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

Abstract

Building on a recent work ofČ. Crnković, M. Douglas and G. Moore, a study of multicritical 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 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 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.

Double scaling limits in gauge theories and matrix models

“…Other reductions, as well as the relationship between the partition function and the tau-functions of the hierarchies, have been discussed in [57]. It is straightforward to check that the free energy scales exactly as in (2.11) with n = 2.…”

“…Remarkably, the corresponding double scaling limits are described in terms of several integrable hierarchies associated to sl(2, C). Following [57], the results can be summarized as follows. First, consider the case of real potentials.…”

“…First, consider the case of real potentials. In the saddle point approximation, their critical behaviours correspond to configurations where an odd number of double points come together with two branch points [54,57]. Their double scaling limits are described in terms of solutions of the non-linear Schrödinger (NLS) hierarchy singled out by string equations associated only to the "even" flows of the hierarchy.…”

“…However, these are not the only two-cut models that have been studied using orthogonal polynomials. Two-cut Hermitian matrix models with both even and odd terms in the potential were considered in [54,56,57]. Remarkably, the corresponding double scaling limits are described in terms of several integrable hierarchies associated to sl(2, C).…”

“…However, as pointed out in [57], the most general critical behaviour, where an arbitrary number of double points collide with two branch points, is obtained by considering complex potentials. Namely, the potential has to be taken such that W * (x) = W (−x), which makes the coefficients σ n pure imaginary.…”