BGWM as Second Constituent of Complex Matrix Model

In [1] 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 Brezin-Gross-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 ofX in the cubic Kontsevich model).