Branes, quivers and wave-functions

TL;DR

This paper explores how brane partition functions in toric geometries can be viewed as quiver generating series and wave-functions, revealing their transformation properties and confirming BPS state integrality.

Contribution

It establishes a correspondence between brane partition functions, quiver series, and wave-functions, and analyzes their transformations under geometric operations.

Findings

Partition functions can be interpreted as quiver generating series.

Transformations of branes correspond to $SL(2,\

) transformations and quiver operations.

Abstract

We consider a large class of branes in toric strip geometries, both non-periodic and periodic ones. For a fixed background geometry we show that partition functions for such branes can be reinterpreted, on one hand, as quiver generating series, and on the other hand as wave-functions in various polarizations. We determine operations on quivers, as well as $SL(2,\mathbb{Z})$ transformations, which correspond to changing positions of these branes. Our results prove integrality of BPS multiplicities associated to this class of branes, reveal how they transform under changes of polarization, and imply all other properties of brane amplitudes that follow from the relation to quivers.