Hodge decomposition theorem for Abelian two-form gauge theory

Abstract

Abstract. We show that the BRST/anti-BRST invariant 3 + 1 dimensional 2-form gauge theory has further nilpotent symmetries (dual BRST /anti-dual BRST) that leave the gauge fixing term invariant. The generator for the dual BRST symmetry is analogous to the coexterior derivative of differential geometry. There exists a bosonic symmetry which keeps the ghost terms invariant and it turns out to be the analogue of the Laplacian operator. The Hodge duality operation is shown to correspond to a discrete symmetry in the theory. The generators of all these continuous symmetries are shown to obey the algebra of the de Rham cohomology operators of differential geometry. We derive the extended BRST algebra constituted by six conserved charges and discuss the Hodge decomposition theorem in the quantum Hilbert space of states. †

Abelian 2-form gauge theory: special features

“…Section 3 is central of our present paper and is devoted to the discussion of some special features associated with the 4D 2-form Abelian gauge theory. These features are mainly (i) the proof of this theory to be a field theoretical model for the Hodge theory [25], and (ii) its quasi-topological nature. Both these issues are so elegantly intertwined with each-other that first we present some of the key cohomological properties [25] and, then only, we discuss the quasi-topological nature.…”

“…These features are mainly (i) the proof of this theory to be a field theoretical model for the Hodge theory [25], and (ii) its quasi-topological nature. Both these issues are so elegantly intertwined with each-other that first we present some of the key cohomological properties [25] and, then only, we discuss the quasi-topological nature. We compare and contrast some of the decisive properties of the 4D 2-form and 2D one-form Abelian gauge theories § We adopt here the conventions and notations in which the 4D flat Minkowski metric is: η µν = diag (+1, −1, −1, −1) and totally anti-symmetric Levi-Civita tensor (ε µνκζ ) is chosen such that: in section 4.…”

“…We provide here a concise synopsis of our earlier work [25] on the existence of the offshell nilpotent (s …”

“…These results are intimately connected with the proof of the fact that the 4D 2-form gauge theory is a field theoretic model for the Hodge theory. In this work [25], all the de Rham cohomology operators (d, δ, ∆) are identified with some specific symmetry transformations (and their generators) for the Lagrangian density of the theory. It is straightforward to check that all the six symmetries (on-shell as well as off-shell versions) obey the following operator form of the algebra…”

“…where the local expressions for the conserved charges Q r can be found in [25]. For our present discussions, these expressions are not required to be quoted here.…”