A boundary-addition identity for Weil-Petersson volumes and their supersymmetric generalizations

Abstract

We consider $V_{g,n}(\{b_i\})$, the Weil-Petersson volumes of the moduli space of Riemann surfaces of genus $g$ with $n$ geodesic boundaries of lengths $b_i$, $(i=1,...,n)$, and various ${ N}{\geq}1$ supersymmetric generalizations of them, using a framework within which they are specializations of a larger family of quantities naturally governed by the integrable KdV hierarchy. An exact boundary-addition identity is derived that directly relates $V_{g,n+1}(\{b_1,b_2,...,b_n,b\})$, for any $b$, to the corresponding $n$-boundary KdV data underlying $V_{g,n}(\{b_1,b_2,...,b_n\})$. For ordinary Weil-Petersson volumes, expansion about the special removable-cone value $b=2\pi i$ reproduces three known results of Do and Norbury, organizing them as the first levels of a complete hierarchy. The higher levels are naturally built from additional KdV data, which we interpret geometrically in intersection theory as a tower of $\kappa$-decorated volumes. Several other recursive relations in the literature are illuminated by the identity, and various new ones are derived. The cases with ${ N}=1,2$ and (small) ${N}=4$ supersymmetry are also covered by the identity, and we exhibit some of the striking special features and results arising in each case. Among many applications presented, we use the identity to uncover the intersection theory descriptions of ${N}=2$ and small ${ N}=4$ Weil-Petersson volumes.