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502
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 ..."
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Cited by 111 (6 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
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.
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 99 (11 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...
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 98 (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.
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 86 (9 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
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 81 (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
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 71 (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
Knot Floer Homology and the fourball genus
 Geom. Topol
"... Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotti ..."
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Cited by 67 (8 self)
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Abstract. We use the knot filtration on the Heegaard Floer complex ĈF to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ gives sharp bounds on the fourball genera of torus knots. As another illustration, we use calculate the invariant for several tencrossing knots. 1.