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Quantum geometric Langlands at irrational level
at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages
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Difficulty:🧠🧠🧠🧠🧠 Ages:13 - ∞
Skills:shapes made of equations Levels:1
Category:Algebraic and complex geometry Lean version:not yet
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Global quantum geometric Langlands at irrational level. Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level $c\in\mathbb C\setminus\mathbb Q$. It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.

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released 2026-10-04  |  7 theorems · 27 lemmas · 64 proofs · 62,458 words  |  PLAY LEVEL 1 »  (pdf)
We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level $c\in\mathbb C\setminus\mathbb Q$, including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on $\mathop{\mathrm{Bun}}\nolimits _G(X)$ and $\mathop{\mathrm{Bun}}\nolimits _{G^\vee}(X)$ at the dual shifted levels c and $-1/(rc)$, where r is the lacing number. The equivalence uses the given global forms and includes all connected components.

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