2026/06/08 by Clifford V. Johnson · 1 voice
Physics and Astronomy · #hep-th #math-ph
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(\bi\), the WP volumes for genus g with n NS-boundaries of geodesic lengths bi (i=1,…,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(\bi\) is demonstrated by using topological recursion.