Symplectic
Topology, Geometry, and Gauge Theory Program
ARTICLE
Newsletter, June 2001
Coxeter Lectures: Alexander Givental
In the middle of March Professor Alexander Givental (UC Berkeley and
CalTech) gave the Coxeter Lecture Series, entitled "Gromov - Witten
invariants in higher genus". In his lectures he outlined his recent
spectacular proof of the so-called Virasoro conjecture for the Gromov-Witten
potential for complex projective spaces. This conjecture claims, roughly
speaking, that the Gromov-Witten potentials are annihilated by some
special differential operators constituting half of the Virasoro algebra.
In the series of three lectures Givental introduced a formula expressing
enumerative information about higher genus holomorphic curves in complex
projective spaces (and conjecturally in many other complex algebraic
manifolds) in terms of such information in genus zero. This subject
intertwines the Witten-Kontsevich intersection theory on Deligne-Mumford
moduli spaces of Riemann surfaces and axiomatic 2-dimensional topological
fields theory with elementary representation theory of loop groups.
Note that the results actually apply to the so-called virtual moduli
space of holomorphic curves. The latter is a regularization of the ordinary
moduli space done in such a way that the resulting space behaves as
if it were Fredholm regular (i.e. as if its actual dimension coincided
with the virtual dimension).
Addressed mostly to specialists, these lectures also served as a natural
extension of another highlight of the Symplectic Topology, Geometry,
and Gauge Theory Program, the intensive course on an intriguing new
subject, symplectic field theory, given two weeks earlier by Yasha Eliashberg
(Stanford University).
B. Khesin, University of Toronto and F. Lalonde, Université de
Montréal
