The Algebra & Geometry of Modern Physics

We begin with combinatorial construction establishing the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model (or, in other words, chord diagrams) and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces of Riemann surfaces with holes given by (quasi)polynomials $N_{g,s}(P_1,\dots,P_s)$ where $(P_1,\dots,P_s)\in\mathbb R_+^s$ are (fixed) perimeters of holes. This generating function thus enjoys the topological recursion, and we demonstrate that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition. Using the topological recursion, we find explicit recurrent relations for general $s$-loop means in all genera proving simultaneously that the corresponding simple linear combinations of ancestor invariants are nonnegative integers. Based on recent joint paper with J.E.Andersen, P.Norbury, and R.C.Penner, arXiv: 1501.05867.