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113
Lectures on 2D YangMills Theory, Equivariant Cohomology and Topological Field Theories
, 1996
"... These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying ..."
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Cited by 99 (7 self)
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These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.
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 78 (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
Holomorphic triangles and invariants for smooth fourmanifolds
"... Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed fourmanifolds, built using the Floer homology theories defined in [8] and [12]. This fourdimensional theory also endows the corresponding threedimensional theories with additional structure: an absolute gradi ..."
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Cited by 75 (22 self)
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Abstract. The aim of this article is to introduce invariants of oriented, smooth, closed fourmanifolds, built using the Floer homology theories defined in [8] and [12]. This fourdimensional theory also endows the corresponding threedimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces. 1.
Higher genus symplectic invariants and sigma model coupled with gravity
"... This paper is a continuation of our previous paper [RT]. In [RT], among other things, we build up the mathematical foundation of quantum cohomology ring on semipositive symplectic manifolds. ..."
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Cited by 71 (7 self)
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This paper is a continuation of our previous paper [RT]. In [RT], among other things, we build up the mathematical foundation of quantum cohomology ring on semipositive symplectic manifolds.
Monopoles and four manifolds
 Math.Res. Lett
, 1994
"... Recent developments in the understanding of N = 2 supersymmetric YangMills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining fourmanifold invariants by counting SU(2) instantons, one can define equivalent fourmanifold invariants by ..."
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Cited by 69 (2 self)
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Recent developments in the understanding of N = 2 supersymmetric YangMills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining fourmanifold invariants by counting SU(2) instantons, one can define equivalent fourmanifold invariants by counting solutions of a nonlinear equation with an abelian gauge group. This is a “dual ” equation in which the gauge group is the dual of the maximal torus of SU(2). The new viewpoint suggests many new results about the Donaldson invariants. November
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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Cited by 67 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
Supersymmetric Yang–Mills theory on a fourmanifold
 Jour. Math. Phys
, 1994
"... By exploiting standard facts about N = 1 and N = 2 supersymmetric YangMills theory, the Donaldson invariants of fourmanifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about ..."
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Cited by 67 (4 self)
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By exploiting standard facts about N = 1 and N = 2 supersymmetric YangMills theory, the Donaldson invariants of fourmanifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric YangMills theory. In fourdimensional supersymmetric YangMills theory formulated on flat R4, certain correlation functions are independent of spatial separation and are hence effectively computable by going to short distances. This is the basis for one of the most fruitful techniques for studying dynamics of those theories [1,2], and
TOPOLOGICAL FIELD THEORY AND RATIONAL CURVES
, 1991
"... We analyze the quantum field theory corresponding to a string propagating on a CalabiYau threefold. This theory naturally leads to the consideration of Witten’s topological nonlinear σmodel and the structure of rational curves on the CalabiYau manifold. We study in detail the case of the world ..."
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Cited by 64 (6 self)
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We analyze the quantum field theory corresponding to a string propagating on a CalabiYau threefold. This theory naturally leads to the consideration of Witten’s topological nonlinear σmodel and the structure of rational curves on the CalabiYau manifold. We study in detail the case of the worldsheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by Candelas, de la Ossa, Green and Parkes.