Abstract
We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly equations. The key examples are smooth elliptic fibrations over (almost) Fano surfaces. We include a detailed review of existing mathematical methods towards defining and calculating the (unrefined) Gopakumar-Vafa invariants (GVI) and the GVNs on compact Calabi-Yau 3-folds using moduli of stable sheaves, in a language that should be accessible to physicists. In particular, we discuss the dependence of the GVNs on the complex structure moduli and on the choice of an orientation. We calculate the GVNs in many instances and compare the B-model predictions with the geometric calculations. We also derive the modular anomaly equations from the holomorphic anomaly equations by analyzing the quasi-modular properties of the propagators. We speculate about the physical relevance of the mathematical choices that can be made for the orientation.
Highlights
We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly equations
We include a detailed review of existing mathematical methods towards defining and calculating the Gopakumar-Vafa invariants (GVI) and the GVNs on compact Calabi-Yau 3folds using moduli of stable sheaves, in a language that should be accessible to physicists
The B-model approach using a refinement [32,33,34] of BCOV (Bershadsky-CecottiOoguri-Vafa) holomorphic anomaly equations [35] supplemented by boundary conditions at the points of parabolic monodromy combined with the modular ansatz can be extended to calculate the refined Gopakumar-Vafa invariants (GVI’s) using the refined holomorphic anomaly equations and refined boundary conditions [32, 34] very efficiently
Summary
The B-model approach using a refinement [32,33,34] of BCOV (Bershadsky-CecottiOoguri-Vafa) holomorphic anomaly equations [35] supplemented by boundary conditions at the points of parabolic monodromy combined with the modular ansatz can be extended to calculate the refined GVI’s using the refined holomorphic anomaly equations and refined boundary conditions [32, 34] very efficiently It applies to local (toric) Calabi-Yau geometries [32, 34], which have B-model mirror whose compact part is a Riemann surface with a meromorphic differential; see [22, 36].
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