Ramond from Random: Weil-Petersson Volumes for Super-Riemann surfaces with NS Boundaries and R Punctures

Abstract

The Weil-Petersson (WP) volumes of the (compactified) moduli space of ${N}=1$ supersymmetric Riemann surfaces with Neveu-Schwarz (NS) boundaries are frequently discussed in the literature. Such surfaces can also have marked points called Ramond (R) punctures, where the superconformal structure degenerates. Computing the volumes when these R punctures are included is more challenging for the usual differential and algebraic geometry approaches, and they are therefore less well explored. In particular, the spectral curve describing the inclusion of R punctures is apparently unknown, so far. However, the right random matrix model approach can handle the NS and~R sectors on an equal footing. Such a construction is presented, showing how to use a recently developed technique to readily compute many closed-form formulae for $V^{(2m)}_{g,n}(\{b_i\})$, the WP volumes for genus $g$ with $n$ NS-boundaries of geodesic lengths $b_i$ ($i{=}1,\ldots,n$), and $2m$ R-punctures. Several striking relations between volumes (and subsectors thereof) emerge naturally in this approach. Moreover, the hitherto missing spectral curve is presented, and its use for (re-)deriving the $V^{(2m)}_{g,n}(\{b_i\})$ is demonstrated by using topological recursion.