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Holomorphic anomaly equations for $\mathbb{C}^5/\mathbb{Z}_5$

Deniz Genlik, Hsian-Hua Tseng

TL;DR

This work extends holomorphic anomaly techniques to the noncompact orbifold target $[oldsymbol{C}^5/oldsymbol{Z}_5]$ by exploiting genus-zero semisimple Frobenius structures and the Givental–Teleman classification. Central to the approach is lifting the modified $R$-matrix to a polynomial ring $\mathbb{F}$ and expressing all genus-$g$ potentials $\mathcal{F}_g$ as decorated graph sums with explicit vertex/edge/leg contributions. The authors derive two holomorphic anomaly equations for $g\ge 2$ after a particular equivariant specialization, tying $ rac{\partial \mathcal{F}_g}{\partial A_2}$ and $\frac{\partial \mathcal{F}_g}{\partial(D^2A_1)}$ to lower-genus correlators. Finite-generation in the ring $\mathbb{F}$ underpins the algebraic structure of the potentials and the validity of the HAE, with the mirror theorem and $I$-function machinery (via the mirror map $T(x)$) providing essential control of the dependence on the mirror parameter. The results broaden the holomorphic anomaly framework to orbifold and toric stacks, offering explicit, computable formulas for higher-genus invariants in this setting."

Abstract

We prove holomorphic anomaly equations for $\mathbb{C}^5/\mathbb{Z}_5$ based on the work arXiv:1707.02910 of Lho.

Holomorphic anomaly equations for $\mathbb{C}^5/\mathbb{Z}_5$

TL;DR

This work extends holomorphic anomaly techniques to the noncompact orbifold target by exploiting genus-zero semisimple Frobenius structures and the Givental–Teleman classification. Central to the approach is lifting the modified -matrix to a polynomial ring and expressing all genus- potentials as decorated graph sums with explicit vertex/edge/leg contributions. The authors derive two holomorphic anomaly equations for after a particular equivariant specialization, tying and to lower-genus correlators. Finite-generation in the ring underpins the algebraic structure of the potentials and the validity of the HAE, with the mirror theorem and -function machinery (via the mirror map ) providing essential control of the dependence on the mirror parameter. The results broaden the holomorphic anomaly framework to orbifold and toric stacks, offering explicit, computable formulas for higher-genus invariants in this setting."

Abstract

We prove holomorphic anomaly equations for based on the work arXiv:1707.02910 of Lho.
Paper Structure (12 sections, 13 theorems, 98 equations)

This paper contains 12 sections, 13 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. On mirror theorem
  4. Frobenius Structures
  5. Lift of modified R-matrix
  6. Canonical Lift
  7. Preparations
  8. Holomorphic anomaly equations
  9. Formula for potentials
  10. Graphs
  11. Formula for $\mathcal{F}_g$
  12. Proof of holomorphic anomaly equations

Key Result

Proposition 1.1

We have the following mirror identity, with the mirror transformationThe gamma function is defined by $\Gamma(z)=\int_0^{\infty} t^{z-1} e^{-t} d t$ where $\Re(z)>0$. One of its fundamental properties is $\Gamma (z+1)=z \Gamma (z)$. We mainly use the gamma function in the expression of mirror transformation (eq:smallmirrorthm) to give a

Theorems & Definitions (20)

  • Proposition 1.1
  • Lemma 1.2
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 4.1
  • Theorem 4.2: Finite generation property
  • ...and 10 more