The symplectic vortex equations and invariants of Hamiltonian group actions

Abstract

In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relation between the invariants introduced here and the Seiberg-Witten invariants of a product of a Riemann surface with a two-sphere.where γ k,d denotes the spin c -structure determined by k and d. Moreover, if k > 2g − 2, then Φ M d,S ,µ d,S k,Σ (c m ) = Φ M k,Σ ,µ k,Σ d,S (c m ). Combining Theorems B and C one can recover the computation of the Seiberg-Witten invariants of product ruled surfaces by Li-Liu [21] and Ohta-Ono [28]. It is also interesting to examine the relation between our invariants and the Gromov-Witten invariants of the symplectic quotient M := M/ /G(τ ) := µ −1 (τ )/G whenever G acts freely on µ −1 (τ ). Such a relation was established in [15] under the hypothesis that the quotient is monotone. Under this condition (and hypotheses (H1 − 3)) it is shown in [15] that there exists a surjective ring homomorphism φ : H * G (M ) → QH * (M ) (with values in the quantum cohomology of the quotient) such that Φ M,µ−τ B,Σ

Vortex Equations in Abelian Gauged σ-Models

“…As a recent mathematical application, the vortex equations have been used to define the so-called Hamiltonian Gromov-Witten invariants [27,13,14,28]. This is described from a topological field theory point of view in [4].…”

Analysis of gauged Witten equation

“…So one can prove the C 0 -bound in the same way as in [CGMS02]. The difficulty lies in the perturbed case, where the perturbation term disturbs the control.…”

“…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…”

“…In gauged Gromov-Witten theory, one can achieve the C 0 -bound by imposing the equivariant convexity assumption on the target space (see [CGMS02,Page 555]). This is an assumption generalizing the convexity condition in [EG91].…”

“…The equation is now called the symplectic vortex equation. 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]).…”