This note is an introduction to our forthcoming paper. There we show how to construct the metaplectic representation using Feynman path integrals. We were led to this by our attempts to understand Atiyah's explanation of topological quantum field theory in.

Like Feynman's original approach in an action integral plays the role of a phase function. Unlike Feynman, we use paths in phase space rather than configuration space and use the symplectic action integral rather than the (classical) Lagrangian integral. We eventually restrict to (inhomogeneous) quadratic Hamiltonians so that the finite dimensional approximation to the path integral is a Gaussian integral. In evaluating this Gaussian integral the signature of a quadratic form appears. This quadratic form is a discrete approximation to the second variation of the action integral.

For Lagrangians of the form kinetic energy minus potential energy, evaluated on curves in configuaration space, the index of the second variation is well-defined and, via the Morse Index Theorem, related to the Maslov Index of the corresponding linear Hamiltonian system. The second variation of the symplectic action has both infinite index and infinite coindex. However, this second variation does have a well-defined signature via the aforementioned discrete approximation. This signature can be expressed in terms of the Maslov index of the corresponding linear Hamiltonian system. This is a symplectic analog of the Morse Index Theorem.

Our treatment is motivated by the formal similarity between Feynman path integrals and the Fourier integral operators of Hörmander.

Recall that a symplectic manifold is a 2n-dimensional smooth manifold M together with a closed nondegenerate 2-form ω. A symplectomorphism is a diffeomorphism of M which preserves ω and an n-dimensional submanifold L ⊂ M is called Lagrangian if ω vanishes on TL. Such structures arise naturally from Hamiltonian dynamics and geometric optics and they have been studied for many decades. The past ten years have seen a number of important developments and major breakthroughs in symplectic geometry as well as the discovery of new links with other subjects such as dynamical systems, topology, Yang-Mills theory, theoretical physics, and singularity theory.

Many of these new developments have been motivated by Gromov's paper on pseudoholomorphic curves in symplectic geometry. The role pseudoholomorphic curves play in Gromov's work is reminiscent of the role of self-dual Yang-Mills instantons in Donaldson's theory on smooth 4-manifolds. Gromov used pseudoholomorphic curves to prove a number of surprising and hitherto inaccessible results in symplectic geometry. For example he proved that there is no symplectic isotopy moving the unit ball in R2n through a hole in a hypersurface whose radius is smaller than 1 (a symplectic camel cannot pass through the eye of a needle). The paper by McDuff and Traynor below gives a proof of this theorem which is based on Eliashberg's techniques of filling by pseudoholomorphic discs.

Moduli spaces of pseudoholomorphic curves also play an important role in McDuff's work on symplectic 4-manifolds.

Introduction

A gradient flow of a Morse function on a compact Riemannian manifold is said to be of Morse-Smale type if the stable and unstable manifolds of any two critical points intersect transversally. For such a Morse-Smale gradient flow there is a chain complex generated by the critical points and graded by the Morse index. The boundary operator has as its (x, y)-entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the Morse indices is 1. The homology of this chain complex agrees with the homology of the underlying manifold M and this can be used to prove the Morse inequalities.

Around 1986 Floer generalized this idea and discovered a powerful new approach to infinite dimensional Morse theory now called Floer homology. He used this approach to prove the Arnold conjecture for monotone symplectic manifolds and discovered a new invariant for homology 3-spheres called instanton homology. This invariant can roughly be described as the homology of a chain complex generated by the irreducible representations of the fundamental group of the homology 3-sphere M in the Lie group SU(2). These representations can be thought of as flat connections on the principal bundle M × SU(2) and they appear as the critical points of the Chern-Simons functional on the infinite dimensional configuration space of connections on this bundle modulo gauge equivalence.