A mathematical perspective on nonplanar on-shell forms
Abstract
On-shell forms are differential forms on the Grassmannian which arise in particle physics. They are defined using bipartite graphs with n distinguished ``boundary'' vertices. Mathematical investigation of on-shell forms has largely focused on the case where the graph is planar, in which case one can utilize combinatorial tools pioneered by Postnikov in the study of the totally nonnegative Grassmannian. In this article, we investigate on-shell forms for arbitrary graphs. We first discuss how to extend various tools from the planar case to arbitrary graphs. We then prove a determinantal formula for a class of on-shell forms which first appeared in physics literature. We explain the relation between this class of forms and the hypertree divisors of M0,n, introduced by Castravet--Tevelev.
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BibTeX
@misc{lisitsyn2026mathematical,
title = {{A mathematical perspective on nonplanar on-shell forms}},
author = {Artyom Lisitsyn and Elizabeth Pratt and Melissa Sherman-Bennett and Jaroslav Trnka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10041},
eprint = {2610.10041},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10041},
}
RIS
TY - UNPB
TI - A mathematical perspective on nonplanar on-shell forms
AU - Artyom Lisitsyn
AU - Elizabeth Pratt
AU - Melissa Sherman-Bennett
AU - Jaroslav Trnka
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10041
UR - https://arxiv.org/abs/2610.10041
AB - On-shell forms are differential forms on the Grassmannian which arise in particle physics. They are defined using bipartite graphs with n distinguished ``boundary'' vertices. Mathematical investigation of on-shell forms has largely focused on the case where the graph is planar, in which case one can utilize combinatorial tools pioneered by Postnikov in the study of the totally nonnegative Grassmannian. In this article, we investigate on-shell forms for arbitrary graphs. We first discuss how to extend various tools from the planar case to arbitrary graphs. We then prove a determinantal formula for a class of on-shell forms which first appeared in physics literature. We explain the relation between this class of forms and the hypertree divisors of M_0,n, introduced by Castravet--Tevelev.
ER -