BGWM as Second Constituent of Complex Matrix Model

TL;DR

The paper demonstrates that four fundamental matrix-model partition functions are intimately connected within the generalized Kontsevich model framework, culminating in decomposition formulas that express complex models as products of simpler building blocks. It identifies the BGW partition function as the second constituent in the complex-matrix model decomposition, Z_C = U_{KBGW}(Z_K ⊗ Z_BGW), and establishes firm links between Hermitian, complex, Kontsevich, and BGW models via Virasoro constraints, determinant representations, and Kontsevich-Penner representations. Utilizing D-module, current-algebra, and spectral-curve formalisms, the work presents explicit Wakimoto-like currents, projection operators, and conjugation mechanisms that realize the decompositions Z_H → Z_K ⊗ Z_K and Z_C → Z_K ⊗ Z_BGW, unifying eigenvalue models under an M-theory of matrix models. The results highlight BGW as a key universal constituent and point toward broader DV-type extensions and check-operator developments, signaling a cohesive, scalable framework for multi-matrix and unitary-matrix ensembles in string/gauge-theory contexts.

Abstract

Earlier 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. However, the less topical complex matrix model appeared to be an exception: its decomposition involved not only the Kontsevich tau-function but also another constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition function. The BGW tau-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).