Gromov-witten Invariants of the Moduli of Bundles on a Surface

Let Σ = Σg be a compact Riemann surface of genus g ≥ 2 and let MΣ stand for the moduli space of rank two stable vector bundles on Σ with odd (and fixed) determinant. In [5] the author produced a presentation for the quantum cohomology ring QH(MΣ) in terms of its natural generators by giving the relations satisfied by them. Here we want to show that this information yields all the multiple-point Gromov-Witten invariants on the generators, or which is equivalent, all 3-point Gromov-Witten invariants on the elements which are quantum products of the generators. On the other hand consider the (instanton) Floer cohomology HF (Y ) of the threemanifold Y = Σ × S endowed with the SO(3)-bundle with w2 = P.D.[S ] ∈ H(Y ;Z/2Z), which is determined in [4]. We show that the ring structure of HF (Y ) yields the Donaldson invariants D1 S of the algebraic surface S = Σ×P , for Kähler metrics whose period point is close enough to Σ in the Kähler cone, U(2)-bundles whose first Chern class c1 ∈ H (S,Z) satisfies c1 · Σ ≡ 1 (mod 2), and on any collection of homology classes coming from Σ ⊂ S. Moreover the isomorphism QH(MΣ) ∼= HF (Σ × S) gives an equality between these Donaldson invariants of S and the multiple-point Gromov-Witten invariants of MΣ on the generators.