Topological quantum field theory and the Gromov-Witten theory of curves in Calabi-Yau threefolds -- Jim Bryan, March 5, 2004

Topological quantum field theory, as formulated by Atiyah, has provided a general framework for understanding invariants of manifolds. The structure of TQFTs in dimension 1+1 (i.e. surfaces with boundary) is completely understood by elementary means -- yet they can still yield surprising results. For each positive integer d, we define a one-parameter family of (1+1)-dimensional TQFTs Z_d(t) which specializes at t=0 to the famous Witten-Dijkgraaf-Freed-Quinn TQFT for gauge theory with gauge group the dth symmetric group. Our family of TQFTs completely encodes all the degree d local Gromov-Witten invariants of a curve (of arbitrary genus) in a Calabi-Yau threefold. This provides us with a "structure theorem" for these local invariants (a.k.a. multiple cover formulas). Using these ideas we completely determine the local invariants for d < 6.