Matrix Models and Geometry of Moduli Spaces

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 N\alpha \tr {\bigl(- \fr 14 \L X\L X -\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 $\cal Z$ of our matrix model expressed in times for d.m.s.: t^\pm_m=\tr \fr{\d^m}{\d\l^m}\fr1{\e^\l-1}. It appears that $\cal Z$ depends only on even times and {\cal Z}[t^\pm_\cdot]=C(\aa N) \e^{\cal A}\e^{F(\{t^{-}_{2n}\}) +F(\{-t^{+}_{2n}\})}, where $F(\{t^\pm_{2n}\})$ is a logarithm of the partition function of the Kontsevich model, $\cal A$ being a quadratic differential operator in \dd{t^\pm_{2n}}.Comment: 40pp., LaTeX, no macros needed, 8 figures in tex

Spectral problem on graphs and L-functions

The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf Q}_p)$. As the homogeneous spaces, they are, in fact, identical to p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the finite subgraph-the reduced graph containing all loops of the generalized tree. We study the spectral problem on these graphs, for which we introduce the notion of spherical functions-eigenfunctions of a discrete Laplace operator acting on the graph. We define the S-matrix and prove its unitarity. We present a proof of the Hashimoto-Bass theorem expressing L-function of any finite (reduced) graph via determinant of a local operator $\Delta(u)$ acting on this graph and relate the S-matrix determinant to this L-function thus obtaining the analogue of the Selberg trace formula. The discrete spectrum points are also determined and classified by the L-function. Numerous examples of L-function calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur

The NBI matrix model of IIB Superstrings

We investigate the NBI matrix model with the potential $X\Lambda+X^{-1}+(2\eta+1)\log X$ recently proposed to describe IIB superstrings. With the proper normalization, using Virasoro constraints, we prove the equivalence of this model and the Kontsevich matrix model for $\eta\ne0$ and find the explicit transformation between the two models.Comment: LaTeX, 11p

Shear coordinate description of the quantised versal unfolding of D_4 singularity

In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingof-Ginzburg Poisson bracket on the affine D_4 cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on the direct sum of three copies of sl_2. We realise the action of the mapping class group by the action of the braid group on the geodesic functions . This action coincides with the procedure of analytic continuation of solutions of the sixth Painlev\'e equation. Finally, we produce the explicit quantisation of the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere and of the braid group action.Comment: 14 pages, 2 picture