Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions

The topological string partition function Z(lambda,t,(t) over bar ) = exp(lambda F-2g-2(g)(t,(t) over bar) is calculated on a compact Calabi-Yau M. The F-g(t,(t) over bar) fulfil the holomorphic anomaly equations, which imply that psi = Z transforms as a wave function on the symplectic space H-3(M,Z). This defines it everywhere in the moduli space. M(M) along with preferred local coordinates. Modular properties of the sections F-g as well as local constraints from the 4d effective action allow us to fix Z to a large extent. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovo's theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic.