Generalized Kazakov-Migdal-Kontsevich Model: Group Theory Aspects

Abstract

The Kazakov-Migdal model, if considered as a functional of external fields, can be always represented as an expansion over characters of GL group. The integration over "matter fields" can be interpreted as going over the model (the space of all highest weight representations) of GL. In the case of compact unitary groups the integrals should be substituted by discrete sums over weight lattice. The D = 0 version of the model is the Generalized Kontsevich integral, which in the above-

On KP-integrable Hurwitz functions

“…Hurwitz τ -function [5][6][7][8] is a new important subject of theoretical physics, which seems relevant to description of non-perturbative phenomena beyond 2d conformal field theory, actually beginning from the 3d Chern-Simons and knot theory, see [9][10][11]. In general, Hurwitz τ -functions do not belong [7,8] to a narrower well-studied class of KP/Toda τ -functions, i.e.…”

“…In general, Hurwitz τ -functions do not belong [7,8] to a narrower well-studied class of KP/Toda τ -functions, i.e. are not straightforwardly reducible to free fermions ( U(1) Kac-Moody algebras) and Plucker relations (the Universal Grassmannian).…”

Matrix models for random partitions

“…As far as concerns integrability: partition function (7) for infinite N is known [36] to be the Todalattice tau-function with times t,t. Both for the second Casimir (90) and for all Casimirs coupled with Miwa variables (136) this integrability is obvious from the matrix model representation for arbitrary finite N ; and N plays the role of discrete Toda time (this can be also derived directly for the sums over partitions from the considerations in [36]). Integrability of another type, namely Toda-chain integrability in times s k for t k =t k = δ k,1 [7][8][9]11], is by no means obvious from our matrix model representations.…”

BGWM as second constituent of complex matrix model

“…These determinant representations are very important, because they are typical for the tau-functions of integrable hierarchies [24,40] -the generalized characters of Lie algebras [11,39]. In other words, partition functions of the matrix models are always the tau-functions [7].…”

“…This equation has its origin in decomposition rule R × R ′ = I R I for representations of Lie algebras and this is an equation for the characters of the algebra [11,39]. For loop algebras the characters can be rather non-trivial, they can be actually labeled by some auxiliary (spectral) Riemann surfaces or, better, by the points of an infinite-dimensional Grassmannian (the universal moduli space of [41]) -what means that the spectral surface can actually have an infinite genus (and this is typically the case for the matrix-model partition functions).…”