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152
Holomorphic Disks and Topological Invariants for Closed ThreeManifolds
 Ann. of Math
, 2000
"... The aim of this article is to introduce certain topological invariants for closed, oriented threemanifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y  U0 tie U1, these theories are variants of the Lagrangian Floer homology for the gfold symmetric product of Y relat ..."
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Cited by 149 (33 self)
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The aim of this article is to introduce certain topological invariants for closed, oriented threemanifolds Y, equipped with a Spin c structure t. Given a Heegaard splitting of Y  U0 tie U1, these theories are variants of the Lagrangian Floer homology for the gfold symmetric product of Y relative to certain totally real subspaces associated to U0 and U1.
Holomorphic disks and threemanifold invariants: properties and applications
"... ̂HF(Y, s),and HFred(Y, s) associated to closed, oriented threemanifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with SeibergWitten theory. The pr ..."
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Cited by 119 (27 self)
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̂HF(Y, s),and HFred(Y, s) associated to closed, oriented threemanifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with SeibergWitten theory. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to threemanifold topology. 1.
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.
Equivariant GromovWitten invariants
 Internat. Math. Res. Notices
, 1996
"... The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (GW) theory, i.e., intersection theory on spaces of (pseudo) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a co ..."
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Cited by 92 (10 self)
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The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (GW) theory, i.e., intersection theory on spaces of (pseudo) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a compact Kähler manifold X, the equivariant GWtheory provides, as we will show in Section 3, the equivariant cohomology space H ∗ G (X) with a Frobenius structure (see [11]). We will discuss applications of the equivariant theory to the computation ([15], [18]) of quantum cohomology algebras of flag manifolds (Section 5), to the simultaneous diagonalization of the quantum cupproduct operators (Sections 7, 8), to the S1equivariant Floer homology theory on the loop space LX (see Section 6 and [14], [13]), and to a “quantum ” version of the Serre duality theorem (Section 12). In Sections 9–11 we combine the general theory developed in Sections 1–6 with the fixedpoint localization technique [21], in order to prove the mirror conjecture (in the form suggested in [14]) for projective complete intersections. By the mirror conjecture, one usually means some intriguing relations (discovered by physicists) between symplectic and complex geometry on a compact Kähler CalabiYau nfold and, respectively, complex and symplectic geometry on another CalabiYau nfold, called the mirror partner of the former one. The remarkable application [8]ofthe mirror conjecture to the enumeration of rational curves on CalabiYau 3folds (1991, see the theorem below) raised a number of new mathematical problems—challenging tests of maturity for modern methods of symplectic topology. On the other hand, in 1993 I suggested that the relation between symplectic and complex geometry predicted by the mirror conjecture can be extended from the class of CalabiYau manifolds to more general compact symplectic manifolds if one admits non
Virtual neighborhoods and pseudoholomorphic curves
 Turkish J. Math
"... Since Gromov introduced his pseudoholomorphic curve theory in 80’s, pseudoholomorphic curve soon became a predominant technique in symplectic topology. Many important theorems in symplectic topology have been proved by the technique of pseudoholomorphic curve, for example, the squeezing theorem [ ..."
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Cited by 83 (10 self)
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Since Gromov introduced his pseudoholomorphic curve theory in 80’s, pseudoholomorphic curve soon became a predominant technique in symplectic topology. Many important theorems in symplectic topology have been proved by the technique of pseudoholomorphic curve, for example, the squeezing theorem [Gr], the rigidity [E], the classifications of rational and ruled symplectic 4
Relative GromovWitten invariants
 Ann. of Math
, 2003
"... We define relative GromovWitten invariants of a symplectic manifold relative to a codimensiontwo symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of ‘Vstable ’ maps. Simple special cases i ..."
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Cited by 81 (8 self)
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We define relative GromovWitten invariants of a symplectic manifold relative to a codimensiontwo symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of ‘Vstable ’ maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris. GromovWitten invariants are invariants of a closed symplectic manifold (X, ω). To define them, one introduces a compatible almost complex structure J and a perturbation term ν, and considers the maps f: C → X from a genus g complex curve C with n marked points which satisfy the pseudoholomorphic map equation ∂f = ν and represent a class A =[f] ∈ H2(X). The set of such maps, together with their limits, forms the compact space of stable maps Mg,n(X, A). For each stable map, the domain determines a point in the DeligneMumford space Mg,n of curves, and evaluation at each marked point determines a point in X. Thus there is a map (0.1) Mg,n(X, A) → Mg,n × X n. The GromovWitten invariant of (X, ω)isthe homology class of the image for generic (J, ν). It depends only on the isotopy class of the symplectic structure. By choosing bases of the cohomologies of Mg,n and X n, the GW invariant can be viewed as a collection of numbers that count the number of stable maps satisfying constraints. In important cases these numbers are equal to enumerative invariants defined by algebraic geometry. In this article we construct GromovWitten invariants for a symplectic manifold (X, ω) relative to a codimension two symplectic submanifold V. These invariants are designed for use in formulas describing how GW invariants The research of both authors was partially supported by the N.S.F. The first author was also
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.
π1 of symplectic automorphism groups and invertibles in quantum cohomology
, 1997
"... The aim of this paper is to establish a connection between the topology of the automorphism group of a symplectic manifold (M,ω) and the quantum product on its homology. More precisely, we assume that M is closed and connected, and consider the group Ham(M,ω) of Hamiltonian automorphisms ..."
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Cited by 68 (0 self)
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The aim of this paper is to establish a connection between the topology of the automorphism group of a symplectic manifold (M,ω) and the quantum product on its homology. More precisely, we assume that M is closed and connected, and consider the group Ham(M,ω) of Hamiltonian automorphisms
SW ⇒ Gr: From the SeibergWitten equations to pseudoholomorphic curves
 J. Amer. Math. Soc
, 1996
"... The purpose of this article is to explain how pseudoholomorphic curves in a symplectic 4manifold can be constructed from solutions to the SeibergWitten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudoholomorphic curves. This article thus provides ..."
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Cited by 66 (2 self)
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The purpose of this article is to explain how pseudoholomorphic curves in a symplectic 4manifold can be constructed from solutions to the SeibergWitten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudoholomorphic curves. This article thus provides a proof of roughly half of