scholarly journals M-theoretic derivations of 4d-2d dualities: from a geometric Langlands duality for surfaces, to the AGT correspondence, to integrable systems

2013 ◽  
Vol 2013 (7) ◽  
Author(s):  
Meng-Chwan Tan
Keyword(s):  
Integrable Systems ◽  
Langlands Duality ◽  
Geometric Langlands ◽  
Agt Correspondence

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 Hitchin and Calabi–Yau Integrable Systems via Variations of Hodge Structures

2020 ◽  
Vol 71 (4) ◽  
pp. 1345-1375
Author(s):  
Florian Beck
Keyword(s):  
Integrable System ◽  
Lie Group ◽  
Integrable Systems ◽  
Hodge Structures ◽  
Langlands Duality ◽  
Hitchin Systems

Abstract Since its discovery by Hitchin in 1987, G-Hitchin systems for a reductive complex Lie group G have extensively been studied. For example, the generic fibers are nowadays well-understood. In this paper, we show that the smooth parts of G-Hitchin systems for a simple adjoint complex Lie group G are isomorphic to non-compact Calabi–Yau integrable systems extending results by Diaconescu–Donagi–Pantev. Moreover, we explain how Langlands duality for Hitchin systems is related to Poincaré–Verdier duality of the corresponding families of quasi-projective Calabi–Yau threefolds. Even though the statement is holomorphic-symplectic, our proof is Hodge-theoretic. It is based on polarizable variations of Hodge structures that admit so-called abstract Seiberg–Witten differentials. These ensure that the associated Jacobian fibration is an algebraic integrable system.

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.

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