Refined matrix models from BPS counting

TL;DR

This paper develops matrix and fermion models for refined BPS generating functions in certain toric manifolds, connecting them to refined topological string amplitudes and explicit examples like the refined resolved conifold.

Contribution

It introduces a novel matrix model and free fermion representation for refined BPS functions, extending the understanding of refined topological string theory in non-compact geometries.

Findings

Derived matrix models with unitary measure and modified potentials for refined BPS functions.

Connected matrix models to refined topological string amplitudes in specific geometries.

Provided explicit examples including the refined resolved conifold and MacMahon function.

Abstract

We construct a free fermion and matrix model representation of refined BPS generating functions of D2 and D0 branes bound to a single D6 brane, in a class of toric manifolds without compact four-cycles. In appropriate limit we obtain a matrix model representation of refined topological string amplitudes. We consider a few explicit examples which include a matrix model for the refined resolved conifold, or equivalently five-dimensional U(1) gauge theory, as well as a matrix representation of the refined MacMahon function. Matrix models which we construct have ordinary unitary measure, while their potentials are modified to incorporate the effect of the refinement.