The Coupling of Chern-Simons Theory to Topological Gravity

TL;DR

This paper explores the coupling of Chern-Simons theory to topological gravity, deriving BRST and BV formalisms, and analyzing anomalies and their cancellation via topological string couplings.

Contribution

It introduces new couplings to the super-ghost in the BV action and investigates topological anomalies in higher ghost deformations of SU(N) Chern-Simons theory.

Findings

Derived BRST rules and BV action for coupled theory

Identified topological anomalies in higher ghost deformations

Proposed anomaly cancellation via topological string couplings

Abstract

We couple Chern-Simons gauge theory to 3-dimensional topological gravity with the aim of investigating its quantum topological invariance. We derive the relevant BRST rules and Batalin-Vilkovisky action. Standard BRST transformations of the gauge field are modified by terms involving both its anti-field and the super-ghost of topological gravity. Beyond the obvious couplings to the metric and the gravitino, the BV action includes hitherto neglected couplings to the super-ghost. We use this result to determine the topological anomalies of certain higher ghost deformations of SU(N) Chern-Simons theory, introduced years ago by Witten. In the context of topological strings these anomalies, which generalize the familiar framing anomaly, are expected to be cancelled by couplings of the closed string sector. We show that such couplings are obtained by dressing the closed string field with topological gravity observables.