Abstract

We give evidence that the all genus amplitudes of topological string theory on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degree. The denominators of these forms have a simple universal form with the property that the poles of the meromorphic form lie only at torsion points. The modular parameter corresponds to the fibre class while the role of the string coupling is played by the elliptic parameter. This leads to very strong all genus results on these geometries, which are checked against results from curve counting. The structure can be viewed as an indication that an N=2 analog of the reciprocal of the Igusa cusp form exists that might govern the topological string theory on these Calabi-Yau manifolds completely.

Highlights

  • Introduction and summaryTopological strings on non-compact Calabi-Yau geometries are solvable and have a very interesting structure with a wealth of connections to gauge theories, integrable models, large N-dualities, Chern-Simons theories, supersymmetric localisation and matrix models

  • We give evidence that the all genus amplitudes of topological string theory on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degree

  • The modular parameter corresponds to the fibre class while the role of the string coupling is played by the elliptic parameter

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Summary

Introduction and summary

Topological strings on non-compact Calabi-Yau geometries are solvable and have a very interesting structure with a wealth of connections to gauge theories, integrable models, large N-dualities, Chern-Simons theories, supersymmetric localisation and matrix models. Knowing these actions we can restrict the possible constants in the rational ambiguity fg to roughly one fourth Using this information, the conifold gap and the regularity at the other points in the moduli space, especially the orbifold point we can solve the topological string for all classes to genus 9 on an elliptic fibration over P2, with one section. The conifold gap and the regularity at the other points in the moduli space, especially the orbifold point we can solve the topological string for all classes to genus 9 on an elliptic fibration over P2, with one section This is already the strongest available result for compact multiparameter Calabi-Yau manifolds and needs only very mild results on the vanishing of the BPS numbers. The formulas we find suggest a possible refinement, which we shortly indicate in (5.1.2)

The elliptic Calabi-Yau manifolds and their mirrors
Construction of toric hypersurface Calabi-Yau spaces
Integral symplectic basis and genus zero topological string amplitudes
Involution symmetry and BCOV formalism
The involution symmetry
Monodromy group versus involution symmetry
The local limit
Involution symmetry at genus one
Involution symmetry at higher genus
The propagators and rigid special geometry
Projective special Kahler manifolds
Involution symmetry on the propagators
Involution symmetry on the higher genus amplitudes
Fiber modularity
The modular anomaly equation
The ring of weak Jacobi forms
Weak Jacobi Forms and holomorphic anomaly equation
Exact formulae for base degree zero
Exact formulae for higher base degrees
BPS invariants
Physical definition of the BPS invariants
Unrefined BPS invariants
Refined BPS invariants
Geometry of curves
Examples and computations
A Gopakumar-Vafa invariants
B Derivation of the involution symmetry on the propagators
C Reducing the ambiguity with the involution symmetry
D Fibre modularity versus involution symmetry
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