Journal ArticleDOI
A relative morse index for the symplectic action
TLDR
The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold.Abstract:
The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define for any pair of critical points of a a relative Morse index, which corresponds to the difference of the two Morse indices in finite dimensions. It is based on the spectral flow of the Hessian of a and can be identified with a topological invariant recently defined by Viterbo, and with the dimension of the space of trajectories between the two critical points.read more
Citations
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Holomorphic disks and topological invariants for closed three-manifolds
Peter Ozsvath,Zoltan Szabo +1 more
TL;DR: In this article, the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain real subspaces associated to U 0 and U 1 is introduced.
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Topological field theory
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The Maslov index for paths
Joel W. Robbin,Dietmar Salamon +1 more
TL;DR: In this paper, the authors define a Maslov index for any path regardless of where its endpoints lie, which is invariant under homotopy with fixed endpoints and additive for catenations.
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Morse theory for periodic solutions of hamiltonian systems and the maslov index
Dietmar Salamon,Eduard Zehnder +1 more
TL;DR: In this article, the authors prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over q(M).
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Symplectic fixed points and holomorphic spheres
TL;DR: In this paper, the authors consider the variational theory of the symplectic action function perturbed by a Hamiltonian term and associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading.
References
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Journal ArticleDOI
Pseudo holomorphic curves in symplectic manifolds
TL;DR: In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).
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Supersymmetry and Morse theory
TL;DR: In this paper, it was shown that the Morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics Hamiltonian, and some of the implications of modern ideas in mathematics for super-ymmetric theories are discussed.