Matrix model tools and geometry of moduli spaces (Q1365514)

This is a survey of the author's results on discrete analogs of the moduli space \(\overline{\mathcal M}_{g,n}\) of compact Riemann surfaces with \(n\) marked points [see \textit{L. Chekhov}, J. Geom. Phys. 12, No. 3, 153-164 (1993; Zbl 0793.14015) and ``Matrix models for moduli spaces'', Commun. Math. Phys. (to appear)]. Recall that the space \(\overline{\mathcal M}_{g,n}\) admits natural line bundles \({\mathcal L}_i\), \(i=1,\ldots,n,\) which define the intersection numbers \(\langle\tau_{d_1}\ldots \tau_{d_n}\rangle=\int_{\overline{\mathcal M}_{g,n}}\prod_{i=1}^nc_1({\mathcal L}_i)^{d_i}\). It was shown by Kontsevich that a certain generating function for the intersection indices with unknowns \(\lambda_1,\ldots,\lambda_n\) can be expressed via the integral \(\int DX\exp(\text{tr } {1\over 2}\Lambda X^2+{1\over 6}X^3)\) over the space of Hermitian \(n\times n\)-matrices with diagonal matrix \(\Lambda= \text{diag}[\lambda_1,\ldots,\lambda_n]\). The main tool in Kontsevich's work was the use of the space \({\mathcal M}_{g,n}^{\text{comb}}\) of equivalence classes of certain connected ribbon graphs with metric. There is a canonical map from \({\mathcal M}_{g,n}\times\mathbb{R}_+^n\to {\mathcal M}_{g,n}^{\text{comb}}\) which assigns to a surface \(C\) and \(n\) positive numbers the critical graph of the canonical Strebel quadratic differential on \(C\). The author introduces a discrete analog of the latter space in which the lengths of edges take values in positive integers. There is an analog of the intersection indices and of a matrix integral expressing the corresponding generating function. The author also describes some relationship between different matrix models.