Canonical formulation of N = 2 supergravity in terms of the Ashtekar variable

TL;DR

This paper develops a canonical formulation of N=2 supergravity using Ashtekar variables, extending the gauge symmetry to a local O(2) invariance, and explores its algebraic structure and quantum solutions.

Contribution

It presents a new Ashtekar-based canonical formulation of N=2 supergravity with a graded algebra structure and no auxiliary fields, linking it explicitly to standard supergravity.

Findings

The algebra of constraints forms a graded G^2SU(2)(Osp(2,2)) algebra.

The theory is quantized in a graded-connection representation.

Explicit solutions to quantum constraints are discussed.

Abstract

We reconstruct the Ashtekar's canonical formulation of N = 2 supergravity (SUGRA) starting from the N = 2 chiral Lagrangian derived by closely following the method employed in the usual SUGRA. In order to get the full graded algebra of the Gauss, U(1) gauge and right-handed supersymmetry (SUSY) constraints, we extend the internal, global O(2) invariance to local one by introducing a cosmological constant to the chiral Lagrangian. The resultant Lagrangian does not contain any auxiliary fields in contrast with the 2-form SUGRA and the SUSY transformation parameters are not constrained at all. We derive the canonical formulation of the N = 2 theory in such a manner as the relation with the usual SUGRA be explicit at least in classical level, and show that the algebra of the Gauss, U(1) gauge and right-handed SUSY constraints form the graded algebra, G^2SU(2)(Osp(2,2)). Furthermore, we introduce the graded variables associated with the G^2SU(2)(Osp(2,2)) algebra and we rewrite the canonical constraints in a simple form in terms of these variables. We quantize the theory in the graded-connection representation and discuss the solutions of quantum constraints.