**7. References**


**1. Introduction**

models of Einsteinian gravity [5, 10, 18, 36].

be effectively reduced to problems in 3D, cf. Refs. [6, 7].

coframe induced from its embedding into three-dimensional spacetime.

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

**S-Duality in Topological Supergravity** 

Eckehard W. Mielke and Alí A. Rincón Maggiolo\*

*Universidad Autónoma Metropolitana–Iztapalapa,* 

**3**

*México* 

*Departamento de Física,* 

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

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

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

\*Permanent address: Departamento de Física, Facultad de Ciencias, Universidad del Zulia, Venezuela

