geometric langlands
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2021 ◽  
Vol 157 (12) ◽  
pp. 2699-2732
Author(s):  
Justin Campbell ◽  
Gurbir Dhillon ◽  
Sam Raskin

In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory and Lurie proposed a conjectural substitute, later termed the fundamental local equivalence. With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.


2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Dario Beraldo

We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding.


2019 ◽  
Vol 155 (12) ◽  
pp. 2235-2262 ◽  
Author(s):  
Tomoyuki Arakawa ◽  
Edward Frenkel

We prove duality isomorphisms of certain representations of ${\mathcal{W}}$-algebras which play an essential role in the quantum geometric Langlands program and some related results.


2019 ◽  
Vol 155 (08) ◽  
pp. 1521-1567 ◽  
Author(s):  
Dario Beraldo

The notion of Hochschild cochains induces an assignment from $\mathsf{Aff}$ , affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor $\mathbb{H}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ , where the latter denotes the category of monoidal DG categories and bimodules. Any functor $\mathbb{A}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ gives rise, by taking modules, to a theory of sheaves of categories $\mathsf{ShvCat}^{\mathbb{A}}$ . In this paper, we study $\mathsf{ShvCat}^{\mathbb{H}}$ . Loosely speaking, this theory categorifies the theory of $\mathfrak{D}$ -modules, in the same way as Gaitsgory’s original $\mathsf{ShvCat}$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $\mathsf{ShvCat}^{\mathbb{H}}$ , its descent properties and the notion of $\mathbb{H}$ -affineness. We then prove the $\mathbb{H}$ -affineness of algebraic stacks: for ${\mathcal{Y}}$ a stack satisfying some mild conditions, the $\infty$ -category $\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is equivalent to the $\infty$ -category of modules for $\mathbb{H}({\mathcal{Y}})$ , the monoidal DG category of higher differential operators. The main consequence, for ${\mathcal{Y}}$ quasi-smooth, is the following: if ${\mathcal{C}}$ is a DG category acted on by $\mathbb{H}({\mathcal{Y}})$ , then ${\mathcal{C}}$ admits a theory of singular support in $\operatorname{Sing}({\mathcal{Y}})$ , where $\operatorname{Sing}({\mathcal{Y}})$ is the space of singularities of ${\mathcal{Y}}$ . As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of $\mathbb{H}(\operatorname{LS}_{{\check{G}}})$ on $\mathfrak{D}(\operatorname{Bun}_{G})$ , thereby equipping objects of $\mathfrak{D}(\operatorname{Bun}_{G})$ with singular support in $\operatorname{Sing}(\operatorname{LS}_{{\check{G}}})$ .


2019 ◽  
Vol 232 (1) ◽  
pp. 299-349
Author(s):  
David Nadler ◽  
Zhiwei Yun

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