A Physics Perspective on Geometric Langlands Duality

2010 ◽  
pp. 219-232
Author(s):  
Karl-Georg Schlesinger
Keyword(s):  
Langlands Duality ◽  
Geometric Langlands

scholarly journals Geometric Langlands duality and the equations of Nahm and Bogomolny

2010 ◽  
Vol 140 (4) ◽  
pp. 857-895 ◽  
Author(s):  
Edward Witten
Keyword(s):  
Gauge Theory ◽  
Lie Group ◽  
Moduli Space ◽  
Dual Group ◽  
Langlands Duality ◽  
Geometric Langlands

Geometric Langlands duality relates a representation of a simple Lie group Gv to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of Gv makes an unexpected appearance. This can be explained using gauge theory, as this paper will show, with the help of the equations of Nahm and Bogomolny.

scholarly journals Langlands duality and global Springer theory

2012 ◽  
Vol 148 (3) ◽  
pp. 835-867 ◽  
Author(s):  
Zhiwei Yun
Keyword(s):  
Chern Class ◽  
Class Action ◽  
Dual Group ◽  
The Other ◽  
Support Theorem ◽  
Langlands Duality ◽  
Dual Groups ◽  
Geometric Langlands ◽  
Springer Theory ◽  
Singular Fibers

AbstractWe compare the cohomology of (parabolic) Hitchin fibers for Langlands dual groups G and G∨. The comparison theorem fits in the framework of the global Springer theory developed by the author. We prove that the stable parts of the parabolic Hitchin complexes for Langlands dual group are naturally isomorphic after passing to the associated graded of the perverse filtration. Moreover, this isomorphism intertwines the global Springer action on one hand and Chern class action on the other. Our result is inspired by the mirror symmetric viewpoint of geometric Langlands duality. Compared to the pioneer work in this subject by T. Hausel and M. Thaddeus, R. Donagi and T. Pantev, and N. Hitchin, our result is valid for more general singular fibers. The proof relies on a variant of Ngô’s support theorem, which is a key point in the proof of the Fundamental Lemma.

scholarly journals Quantum Langlands duality of representations of -algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2235-2262 ◽  
Author(s):  
Tomoyuki Arakawa ◽  
Edward Frenkel
Keyword(s):  
Essential Role ◽  
Langlands Program ◽  
Langlands Duality ◽  
Geometric Langlands Program ◽  
Geometric Langlands ◽  
Representations Of Algebras

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.

scholarly journals The geometric Langlands twist in five and six dimensions

2015 ◽  
Vol 2015 (7) ◽  
Author(s):  
Dongsu Bak ◽  
Andreas Gustavsson
Keyword(s):  
Geometric Langlands ◽  
Six Dimensions

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].

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