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

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.