Chern-Simons supergravity on supergroup manifolds

L. Castellani,C.A. Cremonini,P.A. Grassi

doi:10.1007/jhep05(2020)110

Abstract

We construct N=1 d=3 AdS supergravity within the group manifold approach and compare it with Achucarro-Townsend Chern-Simons formulation of the same theory. We clarify the relation between the off-shell super gauge transformations of the Chern- Simons theory and the off-shell worldvolume supersymmetry transformations of the group manifold action. We formulate the Achucarro-Townsend model in a double supersymmetric action where the Chern-Simons theory with a supergroup gauge symmetry is constructed on a supergroup manifold. This framework is useful to establish a correspondence of degrees of freedom and auxiliary fields between the two descriptions of d=3 supergravity.

Highlights

The action, being the integral of a 3-form on a 3-dimensional manifold, is invariant by construction under 3d diffeomorphisms
We will call it worldvolume supersymmetry to distinguish it from the gauge supersymmetry of the Chern-Simons action
The resulting spacetime action coincides with the Achucarro and Townsend action, and is worldvolume supersymmetric provided some conditions are fulfilled, called “rheonomic” conditions. We show how these conditions can be imposed as constraints on the “outer” components of the 2-form curvatures, and how this leads to a local supersymmetry that not surprisingly coincides with the gauge supersymmetry

Summary

Curvature parametrizations and symmetries of the spacetime action

Where is a tangent vector in fermionic directions is satisfied if all curvatures have no “legs” in fermionic directions, i.e. if i RA = 0. This leads to the parametrizations of the curvatures. The spacetime reduced action being equal to the Achucarro-Townsend action, it has its gauge symmetries. These coincide with the ones expressed by eqs. The action is invariant under the CS gauge symmetry (2.19)–(2.21), and ordinary spacetime diffeomorphisms. The coefficients c1, c2 are fixed by the Bianchi identity (3.6) to the values:. Notice that the origin of supersymmetry is completely algebraic for the Chern-Simons action, while it is geometric (due to superdiffeomorphism invariance of a superspace action) for the rheonomic action

Off-shell degrees of freedom

The extended superAdS algebra

Curvature parametrizations

Off-shell supersymmetry transformations

Equivalence of transformations: trivial gauge transformations

Supersymmetric Achucarro-Townsend model

Supersymmetry

Conclusions and outlook