Gromov-Witten theory is expected to have an "open string" version
of the theory. It should be based on holomorphic maps of Riemann surfaces with boundaries to Calabi-Yau manifolds where the boundary maps to a special Lagrangian submanifold. An example of this theory that has been studied by Katz-Liu and Li-Song is the multiple cover contributions of a disk in a Calabi-Yau 3-fold (a.k.a. the local Gromov-Witten invariants of a disk). We show that local invariants can be defined that describe the multiple cover contributions of an arbitrary Riemann surface with boundary. We show that these invariants can be organized into a 1+1 dimensional Topological Quantum Field Theory and we prove that it is semi-simple. This
leads to a structure theorem for the local invariants and in particular leads to new multiple cover formulas, even for the usual (closed string) local Gromov-Witten invariants.
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