[Paper Review] Open-closed Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric DM stacks

Bohan Fang, Chiu-Chu Melissa Liu+1 more|arXiv (Cornell University)|Dec 25, 2012

Algebraic Geometry and Number Theory6 citations

TL;DR

This paper establishes a comprehensive framework for open-closed Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford stacks relative to Aganagic-Vafa branes, using torus localization and algebraic relative orbifold Gromov-Witten theory. It proves an open mirror theorem linking the genus-zero disk potential to Abel-Jacobi maps of the mirror curve, generalizing earlier results for smooth Calabi-Yau 3-folds to the orbifold setting with non-trivial stabilizers.

ABSTRACT

We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford (DM) stacks (with possibly non-trivial generic stabilizers and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack with boundaries mapped into a Aganagic-Vafa brane. All genus open-closed Gromov-Witten invariants are defined by torus localization and depend on the choice of a framing which is an integer. We also provide another definition of all genus open-closed Gromov-Witten invariants based on algebraic relative orbifold Gromov-Witten theory; this generalizes the definition in Li-Liu-Liu-Zhou [arXiv:math/0408426] for smooth toric Calabi-Yau 3-folds. When the toric DM stack a toric Calabi-Yau 3-orbifold (i.e. when the generic stabilizer is trivial), we define generating functions of open-closed Gromov-Witten invariants or arbitrary genus $g$ and number $h$ of boundary circles; it takes values in the Chen-Ruan orbifold cohomology of the classifying space of a finite cyclic group of order $m$. We prove an open mirror theorem which relates the generating function of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of the toric Calabi-Yau 3-orbifold. This generalizes a conjecture by Aganagic-Vafa [arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa [arXiv:hep-th/0105045] (proved in full generality by the first and the second authors in [arXiv:1103.0693]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.

Motivation & Objective

• To define all-genus open-closed Gromov-Witten invariants for 3-dimensional Calabi-Yau smooth toric Deligne-Mumford stacks relative to Aganagic-Vafa branes.
• To generalize the definition of open-closed invariants using algebraic relative orbifold Gromov-Witten theory, ensuring compatibility with complex structure orientations.
• To construct generating functions $ F_{g,h}^{ ilde{X},( ilde{L},f)} $ with values in Chen-Ruan cohomology for toric orbifolds when the generic stabilizer is trivial.
• To prove an open mirror theorem relating the genus-zero disk potential $ F_{0,1}^{ ilde{X},( ilde{L},f)} $ to Abel-Jacobi maps of the mirror curve.
• To extend the Aganagic-Vafa mirror conjecture to the orbifold case, including non-trivial generic stabilizers and framing choices.

Proposed method

• Define open-closed invariants via torus localization on moduli spaces of stable holomorphic maps from bordered orbifold Riemann surfaces to $ ilde{X} $, with boundaries mapping to an Aganagic-Vafa brane $ ilde{L} $.
• Introduce a framing $ f eq 0 $ as a key parameter in the definition of invariants, affecting the generating functions and orientation data.
• Construct an alternative definition using algebraic relative orbifold Gromov-Witten theory, ensuring canonical orientation compatible with the complex structure on relative stable map moduli.
• For the case where $ ilde{X} $ is a toric orbifold (i.e. trivial generic stabilizer), define generating functions $ F_{g,h}^{ ilde{X},( ilde{L},f)} $ with values in $ H^{*}_{ ext{CR}}( ilde{B}oldsymbol{ u}_{rak{m}};C)^{igotimes h} $, where $ ilde{B}oldsymbol{ u}_{rak{m}} $ is the classifying space of $ oldsymbol{ u}_{rak{m}} $.
• Prove an open mirror theorem by relating $ F_{0,1}^{ ilde{X},( ilde{L},f)} $ to the Abel-Jacobi map of the mirror curve via inverse Laplace transforms and residue analysis of Barnes-type integrals.
• Use the inverse Laplace transform to decompose the generating function into two parts: $ h^+ $ (in powers of $ (-X)^{1/f} $) and $ h^- $ (in powers of $ X $), and show that the coefficients match via analytic continuation and residue computation.

Experimental results

Research questions

• RQ1 How can open-closed Gromov-Witten invariants be consistently defined for 3-dimensional Calabi-Yau smooth toric DM stacks with non-trivial generic stabilizers?
• RQ2 What is the correct algebraic-geometric formulation of relative open-closed invariants in the orbifold setting that respects the canonical complex orientation?
• RQ3 Can the open mirror symmetry conjecture of Aganagic-Vafa be generalized to the case of toric Calabi-Yau 3-orbifolds with non-trivial $ oldsymbol{ u}_{rak{m}} $-gerbes?
• RQ4 How do framing choices $ f eq 0 $ affect the generating functions and the mirror map in the orbifold case?
• RQ5 What is the precise structure of the genus-zero disk potential $ F_{0,1}^{ ilde{X},( ilde{L},f)} $ in terms of special functions and how does it relate to the Abel-Jacobi map of the mirror curve?

Key findings

• The paper constructs all-genus open-closed Gromov-Witten invariants for 3-dimensional Calabi-Yau smooth toric DM stacks relative to Aganagic-Vafa branes using torus localization, with invariants depending on a framing $ f \in \mathbb{Z} $.
• An algebraic definition of the invariants is provided via relative orbifold Gromov-Witten theory, which ensures compatibility with the canonical orientation from the complex structure on the moduli space of relative stable maps.
• For toric orbifolds (i.e. trivial generic stabilizer), the generating function $ F_{g,h}^{ ilde{X},( ilde{L},f)} $ takes values in $ H^{*}_{\text{CR}}(\tilde{B}\boldsymbol{\mu}_{\frak{m}};\mathbb{C})^{\bigotimes h} \cong \mathbb{C}^{\frak{m}} $, generalizing the state space for open invariants.
• The open mirror theorem is proven: $ F_{0,1}^{\tilde{X},(\tilde{L},f)} $ is related to the Abel-Jacobi map of the mirror curve via an inverse Laplace transform of a Barnes-type integral involving Gamma functions.
• The coefficients of the expansion of $ y $ in $ X $ are shown to match the residues of $ \Gamma(u) $ and $ \Gamma(fu + \sum r_a l_a) $, proving that the generating function decomposes into $ h^+ + h^- $, with $ h^+ $ capturing fractional powers and $ h^- $ capturing integer powers.
• The result is extended to all real $ r_a $ via analytic continuation, showing that the coefficient formula holds universally as a rational function of $ r_1, \dots, r_k $.