Topological Gravity on $(D,N)-$Shift Superspace Formulation

In this contribution, we re-assess the subject of topological gravity by following the Shift Supersymmetry formalism. The gauge-fixing of the theory goes under the Batallin-Vilkovisky (BV) prescription based on a diagram that contains both ghost and anti-ghost superfields, associated to the super-vielbein and the super-Lorentz connection. We extend the formulation of the topological gravity action to an arbitrary number of dimensions of the shift superspace by adopting a formulation based on the gauge-fixing for BF-type models.


Introduction
Topological field theories have been introduced by Witten [1] and soon after applied in several areas that describe quantum-mechanical and quantum field-theoretical systems. Over the recent years, they have been applied to study non-perturbative quantum gravity [2], the issues of topological phases, spin foam, Loop Quantum Gravity, a topological approach to the cosmological constant and a number of other relevant applications. One of the basic topics for the construction of Topological Gravity are the topological Yang-Mills theories, by now fairly-well discussed and understood. However, on the other hand, the complexity of the symmetry groups and Lagrangians in topological gravity renders difficult the comparison between the results obtained by using different formalisms. A topological gravity theory may be formulated by gauge-fixing an action that is a topological invariant, which can be achieved by twisting an extended supergravity theory. The latter combines diffeomorphisms and local supersymmetry transformations and exhibit a considerably more complex structure whenever compared with Super-Yang-Mills theories. A good review paper on Topological Gravity, its corresponding physical observables and a number of interesting applications may be found in the work of Ref. [3].
Ever since its formulation, Chern-Simons theory, treated in a supersymmetric formalism [4,5,6], has raised particular interest in the framework of gauge theories as a topological geometric model. Its basic geometrical objects in the principal bundle are the dreibein and the Lorentz connection, where the dreibeins form a basis for the tangent space, while the space-time metric appears as the square dreibein matrix.
The topological supersymmetrization of Yang-Mills [7] and BF -theories by adopting the shift formalism [8,9,10,11] provide a viable way to their extension to any superspace dimensions, with the Wess-Zumino super-gauge choice [12] used in connection with the Batallin-Vilkovisky prescription. In other words, an alternative construction can be used for the BV diagrams, where the latter no longer accommodate fields, but, instead, superfields; they are referred to in the literature as BV Super-diagrams.
Thus, it is possible to formulate an immediate extension to an arbitrary superspace dimension [9]. We think it is possible to proceed seemingly for topological gravity [13], that is, to describe this theory in a topological formalism, based on supersymmetry (SUSY, from now on): a supersymmetric topological gravity [14]. We accomplished this construction by exploiting the shift supersymmetry formalism, and defining the geometric supersymmetric elements of the theory from an extension of the usual elements of the differential Riemannian formulation. For the supersymmetrization, we define these elements in a basis super-manifold M with a mapping to the Euclidian-flat-space carried out by the D − bein. This D − bein and the Lorentz connection describe the geometric sector along with the gauge-fixing fields of the theory in an Euclidean flat-space-time and the Grassmann-valued coordinates; the latter are adjoined to the space-time coordinates to constitute the superspace of the theory.
Our starting point consists in rewriting the Chern-Simons action [15] in the N = 1-shift supersymmetry formalism of a BF-type Model and, next, to carry out its complete gauge-fixing. In the sequel, we shall propose the study of more than a single (simple) SUSY and describe the topological gravity in four space-time dimensions [16]. Consequently, we set up an action for arbitrary superspace dimensions, i.e., with arbitrary D− space-time dimensions and extended N -SUSY. We shall give a brief presentation of the elements of the space-time theory. In the D−dimensional basis manifold M , we define a metric g µν , with index µ, ν, ... = 1, 2, ..., D. The D − bein, e a µ allows the transformation between the basis manifold and the Euclidean flat-space-time whose the flat-metric is δ ab , with the same dimension and index a, b, ... = 1, 2, ..., D. The metric is defined in both spaces, such that e a = e a µ dx µ , represents the basis in the dual T * p M , of the tangent space T p M , with basis elements given by ∂ µ . The transformation properties between the basis manifold M and flat space are v µ = e µ a u a , u a = e a µ v µ (1 .2) and v µ = e a µ u a , u a = e µ a v µ .
where the entry [e a µ ] has its inverse, [e µ a ] −1 . The connection written on the basis manifold, Γ µ νκ , has its correspondent in the Euclidean flatspace-time, ω a µ b = Γ ν µλ e a ν e λ b − e λ b ∂ µ e a λ ,, which is a 1-form in the basis manifold, called Lorentz connection. Inversely, we have, Γ ν µλ = e ν a ∂ µ e a λ + e ν a e b λ ω a µ b .
The objects that compose the dynamical sector are the curvature, R = dω + ω ∧ ω, and the torsion, T = de+ω ∧e. These entities compose the geometrical sector and obeys the Bianch identity relation where the covariant derivative with respect to the connection ω is D ω (·) = d (·)+ω ∧(·) 1 . Following in the incoming Sections, we present the extension of these objects of the Maurer-Cartan-Einstein formalism to the superspace approach.
The paper is outlined as follows: in Section 2, we present a general formulation of gravity as a super-BF model. This Section is split into five subsections to render the presentation clearer. Section 3 presents the so-called super-BF model and an associated on-shell solution. An explicit construction in the D = 3, N = 1-case is worked out to render manifest the whole idea of the method we have followed. Finally, we cast our Concluding Comments in Section 4.
2 Generalized Gravity as a super−BF Model

Preliminary Definitions
The supermanifold we shall work with consists of the basis manifold with the addition of the Grassmann-valued coordinates. The supercoordinates are parametrized as z M = x µ , θ I defined in a superchart of the basis supermanifold 2 M, where θ I is the Grassmannian coordinates [8,9,10,17,18,26] 3 . In the Euclidean flat space-time, the coordinates are represented by z A = (x a , θ I ).
The superderivatives as superforms are given byd = dx µ ∂ µ + dθ I ∂ I . An arbitrary superfield F (x, θ I ), is defined by the action of the transformation generated by the derivatives with respect to the Grassmann coordinates [8,9,12], known as the shift operator, and given according to what follows: where we assign fermionic supersymmetry numbers as follows: [θ I ] = −1 and [Q I ] = +1. In analogy to the Riemannian geometry, we set up a superspace formalism [3] to treat the fundamental elements of the description. In the SUSY formalism, we define the D − bein 1-(super)form asÊ a = E a µ (x, θ)dx µ + E a I (x, θ)dθ I , (2.2) and the 1-(super)form asΩ In both cases, their components are form-superfields (the same being true in all our definitions that involve superforms: superform components are superffields). Then, the dynamical objets of the formalism, the curvature 2-superform and torsion 1-superform, are defined by means of the Cartan algebra associated to the shift SUSY. The (super)curvature as 2-superform reads aŝ where R ab accommodates the genuine 2-form Riemann tensor that we simply write as: Also, the supercovariant derivative is given bŷ (2.8) 2 We stress that the space-time dimension is D, while N is an internal label associated to the number of supersymmetries of the model;N = 1 corresponds to a simple SUSY, N > 1 stands for an N-extended SUSY. 3 The Grassmann Variables of the topological description may be found in [8,9,10], as well as the notations therein. For example for N = 2, Levi-Civita pseudo-tensor ǫ IJ is the antisymmetric metric, ǫ 12 = +1 = −ǫ 12 , where ǫ IJ ǫ J K = δ I K and a field or variable transforms as, ϕ I = ǫ IJ ϕ J and ϕ I = ǫ IJ ϕ J . A scalar product is invariant under SU (2) such that, (θ) 2 = 1 2 θ I θ I = 1 2 ǫ IJ θ I θ J . The derivative is defined as ∂ I = ∂/∂θ I and ∂ I = ∂/∂θ I , and applied to a Grassmann coordinate gives ∂ I θ J = −ǫ IJ . Finally, the Berezin Integral is defined by and (D Ω ) µ = D µ , is the covariant 1-form superfield derivative of the Ω connection. This yields the (super)components of the curvature superfield: we can also write the curvature (2.5) as R = D ω ω.
2.2 Re-assessing D = 3, N = 1 and D = 4, N = 1 Topological Gravities After the papers [4,5] have appeared, a whole line of works has approached supergravity and its topological version with the aim to accomplish the description based on a maximally extended SUSY. The challenge is to conveniently formulate it as a gauges theory, in view of the huge number of degrees of freedom accommodated in all superfields. In the formalism of shift SUSY, or topological supersymmetric formalism, it is understood that (simple) N = 1 superspace is considered [19,8]. We wish, in the present contribution, to formulate, with the help of the shift SUSY [8,9,11,12,3], gravity for any any dimensionality, D, and for a general number of SUSY generators, N . We shall adopt the Batallin-Vilkovisky (BV) prescriptiom [20] combined with the Blau-Thompson minimal action gauge-fixing [10,21], that fix the the Lagrange multipliers associted to the gauge conditions 4 . Finally, we need to add the Fadeev-Popov gauge-fixing action along with the BF-model term, forming the full invariant action.
Our starting point is the gravity formulation for N = 1-SUSY [4,3,22,23,24,25]; for that, we define the 1-form-superfields by means of the following expansionÊ a = E a µ (x, θ)dx µ +E a θ (x, θ)dθ, as having an odd statistics, such that, the component superfields readily follow: and E a θ = χ a + θφ a . (2.11) The super-formÊ a has odd statistics, because e a should be odd − this is a 1-form with SUSY number 1 − with its component e a µ being a bosonic field. To build up the model in D = 3 space-time dimensions, we define the super-connection aŝ where the components 1-form-superfields are The (super)components of the supercurvature can be expanded from expression (2.9).
The invariant (topological) gravity action in the dimension we are now considering is the Chern-Simons model [8,10,1] in the corresponding shift superspace; therefore, here, to ensure the right field content, we need to change the E a given above (which would be compatible with a BF-type action) and re-write it as E a = ψ a + θe a , such that the integrand contains basically the expression for the gravitational Chern-Simons action: where R ab = dω ab + ω ac ω cb = (D ω ω) ab , (D ω ̟) ab = d̟ ab + [ω ac , ̟ cb ] and the super-integral in this formalism is dθ = Q. The final result of this supersymmetrization is an action free from the supersymmetric charges. Then, this formulation for D = 3 dimensions is not interesting to our construction, because, as we have seen above, we have made a new definition of E a , compared with (2.10), so that the action acquires the Chern-Simons form. For that reason, we choose the BF-type formulation to give us the same expected results, and we do not need to change the form of the basic elements of the theory, E a for example, to adapt the supersymmetrization process. Now, that the construction of the action for the geometric sector is totally fixed, we need to describe the BV super-diagram [9] that contains the super-components of the eq. (2.4), using the Blau-Thompson minimal action procedure [10,11]. The super-diagram in D = 3 dimensions is given as below: Let us recall that this gauge-fixing procedure is a shift gauge-fixing rather than the BRST gaugefixing (Faddev-Popov), though it has the similar effect on the shift degrees of freedom (shift ghosts elimination [9,3]. In the some cases, this is named BRST gauge-fixing [8,1]. However, all curvature superfields and Lagrangian multipliers exhibit the same covariant BRST transformation: where C ab is a zero-form superfield ghost associated to Ω ab . According to this diagram, the action can be written as For D = 4 dimensions, we can reproduce the results of (2.16), by generalizing the systematization [19], where the BV super-diagram is now given by In this case, we clearly notice, by means of the Blau-Thompson procedure, that the ghost fields for each Lagrange multiplier are elimined by a simple redefinition in the action, leading to a minimal gauge-fixing action [10,11]. Therefore, the associated action to (2.17) can be written as Now, we are ready to build up the supergravity action in N = 2 superspace. This task is the subject of our next Section.

The Topological Gravity in N = 2, D = 4
Here, it is possible to write down the topological gravity action, using the same construction as the one in the previous Section. Now, the D − bein 1-form-superfield is given by The action for D = 3 space-time dimensions obey the same gauge-fixing systematization as ion the N = 1 case. This super-diagram here can be written as For the invariant topological gravity model in D = 4 dimensions we propose Therefore the action associated to this diagram is given as Once we have understood how an action can be written, we shall define the geometric elements for a general superspace dimension and we shall present the action of topological gravity in the sequel.

The (N, D)−Superspace
We generalize the formulation by expanding the action from N = 2 to an arbitrary number of SUSYs. We need to define the form-superfields for N superspace dimensions. The shift index runs as I = 1, ..., N, and the Levi-Civita pseudo-tensor is defined as: ǫ 1...N = 1. The D − bein (super)component fieds of eq. (2.2) are 1-form superfields and their expansion in component fields is given by In D = 3 dimensions, the construction is the same for all N , i.e., as (2.15). In D = 4 dimensions, the diagram is also similar to (2.17). We can extend the space-time dimension from 4 to any D. We start off the construction with the Levi-Civita symbol, where we assume ε 012...D−1 = 1.
The the Euclidian flat-space-time indices are a 1 , a 2 , ..., a D and they run from 1 to D. The BV super-diagram in the (D, N )−superspace is where each N −power covariant derivative is written as: This action still presents still spurious degrees of freedom; they are localized in the connection Ω µ (x, θ); thus, we need to suppress them from the model by fixing the gauge with the help of the BRST ghosts, as we are next going to do.

The Fadeev-Popov Gauge-Fixing
According to [8,9], the gauge-fixing of topological gravity consists in writing the Batallin-Vilkovisky diagram [3] and defining the respective ghost for each field. The ghost field and the diffeomorphism transformation fix the principal action [27]. Here, we need to fix only the Lorentz gauge condition. The ghost superfield, for N = 1-SUSY, for example, iŝ C ab = C ab = c ab + θc ab θ , (2.29) Associated to this ghost, we need to define the anti-ghosts and their Lagrange multipliers as zeroform-superfields,C ab =c ab + θc ab θ , B ab = b ab + θb ab θ . (2.30) The diffeomorphism ghost field is a super-vector in shift superspace [3], such that, its component fields read as follows: and this superform obeys the following propertieŝ where the BRST operator is nilpotent, s 2 = 0, and anticommutes withd. From the complete set of the BRST transformations described above, we can write the super-diagram of the Batallin-Vilkovisky to fix the spurious degrees of freedom of the Lorentz connection. The super-diagram associated to the general action (2.28), will be The Fadeev-Popov gauge-fixing super-Lagrangian, invariant under the BRST operator, is: so that, according to the BRST transformation above, we have the action where " * " is the dual star transformation, or star product, of a object from a tangent space to a cotangent space. This action in one Grassmann dimension has the same structure, in the superfield form, of the action for N-dimensions, because the star product guarantee this notation. Therefore, the invariant gauge-fixed gravity action for any dimension is determined by the sum of (2.28) and (2.40); this yields S = S N D + S gf . (2.41)

The super−BF Model and an On-shell Solution
The model studied in the previous Section is acceptable, because it correctly describes the particular models previousloy studied; nevertheless, we can formulate a new model whose geometric sector in the action is stable under the BRST invariance of all the other sectors, which we cannot guarantee in the general model of the previous Section. Here, we do not follow any longer a Blau-Thompson minimal action procedure. The diference consists basically in considering the total action written as a BRST invariance, considering also the on-shell instanton solutions of the geometric sector. This means that the first Lagrangian multiplier contains the background zero-modes of the degrees of freedom. Therefore, by virtue of the BRST transformation for the first Lagrangian multiplier −NH ab D−2 of the super-diagram (2.27), the latter gets restricted to the geometric section and the first multipliers according to what follows: 1) and the rest of the gauge-fixing diagram splits from the one above; it involves only the fixation of theH ab and its anti-ghosts. The BRST transformation of the Lagrangian multiplier is then For the other superfields of the super-diagram (3.1), the BRST transformation is covariant: s(·) = −[C, (·)]. To preserve the nilpotency of the BRST operator s, we take the on-shell solution for the new model as an instanton solution, R ab − * R ab = 0 in four dimensions, and for other dimensionalities the null curvature solution, R ab = 0. We now write the BV super-diagram associated to this Lagrangian multiplier: whose the transformations for all ghosts and anti-ghosts read as The super-diagram of the BRST Lagrange multipliers of the anti-ghosts We define only the anti-fieldsH * ,W * of the Lagrangian multipliersH,W in the diagram (3.1) that fix the geometric sector. Those describe the following relation where the geometric sector action is: The N is the counter field operator and the Slavnov-Taylor operator is given by where s = −N and g = 0 if D is even or s = 0 and g = −1 if D is odd. Summarizing, the general form of this new model (3.11) exhibits trivial cohomology for both operators Q and s, S = Q s (L Geometric + L F adeev−P opov ). (3.12) while in the previous Section the total action (2.41) obeys the scheme For the sake of illustrating the gauge-fixing procedure, we can present a simple example in the particular D = 3 and N = 1 case.
Example: Let us reproduce the topological gravity in the case of three dimensions with N = 1-SUSY. The super-diagram is (3.14) Using the defintions (2.12), (2.13), the curvature (2.9) and the Lagrange multiplier superfields are given, component-wise, as follows:

17)
W ab = w ab + θw ab θ . Σ ab =σ ab + θσ ab θ . (3.20) The BRST transformation for all superfields of the Fadeev-Popov gauge-fixing action is given by  (3.25) and the other transformation is given by According to the eq. (3.11) and the BRST transformation above, the explicit topological supergravity invariant action takes the form According to the new term of the transformation (3.25), this requires that the on-shell solution for the superfield curvature be null because we work in a 3-dimensional manifold.
A good example is the topological gravity theory for a two-dimensional Riemannian world-sheet manifold, that usually appears coupled to topological sigma-models. The dimension constrains the connection and curvature superforms not to carry the Euclidean vector index; they should be represented asΩ andR. For the BV super-diagram, we may follow the same systematic construction of example contemplated above, but the Lagrange multiplier associated to R is a zero-form. For that reason, it does not need to have an associated anti-ghost; thus, the gaugefixing systematic procedure does not change.

Concluding Comments
Based on the investigation pursued here, we are able to write down general models over Riemannian manifolds endowed with a metric, both in the topological supersymmetric and in the Witten-type topological formulations, preserving topological invariants like the Euler characteristic, χ, and the correlation functions, while keeping them free from shift spurious degrees of freedom. A good prospective for the application of our results would be the association with twist techniques to re-obtain ordinary supergravity theory in the Weyl representation. This attempt, by using topological supergravity, can be implemented for any superspace dimension because there are no symmetry limitations in superspace that prevents us from following this path. Other possibilities of apllications that we may point out could be in the framework of Loop Quantum Gravity, Spin foam, Global effects with continuous deformations and systems with emergent SUSY, like, for instance, interesting topological materials such as Weyl semi-metals and topological insulators. Finally, an open issue which shall be the subject of a forthcoming work is the coupling of the matter sector in the framework of topological gravity. We shall be soon reporting on that.
The authors state hereby that there is no conflict of interest regarding the publication of this paper.