Abelian 2-form gauge theory: special features

TL;DR

This paper explores the 4D free Abelian 2-form gauge theory, highlighting its role as a model for Hodge theory and a candidate for quasi-topological field theory, while clarifying its non-topological nature.

Contribution

It demonstrates that the 4D free Abelian 2-form gauge theory exhibits features of Hodge theory and quasi-topological field theory but is not an exact topological field theory.

Findings

The theory models aspects of Hodge theory.

It shares some topological features with 2D gauge theories.

The Lagrangian cannot be expressed as a sum of BRST-exact terms.

Abstract

It is shown that the four $(3 + 1)$-dimensional (4D) free Abelian 2-form gauge theory provides an example of (i) a class of field theoretical models for the Hodge theory, and (ii) a possible candidate for the quasi-topological field theory (q-TFT). Despite many striking similarities with some of the key topological features of the two $(1 + 1)$-dimensional (2D) free Abelian (and self-interacting non-Abelian) gauge theories, it turns out that the 4D free Abelian 2-form gauge theory is {\it not} an exact TFT. To corroborate this conclusion, some of the key issues are discussed. In particular, it is shown that the (anti-)BRST and (anti-)co-BRST invariant quantities of the 4D 2-form Abelian gauge theory obey the recursion relations that are reminiscent of the exact TFTs but the Lagrangian density of this theory is not found to be able to be expressed as the sum of (anti-)BRST and (anti-)co-BRST exact quantities as is the case with the {\it topological} 2D free Abelian (and self-interacting non-Abelian) gauge theories.