Hermitian versus anti-hermitian one-matrix models and their hierarchies

Abstract

Building on a recent work of Č. Crnković, M. Douglas and G. Moore, a study of multi-critical multi-cut one-matrix models and their associated sl(2, C ) integrable hirarchies, is further pursued. The double-scaling limits of hermitian matrix models with different scaling ansätze, lead to the KdV hierarchy, to the modified KdV hierarchy and part of the non-linear Schrödinger hierarchy. Instead, the anti-hermitian matrix model, in the 2-arc sector, results in the Zakharov-Shabat hierarchy, which contains both KdV and mKdV as reductions. For all the hierarchies it is found that the Virasoro constraints act on the associated τ-functions. Whereas it is known that the ZS and KdV models lead to the Virasoro constraints of an sl(2, C ) vacuum, we find that the mKdV model leads to the Virasoro constraints of a highest-weight state with arbitrary conformal dimension.