Gromov--Witten invariants of symplectic quotients and adiabatic limits

Abstract

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.

Corrigendum to “Self-dual instantons and holomorphic curves”

“…On page 633 the assertion that the limits A ∞ (θ) and Φ ∞ (θ) exist can be proved by a similar argument as in [2,Prop. 11.1].…”

“…Let Ξ ν = A ν + Φ ν ds + Ψ ν dt be a sequence of solutions of the equation (2), with ( * s , X s ) replaced by ( * νs , X νs ), on Ω ν × P such that…”

Wall crossing for symplectic vortices and quantum cohomology

“…Given λ ∈ Λ(τ ) we denote by GWM λ the genus zero Gromov-Witten invariant ofM with fixed marked points in the homology classλ. In [8,Theorem A] it is proved that, for every λ ∈ Λ(τ ) and every n-tuple ℓ = (ℓ 1 , . .…”

Analysis of gauged Witten equation

“…We recall certain differential operators naturally associated to the triple (u, φ, ψ) (cf. [CGMS02] and [GS05] for more comprehensive treatment of such operators). For any ξ ∈ Γ(U, u * T X), we define…”

“…Using the moduli space of solutions to the symplectic vortex equation, certain invariants of Hamiltonian G-manifolds, called the gauged (or Hamiltonian) Gromov-Witten invariants can be defined (see [Mun03], [CGMS02], [MT] etc.). On the other hand, such invariants are closely related to the Gromov-Witten invariants of the symplectic quotient of X: in the "adiabatic limit" the symplectic vortex equation reduces to J-holomorphic curves in the symplectic quotient (see [GS05]). Therefore, the gauged Gromov-Witten invariants also relate the Gromov-Witten invariants of different symplectic/GIT quotients (cf.…”