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({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.
Related
- Eynard-Orantin Topological Recursions for Higher Weil-Petersson Volumes on Moduli Spaces of Super Riemann Surfaces
- The proper geometric dimension of the mapping class groups of non-orientable surfaces with punctures
- A discrete analogue of 𝒩=2 super Weil-Petersson volumes
- q-deformed Weil-Petersson Volumes from the Spectral Edge of DSSYK and Supersymmetric Extensions
- A boundary-addition identity for Weil-Petersson volumes and their supersymmetric generalizations
BibTeX
@misc{johnson2026ramond,
title = {{Ramond from Random: Weil-Petersson Volumes for Super-Riemann surfaces with NS Boundaries and R Punctures}},
author = {Clifford V. Johnson},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.09990},
eprint = {2606.09990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.09990},
}
RIS
TY - UNPB
TI - Ramond from Random: Weil-Petersson Volumes for Super-Riemann surfaces with NS Boundaries and R Punctures
AU - Clifford V. Johnson
PY - 2026
DA - 2026/06/08
PB - arXiv
DO - 10.48550/arxiv.2606.09990
UR - https://arxiv.org/abs/2606.09990
AB - 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,…,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.
ER -