Recursion between Mumford volumes of moduli spaces
Bertrand Eynard
TL;DR
The paper addresses computing volumes of moduli spaces under general Mumford $\kappa$-class measures, extending Mirzakhani's recursion beyond the Weil-Petersson case.It leverages Kontsevich's matrix integral and its loop equations (EOFg), reframing recursions as expectation values $\left< M_{i_1,i_1}\dots M_{i_n,i_n}\right>_c^{(g)}$ encoded by ribbon-graph decompositions and Laplace transforms to volumes $V_{g,n}$.A central result is a general theorem expressing $V_{g,n}$ in terms of conjugated Kontsevich times $\tilde{t}_b$ and sums over Mumford's $\kappa_b$-classes, recovering the Weil-Petersson recursions and admitting other specializations (e.g., discrete maps, $\kappa_2$).The framework exhibits integrability and Virasoro constraints, suggesting broad applicability to varied spectral curves and promising connections to spin structures and stable maps in 2D quantum gravity contexts.
Abstract
We propose a new proof, as well as a generalization of Mirzakhani's recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich's integral, i.e. we relate them to a Ribbon graph decomposition of Riemann surfaces. We find a generalization of Mirzakhani's recursions to measures containing all higher Mumford's kappa classes, and not only kappa1 as in the Weil-Petersson case.