Topological Conformal Gravity in Four Dimensions

TL;DR

This paper develops a new formulation of topological conformal gravity in four dimensions, focusing on deformations of gravitational instantons, and relates it to topological Yang-Mills theory through index theory and BRST quantization.

Contribution

It introduces a novel approach to topological conformal gravity using elliptic complexes and computes its deformations and invariants, extending the analogy with topological Yang-Mills theory.

Findings

Identifies the elliptic complex describing deformations of gravitational instantons.

Calculates the number of deformations using the Atiyah-Singer index theorem.

Shows the quantum theory's partition function parallels Donaldson invariants.

Abstract

In this paper, we present a new formulation of topological conformal gravity in four dimensions. Such a theory was first considered by Witten as a possible gravitational counterpart of topological Yang-Mills theory, but several problems left it incomplete. The key in our approach is to realise a theory which describes deformations of conformally self-dual gravitational instantons. We first identify the appropriate elliptic complex which does precisely this. By applying the Atiyah-Singer index theorem, we calculate the number of independent deformations of a given gravitational instanton which preserve its self-duality. We then quantise topological conformal gravity by BRST gauge-fixing, and discover how the quantum theory is naturally described by the above complex. Indeed, it is a process which closely parallels that of the Yang-Mills theory, and we show how the partition function generates an uncanny gravitational analogue of the first Donaldson invariant.