Abstract
We revisit the Virasoro constraints and explore the relation to the Hirota bilinear equations. We furthermore investigate and provide the solution to non-homogeneous Virasoro constraints, namely those coming from matrix models whose domain of integration has boundaries. In particular, we provide the example of Hermitean matrices with positive eigenvalues in which case one can find a solution by induction on the rank of the matrix model.
Highlights
Matrix models have established their physical significance through their appearance in the description of quantum gravity in two dimensions
One can conclude that the moduli space of solutions of Virasoro constraints and that of Hirota relations intersect along a subspace containing the matrix model solutions but the precise relation between the two is still not completely understood
In the present article we studied various versions of Virasoro constraints obtained as Ward identities for Hermitean 1-matrix models with polynomial potential
Summary
Matrix models have established their physical significance through their appearance in the description of quantum gravity in two dimensions They appear when transforming the integral over all possible geometries and topologies to its discrete analogue, in other words the summation over random triangulations of surfaces of arbitrary genus (as reviewed in [1]). One can conclude that the moduli space of solutions of Virasoro constraints and that of Hirota relations intersect along a subspace containing the matrix model solutions but the precise relation between the two is still not completely understood (see [3] for a review) Inspired by this classical story of integrable models, we investigate such phenomena for generalizations of the Hermitean matrix model (HMM). In the Appendices we provide definitions for some special functions, we discuss the case of Virasoro constraints for ABJ-like matrix models and we provide a cohomological description of the constraints in terms of Lie algebra cohomology
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