Wall-crossing, Black holes, and Quivers

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.

Pure-Higgs states from the Lefschetz-Sommese theorem

“…Quiver quantum mechanics [1] provides a powerful effective description of BPS states in N = 2 gauge theory and type II string/supergravity theory, see for example [2,3] for reviews. The BPS states correspond to the supersymmetric groundstates of the quiver theory and depending on the coupling can be described geometrically in terms of the Coulomb or Higgs branch [1].…”

“…There are no arrows between pairs of non-abelian nodes, but each of the non-abelian nodes is connected in a symmetric fashion to the two abelian nodes, by κ i arrows respectively, oriented in such a way that a separate loop is formed along the 2 abelian nodes and each one of the non-abelian nodes. 3 See figure 1 for the diagram representing a 2+n node quiver.…”