Ashtekar's formulation for $N=1,2$ supergravities as "constrained" BF theories

TL;DR

This paper extends Ashtekar's formulation of gravity as a constrained BF theory to include N=1 and N=2 supergravities, revealing new topological solutions and symmetry reductions.

Contribution

It introduces a graded Lie group framework for supergravities within the BF theory approach, generalizing Ashtekar's formalism to supersymmetric cases.

Findings

Established BF theory formulations for N=1 and N=2 supergravities.

Provided topological solutions for N=2 supergravity with zero cosmological constant.

Showed the breakdown of Kalb-Ramond symmetry to diffeomorphisms and right-supersymmetry.

Abstract

It is known that Ashtekar's formulation for pure Einstein gravity can be cast into the form of a topological field theory, namely the $SU(2)$ BF theory, with the B-fields subject to an algebraic constraint. We extend this relation between Ashtekar's formalism and BF theories to $N=1$ and $N=2$ supergravities. The relevant gauge groups in these cases become graded Lie groups of $SU(2)$ which are generated by left-handed local Lorentz transformations and left-supersymmetry transformations. As a corollary of these relations, we provide topological solutions for $N=2$ supergravity with a vanishing cosmological constant. It is also shown that, due to the algebraic constraints, the Kalb-Ramond symmetry which is characteristic of BF theories breaks down to the symmetry under diffeomorphisms and right-supersymmetry transformations.