Global quantum geometric Langlands at irrational level

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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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Theorem 1.1

In Global quantum geometric Langlands at irrational level, 4 October 2026

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For every c\in\mathbb C\setminus\mathbb Q there is an equivalence of presentable \mathbb C-linear DG categories

D_c(\operatorname{Bun}_G(X)) \simeq D_{-1/(rc)}(\operatorname{Bun}_{G^\vee}(X)).

The construction uses the given global forms and all connected components. It is characterized by the localization–Whittaker comparison of Section 13, including its compatibility with families of marked points and their collisions. It also intertwines central torsor actions with the corresponding transgression local systems.

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Global quantum geometric Langlands at irrational level

4 October 2026, current edition.

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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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36 references in Global quantum geometric Langlands at irrational level

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