[Paper Review] Open Gromov-Witten theory without Obstruction

Vito Iacovino|arXiv (Cornell University)|Nov 14, 2017

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper introduces a new construction of open Gromov-Witten invariants that count pseudo-holomorphic curves with boundary of fixed Euler characteristic, without requiring the existence of a bounding chain—a key obstruction in prior approaches. By using unconnected decorated graphs and a novel multi-curve homology framework, the method bypasses traditional obstruction conditions, enabling invariants in arbitrary genus and for Lagrangians with non-trivial homology.

ABSTRACT

Open Gromov-Witten invariants are defined as cycles of the multi-curve chain complex, well defined up to isotopy.

Motivation & Objective

• To define open Gromov-Witten invariants for Lagrangian submanifolds without requiring the existence of a bounding chain, which previously obstructed invariants for non-trivial homology classes.
• To overcome the technical limitations of the open Gromov-Witten potential and $A_inity$ structures by directly constructing invariants from moduli spaces of multi-curves.
• To extend the construction to arbitrary genus by avoiding the quantum Maurer-Cartan equation's complications in higher genus.
• To define invariants using a new homology theory—Nice Multi-Curve Homology (NMCH)—that simplifies the technical complexity of the original multi-curve homology.
• To show that the invariants depend only on the symplectic structure and a choice of 4-chain $K$ with $\partial K = L$, enabling a canonical shift in the homology class.

Proposed method

• Constructs invariants via moduli spaces of multi-curves associated to unconnected decorated graphs, which encode boundary components and Euler characteristic.
• Introduces a new homology group, Nice Multi-Curve Homology (NMCH), defined as a quotient of formal linear combinations of disjoint one-currents on $L$, modulo boundary relations from isotopies.
• Uses a 4-chain $K$ with $\partial K = L$ to define a canonical base class $\gamma_0 \in H_1(L,\mathbb{Q})$, shifting the homology class of the invariant.
• Defines moduli spaces $\overline{\mathcal{M}}_{\chi,(n,H)}(\beta)$ as fiber products of main moduli spaces with $K^n$, incorporating boundary behavior via $\partial K$.
• Extends the boundary formula to include both node-type and evaluation-type boundary faces, using perturbations compatible with the graph structure.
• Constructs a multi-curve cycle in $MCH_0(\partial\beta + \gamma_0)$ via a perturbation $\mathfrak{s}$ of the moduli space, and shows equivalence to the NMCH framework under a stronger forgetful condition.

Experimental results

Research questions

• RQ1 Can open Gromov-Witten invariants be defined without requiring the vanishing of obstruction classes (i.e., without a bounding chain) for Lagrangians with non-trivial $H_1(L,\mathbb{Q})$?
• RQ2 How can the obstruction to defining higher-genus open invariants be circumvented, especially when the quantum Maurer-Cartan equation does not yield numerical invariants?
• RQ3 Can unconnected graphs in the multi-curve construction eliminate the need for individual boundary component linking conditions, thereby simplifying the invariant definition?
• RQ4 Is there a homology-theoretic framework that captures the essential topological data of open invariants while being computationally tractable?
• RQ5 What role does the choice of a 4-chain $K$ with $\partial K = L$ play in defining a canonical shift in the homology class of the invariant?

Key findings

• The paper constructs open Gromov-Witten invariants $Z_{\beta,\chi} \in NMCH_0(\partial\beta + \gamma_0)$ for any $\beta \in H_2(M,L,\mathbb{Z})$ and $\chi \in \mathbb{Z}$, without requiring $\partial\beta = 0$ in $H_1(L,\mathbb{Q})$.
• The invariants are defined using unconnected decorated graphs, which allow the use of total boundary linking numbers instead of individual component conditions, thus avoiding obstruction classes.
• The construction is valid in arbitrary genus, as it bypasses the need for the open Gromov-Witten potential and its quantum Maurer-Cartan equation.
• The authors prove that the complex multi-curve homology $MCH_0$ is isomorphic to the simpler $NMCH_0$, which is defined via formal linear combinations of disjoint one-currents modulo isotopy relations.
• The invariant depends only on the symplectic structure $(M,\omega)$ and a choice of 4-chain $K$ with $\partial K = L$, with the class $\gamma_0 = (K \setminus \partial K) \cap L$ providing a canonical shift in the homology class.
• The method ensures compatibility with boundary strata of the moduli space by extending the perturbation $\mathfrak{s}$ to include both node and evaluation-type boundary faces, using $K^n$ and $\partial K$.