Form Factor Alphabets and Antipodal Duality from Cluster Algebras
Abstract
We develop a cluster-algebraic description of symbol alphabets for chiral stress-tensor form factors in planar 𝒩=4 super-Yang--Mills theory. Motivated by unfolding the three-point C2 quiver, we propose a periodic framework for n-point form-factor alphabets based on folding a Gr(4,2n) quiver with two periods of frozen nodes. At four points, the folding leads to the VĨ cluster algebra, which has infinitely many cluster variables. Tropical truncation selects a finite set of rational coordinates and four physical limit rays. The A1(1) mutation sequences approaching these rays generate the four square roots and the corresponding algebraic-letter spaces. Within the resulting finite set of rational candidates, antipodal closure excludes precisely eight additional letters. On the parity-preserving surface, the antipodal map is realized, after a further folding, by two commuting mutations followed by a relabelling. Furthermore, we find an alternative quiver within the same mutation class that becomes self-antipodal after parity folding, mirroring the antipodal self-duality of the four-point MHV form factor. Its double- and triple-collinear boundaries reproduce, respectively, the C2 algebra of the three-point form factor and the A3 algebra of the six-point amplitude; after folding the latter to C2, the antipodal map exchanges the two C2 boundaries.
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BibTeX
@misc{he2026form,
title = {{Form Factor Alphabets and Antipodal Duality from Cluster Algebras}},
author = {Song He and Jiahao Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04839},
eprint = {2609.04839},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04839},
}
RIS
TY - UNPB
TI - Form Factor Alphabets and Antipodal Duality from Cluster Algebras
AU - Song He
AU - Jiahao Liu
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04839
UR - https://arxiv.org/abs/2609.04839
AB - We develop a cluster-algebraic description of symbol alphabets for chiral stress-tensor form factors in planar 𝒩=4 super-Yang--Mills theory. Motivated by unfolding the three-point C_2 quiver, we propose a periodic framework for n-point form-factor alphabets based on folding a Gr(4,2n) quiver with two periods of frozen nodes. At four points, the folding leads to the VĨ cluster algebra, which has infinitely many cluster variables. Tropical truncation selects a finite set of rational coordinates and four physical limit rays. The A_1^(1) mutation sequences approaching these rays generate the four square roots and the corresponding algebraic-letter spaces. Within the resulting finite set of rational candidates, antipodal closure excludes precisely eight additional letters. On the parity-preserving surface, the antipodal map is realized, after a further folding, by two commuting mutations followed by a relabelling. Furthermore, we find an alternative quiver within the same mutation class that becomes self-antipodal after parity folding, mirroring the antipodal self-duality of the four-point MHV form factor. Its double- and triple-collinear boundaries reproduce, respectively, the C_2 algebra of the three-point form factor and the A_3 algebra of the six-point amplitude; after folding the latter to C_2, the antipodal map exchanges the two C_2 boundaries.
ER -