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ChernSimons Gauge Theory as a String Theory”, Prog
 Math
, 1995
"... Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these can sometimes be given spacetime interpretations. For instance, threedimensional ChernSimons gaug ..."
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Cited by 417 (10 self)
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Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these can sometimes be given spacetime interpretations. For instance, threedimensional ChernSimons gauge theory can arise as a string theory. The worldsheet model in this case involves a topological sigma model. Instanton contributions to the sigma model give rise to Wilson line insertions in the spacetime ChernSimons theory. A certain holomorphic analog of ChernSimons theory can also arise as a string theory. In this paper, I will describe how ChernSimons gauge theory in three dimensions can be viewed as a string theory. The string theory in question will be constructed using a topological sigma model [1] (related to Floer/Gromov theory) in which the target space is T ∗M, M being a threemanifold. The perturbation
GromovWitten classes, quantum cohomology, and enumerative geometry
 Commun. Math. Phys
, 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
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Cited by 367 (3 self)
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The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given. Let V be a projective algebraic manifold. Methods of quantum field theory recently led to a prediction of some numerical characteristics of the space of algebraic curves in V, especially of genus zero, eventually endowed with a parametrization and marked points. It turned out that
Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 346 (2 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Hodge integrals and GromovWitten theory
 Invent. Math
"... Let Mg,n be the nonsingular moduli stack of genus g, npointed, DeligneMumford stable curves. For each marking i, there is an associated cotangent line bundle Li → Mg,n with fiber T ∗ C,pi over the moduli point [C, p1,...,pn]. Let ψi = c1(Li) ∈ H ∗ (Mg,n, Q). The integrals of products of the ψ cla ..."
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Cited by 118 (12 self)
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Let Mg,n be the nonsingular moduli stack of genus g, npointed, DeligneMumford stable curves. For each marking i, there is an associated cotangent line bundle Li → Mg,n with fiber T ∗ C,pi over the moduli point [C, p1,...,pn]. Let ψi = c1(Li) ∈ H ∗ (Mg,n, Q). The integrals of products of the ψ classes
The Verlinde Algebra and the Cohomology of the Grassmannian, preprint IASSNSHEP 93/41
, 1993
"... The article is devoted to a quantum field theory explanation of the relationship between the Verlinde algebra of the group U(k) at level N −k and the “quantum” cohomology of the Grassmannian of complex k planes in N space. In §2, I explain the relation between the Verlinde algebra and the gauged WZW ..."
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Cited by 110 (3 self)
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The article is devoted to a quantum field theory explanation of the relationship between the Verlinde algebra of the group U(k) at level N −k and the “quantum” cohomology of the Grassmannian of complex k planes in N space. In §2, I explain the relation between the Verlinde algebra and the gauged WZW model of G/G; in §3, I describe the quantum cohomology and its origin in a quantum field theory; and in §4, I present a path integral argument for mapping between them. My main goal in these lecture notes will be to elucidate a formula of Doron Gepner [1], which relates two mathematical objects, one rather old and one rather new. Along the way we will consider a few other matters as well. The old structure is the cohomology ring of the Grassmannian G(k, N) of
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 97 (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.
Intersection theory, integrable hierarchies and topological field theory
, 1992
"... In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevic ..."
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Cited by 95 (5 self)
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In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevich naturally appear as τfunctions of integrable hierarchies related to topological minimal models.
Symplectic FloerDonaldson theory and quantum cohomology
 in Proceedings of the Symposium on Symplectic Geometry, held at the Isaac Newton Institute in Cambridge in 1994, edited by C.B. Thomas, LMS Lecture Note Series
, 1996
"... The goal of this paper is to give in outline a new proof of the fact that the Floer cohomology groups of the loop space of a semipositive symplectic manifold (M;!) are naturally isomorphic to the ordinary cohomology of M . We shall then outline a proof that this isomorphism intertwines the quantum ..."
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Cited by 87 (10 self)
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The goal of this paper is to give in outline a new proof of the fact that the Floer cohomology groups of the loop space of a semipositive symplectic manifold (M;!) are naturally isomorphic to the ordinary cohomology of M . We shall then outline a proof that this isomorphism intertwines the quantum cupproduct structure on the cohomology of M with the pairofpants product on Floerhomology. One of the key technical ingredients of the proof is a gluing theorem for Jholomorphic curves proved in [20]. In this paper we shall only sketch the proofs. Full details of the analysis will appear elsewhere. 1 Introduction The Floer homology groups of a symplectic manifold (M;!) can intuitively be described as the middle dimensional homology groups of the loop space. The boundary loops of Jholomorphic discs in the symplectic manifold with center in a given homology class ff 2 H (M) (integral homology modulo torsion) form a submanifold of the loop space of roughly half dimension and should ther...