Teichm\"uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders $n_o$. We propose the dual graph description of the corresponding Teichm\"uller spaces, construct the Poisson algebra of the Teichm\"uller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations.Comment: 20 pages, notations and many essential typos corrected, most significantly, formulae 2.3, 2.5, proof of Lemmata 2.6 and 4.5. Journal reference is added (published version contains typos

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

The matrix model for hypergeometric Hurwitz numbers

We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. We take a sum over all possible ramifications at other $n-2$ points with the fixed length of the profile at $z_2$ and with the fixed total length of profiles at the remaining $n-3$ points. All these models belong to a class of hypergeometric Hurwitz models thus being tau functions of the Kadomtsev--Petviashvili (KP) hierarchy. In the case described above, we can present the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type \tr M_iM_{i+1}^{-1}. We describe the technique for evaluating spectral curves of such models, which opens the possibility of applying the topological recursion for developing $1/N^2$-expansions of these model. These spectral curves turn out to be of an algebraic type.Comment: 12 pages, 2 figures in LaTeX, contribution to the volume of TMPh celebrating the 75th birthday of A A Slavno