[Paper Review] Obstruction classes and Open Gromov-Witten invariants

Vito Iacovino|arXiv (Cornell University)|Aug 16, 2011

Geometric and Algebraic Topology2 references3 citations

TL;DR

This paper presents a direct, obstruction-class-based definition of genus-zero open Gromov-Witten invariants for Calabi-Yau threefolds with relatively spin, Maslov index zero Lagrangian submanifolds. By recursively constructing chains using bounding chains and homotopy data, it defines rational invariants without relying on the potential function, showing that invariants are well-defined when obstruction classes vanish in rational homology, and proving their existence via recursive chain-level structures.

ABSTRACT

We define genus zero open Gromov-Witten invariants for Calabi-Yau three folds and relatively spin Lagrangian submanifold of Maslov index zero.

Motivation & Objective

• To provide a direct, potential-free definition of genus-zero open Gromov-Witten invariants for Calabi-Yau threefolds with Maslov index zero, relatively spin Lagrangians.
• To define invariants not only for relative homology classes but also for fixed symplectic areas, reducing the required conditions.
• To clarify the role of obstruction classes and their vanishing in ensuring the consistency and rationality of the invariants.
• To establish a link between open Gromov-Witten invariants and the bounding chains used in Lagrangian Floer homology, enabling a new construction of the latter.

Proposed method

• The method recursively defines open Gromov-Witten invariants by analyzing moduli spaces of multi-disks via decorated trees with vertices labeled by relative homology classes.
• Obstruction classes $ o(A) \in C_1(L, \mathbb{Q}) $ are introduced as singular chains associated to trees with one vertex and one external edge, measuring the failure of boundary conditions in the chain-level gluing.
• Bounding chains $ b(A) \in C_2(L, \mathbb{Q}) $ are used to resolve obstructions, with $ \partial b(A) = o(A) $, and their homotopy classes determine the rational invariants.
• For symplectic energy levels $ E_k $, the invariants $ F_0(E_k) $ are defined using energy-decorated trees, with obstructions $ o(E_k) $ and bounding chains $ b(E_k) $ defined recursively.
• In the presence of an anti-symplectic involution, the symmetry forces all obstruction classes to vanish, allowing $ b(E) = 0 $, and recovering invariants from previous work.
• The construction avoids chain-level structures of the potential, instead relying on homotopy classes of chains and rational coefficients to ensure rationality of the invariants.

Experimental results

Research questions

• RQ1 Can open Gromov-Witten invariants be defined directly without using the generating potential function?
• RQ2 What are the precise obstruction classes that prevent the definition of rational invariants in the open Gromov-Witten theory?
• RQ3 How do bounding chains relate to the consistency of the invariant construction and to Lagrangian Floer homology?
• RQ4 Under what conditions do obstruction classes vanish, and how does this affect the rationality and well-definedness of the invariants?
• RQ5 Can the invariants be defined for fixed symplectic areas rather than relative homology classes, and what are the implications for the obstruction structure?

Key findings

• For each relative homology class $ A \in H_2(X,L) $, the obstruction class $ o(A) \in C_1(L,\mathbb{Q}) $ is well-defined if all smaller classes have defined $ o(A') $ and $ b(A') $, enabling recursive construction.
• The open Gromov-Witten invariant $ F_0(A) \in \mathbb{Q} $ is defined if $ o(A) $ is defined and $ [o(A)] = 0 $ in $ H_1(L,\mathbb{Q}) $, with $ b(A) \in C_2(L,\mathbb{Q}) $ satisfying $ \partial b(A) = o(A) $.
• When the Lagrangian arises as the fixed locus of an anti-symplectic involution, all obstruction classes vanish, and $ b(E) = 0 $, leading to invariants matching those in [5].
• The invariants $ F_0(A) $ are manifestly rational numbers, a key advantage over potential-based definitions.
• The construction provides a new, chain-level approach to Lagrangian Floer homology in any dimension, without requiring the Calabi-Yau condition.
• For symplectic energy levels $ E_k $, the invariants $ F_0(E_k) $ are defined recursively, depending on the homotopy class of $ b(E_{k'}) $ for $ k' < k $.