that a topological sector hides behind the supergravity BRST rules. In the following we will bear this to light.

Topological Yang-Mills coupled to topological gravity
We have seen in the previous Section that some of the supergravity BRST transformation rules take a form which is identical to the BRST transformations of topological gravity. In order to uncover the full topological content of the supergravity BRST rules it is necessary to discuss the coupling of topological gravity to topological Yang-Mills gauge theories. This was derived, in the form we present here, in [6] and [3] in the specific context of 3-dimensional gauge theories. 4 In this Section we will describe this construction in generic dimension and explain its geometric meaning. 5 The coupling of topological gravity to topological Yang-Mills can be useful in different contexts. The first one is when both topological gravity and Yang-Mills fields are dynamical. This gravitational theory is relevant to study the cohomology of the space of space-time metrics. One could also consider the situation in which only the YM degrees of freedom are dynamical while the topological gravity fields play the role of classical backgrounds. In this case the coupling to topological gravity probe the dependence of the quantum physical correlators of the YM theory on the background metric. It might be a useful tool to study, among other things, the occurence of possibile quantum anomalies of the classical topological invariance [6] or wall-crossing phenomena in Donaldson theory [12], [13], [14].
In the following Sections topological YM and topological gravity fields will appear as composites of the "microscopic" supergravity fields. In Section 4 we will explain how such composites emerge from the BRST formulation of supergravity: we will use the existence of this topological structure inside supergravity to derive general information regarding localization of supersymmetric gauge theories coupled to classical supergravity. In this Section -and only in this Section -we think, instead, of topological YM and topological gravity as fundamental "microscopic" theories.
The fields of the topological Yang-Mills theory include, beyond the gauge connection A = A a µ T a dx µ , the topological gaugino λ = λ a µ T a dx µ of ghost number +1, the gauge ghost c = c a T a of ghost number +1 and the super- The BRST rules of "rigid" topological YM -i.e. topological YM before coupling it to topological gravity -read It is convenient to introduce an operator S, whose action is defined on all the fields but not on the ghost field c. S is related to the nilpotent s by the where δ c denotes the gauge transformation with parameter c. We will refer to S as the BRST operator equivariant with respect to local gauge transformations. The nilpotency of s is equivalent to the algebra It is well known that the operator S defined in (3.18)  Coupling topological YM to topological gravity means to find an extension of the nilpotent BRST operator s which includes local diffeomorphism transformations, with a ghost parameter ξ µ that transforms according to the BRST rules of topological gravity (2.17). The request of nilpotency of s dictates the following deformation of (3.18) where F = dA + A 2 is the gauge field strength 2-form, i γ denotes the contraction of a form with the vector field γ µ and S is equivariant both with respect to local gauge transformations and to diffeomorphisms The new equivariant S of topological YM coupled to topological gravity satisfies the relation The BRST transformations of the gauge ghost c are The equivariant S defined in (3.21)-(3.22) is the de Rham differential on the product space A × Met, where Met is the space of space-time metrics, equivariant with respect to the action of both local gauge transformations and diffeomorphisms.
We can re-cast the first equation of (3.21) and the BRST variation of the is nilpotent The rest of the transformations (3.21) are equivalent to the Bianchi identity Therefore the gauge invariant generalized polynomials In the context in which both YM and gravity are dynamical, Eq. In the supergravity context Eq. (3.30) implies -as we will show in the following Sections -cohomological differential equations which are satisfied by localizable backgrounds of classical supergravity.

The topological structure of supergravity
The BRST formulation of supergravity involves both fields with zero ghost number and ghost fields. As recalled in Section 2, one must introduce both anti-commuting ghost fields -which we denoted with ξ and c -associated with bosonic gauge invariance and commuting ghost fields ζ i associated with local supersymmetry. In the rest of this Section we will refer, somewhat unorthodoxly, both to zero ghost number fields and to commuting ghosts ζ i as "matter fields". We will denote them collectively by M .
The BRST transformation rules of the bosonic ghost associated to the gauge symmetries have a slightly different structure Here γ µ ≡ξ µ is the universal bilinear defined in (2.13). The second equation defines insteadĉ which is a function of ghost number 2 of the matter fields: its specific form characterizes the particular supergravity we are considering.
By imposing nilpotency of s on M 0 = s 2 M (4.34) and using the Jacobi identities associated to gauge and diffeomorphisms transformations one obtains the BRST rules for the composite fieldM (M ): The structure of (4.31) and (4.36) makes it convenient to define an operator  Summarizing, the algebra satisfied by S is The consistency condition Sĉ = 0 translates into the equation The composite S A =Â is the topological gaugino, which will be denoted by λ:

.51)
A and λ sit into a multiplet with values in the adjoint of the gauge algebra, Let ω ab = dx µ ω ab µ be the spin-connection and gauge field. The 1-form connection with values in the total bosonic YM Lie algebra is The BRST transformations of the Majorana gravitino field ψ = dx µ ψ µ take the form is a 1-form with values in the total bosonic gauge Lie algebra, H µ dx µ is an auxiliary 1-form field 8 , and H a = e µ a H µ (4.57) 7 In the following paragraphs of this Section we consider the Minkowskian theory and use conventions and notations of [15]. In particular the normalization of the commuting ghosts and gravitinos differ by those used in the rest of our paper. 8 The auxiliary field H a is constrained to have zero divergence, up to fermionic terms: more precisely its Hodge dual H satisfy:

Recalling the Fierz identity valid for Majorana spinors in four dimensions
ζ Γ a ζ Γ a ζ = 0 (4.59) one verifies that S satisfies the algebra (4.49) with When φ is i γ -exact, one can introduce a connection whose associated composite gaugino is From the algebra (4.49) we obtain where is the curvature of the connection A − .
We see therefore that when φ is i γ -trivial, there exists a connection A − and a corresponding topological multiplet F − = F − +λ − +φ − , with vanishing ghost number 2 component, φ − = 0. This is a special feature of N = 1 d = 4 supergravity: we will see in the next Sections that there exist supergravity models, with extended supersymmetry, for which φ is not i γ -exact. where c n = Tr F n + n F n−1 φ + · · · + φ n n = 1, 2, . . . (5.71) and γ µ is the Killing vector satisfying (5.67).
We see therefore that the generalized Chern classes evaluated for localizing backgrounds are closed under the coboundary operator On the other hand it is possible that inequivalent localizable backgrounds give rise to c n 's which are different representatives of the same γ-equivariant class. In the next Section, we will consider more independent, gauge invariant, composite fermions which can be defined for certain supergravity models. Setting to zero their BRST variations one obtains additional topological equations satisfied by supersymmetric backgrounds. In Section 7 we will show that these equations allow for a finer classification of the inequivalent localizable backgrounds.

The gauge invariant ghost bilinears
We have seen in Section 4 that for any supergravity theory there exists a scalar ghost bilinear φ of ghost number +2 with values in the Lie algebra of the bosonic gauge symmetries which characterizes the BRST algebra (4.49).
The ghost bilinear φ is in general a functional of both the supergravity ghosts ζ i and the bosonic fields of ghost number 0 sitting in the supergravity multiplet. The key property of φ, which can be read off Eq. (4.52), is that its BRST variation is i γ -exact: This property ensures that, on the localization locus, φ satisfies the topological equation (5.68).
Let us recall that the BRST variation of the supergravity ghosts ζ i is also It follows that scalar and gauge invariant ghost bilinears which are indepedent of extra bosonic fields automatically satisfy (6.73). Hence, they give rise to cohomological equations of the form (5.68). We will consider in this paper two supergravity models for which ghost bilinears of this sort can be constructed.
The first one is N = (2, 2) supergravity in d = 2, where the gauge symmetry group is SO(2) R . In this case it is convenient to collect the two Majorana ghosts ζ i , with i = 1, 2, into one single Dirac ghost ζ on which the gauge group SO(2) R ∼ U (1) R acts by multiplication by a real phase. 9 Then the two scalar bilinears are gauge invariant and thus S-invariant modulo i γ : where λ 1 ≡ψ ζ +ζ ψ λ 2 ≡ψ Γ 3 ζ +ζ Γ 3 ψ (6.77) It follows from the BRST algebra (4.49) that the generalized forms The 2-formsĤ where H i , with i = 1, 2, are the graphiphoton field strengths. As explained in the previous Section, on the localization locus the following cohomological equations hold Scalar ghost bilinears of the same kind can be constructed also for N = 2 d = 4 supergravity in which the R-symmetry SU (2) R is gauged. In this case we can take the commuting supersymmetry ghosts to be two-components spinors ζ i α where α = 1, 2 is the Lorentz spinorial index and i = 1, 2 is the index of the fundamental representation of the gauge group SU (2) R , together with their conjugateζα i . Two independent scalar and SU (2) R -invariant ghost bilinears are Again, thanks to (6.74), both ϕ andφ are S-invariant modulo i γ where ψ i α andψα i are the gravitinos. The algebra (4.49) ensures that ϕ and ϕ sit in abelian topological gauge multiplets T ≡ ϕ + Λ + T (2) T ≡φ +Λ +T (2) (6.84) Obtaining the 2-forms T (2) andT ( In this Section we will work out the details of the topological structure of N = (2, 2) supergravity in d = 2 with Euclidean signature. 10 The moduli space of localization backgrounds for this supergravity theory has been fully analysed and described in [4], extending the results of [17]. In the following we will see how the analysis of [4] fits into the framework developed in this paper.
The ghost bilinear φ, characterizing the BRST algebra (4.49) of N = (2,2) supergravity in d = 2, has the general form where −i I is the generator of the vectorial U (1) R gauge transformations on Dirac spinors, − i 2 Γ 3 is the generator of the local Lorentz transformations, φ R is the scalar ghost bilinear associated to the U (1) R gauge symmetry and φ Lor the one relative to the Lorentz local transformations. The BRST transformations of the gravitino Dirac field ψ = dx µ ψ µ write where η ij is the Lorentzian metric η 11 = −η 22 = 1 and ij is the Levi Civita tensor in 2 dimensions. The universal topological equations (5.68) for supersymmetric backgrounds read R ) = 0 (7.92) These equations mean that the polyforms of degree 2 associated to the curvature and the field strength of the U (1) R gauge field are γ-equivariantly closed: We explained in Section 6 that, for N = (2, 2) d = 2 supergravity, one can construct, starting from the gauge invariant ghost bilinears (6.75), two more equivariant forms H i of degree 2, which satisfy the γ-equivariant cohomology equations (6.81) on the localization locus.
The relation (7.91) between the scalar components of the equivariantly closed forms H i , R and F R can be extended to the following relations between where we used the Fierz identity ϕ 2 1 − ϕ 2 2 = γ 2 . Note that, as shown in Appendix A, It follows from (7.97) that HenceR is cohomologically trivial 11 and Let us therefore put where σ ± is the relative sign between H 1 (p ± ) and H 2 (p ± ). Moreover, from the second of Eqs (7.96) we obtain Plugging this inside the first of Eqs. (7.96) one arrives to: 12 Eqs. (7.103) and (A.127) show that the γ-equivariant cohomology classes of both the curvatures polyforms F R and R are (semi)-integral. This should be contrasted with the cohomology classes of the H i 's which, as shown in [4], depend on a continuous parameter labelling inequivalent localization backgrounds. We believe that this phenomenon is a general property of the γ-equivariant curvature polyforms of supergravity, although we do not have yet a general proof of it. 11 As mentioned in the Appendix A,R and R = L(R) do not need to be cohomologous. 12 From this relation we derive in particular that S 2 F (2) R = 1 2 σ + + σ − . Therefore, the first Chern class of the gauge field can take values −1, 0, 1 according to the signs σ ± , in agreement with [4] and [17].

Conclusions
In this paper we showed that a topological structure sits inside supergravity.
We did this by writing the BRST algebra of supergravity in the form The vector field γ is given by This expression is universal in the sense that it is valid for any supergravity theory in any dimension. On the other hand, the scalar field φ, which is valued in the adjoint representation of the bosonic YM gauge symmetry group, is a ghost bilinear whose dependence on the bosonic fields of supergravity is non-universal, i.e. it is theory dependent.
Our central observation is that both γ µ and φ have a topological meaning: they can be identified, respectively, with the superghost of topological gravity and the superghost of a topological Yang-Mills theory whose gauge group is the product of the local Lorentz transformations and the Yang-Mills symmetries of the supergravity theory under consideration.
This identification relies on the fact that the supergravity BRST transformations of the ghost bilinears γ µ and φ coincide precisely with the BRST rules of topological gravity S g µν = ψ µν S ψ µν = L γ g µν S γ µ = 0 (8.106) and of topological Yang-Mills coupled to topological gravity: The BRST variations (8.107) are not familiar in the topological Yang-Mills literature, although they already appeared in [3] in a three-dimensional context. In this paper we wrote them down for any dimensions and discussed their geometrical interpretation. We believe that these transformations could be of interest to study, for example, the metric dependence of Donaldson invariants -regardless of the application to supergravity that we explored in this article.
It was observed in [3] and [4] that the conditions for unbroken supersymmetry in certain off-shell supergravities with extended supersymmetry can be recast in terms of cohomological equations. In [3] and [4] these cohomological equations were obtained by setting to zero the BRST variations of fermionic fields of topological gravity coupled to some additional topological gauge multiplets.
In this paper we gave a conceptual explanation of this equivalence, by exploiting the topological structure of supergravity captured by the BRST transformations laws (8.106) and (8.107). By setting to zero the BRST variations of the fermions in (8.106) and (8.107) we obtained equations for supersymmetric backgrounds for which two such bilinears ϕ can be constructed. We have seen how the γ-equivariant cohomological equations (8.108) and (8.109) are related to each other, thus providing an a priori explanation of the results presented in [4].
Another theory for which two gauge invariant scalar bilinears ϕ can be constructed is N = 2 d = 4 supergravity. The study of the cohomological equations (8.109) and (8.108) for this theory might lead to the classification of localizing backgrounds for this theory -a long standing problem to which we hope to come back in the future.
We found two kinds of topological multiplets inside supergravity: the "universal" ones, whose 2-form components are the curvatures and whose 0-form components are the model dependent scalars φ which appear in the BRST algebras; and the gauge invariant scalars multiplets, which exist only for certain supergravities, whose 0-form components are made entirely of commuting ghosts and whose 2-forms components are model dependent. For N = (2, 2) in d = 2 supergravity we found compact quadratic relations, ultimately descending from the Fierz identities, between these topological multiplets. It would be interesting to understand if generalizations of these relations exists for other models, notably for N = 2 d = 4 supergravity: beyond their application to localization these might give important informations about the dynamics of supergravity itself. Let us also observe that the product of two γ-equivariant forms of degree 2 in 2 dimensions is again a γ-equivariant form of degree 2, i. e. the set of γ-equivariant forms of degree 2 in 2 dimensions has a ring structure: Moreover, given a metric, one can introduce a linear operation L which when acting on a γ-equivariant form of degree 2 gives another γ-equivariant form of degree 2: where we introduced the γ-dependent Laplacian for 0-forms 14 . L acts as a derivative with respect to the product (A.116) 14 Since γ µ may in general have zeros, the action of ∆ γ on a generic 0-form is not always well-defined. However it can be shown that ∆ γ Φ (0) is regular when Φ (0) is the 0-form component of a γ-equivariant form of degree 2.
It should be kept in mind the L does not map γ-equivariant forms to cohomologous ones.
Let us observe the cohomology class of a γ-equivariant form Φ of degree 2 on the 2-dimensional sphere is parametrized by the values of the polyform at the two zeros p ± of the Killing vector γ µ : The cohomological invariant obtained by evaluating Φ on the sphere is related to Φ(p ± ) by the localization formula Let us introduce the equivariantly closed polyform where we used the Fierz identity ϕ 2 1 − ϕ 2 2 = γ 2 . Note that Therefore the γ-equivariant cohomology class of R is independent of the metric and can be computed, for example, using the round metric: R(p ± ) = ∓1 (A.127)