The BRST--antibracket cohomology of $2d$ gravity conformally coupled to scalar matter

TL;DR

This paper thoroughly computes the BRST--antibracket cohomology for 2D Weyl invariant gravity coupled with scalar matter, revealing detailed structures of anomalies, symmetries, and conditions for anomaly absence.

Contribution

It provides a complete classification of classical actions, symmetries, background charges, and anomalies in 2D gravity with scalar matter, including new insights into anomaly conditions.

Findings

Antifield dependence can be removed from anomalies.

Conditions for absence of matter-dependent Weyl anomalies are more general than dilaton equations.

All classical actions and symmetries are explicitly characterized.

Abstract

We compute completely the BRST--antibracket cohomology on local functionals in two-dimensional Weyl invariant gravity for given classical field content (two dimensional metric and scalar matter fields) and gauge symmetries (two dimensional diffeomorphisms and local Weyl transformations). This covers the determination of all classical actions, of all their rigid symmetries, of all background charges and of all candidate gauge anomalies. In particular we show that the antifield dependence can be entirely removed from the anomalies and that, if the target space has isometries, the condition for the absence of matter field dependent Weyl anomalies is more general than the familiar `dilaton equations'.