Abstract
I provide an introduction to the recent work on the Montonen–Olive duality of cN=4 super-Yang–Mills theory and the Geometric Langlands Program.
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References
E. Frenkel, “Lectures on the Langlands Program and Conformal Field Theory,” (2005) [arXiv:hep-th/0512172].
G. Laumon, “Correspondance Langlands Geometrique Pour Les Corps Des Fonctions,” Duke Math. J. 54, 309–359 (1987).
P. Goddard, J. Nuyts and D. I. Olive, “Gauge Theories and Magnetic Charge,” Nucl. Phys. B 125, 1 (1977).
C. Montonen and D. I. Olive, “Magnetic Monopoles as Gauge Particles?,” Phys. Lett. B 72, 117 (1977).
H. Osborn, “Topological Charges for N=4 Supersymmetric Gauge Theories and Monopoles of Spin 1,” Phys. Lett. B 83, 321 (1979).
A. Kapustin and E. Witten, “Electric-Magnetic Duality and the Geometric Langlands Program,” (2006) [arXiv:hep-th/0604151].
S. Gukov and E. Witten, “Gauge Theory, Ramification, and the Geometric Langlands Program,” (2006) [arXiv:hep-th/0612073].
N. Dorey, C. Fraser, T. J. Hollowood and M. A. C. Kneipp, “S-Duality In N=4 Supersymmetric Gauge Theories,” Phys. Lett. B 383, 422 (1996) [arXiv:hep-th/9605069].
P. C. Argyres, A. Kapustin and N. Seiberg, “On S-duality for Non-Simply-Laced Gauge Groups,” J. High Energy Phys. 0606, 043 (2006) [arXiv:hep-th/0603048].
E. Witten and D. I. Olive, “Supersymmetry Algebras that Include Topological Charges,” Phys. Lett. B 78, 97 (1978).
A. Sen, “Dyon – Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and SL(2,Z) Invariance in String Theory,” Phys. Lett. B 329, 217 (1994) [arXiv:hep-th/9402032].
C. Vafa and E. Witten, “A Strong Coupling Test of S-Duality,” Nucl. Phys. B 431, 3 (1994) [arXiv:hep-th/9408074].
C. Vafa, “Geometric Origin of Montonen-Olive Duality,” Adv. Theor. Math. Phys. 1, 158 (1998) [arXiv:hep-th/9707131].
E. Witten, “Topological Quantum Field Theory,” Commun. Math. Phys. 117, 353 (1988).
N. Hitchin, “The Self-Duality Equations on the Riemann Surface,” Proc. Lond. Math. Soc. (3) 55, 59–126 (1987).
E. B. Bogomolny, “Stability of Classical Solutions,” Sov. J. Nucl. Phys. 24, 449 (1976) [Yad.\ Fiz.\ 24, 861 (1976)].
K. Corlette, “Flat G-Bundles with Canonical Metrics,” J. Diff. Geom. 28, 361–382 (1988).
E. Witten, “Mirror Manifolds And Topological Field Theory,” (1991) [arXiv:hep-th/9112056].
A. Kapustin, “Topological Strings on Noncommutative Manifolds,” Int. J. Geom. Meth. Mod. Phys. 1, 49 (2004) [arXiv:hep-th/0310057].
A. Kapustin and Y. Li, “Topological Sigma-Models with H-Flux and Twisted Generalized Complex Manifolds,” (2004) [arXiv:hep-th/0407249].
N. Hitchin, “Generalized Calabi-Yau Manifolds,” Q. J. Math. 54, 281–308 (2003).
S. K. Donaldson, “Twisted Harmonic Maps and the Self-Duality Equations,” Proc. Lond. Math. Soc. (3), 55, 127–131 (1987).
A. Kapustin, “Holomorphic Reduction of N = 2 Gauge Theories, Wilson-’t Hooft Operators, and S-Duality,” (2006)[arXiv:hep-th/0612119].
M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, “Topological Reduction of 4-d SYM to 2-d Sigma Models,” Nucl. Phys. B 448, 166 (1995) [arXiv:hep-th/9501096].
T. Hausel and M. Thaddeus, “Mirror Symmetry, Langlands Duality, and the Hitchin System,” Invent. Math. 153, 197–229 (2003) [arXiv:math.AG/0205236].
A. Strominger, S. T. Yau and E. Zaslow, “Mirror symmetry is T-duality,” Nucl. Phys. B 479, 243 (1996) [arXiv:hep-th/9606040].
N. Hitchin, “Stable Bundles and Integrable Systems,” Duke Math. J. 54, 91–114 (1987).
A. Kapustin and D. Orlov, “Remarks on A-branes, Mirror Symmetry, and the Fukaya Category,” J. Geom. Phys. 48, 84 (2003) [arXiv:hep-th/0109098].
F. Malikov, V. Schechtman, and A. Vaintrob, “Chiral De Rham Complex,” Comm. Math. Phys. 204, 439–473 (1999) [arXiv:math.AG/9803041].
A. Kapustin, “Chiral De Rham Complex and the Half-Twisted Sigma-Model,” (2005) [arXiv:hep-th/0504074].
E. Witten, “Two-Dimensional Models with (0,2) Supersymmetry: Perturbative Aspects,” (2005) [arXiv:hep-th/0504078].
A. Kapustin and Y. Li, “Open String BRST Cohomology for Generalized Complex Branes,” Adv. Theor. [arXiv:hep-th/0501071].
K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D 10, 2445 (1974).
A. Kapustin, “Wilson-’t Hooft Operators in Four-Dimensional Gauge Theories And S-Duality,” Phys. Rev. D 74, 025005 (2006) [arXiv:hep-th/0501015].
G. ’t Hooft, “On the Phase Transition Towards Permanent Quark Confinement,” Nucl. Phys. B 138, 1 (1978); “A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories,” Nucl. Phys. B 153, 141 (1979).
G. Lusztig, “Singularities, Character Formula, and a q-Analog of Weight Multiplicities,” in Analyse Et Topolgie Sur Les Espaces Singuliers II-III, Asterisque vol. 101–2, 208–229 (1981).
V. Ginzburg, “Perverse Sheaves on a Loop Group and Langlands Duality,” (1995) [arXiv:alg-geom/9511007].
I. Mirkovic and K. Vilonen, “Perverse Sheaves on Affine Grassmannians and Langlands Duality,” Math. Res. Lett. 7, 13–24 (2000) [arXiv:math.AG 9911050].
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Kapustin, A. (2008). Gauge Theory, Mirror Symmetry, and the Geometric Langlands Program. In: Homological Mirror Symmetry. Lecture Notes in Physics, vol 757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68030-7_4
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