Polyhedral surfaces and the Weil–Petersson form

LetMg;N0 denote the Deligne–Mumford compactification of the moduli spaceMg;N0 of N0–pointed Riemann surfaces of genus g, (see Appendix B). It is well–known that the Chern classes fc1.Lk/g introduced in the previous chapter can be used to define the Witten–Kontsevich intersection theory over Mg;N0 . In such a setting it is also possible [9, 20] to characterize various relevant properties of the Weil–Petersson volume of Mg;N0 . Such a connection is rather involved and deeply related to the algebraic-geometrical subtleties of Witten–Kontsevich theory. Thus, it comes as a pleasant surprise that the conical geometry of polyhedral surface allows to explicitly construct a representative of the Weil-Petersson form !WP on the space of polyhedral structures with given conical singularities POLg; N0 .M; f.k/g; A.M//, (to our knowledge this connection first appeared in [4]; a similar property has been proved for ribbon graphs by G. Mondello in the remarkable papers [11, 12], and recently by other authors, see e.g. [6]). In order to construct such a combinatorial representative of !WP we exploit the connection between similarity classes of Euclidean triangles and the triangulations of 3–manifolds by ideal tetrahedra. This is a well–known property in hyperbolic geometry, (see e.g. [3]), that we are going to describe in some detail since it will play a basic role in connecting the quantum geometry of polyhedral surfaces to 3–dimensional manifolds.