Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory

TL;DR

This paper explores the connection between Donaldson-Thomas invariants of Calabi-Yau threefolds and a six-dimensional topological gauge theory, using localization and noncommutative instantons classified by colored Young diagrams.

Contribution

It introduces a geometric description of noncommutative instantons via stable sheaves and develops a matrix model to compute BPS state indices in this context.

Findings

Partition function localizes on noncommutative instantons

Instantons classified by N-colored three-dimensional Young diagrams

A new matrix model computes BPS state indices

Abstract

We study the relation between Donaldson-Thomas theory of Calabi-Yau threefolds and a six-dimensional topological Yang-Mills theory. Our main example is the topological U(N) gauge theory on flat space in its Coulomb branch. To evaluate its partition function we use equivariant localization techniques on its noncommutative deformation. As a result the gauge theory localizes on noncommutative instantons which can be classified in terms of N-coloured three-dimensional Young diagrams. We give to these noncommutative instantons a geometrical description in terms of certain stable framed coherent sheaves on projective space by using a higher-dimensional generalization of the ADHM formalism. From this formalism we construct a topological matrix quantum mechanics which computes an index of BPS states and provides an alternative approach to the six-dimensional gauge theory.