BRST and Superfield Formalism—A Review

Abstract

This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the N=1 super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well.

Superfield Approaches to a Model of Bosonic String: Curci-Ferrari-Type Restrictions

“…These 2D restrictions are the limiting case of the Ddimensional diffeomorphism invarant theory where the superfield approach (developed by us [13,18]) leads to the existence of the D-dimensional CF-type restrictions…”

“…0 is universal, and so far, their presence has been shown explicitly in the cases of 2D and 1D diffeomorphism invariant theories (see [14,17] for details). Within the ambit of MBTSA, it becomes evident that we have to take, at least, the help of the (anti-)chiral superfield expansions (compare equation ( 26)) so that we can obtain s b C μ = i B μ and s ab C μ = i B μ for the D-dimensional diffeomorphism invariant theory (see [13,18] for details) in addition to the validity of off-shell nilpotency property so that we can obtain:…”

“…According to the basic tenets of MBTSA to BRST formalism, first of all, we generalize the 2D infinitesimal diffeomorphism transformations ξ a ⟶ ξ ′ a = g a ðξÞ = ξ a − ε a ðξÞ to its counterpart onto the ð2, 2Þ-dimensional supermanifold as follows (see [13,18] for details):…”

“…This is what has been called as the horizontality condition (HC) in [13,18]. This HC (compare equation ( 14)) amounts to setting the coefficients of θ, θ and θ θ in expression (13) equal to zero.…”

“…In this section, we exploit the theoretical strength of ACSA to BRST formalism (see [15] and the reference therein) to derive all the (anti-)BRST symmetry transformations ( 6) and ( 5) except such transformations for the target space coordinates X μ and ðdet gÞ which have already been derived in the previous section by using MBTSA to BRST formalism [13,18]. We are inspired to use, in our present section, ACSA to BRST formalism because of our observations in equation ( 26).…”