Valuative independence and cluster theta reciprocity

Abstract

We prove that theta functions constructed from positive scattering diagrams satisfy valuative independence. That is, for certain valuations ${\mathrm{val}}_v$, we have ${\mathrm{val}}_v({\sum }_u{c}_u{\vartheta }_u)={\min }_{c_u\ne 0}{\mathrm{val}}_v({\vartheta }_u)$. 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, ${\mathrm{val}}_v({\vartheta }_u)={\mathrm{val}}_u({\vartheta }_v)$. 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.

56 pages