Notes on Worldsheet-Like Variables for Cluster Configuration Spaces

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.

References in corpus (7)

• Cluster algebras for Feynman integrals
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Truncated cluster algebras and Feynman integrals with algebraic letters
• Symbol Alphabets from Tensor Diagrams
• Multiple zeta values and periods of moduli spaces ${\mathfrak{M}}_{0,n}$
• Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals

Cited by in corpus (3)

• Generalized Color Orderings: CEGM Integrands and Decoupling Identities
• Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity
• Cluster Alphabets from Generalized Worldsheets: A Geometric Approach to Finite Types