Valuative independence and cluster theta reciprocity
Man-Wai Cheung, Timothy Magee, Travis Mandel, Greg Muller
TL;DR
The paper develops and proves a general Valuative Independence Theorem (VIT) and a Theta Reciprocity principle for theta functions built from positive scattering diagrams, with a unified seed-data framework for cluster-type varieties. It shows that valuative minimizers of theta-function combinations are unique generically and remain robust under coefficient specialization, while a duality symmetry val_v(θ_u)=val_u(θ_v) ties valuations on mirror sides. The Theta Extension Theorem guarantees stability of theta functions under diagram extensions, enabling gluing results for moduli spaces of local systems on marked surfaces and leading to canonical theta bases for global sections and Cox rings. The work also introduces seed data and Λ-structures to prove reciprocity within and across dual seeds, and it provides explicit constructions and examples (including loop elements and Log Calabi–Yau surfaces) illustrating how these bases behave under extensions, gluing, and dualities. Overall, the results unify and extend previous GHKK-type results, Keel–Yu geometries, and Langlands dualities, yielding robust tools for constructing and identifying theta-function bases in cluster and mirror-symmetric settings.
Abstract
We prove that theta functions constructed from positive scattering diagrams satisfy valuative independence. That is, for certain valuations $\operatorname{val}_{v}$, we have $\operatorname{val}_v(\sum_u c_u \vartheta_u)=\min_{c_u\neq 0} \operatorname{val}_v(\vartheta_u)$. As applications, we prove linear independence of theta functions with specialized coefficients and characterize when theta functions for cluster varieties are unchanged by the unfreezing of an index. This yields a general gluing result for theta functions from moduli of local systems on marked surfaces. We then prove that theta functions for cluster varieties satisfy a symmetry property called theta reciprocity: briefly, $\operatorname{val}_v(\vartheta_u)=\operatorname{val}_u(\vartheta_v)$. For this we utilize a new framework called a "seed datum" for understanding cluster-type varieties. One may apply valuative independence and theta reciprocity together to identify theta function bases for global sections of line bundles on partial compactifications of cluster varieties.