Landau singularities and convex geometry
Abstract
We revisit the classic 1959 paper of L. D. Landau on possible singularities of the integral associated to a Feynman graph. Focusing on the real setup, we relate it to convex geometry by representing a collection p of incoming momenta as the weighted normals of a convex polytope Q via the Minkowski problem. Then, a regular polyhedral subdivision 𝒫 of Q exhibits p as a Landau singularity labelled by the dual graph of 𝒫 with masses being the areas of the faces. Positivity of the Landau/Feynman/Schwinger multipliers is interpreted as strict convexity of a PL-function. This gives an interesting class of ``polyhedral'' Landau singularities. In the planar case going back to the original 1959 paper, Landau graphs can also be identified with plane webs of Gaiotto-Moore-Witten that provide a language dual to that of regular polygonal subdivisions.
Related
- Landau--Ginzburg mirror symmetry for ADE singularities
- The Landau equation: Well-posedness and singularities of homoenergetic solutions
- Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds
- Pfaffian Geometry and Indicial Singularities of Third-Order Linear ODEs
- Invisible singularities in complex algebraic geometry
BibTeX
@misc{kapranov2026landau,
title = {{Landau singularities and convex geometry}},
author = {Mikhail Kapranov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27508},
eprint = {2609.27508},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27508},
}
RIS
TY - UNPB
TI - Landau singularities and convex geometry
AU - Mikhail Kapranov
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.27508
UR - https://arxiv.org/abs/2609.27508
AB - We revisit the classic 1959 paper of L. D. Landau on possible singularities of the integral associated to a Feynman graph. Focusing on the real setup, we relate it to convex geometry by representing a collection p of incoming momenta as the weighted normals of a convex polytope Q via the Minkowski problem. Then, a regular polyhedral subdivision 𝒫 of Q exhibits p as a Landau singularity labelled by the dual graph of 𝒫 with masses being the areas of the faces. Positivity of the Landau/Feynman/Schwinger multipliers is interpreted as strict convexity of a PL-function. This gives an interesting class of ``polyhedral'' Landau singularities. In the planar case going back to the original 1959 paper, Landau graphs can also be identified with plane webs of Gaiotto-Moore-Witten that provide a language dual to that of regular polygonal subdivisions.
ER -