Generalized Lagrangian of N = 1 supergravity and its canonical constraints with the real Ashtekar variable

TL;DR

This paper extends N=1 supergravity by introducing a parameter that unifies different formulations, deriving a canonical structure with real Ashtekar variables, and connecting it to existing formulations via canonical transformations.

Contribution

It generalizes the N=1 supergravity Lagrangian using an arbitrary parameter, leading to a canonical formulation with real Ashtekar variables and linking it to previous models through canonical transformations.

Findings

Generalized Lagrangian with arbitrary parameter  includes chiral case as a special instance.

Derived canonical formulation with real Ashtekar variables after 3+1 decomposition.

Connected the generalized formulation to standard N=1 SUGRA via Barbero-type canonical transformation.

Abstract

We generalize the Lagrangian of N = 1 supergravity (SUGRA) by using an arbitrary parameter $\xi$, which corresponds to the inverse of Barbero's parameter $\beta$. This generalized Lagrangian involves the chiral one as a special case of the value $\xi = \pm i$. We show that the generalized Lagrangian gives the canonical formulation of N = 1 SUGRA with the real Ashtekar variable after the 3+1 decomposition of spacetime. This canonical formulation is also derived from those of the usual N = 1 SUGRA by performing Barbero's type canonical transformation with an arbitrary parameter $\beta (=\xi^{-1})$. We give some comments on the canonical formulation of the theory.