Phase Space on a Surface with Boundary via Symplectic Reduction

TL;DR

This paper develops a symplectic reduction framework for gauge theories with boundaries, applying it to BF theory, and reveals topological features via cohomology, offering a structured approach to phase space analysis.

Contribution

It extends symplectic reduction to phase spaces with boundaries and connects it with cohomological descriptions, enhancing understanding of gauge theories with boundary conditions.

Findings

Symplectic reduction applies to boundary phase spaces in gauge theories.

Phase space description involves generalized de Rham cohomology.

Multiple reduction steps reveal finite-dimensional intermediate spaces.

Abstract

We describe the symplectic reduction construction for the physical phase space in gauge theory and apply it for the BF theory. Symplectic reduction theorem allows us to rewrite the same phase space as a quotient by the gauge group action, what matches with the covariant phase space formalism. We extend the symplectic reduction method to describe the phase space of the initial data on a slice with boundary. We show that the invariant phase space has description in terms of generalized de Rham cohomology, what makes the topological properties of BF theory manifest. The symplectic reduction can be done in multiple steps using different decompositions of the gauge group with interesting finite-dimensional intermediate symplectic spaces.