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35
Lagrangian Intersections in Contact Geometry
, 1995
"... this paper is to show that this problem can be successfully overcome by using an idea from [11]. We begin with an exposition of the main notions of contact geometry and their symplectic analogs. We develop then an analog of Floer homology theory for the Lagrangian intersection problem in symplectiza ..."
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Cited by 20 (5 self)
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this paper is to show that this problem can be successfully overcome by using an idea from [11]. We begin with an exposition of the main notions of contact geometry and their symplectic analogs. We develop then an analog of Floer homology theory for the Lagrangian intersection problem in symplectizations of contact manifolds and give applications of this theory to contact geometry. There exist other methods for handling similar problem in contact geometry. Let us mention here Givental's approach through, socalled, nonlinear Maslov index (see [9]), as well as the approach based on the theory of generating functions and hypersurfaces as it is described in [3]. All these methods, and the method considered in this paper, have common as well as complementary areas of applications. A part of this paper was written while first and third authors visited IHES. They thank the institute for the hospitality. 2 Contact geometry 2.1 Contact manifolds and their symplectizations
Coherent orientations in symplectic field theory
 Math. Z
"... Abstract. We study the coherent orientations of the moduli spaces of ‘trajectories ’ in Symplectic Field Theory, following the lines of [3]. In particular we examine their behavior at multiple closed Reeb orbits under change of the asymptotic direction. Analogous to the orientation of the unstable t ..."
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Abstract. We study the coherent orientations of the moduli spaces of ‘trajectories ’ in Symplectic Field Theory, following the lines of [3]. In particular we examine their behavior at multiple closed Reeb orbits under change of the asymptotic direction. Analogous to the orientation of the unstable tangent spaces of critical points in finite–dimensional Morse theory, the orientations are determined by a certain choice of orientation at each closed Reeb orbit.
Invariants of Legendrian knots and coherent orientations
 J. SYMPLECTIC GEOM
, 2001
"... We provide a translation between Chekanov’s combinatorial theory for invariants of Legendrian knots in the standard contact R 3 and a relative version of Eliashberg and Hofer’s Contact Homology. We use this translation to transport the idea of “coherent orientations ” from the Contact Homology world ..."
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Cited by 19 (8 self)
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We provide a translation between Chekanov’s combinatorial theory for invariants of Legendrian knots in the standard contact R 3 and a relative version of Eliashberg and Hofer’s Contact Homology. We use this translation to transport the idea of “coherent orientations ” from the Contact Homology world to Chekanov’s combinatorial setting. As a result, we obtain a lifting of Chekanov’s differential graded algebra invariant to an algebra over Z[t, t −1] with a full Z grading.
Homological mirror symmetry for toric del Pezzo surfaces
 Comm. Math. Phys
"... We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an ..."
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Cited by 12 (3 self)
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We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an
Orientations for pseudoholomorphic quilts
, 2007
"... We construct coherent orientations on moduli spaces of quilted pseudoholomorphic surfaces and determine the effect of various gluing operations on the orientations. We also investigate the behavior of the orientations under composition of Lagrangian correspondences. ..."
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Cited by 8 (7 self)
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We construct coherent orientations on moduli spaces of quilted pseudoholomorphic surfaces and determine the effect of various gluing operations on the orientations. We also investigate the behavior of the orientations under composition of Lagrangian correspondences.
HOW TO (SYMPLECTICALLY) THREAD THE EYE OF A (LAGRANGIAN) NEEDLE
, 2002
"... Abstract. We show that there exists no Lagrangian embeddings of the Klein bottle into C 2. Using the same techniques we also give a new proof that any Lagrangian torus in C 2 is smoothly isotopic to the Clifford torus. 1. Lagrangian Embeddings in C 2 The topology of closed Lagrangian embeddings into ..."
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Cited by 7 (0 self)
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Abstract. We show that there exists no Lagrangian embeddings of the Klein bottle into C 2. Using the same techniques we also give a new proof that any Lagrangian torus in C 2 is smoothly isotopic to the Clifford torus. 1. Lagrangian Embeddings in C 2 The topology of closed Lagrangian embeddings into C n (see [1]) is still an elusive problem in symplectic topology. Before Gromov invented the techniques of pseudo– holomorphic curves it was almost intractable and the only known obstructions came from the fact that such a submanifold has to be totally real. Then in [7] he showed that for any such closed, compact, embedded Lagrangian there exists a holomorphic disk with boundary on it. Hence the integral of a primitive over the boundary is different from zero and the first Betti number of the Lagrangian submanifold cannot vanish, excluding the possibility that a three–sphere can be embedded into C 3 as a Lagrangian. A further analysis of these techniques led to more obstructions for the topology of such embeddings in [18] and [20]. For C 2 the classical obstructions restrict the classes of possible closed, compact surfaces
The Semiclassical VanVleck Formula. Application to the AharonovBohm Effect
, 2001
"... At the very beginning of the quantum theory, VanVleck (1928) proposed a nice approximation formula for the integral kernel of the time dependent propagator for the Schrodinger equation. This formula can be deduced from the Feynman path integral by a formal stationary phase argument. After the fonda ..."
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Cited by 7 (2 self)
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At the very beginning of the quantum theory, VanVleck (1928) proposed a nice approximation formula for the integral kernel of the time dependent propagator for the Schrodinger equation. This formula can be deduced from the Feynman path integral by a formal stationary phase argument. After the fondamental works by Hormander and Maslov on Fourierintegral operators, it became possible to give a rigorous mathematical proof of the Van Vleck formula. We present here a more direct and elementary proof, using propagation of coherent states. We apply this result to give a mathematical proof of the AharonovBohm effect observed on the time dependent propagator. This effect concerns a phase factor depending on the flux of a magnetic field, which can be non trivial, even if the particle never meets the magnetic field. 1 Introduction Let us consider the time dependent Schrodinger equation ih @/(t) @t = H(t)/(t); /(t 0 ) = f (1) t 0 is the initial time, f an initial state, H(t) a quantum Hami...
On Equivalence of Two Constructions of Invariants of Lagrangian Submanifolds
 Pacific J. Math
"... We give the construction ofsymplectic invariants which incorporates both the “infinite dimensional ” invariants constructed by Oh in 1997 and the “finite dimensional ” ones constructed by Viterbo in 1992. 1. Introduction. Let M be a compact smooth manifold. Its cotangent bundle T ∗M carries a natura ..."
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Cited by 6 (0 self)
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We give the construction ofsymplectic invariants which incorporates both the “infinite dimensional ” invariants constructed by Oh in 1997 and the “finite dimensional ” ones constructed by Viterbo in 1992. 1. Introduction. Let M be a compact smooth manifold. Its cotangent bundle T ∗M carries a natural symplectic structure associated to a Liouville form θ = pdq. For a given compactly supported Hamiltonian function H: T ∗M → R and a closed submanifold N ⊂ M Oh [30, 27] defined a symplectic invariants of