Ashtekar's Formulation for N = 1, 2 Supergravities as "Constrained" BF Theories

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 two-form 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.