×

Perturbative series and the moduli space of Riemann surfaces. (English) Zbl 0608.30046

We introduce a combinatorial formalism for a certain cell-decomposition of a principal bundle over the moduli space of Riemann surfaces: cells in the decomposition are indexed by (classes) of ”fatgraphs”, which are defined as graphs (in the usual sense) together with some extra structure. The computation of the ”virtual Euler characteristics” of various moduli spaces is formulated as a counting problem involving fatgraphs, and the counting problem is in turn reformulated analytically using the techniques of perturbative series from particle physics. The original counting problem is finally solved by applying Dirichlet series and orthogonal polynomial methods to the analytic formulation: our solution takes the form of an asymptotic series of an otherwise analytic function.
The combinatorics of fatgraphs developed herein forms the basis for an explicit scheme (pursued elsewhere) of numerical integration over moduli space. Furthermore, the analytical and perturbation techniques of this work should apply more generaly to this integration scheme.

MSC:

30Fxx Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
PDFBibTeX XMLCite
Full Text: DOI