N=1 Supersymmetry, Weil-Petersson Volume Recursion, and a Spectral Curve
Abstract
The Stanford-Witten-Norbury generalization of Mirzakhani's volume recursion computes V(2m)g,n({bi}), the Weil-Petersson volumes of the moduli space of N=1 supersymmetric Riemann surfaces of genus g with n Neveu-Schwarz boundaries of geodesic lengths bi (i=1,…,n), and 2m Ramond punctures. Recently, a spectral curve has been derived that allows their Laplace transforms W(2m)g,n({ẑi}) to be computed using topological recursion. We prove that the Stanford-Witten-Norbury volume recursion is directly derivable from the spectral curve. An alternative volume recursion can also be derived from it. The difference comes from whether the Ramond information is in the initial data, or in the volume recursion's kernels. The latter invites a geometrical understanding.
Related
- Scalar curvature, volume, and Weil-Petersson translation length
- Two Recursions for the Seiberg-Witten Prepotential: Zamolodchikov's Recursion, the AGT Correspondence and Topological Recursion on the Seiberg-Witten Curve
- Comparison between renormalized volume and area in the Weil--Petersson topology
- Spectral gaps near one quarter for Weil--Petersson random surfaces with growing cusp number
- q-deformed Weil-Petersson Volumes from the Spectral Edge of DSSYK and Supersymmetric Extensions
BibTeX
@misc{johnson2026supersymmetry,
title = {{$N=1$ Supersymmetry, Weil-Petersson Volume Recursion, and a Spectral Curve}},
author = {Clifford V. Johnson},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.20796},
eprint = {2606.20796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.20796},
}
RIS
TY - UNPB
TI - N=1 Supersymmetry, Weil-Petersson Volume Recursion, and a Spectral Curve
AU - Clifford V. Johnson
PY - 2026
DA - 2026/06/18
PB - arXiv
DO - 10.48550/arxiv.2606.20796
UR - https://arxiv.org/abs/2606.20796
AB - The Stanford-Witten-Norbury generalization of Mirzakhani's volume recursion computes V^(2m)_g,n({b_i}), the Weil-Petersson volumes of the moduli space of N=1 supersymmetric Riemann surfaces of genus g with n Neveu-Schwarz boundaries of geodesic lengths b_i (i=1,…,n), and 2m Ramond punctures. Recently, a spectral curve has been derived that allows their Laplace transforms W^(2m)_g,n({ẑ_i}) to be computed using topological recursion. We prove that the Stanford-Witten-Norbury volume recursion is directly derivable from the spectral curve. An alternative volume recursion can also be derived from it. The difference comes from whether the Ramond information is in the initial data, or in the volume recursion's kernels. The latter invites a geometrical understanding.
ER -