Generalized Lagrangian of $N=1\mathrm{}$ supergravity and its canonical constraints with the real Ashtekar variable

We generalize the Lagrangian of $N=1\mathrm{}$ 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 $\xi =\pm i.\mathrm{}$ We show that the generalized Lagrangian gives the canonical formulation of $N=1\mathrm{}$ SUGRA with the real Ashtekar variable after the $3+1$ decomposition of spacetime. This canonical formulation is also derived from those of the usual $N=1\mathrm{}$ SUGRA by performing Barbero’s type canonical transformation with an arbitrary parameter $\beta (={\xi }^{-1}).\mathrm{}$ We give some comments on the canonical formulation of the theory.