Gromov-Witten invariants of symplectic quotients and adiabatic limits

• Main Authors: Gaio, A. Rita, Salamon, Dietmar A
• Format: Journal Article
• Language: English
• Published: 19-06-2001
• Subjects: Mathematics - Symplectic Geometry
• DOI: 10.48550/arxiv.math/0106157

We study pseudoholomorphic curves in symplectic quotients as adiabatic limits
of solutions of a system of nonlinear first order elliptic partial differential
equations in the ambient symplectic manifold. The symplectic manifold carries a
Hamiltonian group action. The equations involve the Cauchy-Riemann operator
over a Riemann surface, twisted by a connection, and couple the curvature of
the connection with the moment map. Our main theorem asserts that the genus
zero invariants of Hamiltonian group actions defined by these equations are
related to the genus zero Gromov--Witten invariants of the symplectic quotient
(in the monotone case) via a natural ring homomorphism from the equivariant
cohomology of the ambient space to the quantum cohomology of the quotient.