Geometric Langlands duality and the equations of Nahm and Bogomolny

Edward Witten

doi:10.1017/S0308210509000882

Geometric Langlands duality and the equations of Nahm and Bogomolny

Edward Witten

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

Geometric Langlands duality relates a representation of a simple Lie group G^∨ to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL₂ subgroup of G^∨ 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.

Original language	English (US)
Pages (from-to)	857-895
Number of pages	39
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume	140
Issue number	4
DOIs	https://doi.org/10.1017/S0308210509000882
State	Published - Aug 2010
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1017/S0308210509000882

Cite this

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title = "Geometric Langlands duality and the equations of Nahm and Bogomolny",

abstract = "Geometric Langlands duality relates a representation of a simple Lie group G∨ to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of G∨ 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.",

author = "Edward Witten",

note = "Funding Information: The idea of connecting the Langlands correspondence with gauge theory was first proposed in the 1970s by M. F. Atiyah (motivated by the observation that the Goddard–Nuyts–Olive dual group is the same as the Langlands dual group). I am grateful to him for introducing me to those ideas at that time. I also thank S. Cherkis and the referee for a close reading of the manuscript and careful comments, E. Frenkel for some helpful questions and D. Gaiotto for collaboration on electric–magnetic duality of boundary conditions. This paper is based on a lecture at the ICMS conference {\textquoteleft}Geometry and Physics: Atiyah80{\textquoteright} in Edinburgh, April 2009. Supported in part by NSF Grant no. Phy-0503584.",

year = "2010",

month = aug,

doi = "10.1017/S0308210509000882",

language = "English (US)",

volume = "140",

pages = "857--895",

journal = "Proceedings of the Royal Society of Edinburgh Section A: Mathematics",

issn = "0308-2105",

publisher = "Cambridge University Press",

number = "4",

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T1 - Geometric Langlands duality and the equations of Nahm and Bogomolny

AU - Witten, Edward

N1 - Funding Information: The idea of connecting the Langlands correspondence with gauge theory was first proposed in the 1970s by M. F. Atiyah (motivated by the observation that the Goddard–Nuyts–Olive dual group is the same as the Langlands dual group). I am grateful to him for introducing me to those ideas at that time. I also thank S. Cherkis and the referee for a close reading of the manuscript and careful comments, E. Frenkel for some helpful questions and D. Gaiotto for collaboration on electric–magnetic duality of boundary conditions. This paper is based on a lecture at the ICMS conference ‘Geometry and Physics: Atiyah80’ in Edinburgh, April 2009. Supported in part by NSF Grant no. Phy-0503584.

PY - 2010/8

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N2 - Geometric Langlands duality relates a representation of a simple Lie group G∨ to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of G∨ 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.

AB - Geometric Langlands duality relates a representation of a simple Lie group G∨ to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of G∨ 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.

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UR - https://www.scopus.com/pages/publications/77957286333#tab=citedBy

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DO - 10.1017/S0308210509000882

M3 - Article

AN - SCOPUS:77957286333

SN - 0308-2105

VL - 140

SP - 857

EP - 895

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

IS - 4

ER -

Geometric Langlands duality and the equations of Nahm and Bogomolny

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