Space Of Lagrangian Submanifolds Research Articles

Integration over families of Lagrangian submanifolds in BV formalism

Gauge fixing is interpreted in BV formalism as a choice of Lagrangian submanifold in an odd symplectic manifold (the BV phase space). A natural construction defines an integration procedure on families of Lagrangian submanifolds. In string perturbation theory, the moduli space integrals of higher genus amplitudes can be interpreted in this way. We discuss the role of gauge symmetries in this construction. We derive the conditions which should be imposed on gauge symmetries for the consistency of our integration procedure. We explain how these conditions behave under the deformations of the worldsheet theory. In particular, we show that integrated vertex operator is actually an inhomogeneous differential form on the space of Lagrangian submanifolds.

Families of gauge conditions in BV formalism

In BV formalism we can consider a Lagrangian submanifold as a gauge condition. Starting with the BV action functional we construct a closed form on the space of Lagrangian submanifolds. If the action functional is invariant with respect to some group H and Λ is an H-invariant family of Lagrangian submanifold then under certain conditions we construct a form on Λ that descends to a closed form on Λ/H. Integrating the latter form over a cycle in Λ/H we obtain numbers that can have interesting physical meaning. We show that one can get string amplitudes this way. Applying this construction to topological quantum field theories one obtains topological invariants.

Hofer’s metric on the space of Lagrangian submanifolds and wrapped Floer homology

In this paper, we study the relationship between wrapped Floer homology and displaceability of a Lagrangian submanifold which we call vanishing theorem of wrapped Floer homology. We also use this theorem to study Hofer’s pseudometric on the space of Lagrangian submanifolds. We prove an inequality, the Lagrangian version of the inequality of Gromov width and displacement energy, which is called energy-capacity inequality.

Geodesics of Hofer’s metric on the space of Lagrangian submanifolds

We study geodesics of Hofer?s metric on the space of Lagrangian submanifolds in arbitrary symplectic manifolds from the variational point of view. We give a characterization of length?critical paths with respect to this metric. As a result, we see that if two Lagrangian submanifolds are disjoint then we cannot join them by length-minimizing geodesics.

Geodesics on the space of Lagrangian submanifolds in cotangent bundles

We prove that the space of Hamiltonian deformations of zero section in a cotangent bundle of a compact manifold is locally flat in the Hofer metric and we describe its geodesics.

Connections of Berry and Hannay type for moving Lagrangian submanifolds

Berry's phase is the holonomy of the natural connection on the canonical circle bundle over projectivized quantum Hilbert space. Given a symplectic manifold P, a classical limit of this phase is constructed as the holonomy of a Berry connection over a finite-codimensional isodrastic foliation (defined by constancy of action integrals) on the space of lagrangian submanifolds in P equipped with smooth densities of total measure 1. If the densities are determined by a Kähler metric compatible with the symplectic structure, the curvature of the Berry connection at the lagrangian submanifold L is given by a simple formula involving curvature of L. In particular, the Berry connection defines a homotopy invariant for isodrastic loops of minimal lagrangian submanifolds in a simply connected Kähler manifold. A similar invariant constructed by the author for loops of symplectomorphisms is also a special case of the classical Berry phase. A classical analogue of Berry's phase was discovered by Hannay for moving families of completely integrable systems. Following Berry and Hannay, we interpret Hannay's angles as the holonomy of a Hannay connection on a bundle of tori over the isodrastic foliation on the space of lagrangian toral layers consisting of lagrangian tori with a flat affine structure and an extension of this structure to the first infinitesimal neighborhood. Finally, we show that the Hannay angles are the derivatives with respect to action variables of the classical Berry phase.

Cohomology of symplectomorphism groups and critical values of hamiltonians

Let (P, co) be a compact, simply connected symplectic manifold, Sym (P, co) its group of automorphisms. The period group F(P, co)c R is defined to be the image of [co] x H2 (P; Z) under the integration pairing H 2 (P; R) x H2 (P; Z) ~ R. Since [co] 4:0, the quotient G(P, co)=R/F(P, co) is either a circle or the quotient of a circle by a countable dense subgroup. The purpose of this paper is to define and study a natural homomorphism A from 7h (Sym (P, co)) to G (P, co). Simply stated, A measures the average over P (with respect to the symplectic measure) of the action integral around the trajectories of a loop of symplectomorphisms. In terms of hamiltonians, we give a formula for A which becomes especially simple when the loop is a 1-parameter subgroup, in which case we can make some concrete computations. These are related to the work of Guillemin-Sternberg [9], and Duistermaat-Heckmann [7] on momentum mappings. We also give an interpretation of A in terms of lifting symplectomorphisms from P to a prequantization, i.e. a G(P, co) bundle over P with a connection whose curvature form is co. Here, we extend the theory from the usual case where G(P, co) is a circle by using the concept of diffeological structure introduced by Souriau [16]. In a sequel to this paper [19], taking as a starting point the observation that the graphs of symplectomorphisms are lagrangian submanifolds in a product manifold, we shall study the possible extension of A to an invariant on loops in the space of lagrangian submanifolds in a symplectic manifold. It turns out that this invariant should be defined as the holonomy of a connection whose definition involves choosing a probability measure on each lagrangian submanifold. The connection is flat only for certain ways of choosing these measures, and the curvature in the general case is closely related to the "geometric phase" introduced by Berry [4], which is a subject of current interest among theoretical and experimental physicists as well as mathematicians [1, 6, I0, 11, 13, 173.