Canonical formulation of $N=2\mathrm{}$ supergravity in terms of the Ashtekar variable

We reconstruct Ashtekar’s canonical formulation of $N=2\mathrm{}$ supergravity (SUGRA) starting from the $N=2\mathrm{}$ chiral Lagrangian derived by closely following the method employed in the usual SUGRA. In order to obtain the full graded algebra of the Gauss $\mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ gauge and right-handed supersymmetry (SUSY) constraints, we extend the internal, global $\mathrm{O}\mathrm{}\left(2\right)\mathrm{}$ invariance to a local one by introducing a cosmological constant to the chiral Lagrangian. The resultant Lagrangian does not contain any auxiliary fields in contrast with the two-form SUGRA, and the SUSY transformation parameters are not constrained at all. We derive the canonical formulation of the $N=2\mathrm{}$ theory in such a manner that the relation with the usual SUGRA is explicit, at least at the classical level, and show that the algebra of the Gauss $\mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ gauge, and right-handed SUSY constraints form the graded algebra, $G^2\mathrm{SU}\mathrm{}\left(2\right).$ Furthermore, we introduce the graded variables associated with the $G^2\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}$ 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.