N ov 2 00 4 The local Gromov - Witten theory of curves

We study the equivariant Gromov-Witten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the Gromov-Witten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven jointly with C. Faber and A. Okounkov in the appendix. X and N is equipped with the anti-diagonal C *-action, the partition function is ρ⊢d d! dim Q ρ 2g−2 where Q = e iu , u is the genus parameter, and the sum is over irre-ducible representations of the symmetric group S d. The formula is a Q-deformation of the classical Hurwitz formula for counting unramified covers. (iii) An equivariant version of the Gromov-Witten/Donaldson-Thomas correspondence is formulated and discussed in detail for the case of N. The theory generalizes the local Calabi-Yau theory of X defined and studied in [2, 4].