**1. Introduction**

10 Will-be-set-by-IN-TECH

36 Quantum Gravity

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One of the main motivations for constructing a model of topological gravity in *three dimensions* (3D) is that it might serve as a 'laboratory' for applying techniques appearing rather awkward or even intractable in four dimensions. This stems from the fact that a Riemannian spacetime is Ricci-flat, i.e., the Ricci tensor determines the Riemann tensor in 3D and as a result, the only vacuum solutions of the Einstein equations with vanishing cosmological constant are flat. This result implies that the dynamical properties may not be attributed to the metric but rather to the coframe. When matter is included there are nontrivial solutions to the Einstein equations and if topological terms are included, these may induce dynamical properties in 3D. Such a 'laboratory' may no longer be a suitable testing ground for higher–dimensional models of Einsteinian gravity [5, 10, 18, 36].

There are other reasons for studying the dynamical aspects of topological gravity in three dimensions: Some problems in 4D gravity reduce to an effective 3D theory, such as cosmic strings, the high–temperature behavior of 4D theories and some membrane models of extended relativistic systems. Moreover, many aspects of black hole thermodynamics can be effectively reduced to problems in 3D, cf. Refs. [6, 7].

Outside of quantum gravity, the continuum theory of lattice defects in crystal physics can be regarded as 'analogue gravity' with Cartan's torsion in 3D, where such defects are modeled by connections in the orthonormal frame bundle and the Chern-Simons type free-energy integral by Riemann–Cartan (RC) spaces with constant torsion [11, 26]. Recently, flexural modes of graphene have also been considered as membranes with a 'gravitational' metric [25] or coframe induced from its embedding into three-dimensional spacetime.

Our paper is organized as follow: In Section 2, we give a brief introduction to the Mielke-Baekler (MB) model of toplogical gravity in 3D, in which the Einstein-Cartan Lagrangian is substituted by a *mixed* topological term, the so-called *mix*-model. The coupling of Rarita-Schwinger fields to topological gravity is presented in Section 3, whereas in Section 4 we deduce the restrictions on the coupling parameters in order to ensure that the model is supersymmetric. The particular dynamical symmetry of the MB model, in Ref. [32] dubbed "S–duality", is generalized in Section 5 to our topological supergravity model. In Section 6 and in an Outlook, we consider the still speculative applicability of this model to the

<sup>\*</sup>Permanent address: Departamento de Física, Facultad de Ciencias, Universidad del Zulia, Venezuela

Allowing for arbitrary "vacuum angles" *θ*T, *θ*<sup>L</sup> and *θ*TL = −*χ*, the most general purely

S-Duality in Topological Supergravity 39

respectively, are the translational, rotational and 'mixed' Chern-Simons type three forms of gauge type *C* =Tr{*A* ∧ *F*} in RC spacetime [8, 12, 43]. The equation (5) is the known topological Lagrangian of the Mielke-Baekler (MB) *mix*-model [28, 31]. Since the translational term *C*<sup>T</sup> is covariant, it appears that the MB model is semi-topological, with interesting

Varying the Lagrangian (5) with respects to *ϑ<sup>α</sup>* and Γ*<sup>α</sup>* and employing the results of Appendix

*<sup>α</sup>* <sup>−</sup> *<sup>θ</sup>*<sup>T</sup>

cf. Eq. (6.9) of Ref. [2]. Observe that the translational CS term proportional to *θ*<sup>T</sup> induces in the second field equation a constant term, familiar in 4D from Einstein's equation with

Thereby, combining the *vacuum* field equations (9) and (8) yield for the torsion and the RC

*ηα* , *<sup>R</sup>*

*<sup>A</sup>* (*θ*TL*<sup>τ</sup>*

Commonly, *supergravity* [15, 19] with one supersymmetry generator, i.e. N =1, represents the

*<sup>A</sup>* (*θ*TL<sup>Σ</sup>*<sup>α</sup>* <sup>−</sup> *<sup>θ</sup>*T*<sup>τ</sup>*

<sup>2</sup> *ηα* <sup>−</sup> *<sup>θ</sup>*<sup>L</sup> *<sup>R</sup>*

*<sup>α</sup>* <sup>=</sup> *<sup>ρ</sup>*

*<sup>α</sup>*) = *θ*T*C*<sup>T</sup> + *θ*L*C*<sup>L</sup> + *θ*TL*C*TL , (5)

3! *ηαβγ* <sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> <sup>Γ</sup>*<sup>β</sup>* <sup>∧</sup> <sup>Γ</sup>*<sup>γ</sup>* (6)

, (7)

*<sup>T</sup><sup>α</sup>* <sup>=</sup> <sup>Σ</sup>*<sup>α</sup>* , (8)

*<sup>α</sup>* , (9)

<sup>2</sup> *ηα* (10)

TL + 2*θ*T*θ*<sup>L</sup> �= 0.

*<sup>α</sup>* − *θ*<sup>L</sup>Σ*α*) , (11)

<sup>2</sup> field [35] to gravity.

<sup>T</sup>/*A* are related to the vacuum

*<sup>α</sup>* ), (12)

*<sup>α</sup>* <sup>=</sup> *<sup>τ</sup>*

*<sup>α</sup>* <sup>−</sup> <sup>1</sup>

<sup>2</sup> *ηαβγ* <sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> <sup>Γ</sup>*<sup>β</sup>* <sup>∧</sup> *<sup>ϑ</sup><sup>γ</sup>*

*topological* gravity Lagrangian in 3D, in first order formalism, takes the form

<sup>2</sup><sup>2</sup> *<sup>ϑ</sup><sup>α</sup>* <sup>∧</sup> *<sup>T</sup>α*, *<sup>C</sup>*<sup>L</sup> := (−1)*<sup>s</sup>* <sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> *<sup>R</sup>*

consequence on the degrees of propagating modes, cf. Ref. [4, 32, 36].

<sup>−</sup> (−1)*<sup>s</sup>*

where the contortional constants *<sup>κ</sup>* <sup>=</sup> *<sup>θ</sup>*TL*θ*T/2*<sup>A</sup>* and *<sup>ρ</sup>* <sup>=</sup> <sup>−</sup>*θ*<sup>2</sup>

angles. A singular case is exclude by assuming that *<sup>A</sup>* <sup>=</sup>: <sup>−</sup>(−1)*sθ*<sup>2</sup>

When including matter couplings, we explicitly find for the torsion

*<sup>T</sup><sup>α</sup>* <sup>−</sup> <sup>2</sup>*<sup>κ</sup>*

*R <sup>α</sup>* <sup>−</sup> *<sup>ρ</sup>*

simplest consistent coupling of a Rarita–Schwinger (RS) spin– <sup>3</sup>

**3. Rarita–Schwinger Lagrangian in 3D**

<sup>−</sup> *<sup>θ</sup>*TL *<sup>R</sup>*

*<sup>T</sup><sup>α</sup>* <sup>=</sup> <sup>2</sup>*<sup>κ</sup>*

*ηα* <sup>=</sup> <sup>2</sup>

<sup>2</sup> *ηα* <sup>=</sup> <sup>2</sup>

*<sup>θ</sup>*TL *<sup>T</sup><sup>α</sup>* <sup>−</sup> *<sup>θ</sup>*<sup>T</sup>

<sup>Γ</sup>*<sup>α</sup>* <sup>∧</sup> *<sup>T</sup><sup>α</sup>* <sup>−</sup> (−1)*<sup>s</sup>*

*L*MB(*ϑα*, Γ

*<sup>C</sup>*TL :<sup>=</sup> <sup>1</sup> 

where

and

and

*<sup>C</sup>*<sup>T</sup> :<sup>=</sup> <sup>1</sup>

A, yields the topological field equations

cosmological constant Λ.

curvature the constrictions:

and the RC curvature

cf. Ref. [31].


Table 1. Geometrical objects and fields

flexural modes of corrugated surfaces (2D membranes) embedded in 3D spacetime, as recently realized by the rather prospective new material called *graphene*.
