INTRODUCTION 235by E. Witten and motivated by ideas of 1 + 1-dimensional conformal field theory.
Witten showed that various enumerative invariants proposed by Gromov in order
to distinguish symplectic structures actually obey numerous universal identities -
to regrets of symplectic topologists and benefits of algebraic geometers.- Several remarkable applications of such identities to enumeration of holomor-
phic curves and especially the so called mirror conjecture inspired an algebraic-
geometrical approach to Gromov-Witten invariants, namely-Kontsevich's project
(1994) of stable maps. The successful completion of the project in 1996 by several
(groups of) authors (K. Behrend and B. Fantechi, J. Li & G. Tian, Y. Ruan, ... )
and the proof of the Arnold-Morse inequality in general symplectic manifolds (K.
Fukaya & K. Ono, 1996) based on similar ideas make intersection theory in moduli
spaces of stable maps the most efficient technique in symplectic topology.
Exercise. Let z = p + iq be a complex variable and z(t) = I:kEZ Zk expikt be
the Fourier series of a p eriodic function. Show that the symplectic area f pdq is
the indefinite quadratic form f pdq = 7f I: klzk 12 on the loop space DC. Deduce
that gluing Morse cell complexes from unstable disks of critical points in the case
of action functionals on loop spaces LX would give rise to contractible topological
spaces. (This exercise shows that Morse-Floer theory has to deal with cycles of
infinite dimension and codimension rather then with usual homotopy invariants of
loop spaces.)