ABCD of Beta Ensembles and Topological Strings

TL;DR

This paper explores the connections between beta-ensembles of eigenvalues associated with classical Lie groups and refined topological string theories, revealing a web of relations through deformations, specializations, and quantum shifts.

Contribution

It generalizes the link between matrix models and topological strings to Bn, Cn, Dn ensembles, and computes partition functions for polynomial potentials beyond the leading order.

Findings

Classical eigenvalue ensembles are interconnected via deformations and quotients.

Partition functions for polynomial potentials are calculated perturbatively in the multi-cut phase.

Relations to refined topological string orientifolds are established.

Abstract

We study beta-ensembles with Bn, Cn, and Dn eigenvalue measure and their relation with refined topological strings. Our results generalize the familiar connections between local topological strings and matrix models leading to An measure, and illustrate that all those classical eigenvalue ensembles, and their topological string counterparts, are related one to another via various deformations and specializations, quantum shifts and discrete quotients. We review the solution of the Gaussian models via Macdonald identities, and interpret them as conifold theories. The interpolation between the various models is plainly apparent in this case. For general polynomial potential, we calculate the partition function in the multi-cut phase in a perturbative fashion, beyond tree-level in the large-N limit. The relation to refined topological string orientifolds on the corresponding local geometry is discussed along the way.