Corfu lectures on wall-crossing, multi-centered black holes, and quiver invariants

TL;DR

This paper explains how BPS state indices in N=2 theories change across walls in moduli space using a physical picture of bound states, and introduces a Coulomb branch formula for quiver moduli spaces.

Contribution

It provides a physical derivation of the universal wall-crossing formula and introduces the Coulomb branch formula linking quiver invariants to black hole microstates.

Findings

Derived the wall-crossing formula from a bound state perspective.

Proposed the Coulomb branch formula for quiver moduli spaces.

Connected quiver invariants to single-centered black hole microstates.

Abstract

The BPS state spectrum in four-dimensional gauge theories or string vacua with N=2 supersymmetries is well known to depend on the values of the parameters or moduli at spatial infinity. The BPS index is locally constant, but discontinuous across real codimension-one walls where some of the BPS states decay. By postulating that BPS states are bound states of more elementary constituents carrying their own degrees of freedom and interacting via supersymmetric quantum mechanics, we provide a physically transparent derivation of the universal wall-crossing formula which governs the jump of the index. The same physical picture suggests that at any point in moduli space, the total index can be written as a sum of contributions from all possible bound states of elementary, absolutely stable constituents with the same total charge. For D-brane bound states described by quivers, this `Coulomb branch formula' predicts that the cohomology of quiver moduli spaces is uniquely determined by certain `pure-Higgs' invariants, which are the microscopic analogues of single-centered black holes. These lectures are based on joint work with J. Manschot and A. Sen.