On the M2-Brane Index on Noncommutative Crepant Resolutions

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.

Automated summary

The quantum theory of M2 branes can be studied using the K-theoretic Donaldson-Thomas theory on certain M-theory backgrounds. This relation is extended to noncommutative crepant resolutions, where the classical smooth geometry is replaced by the algebra of paths of a specific quiver. K-theoretic quantities on the quiver representation moduli space can be computed using toric localization and result in rational functions of the toric parameters. We discuss the case of the conifold and certain orbifold singularities.