Geometric Langlands Program and Dualities in Quantum Physics

2009 ◽  
Author(s):  
Edward Frenkel
Keyword(s):  
Quantum Physics ◽  
Langlands Program ◽  
Geometric Langlands Program ◽  
Geometric Langlands

scholarly journals A spectral incarnation of affine character sheaves

2017 ◽  
Vol 153 (9) ◽  
pp. 1908-1944
Author(s):  
David Ben-Zvi ◽  
David Nadler ◽  
Anatoly Preygel
Keyword(s):  
Integral Transforms ◽  
Companion Paper ◽  
Langlands Program ◽  
Coherent Sheaves ◽  
Singular Support ◽  
Geometric Langlands Program ◽  
Descent Theory ◽  
Geometric Langlands ◽  
Support Conditions ◽  
Character Sheaves

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions, which appear in recent advances in the geometric Langlands program. The key technical tools in our arguments are a new descent theory for coherent sheaves or ${\mathcal{D}}$-modules with prescribed singular support and the theory of integral transforms for coherent sheaves developed in the companion paper by Ben-Zvi et al. [Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS), to appear].

scholarly journals Springer theory via the Hitchin fibration

2011 ◽  
Vol 147 (5) ◽  
pp. 1635-1670 ◽  
Author(s):  
David Nadler
Keyword(s):  
Fukaya Category ◽  
Group Representations ◽  
Langlands Program ◽  
Geometric Langlands Program ◽  
Constructible Sheaves ◽  
Geometric Langlands ◽  
Springer Theory ◽  
Hitchin Fibration ◽  
Semisimple Complex ◽  
The Fourier Transform

AbstractWe develop the Springer theory of Weyl group representations in the language of symplectic topology. Given a semisimple complex group G, we describe a Lagrangian brane in the cotangent bundle of the adjoint quotient 𝔤/G that produces the perverse sheaves of Springer theory. The main technical tool is an analysis of the Fourier transform for constructible sheaves from the perspective of the Fukaya category. Our results can be viewed as a toy model of the quantization of Hitchin fibers in the geometric Langlands program.

scholarly journals Geometric Waldspurger periods

2008 ◽  
Vol 144 (2) ◽  
pp. 377-438 ◽  
Author(s):  
Sergey Lysenko
Keyword(s):  
Derived Categories ◽  
Hecke Operators ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Langlands Program ◽  
Geometric Langlands Program ◽  
Geometric Langlands ◽  
Langlands Functoriality

AbstractLet X be a smooth projective curve. We consider the dual reductive pair $H=\mathrm {G\mathbb {O}}_{2m}$, $G=\mathrm {G\mathbb {S}p}_{2n}$ over X, where H splits on an étale two-sheeted covering $\pi :\tilde X\to X$. Let BunG (respectively, BunH) be the stack of G-torsors (respectively, H-torsors) on X. We study the functors FG and FH between the derived categories D(BunG) and D(BunH), which are analogs of the classical theta-lifting operators in the framework of the geometric Langlands program. Assume n=m=1 and H nonsplit, that is, $H=\pi _*{\mathbb {G}_m}$ with $\tilde X$ connected. We establish the geometric Langlands functoriality for this pair. Namely, we show that FG :D(BunH)→D(BunG) commutes with Hecke operators with respect to the corresponding map of Langlands L-groups LH→LG. As an application, we calculate Waldspurger periods of cuspidal automorphic sheaves on BunGL2 and Bessel periods of theta-lifts from $\mathrm {Bun}_{\mathrm {G\mathbb {O}}_4}$ to $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$. Based on these calculations, we give three conjectural constructions of certain automorphic sheaves on $\mathrm {Bun}_{\mathrm {G\mathbb {S}p}_4}$ (one of them makes sense for ${\mathcal D}$-modules only).

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