The role of the Legendre transform in the study of the Floer complex of cotangent bundles

Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM -> R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Phi from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Psi of Phi. Contrary to other previously defined maps going in the same direction, Psi is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Phi and Psi are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of TM on 2-tori in M

Unstable geodesics and topological field theory

A topological field theory is used to study the cohomology of mapping space. The cohomology is identified with the BRST cohomology realizing the physical Hilbert space and the coboundary operator given by the calculations of tunneling between the perturbative vacua. Our method is illustrated by a simple example.Comment: 28 pages, OCU-15

Squeezing in Floer theory and refined Hofer-Zehnder capacities of sets near symplectic submanifolds

We use Floer homology to study the Hofer-Zehnder capacity of neighborhoods near a closed symplectic submanifold M of a geometrically bounded and symplectically aspherical ambient manifold. We prove that, when the unit normal bundle of M is homologically trivial in degree dim(M) (for example, if codim(M) > dim(M)), a refined version of the Hofer-Zehnder capacity is finite for all open sets close enough to M. We compute this capacity for certain tubular neighborhoods of M by using a squeezing argument in which the algebraic framework of Floer theory is used to detect nontrivial periodic orbits. As an application, we partially recover some existence results of Arnold for Hamiltonian flows which describe a charged particle moving in a nondegenerate magnetic field on a torus. We also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper40.abs.htm

The Physics Inside Topological Quantum Field Theories

We show that the equations of motion defined over a specific field space are realizable as operator conditions in the physical sector of a generalized Floer theory defined over that field space. The ghosts associated with such a construction are found not to be dynamical. This construction is applied to gravity on a four dimensional manifold, $M$; whereupon, we obtain Einstein's equations via surgery, along $M$, in a five-dimensional topological quantum field theory.Comment: LaTeX, 7 page

The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems

An equivariant version of Conley's homotopy index theory for flows is described and used to find periodic solutions of a Hamiltonian system locally near an equilibrium point which is at resonanc

Symplectic Floer homology of area-preserving surface diffeomorphisms

The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class.Comment: 57 pages, 4 figures. Revision for publication, with various minor corrections. Adds results on the module structure and invariance thereo

Morse homology for the heat flow

We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically to nondegenerate closed geodesics of Morse index difference one.Comment: 89 pages, 3 figure

Quantum cohomology of flag manifolds and Toda lattices

We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.Comment: 35 page

Anti-self-dual instantons with Lagrangian boundary conditions II: Bubbling

We study bubbling phenomena of anti-self-dual instantons on \H^2\times\S, where $\S$ is a closed Riemann surface. The restriction of the instanton to each boundary slice $\{z\}\times\S$, z\in\pd\H^2 is required to lie in a Lagrangian submanifold of the moduli space of flat connections over $\S$ that arises from the restrictions to the boundary of flat connections on a handle body. We establish an energy quantization result for sequences of instantons with bounded energy near $\{0\}\times\S$: Either their curvature is in fact uniformly bounded in a neighbourhood of that slice (leading to a compactness result) or there is a concentration of some minimum quantum of energy. We moreover obtain a removable singularity result for instantons with finite energy in a punctured neighbourhood of $\{0\}\times\S$. This completes the analytic foundations for the construction of an instanton Floer homology for 3-manifolds with boundary. This Floer homology is an intermediate object in the program proposed by Salamon for the proof of the Atiyah-Floer conjecture for homology-3-spheres. In the interior case, for anti-self-instantons on $\R^2\times\S$, our methods provide a new approach to the removable singularity theorem by Sibner-Sibner for codimension 2 singularities with a holonomy condition.Comment: 44 pages. Some corrections and rearrangements in section 5: Theorem 5.1 (now 5.3) was previously stated with incorrect assumption

Seiberg-Witten-Floer Homology and Gluing Formulae

This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion \spinc structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.Comment: 63 pages, LaTe