On the equivariant Gromov-Witten Theory of P^2-bundles over curves

Abstract

We compute section class relative equivariant Gromov-Witten invariants of the total space of P^2-bundles of the form P(O+L1+L2)-->C where C is a genus g curve, O is the trivial bundle, and L1 (resp. L2) is an arbitrary line bundle of degree k1 (resp. k2) over C. We prove a gluing formula for the partition functions of these invariants. Using this gluing formula together with localization techniques, we construct three explicit 3x3 matrices G, U1 and U2 with entries in Q((u))(t0,t1,t2), where u is the genus parameter, and t0,t1,t2 are the equivariant parameters. Then we prove that the partition function of the section class, ordinary equivariant Gromov-Witten invariants of X is given by trace(G^(g-1).U1^k1.U2^k2). As an application, we establish a formula for the partition function of the ordinary Gromov-Witten invariants of any P^2-bundle X over a curve of genus g for any class which is a Calabi-Yau section class. We prove that this partition function is given by 3^g(2sin u/2)^(2g-2).