Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
S. Kharchev, A. Marshakov, A. Mironov, A. Morozov
TL;DR
The paper develops a group-theoretic framework for the Generalized Kazakov-Migdal-Kontsevich Model (G(KM)^2), showing that the model can be expanded in GL group characters and interpreted as a sum over representations in the weight space, with discrete sums replacing integrals for unitary groups. In the D=0 limit, the model reduces to a Generalized Kontsevich integral, which yields a Toda-lattice τ-function and a simple Grassmannian point in the Kontsevich phase, while a steepest-descent regime leads to a more intricate Grassmannian element in a distinct Kontsevich phase. The work develops determinant (fermionic) representations of τ-functions, elucidates singular KP τ-functions associated with GL characters, and presents a bilinear character construction that connects to known τ-functions such as the Itzykson-Zuber formula. It also discusses vertex-operator sewing and topology-changing operations, their relation to 2d Yang-Mills theory, and the dichotomy between character and Kontsevich phases in both continuous and discretized (unitary) Kontsevich models. The study highlights open questions about the proper group-theoretic interpretation of integrals over moduli and identifies multiple directions for future research in the interplay between representation theory, integrable hierarchies, and matrix-model phases.
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 {\it model} (the space of all highest weight representations) of $GL$. In the case of compact unitary groups the integrals should be substituted by {\it discrete} sums over weight lattice. The $D=0$ version of the model is the Generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with partition function of the $2d$ Yang-Mills theory with the target space of genus $g=0$ and $m=0,1,2$ holes. This particular quantity is always a bilinear combination of characters and appears to be a Toda-lattice $τ$-function. (This is generalization of the classical statement that individual $GL$ characters are always singular KP $τ$-functions.) The corresponding element of the Universal Grassmannian is very simple and somewhat similar to the one, arising in investigations of the $c=1$ string models. However, under certain circumstances the formal sum over representations should be evaluated by steepest descent method and this procedure leads to some more complicated elements of Grassmannian. This "Kontsevich phase" as opposed to the simple "character phase" deserves further investigation.