Consistent anomalies of the induced W gravities

TL;DR

This paper evaluates the BRST anomaly in induced W gravity theories using a geometric approach, providing a covariant formulation and analyzing the specific case of W3 theory.

Contribution

It introduces a geometric framework for calculating BRST anomalies in induced W gravity and compares different anomaly candidates, including the W3 case.

Findings

Calculated BRST anomalies using a geometric approach

Provided a conformally covariant formulation valid on Riemann surfaces

Analyzed the W3 theory and compared with existing anomaly candidates

Abstract

The BRST anomaly which may be present in the induced $W_n$ gravity quantized on the light-cone is evaluated in the geometrical framework of Zucchini. The cocycles linked by the cohomology of the BRST operator to the anomaly are straightforwardly calculated thanks to the analogy between this formulation and the Yang-Mills theory. We give also a conformally covariant formulation of these quantities including the anomaly, which is valid on arbitrary Riemann surfaces. The example of the $W_3$ theory is discussed and a comparison with other candidates for the anomaly available in the literature is presented.