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Topological String on elliptic CY 3-folds and the ring of Jacobi forms

Min-xin Huang, Sheldon Katz, Albrecht Klemm

TL;DR

The paper analyzes topological string theory on compact elliptically fibered Calabi–Yau threefolds, revealing that all-genus amplitudes organize into meromorphic Jacobi forms with weights growing linearly and indices growing quadratically with base degree. A Z2 involution provides a fibre-modularity constraint that, together with BCOV holomorphic anomaly equations, sharply reduces the holomorphic ambiguity and enforces a Jacobi-form structure for the base-degree sector, yielding exact base-zero results and a universal formula for higher base degrees. The authors derive and test explicit base-degree formulae Z_{d_B}(τ,z) = φ_{d_B}(τ,z) / [η(τ)^{36 d_B} ∏_{s=1}^{d_B} φ_{-2,1}(τ, s z)], where φ_{d_B} is a weak Jacobi form of weight 16 d_B and quadratic-in-d_B index, connecting to GV/BPS invariants via an extended modular framework. They compute GV invariants from geometry for the X18 model (elliptic fibration over P^2), validate numerous cases up to d_B = 5 and genus up to 8, and discuss the implications for an N=2 analog of the reciprocal Igusa cusp form, highlighting deep connections between modular objects, curve counting, and topological strings on elliptic CYs.

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.

Topological String on elliptic CY 3-folds and the ring of Jacobi forms

TL;DR

The paper analyzes topological string theory on compact elliptically fibered Calabi–Yau threefolds, revealing that all-genus amplitudes organize into meromorphic Jacobi forms with weights growing linearly and indices growing quadratically with base degree. A Z2 involution provides a fibre-modularity constraint that, together with BCOV holomorphic anomaly equations, sharply reduces the holomorphic ambiguity and enforces a Jacobi-form structure for the base-degree sector, yielding exact base-zero results and a universal formula for higher base degrees. The authors derive and test explicit base-degree formulae Z_{d_B}(τ,z) = φ_{d_B}(τ,z) / [η(τ)^{36 d_B} ∏_{s=1}^{d_B} φ_{-2,1}(τ, s z)], where φ_{d_B} is a weak Jacobi form of weight 16 d_B and quadratic-in-d_B index, connecting to GV/BPS invariants via an extended modular framework. They compute GV invariants from geometry for the X18 model (elliptic fibration over P^2), validate numerous cases up to d_B = 5 and genus up to 8, and discuss the implications for an N=2 analog of the reciprocal Igusa cusp form, highlighting deep connections between modular objects, curve counting, and topological strings on elliptic CYs.

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.

Paper Structure

This paper contains 36 sections, 6 theorems, 274 equations, 1 figure, 13 tables.

Table of Contents

  1. Introduction and Summary
  2. The elliptic Calabi-Yau manifolds and their mirrors
  3. Construction of toric hypersurface Calabi-Yau spaces
  4. The $X_{18}(1,1,1,6,9)$ 3-fold, an elliptic fibration over $\mathbb{P}^2$
  5. Integral symplectic basis and genus zero topological string amplitudes
  6. Involution symmetry and BCOV formalism
  7. The involution symmetry
  8. Monodromy group versus involution symmetry
  9. The local limit
  10. Involution symmetry at genus one
  11. Involution symmetry at higher genus
  12. The propagators and rigid special geometry
  13. Projective special Kähler manifolds
  14. Involution symmetry on the propagators
  15. Involution symmetry on the higher genus amplitudes
  16. ...and 21 more sections

Key Result

Lemma 1

Let $C$ have degree $(d_E,d_B)$. Then $\pi(C)$ is a plane curve of degree $d_B$, including multiplicity.

Figures (1)

  • Figure 1: The figure shows the boundary of non-vanishing curves for the values of $d_B=1,2,3,4,5$. The dot on each $d_B$ line at $(g^c(d_B)=\frac{1}{2}(3 d_B^2-d_B+2),d^c_E(d_B)=3 d_B-1)$ indicates the value at which the boundary slope becomes generic and the numbers of BPS states on the line are given by (\ref{['genericvalue']}).

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • ...and 4 more