On Framed Quivers, BPS Invariants and Defects

TL;DR

This paper reviews how framed quivers are used to study BPS invariants in non-compact Calabi-Yau threefolds, connecting them to generalized instanton problems and exploring the impact of defects on Donaldson-Thomas invariants.

Contribution

It provides a comprehensive overview of the application of framed quivers to BPS invariants and introduces the role of defects in modifying Donaldson-Thomas problems.

Findings

Framed quivers describe moduli spaces of BPS states in Calabi-Yau threefolds.

Defects lead to a modified Donaldson-Thomas counting problem.

Examples include affine space and noncommutative resolutions.

Abstract

In this note we review some of the uses of framed quivers to study BPS invariants of Donaldson-Thomas type. We will mostly focus on non-compact Calabi-Yau threefolds. In certain cases the study of these invariants can be approached as a generalized instanton problem in a six dimensional cohomological Yang-Mills theory. One can construct a quantum mechanics model based on a certain framed quiver which locally describes the theory around a generalized instanton solution. The problem is then reduced to the study of the moduli spaces of representations of these quivers. Examples include the affine space and noncommutative crepant resolutions of orbifold singularities. In the second part of the survey we introduce the concepts of defects in physics and argue with a few examples that they give rise to a modified Donaldson-Thomas problem. We mostly focus on divisor defects in six dimensional Yang-Mills theory and their relation with the moduli spaces of parabolic sheaves. In certain cases also this problem can be reformulated in terms of framed quivers.