{$\mathcal{W}$}-Gauge Structures and their Anomalies:An Algebraic Approach

Abstract

Starting from flat two-dimensional gauge potentials we propose the notion of $\mathcal{W}$-gauge structure in terms of a nilpotent BRS differential algebra. The decomposition of the underlying Lie algebra with respect to an $SL(2)$ subalgebra is crucial for the discussion conformal covariance, in particular the appearance of a projective connection. Different $SL(2)$ embeddings lead to various $\mathcal{W}$-gauge structures. We present a general soldering procedure which allows to express zero curvature conditions for the $\mathcal{W}$-currents in terms of conformally covariant differential operators acting on the $\mathcal{W}$ gauge fields and to obtain, at the same time, the complete nilpotent BRS differential algebra generated by $\mathcal{W}$-currents, gauge fields and the ghost fields corresponding to $\mathcal{W}$-diffeomorphisms. As illustrations we treat the cases of $SL(2)$ itself and to the two different $SL(2)$ embeddings in $SL(3)$, {\it viz.} the ${\mathcal{W}}_3^{(1)}$- and ${\mathcal{W}}_3^{(2)}$-gauge structures, in some detail. In these cases we determine algebraically $\mathcal{W}$-anomalies as solutions of the consistency conditions and discuss their Chern-Simons origin.

46 pages,LaTeX

References in corpus (1)

• Deformation Theory of Holomorphic Vector Bundles, Extended Conformal Symmetry and Extensions of 2D Gravity

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