On Unitary/Hermitian Duality in Matrix Models

TL;DR

This paper establishes an exact duality between unitary and hermitian matrix models, enabling the transfer of techniques and results between these models, with applications to gauge theories and topological strings.

Contribution

It demonstrates a precise equivalence between unitary and hermitian matrix models, allowing the use of hermitian model techniques for unitary models and related applications.

Findings

Confirmed the duality reproduces genus-1 topological string amplitudes

Derived special geometry relations in unitary and Chern-Simons matrix models

Enabled transfer of techniques between matrix model types

Abstract

Unitary 1-matrix models are shown to be exactly equivalent to hermitian 1-matrix models coupled to 2N vectors with appropriate potentials, to all orders in the 1/N expansion. This fact allows us to use all the techniques developed and results obtained in hermitian 1-matrix models to investigate unitary as well as other 1-matrix models with the Haar measure on the unitary group. We demonstrate the use of this duality in various examples, including: (1) an explicit confirmation that the unitary matrix formulation of the N=2 pure SU(2) gauge theory correctly reproduces the genus-1 topological string amplitude (2) derivations of the special geometry relations in unitary as well as the Chern-Simons matrix models.