Quantum geometry of elliptic Calabi-Yau manifolds

TL;DR

The paper develops a global quantum geometry for elliptic Calabi–Yau threefolds by formulating a holomorphic anomaly equation that is iterative in genus and base class, with amplitudes expressed in terms of (quasi-)modular forms. It then connects topological string theory to D4-brane BPS invariants via T-duality on the elliptic fiber, and confirms this through explicit BPS calculations on the rational elliptic surface, matching modular predictions from mirror symmetry and BCOV theory. The authors provide concrete toric realizations, derive the genus-zero anomaly from the B-model, and analyze higher-genus structure and dualities, including monodromy and modular properties. They also compute and validate BPS invariants for small D4-brane ranks on the rational elliptic surface, using lattice theta decompositions and wall-crossing, thereby linking geometric data, modular forms, and brane counting in a coherent framework.

Abstract

We study the quantum geometry of the class of Calabi-Yau threefolds, which are elliptic fibrations over a two-dimensional toric base. A holomorphic anomaly equation for the topological string free energy is proposed, which is iterative in the genus expansion as well as in the curve classes in the base. T-duality on the fibre implies that the topological string free energy also captures the BPS-invariants of D4-branes wrapping the elliptic fibre and a class in the base. We verify this proposal by explicit computation of the BPS invariants of 3 D4-branes on the rational elliptic surface.

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