On the M2-Brane Index on Noncommutative Crepant Resolutions

TL;DR

This paper explores the extension of M2-brane index calculations to noncommutative crepant resolutions, replacing smooth geometries with algebraic structures, and computes related K-theoretic invariants using toric localization.

Contribution

It introduces a framework for studying M2-brane indices on singular spaces via noncommutative resolutions and computes K-theoretic quantities using toric localization techniques.

Findings

Derived rational functions for K-theoretic invariants on quiver moduli spaces

Extended the M2-brane index relation to noncommutative resolutions

Analyzed specific cases like the conifold and orbifold singularities

Abstract

On certain M-theory backgrounds which are a circle fibration over a smooth Calabi-Yau, the quantum theory of M2 branes can be studied in terms of the K-theoretic Donaldson-Thomas theory on the threefold. We extend this relation to noncommutative crepant resolutions. In this case the threefold develops a singularity and classical smooth geometry is replaced by the algebra of paths of a certain quiver. K-theoretic quantities on the quiver representation moduli space can be computed via toric localization and result in certain rational functions of the toric parameters. We discuss in particular the case of the conifold and certain orbifold singularities.