Topological Conformal Gravity in Four Dimensions

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.

35 pages, harvmac, DAMTP R92/42

Cited by in corpus (12)

• Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance
• Topological Sigma-Models in Four Dimensions and Triholomorphic Maps
• Predictivity and Nonrenormalizability
• Einsteinian gravity from a topological action
• Gravitational Instantons and Moduli Spaces of Topological 2-form Gravity
• The Gauss-Bonnet Coupling Constant in Classically Scale-Invariant Gravity
• Self-dual Gravitational Instantons in Conformal Gravity: Conserved Charges and Thermodynamics
• Anomalies in Instanton Calculus
• Dimension of the moduli space and Hamiltonian analysis of BF field theories
• Symmetries and observables in topological gravity
• BRST Approach to Hamiltonian Systems
• Noncommutative Topological Half-flat Gravity