Defects in Cohomological Gauge Theory and Donaldson-Thomas Invariants

TL;DR

This paper introduces a new class of defects in six-dimensional cohomological gauge theory related to Donaldson-Thomas invariants, generalizing surface defects and enabling explicit computations via quiver representations.

Contribution

It defines defects associated with divisors in cohomological gauge theory, generalizes Donaldson-Thomas invariants, and connects them to parabolic sheaves and quiver representations for explicit calculations.

Findings

Defined boundary conditions for gauge fields with defects.

Proposed a generalized Donaldson-Thomas invariant.

Explicit computation of invariants via quiver representation theory.

Abstract

Donaldson-Thomas theory on a Calabi-Yau can be described in terms of a certain six-dimensional cohomological gauge theory. We introduce a certain class of defects in this gauge theory which generalize surface defects in four dimensions. These defects are associated with divisors and are defined by prescribing certain boundary conditions for the gauge fields. We discuss generalized instanton moduli spaces when the theory is defined with a defect and propose a generalization of Donaldson-Thomas invariants. These invariants arise by studying torsion free coherent sheaves on Calabi-Yau varieties with a certain parabolic structure along a divisor, determined by the defect. We discuss the case of the affine space as a concrete example. In this case the moduli space of parabolic sheaves admits an alternative description in terms of the representation theory of a certain quiver. The latter can be used to compute the invariants explicitly via equivariant localization. We also briefly discuss extensions of our work to other higher dimensional field theories.