Hermitian vs. Anti-Hermitian 1-Matrix Models and Their Hierarchies

TL;DR

This paper explores the relationships between Hermitian and anti-Hermitian one-matrix models and their associated integrable hierarchies, revealing how different scaling limits lead to various well-known hierarchies like KdV, mKdV, and Zakharov-Shabat.

Contribution

It demonstrates how different matrix models correspond to specific integrable hierarchies and how Virasoro constraints act on their tau-functions, extending understanding of their algebraic structures.

Findings

Hermitian models lead to KdV and mKdV hierarchies.

Anti-Hermitian models in the two-arc sector lead to Zakharov-Shabat hierarchy.

Virasoro constraints act on tau-functions, with mKdV associated with a highest weight state.

Abstract

Building on a recent work of \v C. Crnkovi\'c, M. Douglas and G. Moore, a study of multi-critical multi-cut one-matrix models and their associated $sl(2,C)$ integrable hierarchies, is further pursued. The double scaling limits of hermitian matrix models with different scaling ans\"atze, lead, to the KdV hierarchy, to the modified KdV hierarchy and part of the non-linear Schr\"odinger hierarchy. Instead, the anti-hermitian matrix model, in the two-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 tau-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.