Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

Constantin Teleman; Christopher Woodward

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Let X be a compact Riemann surface and associated to each point p-i of a finite subset S of X is a positive integer m-i. Fix an elliptic curve C. To this data we associate a smooth elliptic surface Z fibered over X. The group C acts on Z with X as the quotient. It is shown that the space of all vector bundles over Z equipped with a lift of the action of C is in bijective correspondence with the space of all parabolic bundles over X with parabolic structure over S and the parabolic weights...

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Let $G$ be a complex algebraic group, simple and simply connected, $T$ a maximal torus and $W$ the Weyl group. One shows that the coarse moduli space $M_{G} (X)$ parametrizing $S$ -equivalence classes of semistable $G$ -bundles over an elliptic curve $X$ is isomorphic to $[Γ (T) \otimes_{Z} X] / W$ . By a result of Looijenga, this shows that $M_{G} (X)$ is a weighted projective space.