BGWM as second constituent of complex matrix model

Abstract

BGWM as second constituent of complex matrix modelTo cite this article: A. Alexandrov et al JHEP12(2009) we explained that partition functions of various matrix models can be constructed from that of the cubic Kontsevich model, which, therefore, becomes a basic elementary building block in "M-theory" of matrix models [2]. However, the less topical complex matrix model appeared to be an exception: its decomposition involved not only the Kontsevich τ -function but also another constituent, which we now identify as the BrezinGross-Witten (BGW) partition function. The BGW τ -function can be represented either as a generating function of all unitary-matrix integrals or as a Kontsevich-Penner model with potential 1/X (instead of X 3 in the cubic Kontsevich model).

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

“…As the first (simplest) part of solution to this problem, in this paper we demonstrate that Z (0) * , the PGL of the Selberg partition function Z (0) S is actually the partition function of the (β-deformed) celebrated BGW model [98][99][100][101] of size n and in the character phase [101]:…”

“…the DotsenkoFateev-like β-ensemble) representations are already known both for B (0) [18][19][20][21][22][23][24]70] and for B (1) [64,65]. The first one is represented [53] as an AMM decomposition [90,91,98] into two spherical Selberg integrals…”

“…This is just an algebraic exercise, which still may be not too much transparent. A more conceptual way may be the method of section 4, where one instead takes the PGL of the Virasoro constraints (Ward identities) for the spherical Selberg model, and then shows that they coincide with the known Virasoro constraints [98,101] for the character phase of the BGW β-ensemble.…”

“…A promising approach to origins of the AGT relations is through their reformulation as relations between matrix models [76][77][78][79] and Seiberg-Witten theory [80][81][82][83], see [84,85] for a concise review of this idea (which is a new application of the topological recursion [86][87][88][89]- [98]). …”

Checks of integrality properties in topological strings

“…Further localization ideas a la [89][90][91][92][93][94][95][96] can convert these averages into a finite-dimensional matrix model integral satisfying the AMM/EO topological recursion [97][98][99][100]. This has been achieved for the torus knots [101,102].…”