Discrete and Continuum Virasoro Constraints in Two-Cut Hermitian Matrix Models

TL;DR

This paper derives continuum Virasoro constraints from discrete Ward identities in two-cut Hermitian matrix models, linking the $GL( olinebreak ext{infinity})$ Toda hierarchy to the nonlinear Schrödinger hierarchy, and explains the quantization of an integration constant.

Contribution

It introduces a novel derivation of continuum Virasoro constraints from discrete identities via hierarchy mappings and clarifies the quantization of an integration constant.

Findings

Continuum Virasoro constraints are derived from discrete Ward identities.

The invariance of the string equation under NLS flows is established.

The quantization of the integration constant is explained through analyticity.

Abstract

Continuum Virasoro constraints in the two-cut hermitian matrix models are derived from the discrete Ward identities by means of the mapping from the $GL(\infty )$ Toda hierarchy to the nonlinear Schr\"odinger (NLS) hierarchy. The invariance of the string equation under the NLS flows is worked out. Also the quantization of the integration constant $\alpha$ reported by Hollowood et al. is explained by the analyticity of the continuum limit.