Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 045, 24 pages      arXiv:2109.13900      https://doi.org/10.3842/SIGMA.2023.045

Notes on Worldsheet-Like Variables for Cluster Configuration Spaces

Song He abcde, Yihong Wang abdf, Yong Zhang gh and Peng Zhao b
a) School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, 310024 Hangzhou, P.R. China
b) CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, P.R. China
c) International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, P.R. China
d) School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, 100049 Beijing, P.R. China
e) Peng Huanwu Center for Fundamental Theory, Hefei, 230026 Anhui, P.R. China
f) Laboratoire d'Annecy-le-Vieux de Physique Théorique, Université Savoie Mont Blanc, 9 Chemin de Bellevue, 74941 Annecy-le-Vieux, France
g) Department of Physics and Astronomy, Uppsala University, 75108 Uppsala, Sweden
h) Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada

Received November 21, 2022, in final form June 29, 2023; Published online July 12, 2023

Abstract
We continue the exploration of various appearances of cluster algebras in scattering amplitudes and related topics in physics. The cluster configuration spaces generalize the familiar moduli space ${\mathcal M}_{0,n}$ to finite-type cluster algebras. We study worldsheet-like variables, which for classical types have also appeared in the study of the symbol alphabet of Feynman integrals. We provide a systematic derivation of these variables from $Y$-systems, which allows us to express the dihedral coordinates in terms of them and to write the corresponding cluster string integrals in compact forms. We mainly focus on the $D_n$ type and show how to reach the boundaries of the configuration space, and write the saddle-point equations in terms of these variables. Moreover, these variables make it easier to study various topological properties of the space using a finite-field method. We propose conjectures about quasi-polynomial point count, dimensions of cohomology, and the number of saddle points for the $D_n$ space up to $n=10$, which greatly extend earlier results.

Key words: cluster algebras; generalized associahedra; $Y$-systems; string amplitudes.

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