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Topological string theory on compact Calabi-Yau: Modularity and boundary conditions. (English) Zbl 1166.81358

Kapustin, Anton (ed.) et al., Homological mirror symmetry. New developments and perspectives. Berlin: Springer (ISBN 978-3-540-68029-1/hbk). Lecture Notes in Physics 757, 45-102 (2009).
Summary: The topological string partition function \(Z(\lambda ,t,\bar t) =\exp(\lambda ^{2g-2} F_{g}(t, \bar t))\) is calculated on a compact Calabi-Yau \(M\). The \(F_{g}(t, \bar t)\) fulfil the holomorphic anomaly equations, which imply that \(\psi =Z\) transforms as a wave function on the symplectic space \(H^3(M, \mathbb Z)\). This defines it everywhere in the moduli space \(\mathcal M(M)\) 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.
For the entire collection see [Zbl 1151.81001].

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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