Gravitational Instantons and Moduli Spaces of Topological 2-form Gravity

TL;DR

This paper develops a topological 4D gravity theory based on anti-self-dual 2-forms, analyzing its moduli spaces for different cosmological constants and computing related indices and partition functions.

Contribution

It introduces a novel topological gravity framework focusing on anti-self-dual forms and explores the structure of moduli spaces in both zero and non-zero cosmological constant scenarios.

Findings

Calculated the index of the elliptic complex for non-zero cosmological constant.

Evaluated the partition function for the moduli space.

Clarified the moduli space structure for zero cosmological constant, linked to Plebansky's heavenly equations.

Abstract

A topological version of four-dimensional (Euclidean) Einstein gravity which we propose regards anti-self-dual 2-forms and an anti-self-dual part of the frame connections as fundamental fields. The theory describes the moduli spaces of conformally self-dual Einstein manifolds for the non-zero cosmological constant case and Einstein-Kahlerian manifold with the vanishing real first Chern class for the zero cosmological constant. In the non-zero cosmological constant case, we evaluate the index of the elliptic complex associated with the moduli space and calculate the partition function. We also clarify the moduli space and its dimension for the zero cosmological constant case which are related to the Plebansky's heavenly equations.