Formula omitted]-gravity anomalies (I). Induced quantum [formula omitted]-gravity

Abstract

For any theory of W N or W ∞ gravity coupled to matter, there are W -gravity anomalies depending only on the W -gravity gauge fields, corresponding to the central charge terms in the current algebra. For left-right symmetric theories, these can be cancelled at the expense of introducing anomalies in symmetries which are higher-spin generalisations of Weyl symmetry. Integrating out the matter fields gives a term in the effective action proportional to Σ s R ( s) □ −1 R ( s) where R ( s) is the curvature scalar for the spin- s gauge field and this is the induced effective action for non-critical quantum W -gravity. In a conformal gauge, this reduces to a generalisation of the Liouville action involging N − 1 scalar fields. In a chiral gauge, there is a Kac-Moody symmetry corresponding to a group G, where G = SL(∞) for W ∞ and G is a group contraction of SL(N) for W N (e.g. ISL(2) for W 3). If the W -gravity symmetries are non-linearly realised, there are in addition anomalies depending on the matter fields. These are discussed in a separate paper.