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Topological String Partition Functions as Polynomials

Satoshi Yamaguchi, Shing-Tung Yau

TL;DR

<3-5 sentence high-level summary> The paper demonstrates that higher-genus topological string partition functions on the quintic Calabi–Yau can be organized into degree-(3g−3) quasi-homogeneous polynomials in a finite set of generators, derived from mirror symmetry and the holomorphic anomaly equation. It establishes a systematic framework where genus-zero data feed the holomorphic anomaly calculations, proves the polynomial structure using an infinite generator set reduced to a finite basis, and provides explicit genus-2, genus-3, and genus-4 polynomial forms along with partial coefficients for all genus. The authors also analyze the v_3^n sector via a generating-function approach, exposing a nonperturbative facet linked to boundary data and Bernoulli-number contributions. They discuss how the holomorphic ambiguity can be constrained by additional dualities and consider generalizations to other CY hypersurfaces.

Abstract

We investigate the structure of the higher genus topological string amplitudes on the quintic hypersurface. It is shown that the partition functions of the higher genus than one can be expressed as polynomials of five generators. We also compute the explicit polynomial forms of the partition functions for genus 2, 3, and 4. Moreover, some coefficients are written down for all genus.

Topological String Partition Functions as Polynomials

TL;DR

<3-5 sentence high-level summary> The paper demonstrates that higher-genus topological string partition functions on the quintic Calabi–Yau can be organized into degree-(3g−3) quasi-homogeneous polynomials in a finite set of generators, derived from mirror symmetry and the holomorphic anomaly equation. It establishes a systematic framework where genus-zero data feed the holomorphic anomaly calculations, proves the polynomial structure using an infinite generator set reduced to a finite basis, and provides explicit genus-2, genus-3, and genus-4 polynomial forms along with partial coefficients for all genus. The authors also analyze the v_3^n sector via a generating-function approach, exposing a nonperturbative facet linked to boundary data and Bernoulli-number contributions. They discuss how the holomorphic ambiguity can be constrained by additional dualities and consider generalizations to other CY hypersurfaces.

Abstract

We investigate the structure of the higher genus topological string amplitudes on the quintic hypersurface. It is shown that the partition functions of the higher genus than one can be expressed as polynomials of five generators. We also compute the explicit polynomial forms of the partition functions for genus 2, 3, and 4. Moreover, some coefficients are written down for all genus.

Paper Structure

This paper contains 14 sections, 3 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Calculation of topological string amplitudes by mirror symmetry and the holomorphic anomaly equation
  3. Genus zero
  4. Genus one and higher
  5. Generators for the higher genus amplitudes
  6. Expression of amplitudes by infinite number of generators
  7. Relation between generators
  8. Back to holomorphic anomaly equation
  9. Some results for the coefficients of the polynomial representation
  10. Lower genus partition functions
  11. Coefficients of $v_3^n$
  12. Conclusion and Discussion
  13. Polynomial form of genus 3 and 4 partition function
  14. Generalization to the hypersurfaces in weighted projective space

Key Result

Proposition 1

Each $P_g^{(n)}$ is an degree $(3g-3+n)$ inhomogeneous polynomial of $A_p,B_p,X,\ (p=1,2,3,\dots)$, where we assign "degree" $p$ to $A_p$ and $B_p$, and $1$ to $X$.

Figures (1)

  • Figure 1: A diagram which contribute to $F_{4}$ the symmetric factor is $s_{\cal D}=2\times 2 \times 3! \times 2=48$.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Theorem 1