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

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.

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Unhealthier behaviors and lower socioeconomic position were the main factors associated with the unhealthier dietary pattern (fast food and sweets). This information should be considered in the development of health-promotion interventions.