Continuum versus discrete Virasoro in one-matrix models

Abstract

Virasoro constraints are imposed on the partition function of the one-matrix models at the discrete level before the continuum limit is taken. After a preliminary discussion of the role of the Virasoro constraints and the associated Virasoro group orbit structure in the proof of universality, the interrelation between discrete and continuum Virasoro constraints is considered. As an intermediate step, the model of complex matrices is discussed. An appropriate change of time variables for the continuum limit (Kazakov's or “admissible”) as well as time-dependent rescales of the partition function are introduced in the approaches of orthogonal polynomials and loop equations. Even after these modifications, Virasoro constraints do not possess a nice continuum limit in the model of complex matrices because of additional constant terms in the Virasoro generators. These terms are eliminated in the case of the reduced hermitean matrix model so that the results of Fukuma et al. and Dijkgraaf et al. are rigorously derived.