The global quantum geometric Langlands conjecture at irrational level

Theorem 1.1: for every connected simple complex algebraic group $G$, every smooth projective connected complex curve $X$ and every $c\in\mathbb C\setminus\mathbb Q$ (including non-real $c$) there is an equivalence $D_c(\mathrm{Bun}_G(X))\simeq D_{-1/(rc)}(\mathrm{Bun}_{G^\vee}(X))$ of presentable DG categories, characterized by its localization-Whittaker comparison and intertwining central torsor actions with transgression local systems. It does not treat rational levels, ramification, the Betti setting, or non-simple reductive groups.

Theorem 1.1: for every connected simple complex algebraic group $G$, every smooth projective connected complex curve $X$ and every $c\in\mathbb C\setminus\mathbb Q$ (including non-real $c$) there is an equivalence $D_c(\mathrm{Bun}_G(X))\simeq D_{-1/(rc)}(\mathrm{Bun}_{G^\vee}(X))$ of presentable DG categories, characterized by its localization-Whittaker comparison and intertwining central torsor actions with transgression local systems. It does not treat rational levels, ramification, the Betti setting, or non-simple reductive groups.