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431
A holomorphic Casson invariant for CalabiYau 3folds, and bundles on K3 fibrations
 J. Differential Geom
, 2000
"... We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fol ..."
Abstract

Cited by 105 (5 self)
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We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [LT], [BF] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3, and Donaldson – and GromovWitten – like invariants of Fano 3folds. It also allows us to define the holomorphic Casson invariant of a CalabiYau 3fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to GromovWitten invariants of the “Mukaidual ” 3fold for others. As an example the invariant is shown to distinguish Gross ’ diffeomorphic 3folds. Finally the Mukaidual 3fold is shown to be CalabiYau and its cohomology is related to that of X. 1
The spectrum of BPS branes on a noncompact CalabiYau
, 2000
"... We begin the study of the spectrum of BPS branes and its variation on lines of marginal stability on OIP2(−3), a CalabiYau ALE space asymptotic to C 3 /Z3. We show how to get the complete spectrum near the large volume limit and near the orbifold point, and find a striking similarity between the de ..."
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Cited by 97 (10 self)
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We begin the study of the spectrum of BPS branes and its variation on lines of marginal stability on OIP2(−3), a CalabiYau ALE space asymptotic to C 3 /Z3. We show how to get the complete spectrum near the large volume limit and near the orbifold point, and find a striking similarity between the descriptions of holomorphic bundles and BPS branes in these two limits. We use these results to develop a general picture of the spectrum. We also suggest a generalization of some of the ideas to the quintic CalabiYau.
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.
Rational blowdowns of smooth 4manifolds
 Jour. Diff. Geom
, 1997
"... The invariants of Donaldson and of Seiberg and Witten are powerful tools for studying smooth 4manifolds. A fundamental problem is to determine procedures which relate smooth 4manifolds in such a fashion that their effect on both the Donaldson and SeibergWitten invariants can be computed. The purp ..."
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Cited by 91 (11 self)
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The invariants of Donaldson and of Seiberg and Witten are powerful tools for studying smooth 4manifolds. A fundamental problem is to determine procedures which relate smooth 4manifolds in such a fashion that their effect on both the Donaldson and SeibergWitten invariants can be computed. The purpose of this paper is to initiate this study by introducing a surgical procedure,
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...
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
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 68 (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
Selfdual instantons and holomorphic curves
 Annals of Mathematics 139
, 1994
"... 2. Floer homology for symplectic fixed points ..."