Cohomological aspects of Abelian gauge theory

Abstract

We discuss some aspects of cohomological properties of a two-dimensional free Abelian gauge theory in the framework of BRST formalism. We derive the conserved and nilpotent BRST-and co-BRST charges and express the Hodge decomposition theorem in terms of these charges and a conserved bosonic charge corresponding to the Laplacian operator. It is because of the topological nature of free U(1) gauge theory that the Laplacian operator goes to zero when equations of motion are exploited. We derive two sets of topological invariants which are related to each-other by a certain kind of duality transformation and express the Lagrangian density of this theory as the sum of terms that are BRST-and co-BRST invariants. Mathematically, this theory captures together some of the key features of Witten-and Schwarz type of topological field theories

Abelian 2-form gauge theory: special features

“…Because of the existence of the above mapping, the analogue of the Hodge decomposition theorem can be defined in the quantum Hilbert space of states [25]. In fact, all the above cited properties of this section are common to the 2D one-form free Abelian (and self-interacting non-Abelian) as well as 4D free Abelian 2-form gauge theories [25,[31][32][33][34][35][36] except for the sign difference in the inverse relationship (3.6). To be more precise, the analogues of (3.2) and (3.6) for the 2D theories bear the same signs on the r.h.s.…”

“…The latter 2D theories are also exact TFTs because, the existence of (anti-)BRST and (anti-)co-BRST symmetries in the theory, turns out to be responsible for gauging out both the propagating degrees of freedom of 2D photon (and/or 2D gluon) (see, e.g., [31][32][33]37] for details). In the recent past, there has been some interest in studying almost TFTs (or quasi-TFTs) which involve constraints that leave out only a finite number of degrees of freedom in the theory [37,46].…”

“…From these, the other two (w.r.t. anti-BRST and anti-co-BRST charges) can be computed by exploiting the discrete ghost symmetries of the theory [31][32][33][34]. In contrast, in the present 2-form 4D Abelian gauge theory, there are three sets of (anti-)ghost fields, namely; (β)β, (C µ )C µ , (ρ)λ with ghost numbers: (−2)2, (−1)1, (−1)1, respectively.…”

“…For the sake of completeness and clarity, we first recapitulate some of the key properties of the 2D one-form Abelian gauge theory † † [31][32][33]. Before we embark on addressing the difference(s) between these theories, it is important to emphasize that these theories have something in common.…”

“…There exists a ghost scale symmetry transformation for this theory where only the ghost fields undergo a scale transformation and other fields remain untransformed. Together, the above symmetries obey an algebra that is reminiscent of the algebra obeyed by the de Rham cohomology operators [31][32][33][34][35][36]. The kinetic energy-and gauge-fixing terms of the Lagrangian density (4.1) can be linearized by introducing the auxiliary fields B and B as [31][32][33] …”