Noncommutative Topological Theories of Gravity

TL;DR

This paper explores the formulation of noncommutative topological gravity using the Seiberg-Witten map, aiming to extend topological invariants and instanton concepts into noncommutative geometry.

Contribution

It introduces a novel approach to noncommutative topological gravity based on SL(2,C) connections, connecting it with classical invariants like Euler characteristic and signature.

Findings

Construction of noncommutative topological gravity via Seiberg-Witten map

Derivation of noncommutative Euler characteristic and signature

Potential framework for noncommutative gravitational instantons

Abstract

The possibility of noncommutative topological gravity arising in the same manner as Yang-Mills theory is explored. We use the Seiberg-Witten map to construct such a theory based on a SL(2,C) complex connection, from which the Euler characteristic and the signature invariant are obtained. This gives us a way towards the description of noncommutative gravitational instantons as well as noncommutative local gravitational anomalies.