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Motivic mirror symmetry for character stacks

Let GG be a connected, simply connected, and semisimple complex Lie group. Let FF be a finite subgroup of its center Z(G)Z(G), and let LG~{}^L\tilde{G} be the universal covering of its Langlands dual group. For generic tuples of nice semisimple conjugacy classes C\mathcal{C} and Cˇ\check{\mathcal{C}} of GG and LG~{}^L\tilde{G}, respectively, and any g≥0g\geq 0, write [X]st[\mathcal{X}]_{st} for the stringy motive of a Deligne–Mumford character stack X\mathcal{X}. Motivic mirror symmetry conjecture.

[MBC(G/F)]st=[MBCˇ(LG~/(Z(G)/F)∨)]st.[\mathcal{M}_B^{\mathcal{C}}(G/F)]_{st}=[\mathcal{M}_B^{\check{\mathcal{C}}}({}^L\tilde{G}/(Z(G)/F)^\vee)]_{st}.

This conjecture proposes a motivic version of topological mirror symmetry for character stacks, extending the corresponding duality phenomena for moduli spaces of Higgs bundles. The paper verifies it for the special linear group of rank two as a toy example, while the general statement remains open.

Sources & referencesView supporting material

Primary source

Lucas de Amorin, “Motivic Mirror Symmetry for Character Stacks”, arXiv:2508.12995 (2025).

character-stackslanglands-dualitymotivic-mirror-symmetry

Progress summary

Refreshed
Partially solved

The conjecture is proved only for the rank-two special-linear toy case; a 2025 paper develops useful reductions, but the general claim remains open.

Lucas de Amorin proposed a motivic mirror-symmetry conjecture for character stacks associated with connected, simply connected, semisimple GG, finite central subgroups FF, all g≥0g \geq 0, and generic semisimple conjugacy classes. It predicts equality of the corresponding stringy motives on Langlands-dual sides.

2025 structural reduction and toy case

De Amorin generalizes Mellit’s cell decomposition from GLn\mathrm{GL}_n to arbitrary connected reductive groups, analyzes cell automorphisms, and proves a duality exchanging those automorphisms with connected-component data. The general conjecture is reduced to a cellwise Conjecture 2, and the full statement is proved only for G=SL2G = \mathrm{SL}_2; no general proof or counterexample is reported.

Current status (as of August 2026): The SL2\mathrm{SL}_2 toy case and the stated structural reductions are established, but the general motivic mirror-symmetry conjecture remains open.

Sources

Solutions 0

No solutions have been posted yet.