Covariant canonical formalism for four-dimensional BF theory

TL;DR

This paper develops a covariant canonical formalism for four-dimensional BF theory, analyzing constraints, topological terms, and the effects of additional terms like the second Chern character, revealing differences in symplectic structure and gauge symmetries.

Contribution

It provides a detailed covariant canonical analysis of 4D BF theory, including topological terms and their impact on constraints and symplectic structure, which was not previously explored.

Findings

The presymplectic 3-form differs when the second Chern character is added.

The equations of motion remain the same despite different presymplectic forms.

Differences between diffeomorphisms and gauge symmetries are identified.

Abstract

The covariant canonical formalism for four-dimensional BF theory is performed. The aim of the paper is to understand in the context of the covariant canonical formalism both the reducibility that some first class constraints have in Dirac's canonical analysis and also the role that topological terms play. The analysis includes also the cases when both a cosmological constant and the second Chern character are added to the pure BF action. In the case of the BF theory supplemented with the second Chern character, the presymplectic 3-form is different to the one of the BF theory in spite of the fact both theories have the same equations of motion while on the space of solutions they both agree to each other. Moreover, the analysis of the degenerate directions shows some differences between diffeomorphisms and internal gauge symmetries.