Hodge and Prym Tau Functions, Strebel Differentials and Combinatorial Model of ${\mathcal {M}}_{g,n}$

Abstract

The goal of the paper is to apply the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on Strebel differentials. These line bundles are tensor products of the determinants of the Hodge or Prym vector bundles with the standard tautological line bundles $\mathcal {L}_j$, and the sections are constructed in terms of tau functions. The combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials where the phase of each tau-function provides a section of a circle bundle. The phase of the ratio of the Prym and Hodge tau functions gives a section of the $\kappa _1$-circle bundle. By evaluating the increment of the phase around co-dimension 2 sub-complexes, we identify the Poincaré dual cycles to the Chern classes of the corresponding line bundles: they are expressed explicitly as combination of Witten’s cycle $W_{5}$ and Kontsevich’s boundary. This provides combinatorial analogues of Mumford’s relations on ${\mathcal {M}}_{g,n}$ and Penner’s relations in the hyperbolic combinatorial model. The free homotopy classes of loops around $W_{5}$ are interpreted as pentagon moves while those of loops around Kontsevich’s boundary as combinatorial Dehn twists. Throughout the paper we exploit the classical description of the combinatorial model in terms of Strebel differentials, parametrized in terms of period, or homological coordinates; we show that they provide Darboux coordinates for the symplectic structure introduced by Kontsevich. We also express the latter as the intersection pairing in the odd homology of the canonical double cover.