The Quantum Theory of Chern-Simons Supergravity

We consider $AdS_3$ $N$-extended Chern-Simons supergravity (\`a la Achucarro-Tonswend) and we study its gauge symmetries. We promote those gauge symmetries to a BRST symmetry and we perform its quantization by choosing suitable gauge-fixings. The resulting quantum theories have different features which we discuss in the present work. In particular, we show that a special choice of the gauge-fixing correctly reproduces the Ansatz by Alvarez, Valenzuela and Zanelli for the graphene fermion.


Introduction
It is tantalising to relate models which are apparently very different. Some years ago, Gaiotto and Witten, in [1], considered a Chern-Simons (CS) gauge theory in three dimensions coupled to N = 2 supersymmetric multiplets whose scalar components are coordinates of a Hyper-Kähler manifold. The model descends from a four dimensional one in the presence of a defect and the potential is chosen to enhance the supersymmetry from N = 2 to N = 4.
The conditions under which this is possible are certain relations between the moment maps associated with the linear action of the gauge group on the Hyper-Kähler manifold which unveil a superalgebra hidden in the model.
They argued that the supersymmetric Wilson loops can be constructed in terms of that super algebra, but they did not elaborate further along these lines. In another paper [2], Kapustin and Saulina showed that Rozansky-Witten theory [3] coupled to a Chern-Simons gauge field can be written (up to an exact BRST term) as a Chern-Simons gauge theory on a supergroup. In particular they showed the following duality relation Chern-Simons theory| SG + s gauge fixing of fermion gauge symmetries = super-Chern-Simons theory| G + matter fields . (1.1) where Chern-Simons theory| SG is a Chern-Simons theory on the supergroup SG (see also [4] for a complete discussion) plus a gauge-fixing and it can be recast in the form of a N = 4 super-Chern-Simons theory on the group G (which is the bosonic subgroup of SG) coupled to N = 4 supermultiplets. The gauginos and the scalar fields of the N = 4 Chern-Simons multiplet are not dynamical and their equations of motion relate the gaugino fields to the fermionic currents and the scalar (partners of the CS gauge field) fields to the scalar potential.
By inserting this solution back in the action, one gets additional non-trivial terms for the potential. The complete scalar potential is incorporated into the gauge-fixing fermion on the left-hand side of the correspondence. The mapping from the right-hand side to the lefthand side has been used in [5][6][7][8], to compute supersymmetric Wilson loops in terms of the Chern-Simons theory on the supergroup instead of the supersymmetric Chern-Simons theory.
From another point of view, some years ago, Achucarro and Townsend [9] observed that, in three dimensions, N -extended supergravity in the presence of a negative cosmological constant can be rewritten as a Chern-Simons theory whose gauge fields take values in the superalgebra osp(p|2) × osp(q|2) where p + q = N . The bosonic subalgebra is so(p) ⊗ so(q) ⊗ sp(2) ⊗ sp (2) and the gauge fields associated with the two sp(2)'s are given in terms of the vielbein and the spin connection of the three dimensional manifold. The "gravitinos" (the gauge fields associated with the fermionic gauge generators) are in the bifundamental representations of SO(p) × Sp (2) and/or SO(q) × Sp (2) subgroups. The supersymmetry is realized as a gauge symmetry and therefore the fermionic charges are related to the fermionic (anticommuting) gauge fields. Being a Chern-Simons theory it does not depend on the 3-dimensional metric on the world-volume which however can be inferred by identifying the vielbein on the worldvolume with the difference of the two sp(2)'s gauge fields. We point out that, from this point of view, the fermionic gauge fields have to be identified with world-volume gravitinos, again by identifying the spinorial index of the latter with the fundamental representation of the two sp(2)'s that they carry as gauge fields. Therefore, this interpretation of the supergravity as Chern-Simons gauge theory on a supergroup exactly fits the relation proposed by [1,2].
The structure of Achucarro and Townsend supergravity from a mathematical point of view is reminiscent of the ABJM model [10], which is a difference of two CS actions as well.
In the above mentioned papers, the duality is based on a topological twist, where the twist is realized by taking the diagonal subgroup of the product of the original Lorentz group with an Sp(2) part of the R-symmetry. On a AdS background, instead, the analogous topological twist is naturally related to the (non-unique) choice of the Lorentz subalgebra inside the antide Sitter isometry so(2, 2) sp(2) × sp (2). Indeed, in this framework this topological twist corresponds to trading the antisymmetric part of the torsion τ for a cosmological constant Λ.
More precisely, choosing as Lorentz group one of the two Sp(2) factors, which corresponds to the untwisted theory in the flat background, is here associated with a non vanishing spacetime torsion τ = 0. On the other hand, choosing instead the diagonal subgroup SO(1, 2) D ⊂ SO(2, 2) as Lorentz group, which corresponds to the topologically twisted theory in a flat background, is here associated with the choice of a torsionless spin-connection in a background with cosmological constant Λ = −τ 2 = 0.
The first step towards constructing the super-Chern-Simons theory coupled to matter fields for supergravity is to perform a suitable gauge-fixing of the super-gauge symmetry. Let us focus (as we will do it in the main text) only on the fermionic generators. The gauge symmetry at the quantum level is replaced by the BRST symmetry and the gauge parameters are replaced by the ghost fields, which, in the present case, are commuting scalar fields. To complete the gauge-fixing procedure, one needs an auxiliary sector, also known as an anti-ghost sector, which consists of a set of commuting scalar fields and a set of fermionic Nakanishi-Lautrup fields [2]. The ghosts and the anti-ghosts are commuting scalar fields and they belong to conjugated representations, thus they form a complex vector space carrying a quaternionic representation of the bosonic gauge group. In fact, they even span a Hyper-Kähler manifold.
Finally, using the work of Kapustin and Saulina [2], we translate the degrees of freedom of the Chern-Simons supergravity in terms of the ones of a super-Chern-Simons theory with N = 4 extended supersymmetry coupled to matter. That theory has a superpotential, which can be related to the gauge-fixing of the Chern-Simons on the supergroup.
Of course, there are several gauge-fixing choices and we will explore them, pointing out the relevant features of the corresponding quantized theories. In the BRST formalism the gauge-fixing is chosen by adding to the action the BRST variation of the gauge-fixing fermion Ψ. The latter has to carry negative ghost number, should be Lorentz invariant and, since at this stage we are gauge-fixing only the fermionic gauge symmetries, it has to be gauge invariant under the bosonic symmetries.
Given these constraints, we can select several types of gauge-fixing. In particular, we point out that, among the possible choices, there is an unconventional one, whose propagating degrees of freedom correspond to a propagating massive Dirac spinor, which reproduces the degrees of freedom of the model described by Alvarez, Valenzuela and Zanelli [11,12], to be referred to in the sequel as AVZ model. The latter is based on an N = 2 supergroup and provides a phenomenological description of graphene. In that case, a three dimensional Chern-Simons theory with OSp(2|2) gauge group, and the fermionic 1-forms ψ α I , I = 1, 2, in the bifundamental of the Sp(2) × SO(2) group are written in terms of spin-1/2 fields χ α I through of the Ansatz: where e i , i = 0, 1, 2, are the vielbein 1-forms of the three dimensional spacetime and γ i are the corresponding gamma matrices. Since e i and χ β I only enter the action through the above Ansatz, the theory is invariant under the local rescaling symmetry [11]: e i → λ(x)e i , χ β I → 1 λ(x) χ β I , λ(x) being a real function. The Sp(2)-connection is identified with the Lorentz one ω ab , and a space-time torsion T i = De i is allowed for. By suitably fixing the local rescaling symmetry of e i , T i can be made constant of the form: τ being a dimensionful constant.
The Ansatz (1.2) amounts to setting the spin-3/2 component of the gravitino fields to zero, keeping however a non-zero spin-1/2 component. As a consequence of this choice, the original Chern-Simons theory yields an effective model describing a propagating massive spin-1/2 Dirac field χ α = χ α 1 + i χ α 2 , whose mass is related to the spacetime torsion τ . This model is suited to describe graphene in the presence of space-time curvature and torsion.
In [13] the model of [11] is embedded in supergravity. It is first of all embedded in AdS 3 supergravity by identifying its gauge supergroup with the OSp(2|2)-factor of the super-AdS 3 symmetry OSp(2|2) × SO(2, 1). The D = 3 supergravity is then characterized as the boundary theory of an AdS 4 supergravity with N = 2 supersymmetry [14]. In this holographic correspondence, an appropriate parametrization of the AdS 4 space is chosen, which corresponds to an AdS 3 -slicing of the same space. Furthermore, by choosing suitable boundary conditions for the four-dimensional fields, the model of [11] is retrieved at the AdS 3 boundary. In this picture the spin-1/2 field χ α , which ought to describe the collective electron modes in the graphene, originates from the radial component of the D = 4 gravitino field (i.e. the component of the gravitino 1-form along the direction perpendicular to the boundary), and the torsion parameter τ , which behaves as a mass term for the spinor χ, is naturally related to the curvature of the AdS 3 spacetime. However, while in the AVZ model of [11,12] the presence of a cosmological constant, with the corresponding enhancement of the gauge symmetry to OSp(2|2) × SO(1, 2), is optional, this is not the case if one aims to identify the CS model with D=3 supergravity, since the Achucarro-Townsend map [9], on which the identification in [13] is based, requires a non-vanishing cosmological constant, which induces a non-vanishing mass term for the Dirac spinor χ.
In both the constructions in [11] and [13], the condition (1.2) is put by hand. An important goal of the present paper is to retrieve it dynamically within a covariant BRST-quantized setting. As said above, by a Landau-type gauge-fixing of the (gauge) supersymmetry, we are able to see that there is one massless Dirac spinor propagating. However, to compare it with the Ansatz (1.2) for the case of a massive spinor, as it is the case in [13], we have to modify the gauge-fixing term by adding a first order differential and a vorticity term to the action.
By a simple analysis of the quadratic part of the action, it is then easy to show that the Dirac spinor has a non-vanishing mass related to the cosmological constant.
Another outcome of our analysis is the study of the symmetry properties of the quantum model on the right-hand side of the duality in (1.1). In particular we find, at least for the conventional Landau gauge-fixing, that the quantum CS-theory exhibits a rigid N = 4 world-volume supersymmetry on AdS 3 , which comprises, besides the BRST symmetry, also an emerging "vector"-supersymmetry [15] - [18]. We shall expand on this particular issue in a forthcoming paper [19].
The paper is organized as follows: in subsect. 2.1 we recall the basic facts about three dimensional AdS N -extended supergravity. In subsect. 2.2 we review its interpretation as describing the graphene in the AVZ Ansatz. In subsect. 2.3, we present the BRST formulation.
In subsect. 2.4, a secondary BRST symmetry is discussed. In sect. 3, we finally quantize the model by considering different types of gauge-fixing: 1) a conventional gauge-fixing leading to a massless Dirac spinor (Landau gauge-fixing) and yielding a dual N = 4 supersymmetric model, 2) Landau gauge-fixing with additional non-linear terms to reproduce the scalar potential in the dual theory, 3) an additional term (Nakanishi-Lautrup term) to allow for mass deformations of the model, 4) an ss gauge-fixing based on the presence of a secondary BRST symmetrys and, finally, 5) an unconventional gauge-fixing, which reproduces the AVZ Ansatz with a non-vanishing mass.
We conclude with the summary and with the future perspectives.

Basic Facts
We first recall some basic facts about D = 3 supergravity with negative cosmological constant.
denote the corresponding gauge connections, the last being associated with Majorana spinor 1-forms.
The relation between the topological CS theory and D=3 N -extended supergravity (which does not have local propagating degrees of freedom) with AdS 3 radius , is found by introducing the fields: 1 We denote with a bar the adjoint fermion:Q ≡ Q t C, C being the charge conjugation matrix.
where ω ij is identified with the (torsionless) spin connection of the Lorentz algebra: and E i as the bosonic components of the supervielbein of the N -extended superspace. Note Recently, some of us reconsidered, in [13], the Achucarro-Townsend theory [9] and the correspondence (2.1) for the special case N = p = 2, q = 0. In particular, the field equations of the N = 2 AdS 3 supergravity were found as asymptotic boundary conditions on the supergravity field-strengths of N = 2 AdS 4 pure supergravity, along the lines discussed in [14].
In that framework, OSp(2|4)-invariant Neumann conditions are recovered on the boundary as consistency conditions for supersymmetry of the full action. An N = p = 2 AdS 3 description of those Neumann conditions was found, in [13], for an asymptotic boundary located at r → ∞ as a particular asymptotic limit, inspired by the so-called "ultraspinning limit" [20] in the Fefferman-Graham parametrization of the D = 4 superfields.

A Model for Graphene from D=3 Supergravity
The aim of the analysis in [13] was to make contact with the results of [11], where the D = 3 Chern-Simons theory of the supergroup OSp(2|2) (+) was considered, assuming however a peculiar Ansatz for the odd component of the gauge connection 1-form: Here, e i , γ i are, respectively, the dreibein and a set of gamma matrices on the D = 3 worldvolume where the Chern-Simons theory is defined. With the assumption (2.9), the CS Lagrangian turns out to describe the local dynamics of the spin-1/2 field χ ≡ χ I=1 + iχ I=2 .
More precisely, χ is a Dirac spinor satisfying in general the massive Dirac equation, with mass given in terms of the contorsion τ = 1 For non-zero τ , as discussed in [11], the contorsion can be set to zero by a redefinition of the spin connection, and with that choice the background space-time turns out to be AdS 3 , with cosmological constant −τ 2 , and the world-volume symmetry is enhanced to SO(2, 2) . 2 In [13] it was shown that, in the case of contorsion τ = − 1 , the model of [11] can be recovered at the asymptotic boundary of AdS 4 , N = 2 supergravity. It corresponds to imposing in a non-trivial way the condition that in D = 4 supergravity projects out the spin-1/2 part of the gravitino field: where Γ, Ψ denote D = 4 gamma-matrices and gravitino, respectively,μ = (µ, r) = 0, 1, 2, 3 being holonomic world indices.
The Ansatz (2.9) of [11], in light of its relation with supergravity in D = 3 [9] and in D = 4 [13], is remarkable in several respects: • It introduces in the topological Chern-Simons Lagrangian a dependence on the spacetime background and a local dynamics for the spinor χ.
• It implies that the radial component of the D = 4 gravitino is not suppresed in the asymptotic limit.
• It also implies a non-trivial relation between the world-volume dreibein e i and the bosonic part of the super-dreibein E i , which is discussed in detail in [13].
When writing the Ansatz (2.9), a clear distinction has to be made between the target space quantities, which are connections of the superalgebra, and the world-volume quantities e i , χ, and the world-volume Lorentz connection. It is appropriate, therefore, to introduce a different notation for the spinorial index on the world-volume, denoting it by "primed" greek letters.
Accordingly, eq. (2.9) will then be written, in the following, as: In this expression it is manifest that (γ i ) α β is an intertwiner between the spinor representation of the target space, labeled by α, and the one on the world-volume, labeled by β . An identification between the Lorentz groups on the target space and on the world-volume is implicit. In flat space, this identification would be unambiguous. However, in the Achucarro-Townsend model a negative cosmological constant is present on the target space, and this naturally induces the same anti-de Sitter geometry also on the world-volume. This allows multiple choices for the identification of the two Lorentz groups inside the two SO(2, 2) AdS 3 symmetry groups, which, due to the non trivial relationship between the E i and e i , are distinct.
We can identify the Lorentz group, both in the target space and in the world-volume, with the diagonal SL(2) D ⊂ SO(2, 2), which is associated with a Riemannian spin-connection.
Alternatively, we can identify the common Lorentz group with one of the two SL(2) ± factors.
The corresponding spin-connection is torsionful; this is the choice made in [13]. In this latter case, the SL(2) ⊂ SO(2, 2) factor which is not identified with the Lorentz group can be interpreted as an internal symmetry, associated with new spinorial indices:α in the target space andα on the world-volume. This observation will be relevant for the discussion in the next sections.
In the forthcoming section, we will show that the condition (2.9), with all its peculiar properties discussed above, can be naturally reproduced as a (non-standard) gauge-fixing of the gauged supersymmetry of the Chern-Simons theory, or equivalently, in light of the correspondence in (2.1), of D = 3 supergravity. In the following we are going to reformulate the theory in a BRST covariant framework in order to set up the gauge-fixing properly.
The equations of motion then read: where the Lorentz covariant derivative is Dψ α I = dψ α I + 1 2 (ω + ) α β ψ β I . 3 The supersymmetry transformations can be cast in the following way 14) The supersymmetry parameter α I is a local fermionic real parameter. Although we are interested in the quantization of the full gauge symmetry, which requires the gauge-fixing of the full superalgebra, in the present paper we only focus on the generators associated with the supercharges. The corresponding ghosts will be interpreted as scalar fields in the dual picture. The gauge-fixing of the rest of the gauge symmetry is performed along the conventional procedure.
If we promote the local supersymmetry parameter α I to a quantum field, it becomes a ghost field that we denote by φ α I . Note that, since φ α I is the ghost field of the supersymmetry, it has opposite statistics and, therefore, it is a commuting scalar field. On the other hand, it carries a positive ghost charge with respect to a corresponding U (1) group. Then, φ α I is intrinsically complex, but appears only holomorphically in the action.
We can translate (2.14) into BRST transformation rules: where we set the BRST transform of the ghost field φ α I to zero, since we are only dealing with fermionic gauge symmetries.
Let us also check the nilpotency of the BRST transformations. By introducing the two composite fields (one for each bosonic generator of the supergroup, namely Sp(2) and SO (2)) (the notation is adopted from [2] where µ (αβ) + and µ + denote the holomorphic moment maps of the action of the gauge group Sp(2) × SO(2) on the vector space of the ghost fields φ α I ) we find Note that the above BRST transformations are not nilpotent (except those on ω − and on φ α I ), but they yield bosonic gauge transformations of the Lie algebra osp(2|2) (+) with local parameters −i µ (αβ) + and −i µ + . A nilpotent BRST symmetry is attainable by adding the ghosts c (αβ) and c of the bosonic symmetry Sp(2) × SO (2). That follows the conventional procedure and we refer to the vast literature on the subject, see for instance [2]. For the purpose of the present paper, we do not need to describe this sector and therefore we omit it.
To set up the gauge-fixing, one needs also some BRST doublets 4 which transform in the conjugate representation with respect to ψ α I and φ α I . To this aim we introduce the set (φ I α ,η I α ) 5 with the BRST tranformations: One can verify that, acting twice with the BRST differential s on the latter fields, one has again nilpotency up to gauge transformations (as in eq. (2.17)). With the anti-ghost fields φ I α , we can define the moment maps related to the Kähler structure K = dφ α I ∧ dφ I α as follows: where, again, the notation is adopted form [2]. In addition to the holomorphic moment maps given in (2.16), we can also introduce the anti-holomorphic moment maps: There is a Hyper-Kähler structure underlying the above relations (2.16), (2.19), (2.20). Indeed, the scalar bosonic ghost fields φ andφ introduced for the gauge-fixing have a natural interpretation as coordinates on a Hyper-Kälher manifold, as emphasized in [2], where the SU (2) 4 A BRST doublet is cohomologically trivial and this implies that all observables are independent of it. 5 Beware: the bar over the fields in the BRST-exact sector denotes the anti-ghost sector.
symmetry associated with the Hyper-Kähler structure can be made manifest by arranging φ α I andφ I α in the following doublet Φ α|A I , A = 1, 2: Here A labels the eigenvectors of the U (1) generator having eigenvalues ±i on the ghost and anti-ghost field, respectively. The Hyper-Kähler structure is described by the following triplet of closed 3-forms Ω AB = Ω BA : We shall also denote by Ω = Ω 11 = αβ dφ α I ∧dφ β I and Ω = Ω 22 = αβ dφ α I ∧dφ β I the holomorphic and anti-holomorphic structures, respectively. The Kähler form K introduced earlier, on the other hand, coincides with the remaining component of Ω AB : K = Ω 12 . In terms of Ω AB , the triholomorphic moment maps, associated with the generators of Sp(2) × SO(2) symmetry group, are defined as follows:

23)
V (αβ) , V IJ being the Killing vectors generating Sp(2) and SO(2), respectively. 6 The explicit form of the triholomorphic moment maps is readily computed to be: In particular, with reference to the above definition, we have the following identifications: In addition, the moment maps satisfy the condition: 7 µ (αβ) that is crucial for the closure of the superalgebra of osp(2|2).
The BRST invariant action is:

A secondary BRST symmetry
From [3], trying to understand the origin of the world-volume supersymmetry, we learn that there is a secondary BRST symmetry, that we denote bys. It is obtained by exchanging the role of the ghost field φ α I with that of the anti-ghostφ I α , as follows: In addition, thes transformation of φ α I is This means that the anticommutation of the two BRST transformations yields a gauge transformation with parameters −i µ αβ 3 , −iµ 3 , and therefore they anticommute only when acting on gauge-invariant quantities. This will clarify the correspondence, outlined in the Introduction, between the Achucarro-Townsend model [9] and an N = 4 world-volume supersymmetric Chern-Simons theory coupled to matter, analogous to the model discussed in [1]. In the counting, negative d.o.f.'s mean gauge symmetries.
We have the gauge fields ω In the AVZ model [11], the authors introduce a spinor 0-form χ β I as in eq. (2.11), where the index β is a truly spinorial index on the world-volume, and γ αβ However, it would be desiderable to derive (2.11) in terms of a gauge symmetry of the model (which actually reduces the off-shell degrees of freedom from 8 down to 4). To this end, let us observe that the felds ψ α I are fermionic 1-forms, therefore their components ψ α I,µ are fermionic d.o.f.'s. In the ghost sector we introduce the Nakanishi-Lautrup fieldsη I α needed for the gauge-fixing of the fermionic gauge symmetry. The total amount of fermionic degrees of freedom is then 2 × 2 × 3 for ψ α I,µ 8 and 2 × 2 forη I α .
All the off-shell fermionic d.o.f.'s in our gauge-fixed model can be arranged into a single spinor field of the form Λ α I,α β given by where α and β both refer to the diagonal world-volume symmetry Sp(2) D ⊂ SO(2, 2) . On the other hand, if we decompose Λ α Iα β with respect to the diagonal subgroup of the target space Sp(2) (which the index α refers to) and of the world-volume Sp(2) D , the fermionic d.o.f.'s Λ α I,α β can also be decomposed as follows where χ µβ ,I is identified with the world-volume gravitino, while χ Iβ contains the AVZ fermionic degrees of freedom.
The full world-volume isometry SO(2, 2) can be made manifest by promoting Λ α Iα β to an object in the (1/2, where we recall that α andβ refer to the groups Sp(2) + and Sp(2) − , respectively.
The on-shell analysis is difficult since there are no local degrees of freedom. In addition, in the previous sections we introduced the ghost fields φ α I ,φ I α , that are scalar bosonic degrees of freedom, which are interpreted as the scalar superpartners of Λ α Iα β . As discussed earlier, they naturally parametrise a Hyper-Kähler manifold.
We can then group the d.o.f.'s as follows: the gauge fields ω The N = 4 supersymmetry parameter has the following structure: α α A , where the SU (2) group acts on the index A [19]. We shall further elaborate on the emergence of this supersymmetry at the end of next subsection.

Landau Gauge-Fixing and N = 4 supersymmetry
The gauge-fixed Lagrangian (only for the fermionic gauge symmetry) has the general form where Ψ =φ I α F a I (ω + , ψ α I , A, φ α I ) is known as the gauge-fixing fermion which encodes the gauge-fixing and depends upon the gauge fields (ω + , ψ α I , A) and possibly also on the ghost field φ α I .
Let us discuss the gauge-fixing in detail. As already mentioned in the introduction, we do not introduce here the ghosts for the bosonic part of the gauge symmetry, since we are only interested in the gauge-fixing of the odd part of the superalgebra. The gauge-fixing of the even part of the gauge algebra can be done according to the standard procedure.
Let us start with a gauge-fixing, known also as Landau gauge fxing, of the following form (up to total derivatives) The Hodge dual is needed to write the gauge fixed action on any three dimensional manifold, given a metric on it. In particular, we have: ψ α I ∧ ψ β J = ψ α µ,I g µν ψ β ν,J V (3) , where g µν is the inverse metric on the world-volume M (3) , whose volume form is V (3) = 1. Notice that a world-volume metric is needed only in the gauge-fixing term and it is not present in the gauge-invariant action L (3) .
The first term in (3.5) is an usual kinetic term for complex bosonic fields φ α I ,φ I α . The second term contains a first-order linear differential operator onη I α which, together with the gravitino field equation from eq. (2.28), leads to an invertible wave operator. The last term produces an interaction between ψ and the ghost fields φ andφ. Notice that the ghost fields in the present case are bosonic commuting fields, therefore the kinetic term leads to a positive definite metric for the Hilbert space. Now, we check that the gauge-fixing fixes the fermionic gauge symmetry and we study the wave operator in the fermionic sector. In order to simplify the discussion, we neglect for the time being the interactions among ghost fields (φ α I ,φ I α ). The free equations for ψ α I and forη I α read We re-write these equations in components as follows µνρ αβ ∇ ν ψ α ρ,I + ∇ µη The second equation is the usual Landau-Lorentz gauge-fixing for the gravitino.
Before studying the alternative gauge-fixing, let us clarify the issue of supersymmetry.
Even though a general discussion of that issue for all possible gauges will deserve a longer work [19], let us observe that, in the present case, supersymmetry emerges in a very interesting way. Indeed, the action depends upon the world-volume metric only through the gauge-fixing terms. Therefore, the world-volume energy-momentum tensor satisfies the equation [15][16][17][18] δS δg µν ≡ T µν = sΓ µν (3.10) where Γ µν is the variation of the gauge-fixing fermion Ψ with respect to the world-volume metric g µν . It can be proven that the conservation of Γ µν (up to a suitable redefinition of T µν ) follows from the following equation:

Feynman Gauge Fixing and Mass Deformations
The Feynman gauge-fixing also requires the introduction of the Nakanishi-Lautrup auxiliary fieldη α . It can be added to the gauge-fixing as follows Here the first two terms in the last line are potential terms for the bosonic ghost fields (φ,φ), while the third one is a mass term forη. We used the definition of µ 3 to simplify the expression and β is a gauge-fixing parameter.
The form of the potential generated by the present gauge-fixing is similar to the mass deformations of the Gaiotto-Witten model discussed in [21,22]. As mentioned in the Introduction, the model we are considering bears resemblance with the ABJM model [10] in that it is described by the difference of two Chern-Simons. However, as is apparent from (3.12), the BRST variation of this gauge-fixing term cannot generate higher powers of φ andφ which could reproduce the sextic scalar potential of ABJM models.
The gauge-fixing produces aη 2 term, which serves as the Nakanishi-Lautrup term, modifying the quadratic part of the ψ −η system, whose equations of motion were discussed in the previous section.

Non-linear Feynman gauge-fixing and Gaiotto-Witten model
One can add further interaction terms by introducing a more general gauge-fixing piece of the form where V αβ IJ is a generic tensor built out of the φ's. As can be noticed, this additional term modifies the action by non-quadratic terms and it is responsible of the generation of the scalar potential. As discussed in [2], a suitable choice of V αβ IJ (φ) (compatible with the bosonic gauge invariance of the model) leads to the scalar potential of the dual theory and the coupling with the fermions.

Unconventional Gauge Fixing and AVZ Ansatz
Let us now add a new term to the equations of motion and recompute the wave operator. In particular, we will derive the AVZ Ansatz for a massive Dirac spinor by a suitable choice of the gauge-fixing parameters. The corresponding quadratic part of the Lagrangian is L = 2 αβ µνρ ψ α µ,I ∇ ν ψ β ρ,I +η I α (∇ µ ψ α µ,I − 2iβ(γ i ) α β ψ β i,I ) + 2β αβηI αηβ,I (3.14) where the last term is the usual quadratic term in the Nakanishi-Lautrup fieldη I α . The new equations of motion are Manipulating those equations as above, we finally find the massive Dirac equation for the combination χ I = − 1 4 (i ψ I + 1 2η I ) which should be identified with the AVZ Dirac spinor. Finally, we can write down the complete unconventional gauge-fixing of the CS action with this additional piece as follows No additional term is produced by the BRST variation of the second term of the gauge-fixing.
In addition, no BRST variation is required for the 3d vielbein e i . The new term leading to the unconventional gauge-fixing and to the identification of our fermionic fields with the AVZ Ansatz modifies the kinetic term of the bosonic ghost fields (viewed as coordinates of a Hyper-Kähler manifold) by a vorticity term involving a first order differential.

ss-Gauge Fixing
Finally, let us come to the last example of gauge-fixing that can be constructed. Given the fact that there are two BRST symmetries, instead of writing the gauge-fixing as the s-variation of Ψ, one can construct the gauge-fixing as follows where Σ has no ghost charge, it is Lorentz invariant and it must also be gauge invariant under the bosonic subgroup of the supergroup OSp(2|2). Since s ands are nilpotent, the total action S will be s ands invariant. Acting with s, it is trivially zero, while to check thes-invariance, we have to anticommutes with s and that can be done at the price of a gauge transformation in the subgroup. However, being Σ gauge invariant, this gauge transformation is ineffective and nilpotency ofs does the rest.
The Landau gauge-fixing plus non-linear terms (3.5)-(3.13) can be written as follows L g.f. = ss ψ α I αβ η IJ ∧ ψ β J +η I α αβηI β 1 . Using the present framework, we can rewrite the unconventional gauge-fixing in a similar way by observing that since the group Sp (2) is the spin group associated with SO(1, 2) (the Lorentz group of the world-volume), we can use the intertwiners γ αβ i (or even better Γ α iβ = i αγ γ γβ i ) to extend the above gauge-fixing term as follows where e i is the wordvolume vielbein and e i is the world-volume Hodge dual and β is the To conclude, we observe that we can add a further term in order to build a quadratic term, of the form ss A (αβ) γ iαβ ∧ e i (3.20) which carries zero ghost number and is a 3-form. It is manifestly Lorentz invariant and its variation gives s ψ α ∧φ β (γ i ) αβ e i = ∇φ α ∧φ β (γ i ) αβ e i − ψ α ∧η β (γ i ) αβ e i . (3.21) The first term is a viscosity term modifying the quadratic part of the action for the scalar fields. The second term is an off-diagonal mass term for the fermions.
Therefore, also in the present formulation, there are at least two supercharges in the game and part of the supersymmetry is preserved. We leave further consideration in a forthcoming more detailed work [19].

Conclusions and Outlook
In the present work, we have considered some ways of quantizing Chern-Simons supergravity revealing different important features of the same theory. The ultimate goal, which will be tackled in the forthcoming work, see [19], will be the study of the supersymmetry of the different gauge-fixing choices. Let us summarize the results: 1. Landau Gauge Fixing: this is achieved by setting to zero the Nakanishi-Lautrup terms of the gauge-fixing (3.12) and non-linear gauge fixing terms (3.13). The model has a manifest N = 4 supersymmetry as a result of a combination of BRST symmetries (s ands) and of the vector supersymmetries.

Kapustin-Saulina Gauge
Fixing: adding the non-linear terms (3.13). Those terms are needed to reproduce the scalar potential of the dual theory. In the dual picture, the theory still displays an N = 4 supersymmetry, though realised in a non-linear way.
That supersymmetry is a manifestation of the BRST symmetries of the original Chern-Simons supergravity and of the vector supersymmetris, which however are not evident in the twisted version due to the non-linearities. This leads to an open problem: to show that the CS supergravity with this particular gauge-fixing admits a formulation with a manifest non-linear vector supersymmetry.
3. Gauge Fixing with Mass Terms: in order to reproduce the deformations discussed in [21][22][23] one has to change the gauge-fixing adding a new Nakanishi-Lautrup term. Its BRST variation leads to the expected terms. supergravities; the investigation of the holographic D = 4 duals to these models. As for the latter objective, a natural candidate would be the maximal D = 4 supergravity realized on the N = 4 AdS 4 vacuum [24,25], which describes a class of Type IIB Janus solutions [26,27].
As a concluding remark: we have shown that the N -extended D=3 supergravity can be reformulated in terms of worldvolume supersymmetric models coupled to matter. This might be useful for computation of interesting observables in both sides of this duality. We leave this to further investigations.