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On sinks and mirrors

Mingyuan Hu, Vivek Shende

math.SG Nov 8, 2026 · v1 math.AG math.CO math.RT
formalization mathlib geometry
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TL;DR
The skeleton calculation in Theorem A.2 is formalized in Lean 4 using Mathlib, with a public GitHub repository.
Abstract
Given a full rank sink recurrent cluster $\mathcal{A}$ variety, we prove that its wrapped Fukaya category is equivalent to the category of coherent sheaves on a stacky elaboration of the dual Fock-Goncharov cluster $\mathcal{X}$ variety. In particular, braid varieties for a semisimple group are mirror to the braid varieties for the Langlands dual group. Another example: for compact surfaces with no interior marked points, a least one marked point on each boundary component, and at least two marked points in total, the Fock-Goncharov moduli for SL(2) and PGL(2) are homologically mirror. The results were obtained in communion with ChatGPT Astra.
Problem

Homological mirror symmetry between Fock-Goncharov pairs of Langlands-dual cluster varieties has long been conjectured. A central obstacle is the 'deep locus' outside the cluster charts, which compact real tori in those charts cannot probe.

Approach

A new descent result for partially wrapped Fukaya categories (Sylvan-Orlov-Viterbo descent) is proved via a categorical pullback criterion for spherical functors. Sink covers and freezings of cluster algebras are combined with good Liouville structures on cluster varieties. These are matched with a B-side computation using blowups and coherent sheaves on a stacky dual cluster X variety. The skeleton calculation in an appendix is formalized in Lean 4 with Mathlib.

Results

For any full rank sink-recurrent cluster A variety, the wrapped Fukaya category is equivalent to coherent sheaves on a stacky version of the dual Fock-Goncharov X variety. Consequences include mirror symmetry between braid varieties of Langlands-dual groups, and between the SL(2) and PGL(2) Fock-Goncharov moduli for suitable marked surfaces.