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ElectricMagnetic duality and the geometric Langlands program
, 2006
"... The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super YangMills theory in four dimensions. The key ingredients are electricmagnetic duality of gauge theory, mirror symmetry of sigmamodels, branes, Wilson and ’t H ..."
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Cited by 294 (26 self)
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The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super YangMills theory in four dimensions. The key ingredients are electricmagnetic duality of gauge theory, mirror symmetry of sigmamodels, branes, Wilson and ’t Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlands program, such as Hecke
SelfDual ChernSimons Theories
, 1995
"... In these lectures I review classical aspects of the selfdual ChernSimons systems which describe charged scalar fields in 2 + 1 dimensions coupled to a gauge field whose dynamics is provided by a pure ChernSimons Lagrangian. These selfdual models have one realization with nonrelativistic dynamics ..."
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Cited by 116 (4 self)
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In these lectures I review classical aspects of the selfdual ChernSimons systems which describe charged scalar fields in 2 + 1 dimensions coupled to a gauge field whose dynamics is provided by a pure ChernSimons Lagrangian. These selfdual models have one realization with nonrelativistic dynamics for the scalar fields, and another with relativistic dynamics for the scalars. In each model, the energy density may be minimized by a Bogomol’nyi bound which is saturated by solutions to a set of firstorder selfduality equations. In the nonrelativistic case the selfdual potential is quartic, the system possesses a dynamical conformal symmetry, and the selfdual solutions are equivalent to the static zero energy solutions of the equations of motion. The nonrelativistic selfduality equations are integrable and all finite charge solutions may be found. In the relativistic case the selfdual potential is sixth order and the selfdual Lagrangian may be embedded in a model with an extended supersymmetry. The selfdual potential has a rich structure of degenerate classical minima, and the vacuum masses generated by the ChernSimons Higgs mechanism reflect the selfdual nature of the potential. 1
The SeibergWitten equations and 4–manifold topology
 Bull. Amer. Math. Soc
, 1996
"... Since 1982 the use of gauge theory, in the shape of the YangMills instanton equations, has permeated research in 4manifold topology. At first this use of differential geometry and differential equations had an unexpected and unorthodox flavour, but over the years the ideas have become more familia ..."
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Cited by 95 (0 self)
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Since 1982 the use of gauge theory, in the shape of the YangMills instanton equations, has permeated research in 4manifold topology. At first this use of differential geometry and differential equations had an unexpected and unorthodox flavour, but over the years the ideas have become more familiar; a body of techniques has built up through the efforts of many mathematicians, producing results which have uncovered some of the mysteries of 4manifold theory, and leading to substantial internal conundrums within the field itself. In the last three months of 1994 a remarkable thing happened: this research area was turned on its head by the introduction of a new kind of differentialgeometric equation by Seiberg and Witten: in the space of a few weeks longstanding problems were solved, new and unexpected results were found, along with simpler new proofs of existing ones, and new vistas for research opened up. This article is a report on some of these developments, which are due to various mathematicians, notably Kronheimer, Mrowka, Morgan, Stern and Taubes, building on the seminal work of Seiberg [S] and Seiberg and Witten [SW]. It is written as an attempt to take stock of the progress stemming
Mirror symmetry, Langlands duality and Hitchin systems
 arXiv: math.AG/0205236 56 Hausel, T. and Sturmfels, B
"... Abstract. Among the major mathematical approaches to mirror symmetry are those of BatyrevBorisov and StromingerYauZaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious ..."
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Cited by 55 (9 self)
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Abstract. Among the major mathematical approaches to mirror symmetry are those of BatyrevBorisov and StromingerYauZaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious that mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of examples that can be shown to satisfy the requirements of SYZ, but whose Hodge numbers are also equal. This provides significant evidence in support of SYZ. Moreover, the examples are of great interest in their own right: they are spaces of flat SLrconnections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGLr. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program. When it emerged in the early 1990s, mirror symmetry was an aspect of theoretical physics, and specifically a duality between quantum field theories. Since then, many people have tried to place it on a mathematical foundation. Their labors have built up a fascinating but somewhat unruly subject. It describes some sort of relation between pairs of Calabi
The monodromy groups of Schwarzian equations on closed Riemann surfaces
 ANN. OF MATH
, 2000
"... Let θ: π1(R) → PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either ..."
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Cited by 54 (1 self)
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Let θ: π1(R) → PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π1(R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to