Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gromov-Witten Generating Series of Elliptic Curves and Iterated Integrals of Eisenstein-Kronecker Forms

Jie Zhou

Abstract

We compute various types of iterated integrals of Eisenstein-Kronecker forms that are constructed from the Kronecker theta function. Furthermore, we relate the generating series of Gromov-Witten invariants of elliptic curves to these iterated integrals.

Gromov-Witten Generating Series of Elliptic Curves and Iterated Integrals of Eisenstein-Kronecker Forms

Abstract

We compute various types of iterated integrals of Eisenstein-Kronecker forms that are constructed from the Kronecker theta function. Furthermore, we relate the generating series of Gromov-Witten invariants of elliptic curves to these iterated integrals.
Paper Structure (18 sections, 12 theorems, 119 equations, 1 figure, 1 table)

This paper contains 18 sections, 12 theorems, 119 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jacobi theta functions and Eisenstein-Kronecker series
  4. Laurent coefficients of the Kronecker theta function
  5. Ring of almost-elliptic functions
  6. Singularities and differential relations of Laurent coefficients
  7. Iterated integrals of Eisenstein-Kronecker forms
  8. Eisenstein-Kronecker forms
  9. Iterated residues and iterated regularized integrals
  10. Reduction from $\mathcal{V}_{\mathsf{D}}$ to $\mathcal{V}_{\mathsf{D}}^{\circ}$
  11. Evaluation of iterated residues and iterated regularized integrals
  12. Iterated $A$-cycle integrals and Chen's iterated path integrals
  13. Relation between iterated $A$-cycle integrals and iterated regularized integrals
  14. Chen's iterated path integrals and KZB connection
  15. Gromov-Witten generating series of elliptic curves
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume the following notation. Then the regularized integral Li:2020regularized${ \vcenter{\hbox{$-$}}}\!\int_{E^{n}}\omega_{n}$ satisfies

Figures (1)

  • Figure 1: Indicating graphs that are chain, loop, tree, respectively. Here the labellings of the vertices are omitted.

Theorems & Definitions (42)

  • Theorem 1.1: =Theorem \ref{['mainthm1']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 32 more