Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds

Song Yu, Zhengyu Zong

TL;DR

The paper develops open WDVV theory for toric Calabi-Yau 3-folds with Aganagic-Vafa branes by leveraging an open/closed correspondence with a toric Calabi-Yau 4-fold. It constructs two parallel formalisms: a semi-simple formal Frobenius manifold encoding open and closed Gromov-Witten data and a flat formal $F$-manifold capturing the same data in a different algebraic framework, including a nilpotent open direction. The approach hinges on relating $F_0^{X,T'}$ and $F_{0,1}^{X,(L,f)}$ to the genus-zero potential of the CY 4-fold $\widetilde{X}$ and proving corresponding open WDVV equations. Together, these results provide a structured, recursive, and algebraically robust way to organize and compute open and closed GW invariants in this toric geometric setting, with potential connections to variations of Hodge structures and open-closed maps.

Abstract

Let $X$ be a toric Calabi-Yau 3-fold and let $L\subset X$ be an Aganagic-Vafa outer brane. We prove two versions of open WDVV equations for the open Gromov-Witten theory of $(X,L)$. The first version of the open WDVV equation leads to the construction of a semi-simple (formal) Frobenius manifold and the second version leads to the construction of a flat (formal) $F$-manifold.

Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds

TL;DR

The paper develops open WDVV theory for toric Calabi-Yau 3-folds with Aganagic-Vafa branes by leveraging an open/closed correspondence with a toric Calabi-Yau 4-fold. It constructs two parallel formalisms: a semi-simple formal Frobenius manifold encoding open and closed Gromov-Witten data and a flat formal -manifold capturing the same data in a different algebraic framework, including a nilpotent open direction. The approach hinges on relating and to the genus-zero potential of the CY 4-fold and proving corresponding open WDVV equations. Together, these results provide a structured, recursive, and algebraically robust way to organize and compute open and closed GW invariants in this toric geometric setting, with potential connections to variations of Hodge structures and open-closed maps.

Abstract

Let be a toric Calabi-Yau 3-fold and let be an Aganagic-Vafa outer brane. We prove two versions of open WDVV equations for the open Gromov-Witten theory of . The first version of the open WDVV equation leads to the construction of a semi-simple (formal) Frobenius manifold and the second version leads to the construction of a flat (formal) -manifold.
Paper Structure (41 sections, 15 theorems, 134 equations)

This paper contains 41 sections, 15 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Historical background and motivation
  3. WDVV equations and Frobenius manifolds
  4. Open WDVV equations and $F$-manifolds
  5. Open WDVV equations for toric Calabi-Yau 3-folds
  6. Statement of the main results
  7. A formal Frobenius manifold
  8. A flat formal $F$-manifold
  9. Future works
  10. Recursion for open Gromov-Witten invariants of toric Calabi-Yau 3-folds
  11. Open-closed map and variation of Hodge structures
  12. Outline of the paper
  13. Geometric setup
  14. Notations for toric geometry
  15. Open geometry
  16. ...and 26 more sections

Key Result

Theorem 1.1

For any $i,j,k,l\in\{1,\dots,m\}$, the following WDVV equation holds

Theorems & Definitions (34)

  • Theorem 1.1: MS94RT94
  • Definition 1.2
  • Theorem 1.3: HSST19
  • Definition 1.4
  • Proposition 1.5: See Proposition \ref{['prop:openWDVV1']}
  • Theorem 1.6: See Theorems \ref{['thm:Frob']}, \ref{['thm:decomp']}
  • Remark 1.7
  • Proposition 1.8: See Proposition \ref{['prop:openWDVV2']}
  • Theorem 1.9: See Theorem \ref{['thm:F']}
  • Remark 1.10
  • ...and 24 more