The purpose of this thesis is to study the classical BV-BFV (Batalin–Fradkin–Vilkovisky) structure of gravity coupled to spinors and, specifically, of the simplest case of supergravity, where only one gravitino is introduced in dimension four.
After a synthetic but thorough introduction on the BV-BFV machinery with some simple
examples, this presents a summary of known results on Palatini–Cartan gravity in the BV-BFV
formalism, along with minor redefinitions, serving as a starting point for the further developments and leading to an original description of Palatini–Cartan–Dirac gravity on manifolds with boundary, in which, starting from the study of the boundary structure of the classical fields via the Kijowski–Tulczjiew construction, a BFV formulation is first obtained and then linked to its BV bulk counterpart by means of the 1–dimensional AKSZ construction.
The main body of the present work is a thorough BV-BFV alnalysis of N = 1,D = 4
supergravity. In particular, after studying constraints of the theory and identifying the relevant gauge symmetries, the existence of a BFV structure is established, but not directly computed due to technical difficulties. Such study, along with the simpler case of PCD gravity, provides enough insights to study the BV structure of SuGra in the bulk, where a complete off-shell BV formulation is obtained, generalizing the results of Baulieu et al.
Finally, the last part of the thesis complements the above findings by constructing a BV-BFV
extendable theory of N = 1,D = 4 supergravity, which is obtained by eliminating the degrees
of freedom which are responsible for the obstruction in the BV-BFV extension of the theory.
Such procedure goes by the name of BV–pushforward, a technique that formalizes the concept of "integrating out" certain modes, which is adapted here to the case of classical supergravity.
These results provide a foundational step toward the quantization of supergravity theories in the presence of boundaries.
