Wall Crossing, Quivers and Crystals

TL;DR

This paper explores the spectrum of BPS D-branes on Calabi-Yau manifolds using quiver gauge theories, linking wall crossing phenomena, crystal melting models, and enumerative geometry to understand BPS degeneracies and their transformations.

Contribution

It establishes a connection between Seiberg dualities, wall crossing, and crystal melting models, providing a geometric interpretation of BPS state counting and proving the Kontsevich-Soibelman formula in this context.

Findings

BPS degeneracies are counted by melting crystal configurations.

Seiberg dualities correspond to wall crossing in the crystal model.

The crystal model reproduces the Gromov-Witten/Donaldson-Thomas correspondence.

Abstract

We study the spectrum of BPS D-branes on a Calabi-Yau manifold using the 0+1 dimensional quiver gauge theory that describes the dynamics of the branes at low energies. The results of Kontsevich and Soibelman predict how the degeneracies change. We argue that Seiberg dualities of the quiver gauge theories, which change the basis of BPS states, correspond to crossing the "walls of the second kind." There is a large class of examples, including local del Pezzo surfaces, where the BPS degeneracies of quivers corresponding to one D6 brane bound to arbitrary numbers of D4, D2 and D0 branes are counted by melting crystal configurations. We show that the melting crystals that arise are a discretization of the Calabi-Yau geometry. The shape of the crystal is determined by the Calabi-Yau geometry and the background B-field, and its microscopic structure by the quiver Q. We prove that the BPS degeneracies computed from Q and Q' are related by the Kontsevich Soibelman formula, using a geometric realization of the Seiberg duality in the crystal. We also show that, in the limit of infinite B-field, the combinatorics of crystals arising from the quivers becomes that of the topological vertex. We thus re-derive the Gromov-Witten/Donaldson-Thomas correspondence.