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

TL;DR

This work develops a global solution to the topological B-model on compact Calabi–Yau threefolds by exploiting space-time modularity, the BCOV holomorphic anomaly equations, and boundary data from light BPS states. By recasting higher-genus amplitudes $F^{(g)}$ as polynomials in a finite set of generators and employing a universal conifold gap, orbifold regularity, and Castelnuovo bounds, the authors fix the holomorphic ambiguities up to very high genus (e.g., genus 51 for the quintic). The method extends from the quintic to 13 one-parameter Calabi–Yau spaces with three regular singular points, providing detailed boundary behavior, Picard–Fuchs data, and polynomial structures that yield explicit Gromov–Witten, Gopakumar–Vafa, and Donaldson–Thomas invariants. The results illuminate the full nonperturbative BPS spectrum across moduli, link to metaplectic transformations between bases, and demonstrate a highly efficient, modularly constrained approach to compact Calabi–Yau topological strings with potential extensions to multi-parameter moduli and gravity couplings.

Abstract

The topological string partition function Z=exp(lambda^{2g-2} F_g) is calculated on a compact Calabi-Yau M. The F_g fulfill the holomorphic anomaly equations, which imply that Z transforms as a wave function on the symplectic space H^3(M,Z). This defines it everywhere in the moduli space of 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 extend. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovos theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic.