Generation of matrix models by Ŵ-operators

Abstract

We show that partition functions of various matrix models can be obtained by acting on elementary functions with exponents ofŴ operators. A number of illustrations is given, including the Gaussian Hermitian matrix model, Hermitian model in external field and the Hurwitz-Kontsevitch model, for which we suggest an elegant matrix-model representation. In all these examples, the relevantŴ operators belong to theŴ (3) algebra.

Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

“…The third possibility is to apply the character calculus of [105][106][107][108][109], either to the elliptic Selberg integral itself or to the decomposition formula (1.4). This is what we do in the present paper, in section 5 below.…”

“…Thus, to define Z BGWc (t), it suffices to define the Itzykson-Zuber integral (2.4) in some independent way. Such a possibility is provided by the character calculus [105][106][107][108][109]: for β = 1, the IZ integral can be defined as an expansion…”

“…JHEP03 (2011)102 is of course a β-deformation of the well-known character expansion [105][106][107][108][109] …”

“…(1.23) via Fourier transform Definition (3.7) can be directly used to convert the action of the intertwining operator in (1.4) into a kind of a 2-matrix BGW model. This can be done via the Fourier transform, widely used in the character calculus [105][106][107][108][109]. The basic and most important Fourier relation has the form…”

BGWM as second constituent of complex matrix model

“…A highly non-trivial step here was introduction of "infinite-genus" τ -functions, satisfying the string equations [12] and Virasoro/W -constraints [13]- [21], and it was once again inspired by the study of matrix models. Unfortunately, even the simplest of these τ -functions, associated with Hermitian [3] and Kontsevich [22][23][24][25] matrix models, are not yet systematically studied/tabulated and still can not be included into the special-functions textbookssee [26] for the first attempts in this direction.…”