Peng Zhao: Cluster Configuration Spaces and Hypersurface Arrangements

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Abstract: Cluster algebras have inspired many recent developments in physics. In the other direction, the study of scattering amplitudes motivates us to study certain moduli spaces associated with finite-type cluster algebras. For A-type it is M_{0,n}, the moduli space of n marked points on CP^1. By introducing a gluing construction for other types, we realize such spaces as hypersurface arrangements. We make contact with Fomin-Zelevinsky's work on Y-systems and Fomin-Shapiro-Thurston's work on surface cluster algebras. We study various topological properties using a finite-field method and propose conjectures about quasi-polynomial point count, dimensions of cohomology, and Euler characteristics for the D_n space up to n=10. 


Based on 2109.13900 with Song He, Yihong Wang, Yong Zhang, and upcoming work with Yihong Wang.