Covariant hamiltonian for supergravity in $d=3$ and $d=4$

We extend the covariant canonical formalism recently discussed in ref. [1] to geometric theories coupled to both bosonic and fermionic $p$-forms. This allows a covariant hamiltonian treatment of supergravity theories. As examples we present the covariant hamiltonian formulation for $d=3$ anti-De Sitter supergravity and for the “new minimal” $d=4$, $N=1$ supergravity (with $1$-form and $2$-form auxiliary fields). Form-Poisson brackets and form-Dirac brackets are defined, and used to find the form-canonical generators of all gauge symmetries via an algorithmic procedure.