Geometric recursion

Abstract

We propose a general theory for constructing functorial assignments $\Sigma \longmapsto \Omega_{\Sigma} \in E(\Sigma)$ for a large class of functors $E$ from a certain category of bordered surfaces to a suitable target category of topological vector spaces. The construction proceeds by successive excisions of homotopy classes of embedded pairs of pants, and thus by induction on the Euler characteristic. We provide sufficient conditions to guarantee the infinite sums appearing in this construction converge. In particular, we can generate mapping class group invariant vectors $\Omega_{\Sigma} \in E(\Sigma)$. The initial data for the recursion encode the cases when $\Sigma$ is a pair of pants or a torus with one boundary, as well as the "recursion kernels" used for glueing. We give this construction the name of Geometric Recursion. As a first application, we demonstrate that our formalism produce a large class of measurable functions on the moduli space of bordered Riemann surfaces. Under certain conditions, the functions produced by the geometric recursion can be integrated with respect to the Weil--Petersson measure on moduli spaces with fixed boundary lengths, and we show that the integrals satisfy a topological recursion generalizing the one of Eynard and Orantin. We establish a generalization of Mirzakhani--McShane identities, namely that multiplicative statistics of hyperbolic lengths of multicurves can be computed by the geometric recursion. As a corollary, we show that the systole function can be obtained from the geometric recursion. The theory has however a wider scope than functions on Teichm\"uller space, which will be explored in subsequent papers; one expects that many functorial objects in low-dimensional geometry could be constructed by variants of this geometric recursion.