Matrx Models: a Way to Quantum Moduli Spaces

TL;DR

This paper describes discretized moduli spaces using discrete de Rham cohomologies, demonstrating their intersection indices match those of continuum spaces, and proposes a quantum group structure underlying these models.

Contribution

It introduces a matrix model for discretized moduli spaces and links them to quantum groups, providing a new perspective on their structure and relations to continuum moduli spaces.

Findings

Intersection indices for discretized and continuum moduli spaces coincide.

Matrix models for discretized moduli spaces are explicitly constructed.

Discretized moduli spaces can be represented as cosets of complex tori by symmetry groups.

Abstract

We give the description of discretized moduli spaces (d.m.s.) $\Mcdisc$ introduced in \cite{Ch1} in terms of a discrete de Rham cohomologies for each moduli space $\Mgn$ of a genus $g$, $n$ being the number of punctures. We demonstrate that intersection indices (cohomological classes) calculated for d.m.s. coincide with the ones for the continuum moduli space $\Mc$ compactified by Deligne and Mumford procedure. To show it we use a matrix model technique. The Kontsevich matrix model is a generating function for these indices in the continuum case, and the matrix model with the potential $N\alpha \tr {\bigl(- \fr 12 \L X\L X +\log (1-X)+X\bigr)}$ is the one for d.m.s. In the last case the effects of reductions become relevant, but 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. The coincidence of the cohomological classes for both continuum and d.m.s. models enables us to propose the existence of a quantum group structure on d.m.s. Then d.m.s. are nothing but cyclic (exceptional) representations of a quantum group related to a moduli space $\Mc$. Considering the explicit expressions for integrals of Chern classes over $\Mc$ and $\Mcdisc$ we conjecture that each moduli space $\Mc$ in the Kontsevich parametrization can be presented as a coset $\Mc ={\bf T}^d/G$, $d=3g-3+n$, where ${\bf T}^d$ is some $d$--dimensional complex torus and $G$ is a finite order symmetry group of ${\bf T}^d$.