On free 4D Abelian 2-form and anomalous 2D Abelian 1-form gauge theories

TL;DR

This paper explores the similarities and differences between 4D Abelian 2-form and 2D anomalous Abelian 1-form gauge theories within BRST formalism, revealing their cohomological structures and physical realizations.

Contribution

It shows that the anomalous 2D Abelian 1-form gauge theory can be formulated as a Hodge theory model with cohomological operators realized through symmetry transformations.

Findings

Both theories have similar Lagrangian transformation properties.

The 2D anomalous gauge theory models Hodge theory with cohomological operators.

Conserved charges form an algebra akin to de Rham cohomology algebra.

Abstract

We demonstrate a few striking similarities and some glaring differences between (i) the free four (3 + 1)-dimensional (4D) Abelian 2-form gauge theory, and (ii) the anomalous two (1 + 1)-dimensional (2D) Abelian 1-form gauge theory, within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We demonstrate that the Lagrangian densities of the above two theories transform in a similar fashion under a set of symmetry transformations even though they are endowed with a drastically different variety of constraint structures. Taking the help of our understanding of the 4D Abelian 2-form gauge theory, we prove that the gauge invariant version of the anomalous 2D Abelian 1-form gauge theory is a new field-theoretic model for the Hodge theory where all the de Rham cohomological operators of differential geometry find their physical realizations in the language of proper symmetry transformations. The corresponding conserved charges obey an algebra that is reminiscent of the algebra of the cohomological operators. We briefly comment on the consistency of the 2D anomalous 1-form gauge theory in the language of restrictions on the harmonic state of the (anti-) BRST and (anti-) co-BRST invariant version of the above 2D theory.