Quantization of anomalous gauge field theory and BRST-invariant models of two-dimensional quantum gravity

We analyze the problems with the so-called gauge-invariant quantization of the anomalous gauge field theories originally due to Faddeev and Shatashvili (FS). Our analysis is a generalization of the FS method, which allows us to construct a series of classically equivalent theories that are nonequivalent at the quantum level. We prove that these classical theories are all consistent with the BRST invariance of the original gauge symmetry with a suitably augmented field content. As an example of such a scenario, we discuss the class of physically distinct models of two-dimensional induced gravity, which are a generalization of the David-Distler-Kawai model.