Matrix Models and Geometry of Moduli Spaces

Abstract

We give the description of discretized moduli spaces (d.m.s.) $\Mcdisc$ introduced in \cite{Ch1} in terms of discrete de Rham cohomologies for moduli spaces $\Mgn$. The generating function for intersection indices (cohomological classes) of d.m.s. is found. Classes of highest degree coincide with the ones for the continuum moduli space $\Mc$. To show it we use a matrix model technique. The Kontsevich matrix model is the generating function in the continuum case, and the matrix model with the potential $Nα\tr {\bigl(- \fr 14 ŁXŁX -\fr12\log (1-X)-\fr12X\bigr)}$ is the one for d.m.s. In the latest case the effects of Deligne--Mumford reductions become relevant, and we use the stratification procedure in order to express integrals over open spaces $\Mdisc$ in terms of intersection indices, which are to be calculated on compactified spaces $\Mcdisc$. We find and solve constraint equations on partition function $\mathcal{Z}$ of our matrix model expressed in times for d.m.s.: $t^\pm_m=\tr \fr{\d^m}{\dł^m}\fr1{\e^ł-1}$. It appears that $\mathcal{Z}$ depends only on even times and ${\cal Z}[t^\pm_\cdot]=C(åN) \e^{\cal A}\e^{F(\{t^{-}_{2n}\}) +F(\{-t^{+}_{2n}\})}$, where $F(\{t_{2n}^{\pm }\})$ is a logarithm of the partition function of the Kontsevich model, $\mathcal{A}$ being a quadratic differential operator in $\dd{t^\pm_{2n}}$.

40pp., LaTeX, no macros needed, 8 figures in text

References in corpus (4)

• Symplectic structure of the moduli space of flat connections on a Riemann surface
• Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
• On Equivalence of Topological and Quantum 2d Gravity
• The Multicritical Kontsevich-Penner Model

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