Multicut solutions of the matrix Kontsevich–Penner model

Multicut solutions of the Hermitian one-matrix model parametrized by the recently introduced matrix model [1] with external field and Lagrangian having the form $\,\operatorname {tr}{(\Lambda X\Lambda X)} - \alpha N$ $(\log {(1+X)} -X)$ are considered. A brief review of the model, which describes the discretized moduli space of Riemann surfaces, is given. The general structure of multicut solutions is investigated, and it is shown that there arises an additional symmetry and that $s$ parameters remain free for the $(s+1)$-cut solution. A detailed analysis of the one-cut solution is made. Among other results, all solutions of Kazakov type are reproduced. We also discuss the general form for the two-cut solution which arises as generalization of the string equation to the case of two cuts. The entire treatment is given in the approximation of planar diagrams.