A boundary-addition identity for Weil-Petersson volumes and their supersymmetric generalizations
Abstract
We consider Vg,n({bi}), the Weil-Petersson volumes of the moduli space of Riemann surfaces of genus g with n geodesic boundaries of lengths bi, (i=1,...,n), and various N≥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 Vg,n+1({b1,b2,...,bn,b}), for any b, to the corresponding n-boundary KdV data underlying Vg,n({b1,b2,...,bn}). For ordinary Weil-Petersson volumes, expansion about the special removable-cone value b=2πi 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 κ-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.
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BibTeX
@misc{johnson2026boundaryaddition,
title = {{A boundary-addition identity for Weil-Petersson volumes and their supersymmetric generalizations}},
author = {Clifford V. Johnson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21030},
eprint = {2609.21030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21030},
}
RIS
TY - UNPB
TI - A boundary-addition identity for Weil-Petersson volumes and their supersymmetric generalizations
AU - Clifford V. Johnson
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21030
UR - https://arxiv.org/abs/2609.21030
AB - 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≥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=2πi 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 κ-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.
ER -