Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants

Abstract

Abstract. A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics.We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics c · (πL 2 ). Here we explicitly compute the constant c for a configuration of every type. The constant c is found from a Siegel-Veech formula.To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.

Instantons, fluxons, and open gauge string theory

“…(These agree with the values for low genus computed in [64]). Their general expressions are provided by Theorem 2.…”

“…We will now demonstrate that the double scaling limit of the U (N ) gauge theory on T 2 is in fact, rather remarkably, related to the geometry of some very special moduli spaces [61]- [64], whose "integer lattice" points (and thus their volumes) are "counted" by the strong coupling expansion coefficients of Section 6.1. Let H h be the moduli space of (topological classes of) pairs (Σ, du), where Σ is a compact Riemann surface of genus h and du is a holomorphic one-form on Σ with exactly m = 2h − 2 simple zeroes.…”

The fine structure of the moduli space of abelian differentials in genus 3

“…The possible codimension 1 degenerations within each stratum were analyzed by EskinMasur-Zorich in [3].…”

“…It turns out that they can be given in a somewhat similar spirit as the hyperelliptic strata: some orbifold cover appears as the complement of a locally finite arrangement in a domain and the in principle their fundamental group can be computed. For instance, for H (3,1) resp. H (4) we get the discriminant complement of the root system of type E 7 resp.…”

“…Hence we find: Corollary 2.2 The stratum PH(4) resp. PH (3,1) is an orbifold classifying space for the Artin group of type E 7 resp. E 6 modulo its natural (infinite cyclic) central subgroup.…”