Generalised Moonshine and Holomorphic Orbifolds

TL;DR

This paper explores the role of third cohomology in holomorphic orbifolds and applies these ideas to Mathieu moonshine, revealing a complete classification of twisted twining genera linked to M_24.

Contribution

It introduces a cohomological framework for understanding holomorphic orbifolds and provides a complete classification of twisted twining genera in Mathieu moonshine.

Findings

Identifies the role of H^3(G, U(1)) in orbifold constructions

Classifies all twisted twining genera for Mathieu moonshine

Connects modular properties to cohomology classes

Abstract

Generalised moonshine is reviewed from the point of view of holomorphic orbifolds, putting special emphasis on the role of the third cohomology group H^3(G, U(1)) in characterising consistent constructions. These ideas are then applied to the case of Mathieu moonshine, i.e. the recently discovered connection between the largest Mathieu group M_24 and the elliptic genus of K3. In particular, we find a complete list of twisted twining genera whose modular properties are controlled by a class in H^3(M_24, U(1)), as expected from general orbifold considerations.