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

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

1991-09-24

arxiv.org/abs/hep-th/9109046v1

Nucl.Phys.B373:247-280,1992

Physics

High Energy Physics

High Energy Physics - Theory

31 pages

Scientific paper

10.1016/0550-3213(92)90457-M

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.

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