Quantum Cohomologies of Symmetric Products

Abstract

In this paper we investigate the quantum cohomologies of symmetric products of Kähler manifolds. To do this we study the moduli space of product space and symmetric group action on it, Gromov-Witten invariant and relative Gromov-Witten invariant. Also we investigate the relations between symmetric invariant properties on the products space and the corresponding ones on the symmetric product. As an example we examine the symmetric product of k copies complex projective line P 1 , which is the k-dimensional complex projective space P k .

Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds

“…The moduli space [4,5]. The quantum cohomology of N is QH Ã ðNÞ¼H Ã ðNÞ⊗Λ where Λ is the universal Novikov ring [5].…”

“…In Section 2, we introduce basic definitions on symplectic manifolds [1][2][3][4][5][10][11][12][13] and flux homomorphism. In Section 2.1, we recall J-holomorphic curve, moduli space of J-holomorphic curves, Gromov-Witten invariant and Gromov-Witten potential, quantum product and quantum cohomology, and in Section 2.2, symplectic action functional and its gradient flow line, Maslov type index of critical loop, Floer cochain complex and Floer cohomology, and theorem of Arnold conjecture.…”

“…In Section 3, we introduce almost contact metric manifolds ðM, g, ϕ, η, ξ, φÞ with a closed fundamental 2-form φ and their product [4,7,8]. In Section 3.1, we study ϕ-coholomorphic map, moduli space of ϕ-coholomorphic maps which represent a homology class of dimension two, Gromov-Witten type cohomology, quantum type product and quantum type cohomology, Gromov-Witten type potentials on the product of a symplectic manifold, and an almost contact metric manifold [5,6,13].…”