𝒲-gauge structures and their anomalies: An algebraic approach

Abstract

Starting from flat two-dimensional gauge potentials we propose the notion of 𝒲-gauge structure in terms of a nilpotent Becchi–Rouet–Stora (BRS) differential algebra. The decomposition of the underlying Lie algebra with respect to an SL(2) subalgebra is crucial for the discussion of conformal covariance, in particular the appearance of a projective connection. Different SL(2) embeddings lead to different 𝒲-gauge structures. We present a general soldering procedure which allows one to express zero curvature conditions for the 𝒲-currents in terms of conformally covariant differential operators acting on the 𝒲-gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by 𝒲-currents, gauge fields, and the ghost fields corresponding to 𝒲-diffeomorphisms. As illustrations we treat the cases of SL(2) itself and the two different SL(2) embeddings in SL(3), viz., the 𝒲 (1)3- and 𝒲 (2)3-gauge structures, in some detail. In these cases we determine algebraically 𝒲-anomalies as solutions of the consistency conditions and discuss their Chern–Simons origin.