Genus-Zero Open Gromov–Witten Invariants

• Genus-zero open Gromov–Witten invariants are rational curve-counting invariants that quantify holomorphic disks mapped into Calabi–Yau threefolds with boundaries on Lagrangian submanifolds.
• The construction relies on moduli spaces decorated with Kuranishi structures and bounding chains to manage obstruction theory and guarantee well-defined virtual fundamental chains.
• These invariants connect to knot theory and mirror symmetry by linking augmentation polynomials from Legendrian contact homology with recursive, integrality-based disk counts.

Genus-zero open Gromov–Witten (OGW) invariants are rational curve-counting invariants associated with holomorphic disks mapped into Calabi–Yau threefolds with boundary on Lagrangian submanifolds. In the foundational approach of Iacovino, and in subsequent developments relating OGW invariants to knot theory and mirror symmetry, these quantities encode subtle information about the symplectic topology of the pair $(X, L)$, obstruction theory, and the algebraic geometry of moduli spaces. Their construction fundamentally relies on Kuranishi models, obstruction classes, and, in certain cases, connections to the augmentation polynomial from Legendrian knot contact homology.

1. Geometric and Moduli-Theoretic Foundations

Let $X$ be a Calabi–Yau threefold: a compact complex threefold equipped with a symplectic form $\omega$ and a nowhere-vanishing holomorphic volume form $\Omega$, with $c_1(TX)=0$. The relevant Lagrangian submanifolds $L\subset X$ are assumed oriented, relatively spin, and of Maslov index zero. The Maslov index zero condition is enforced so that the virtual dimension of the moduli space of disks is independent of the homology class $A\in H_2(X,L)$ and depends only on the number of boundary marked points $k$: $\dim^{\vir}_\R \Mbar_{0,k}(X,L;A) = \dim_\R L + \mu(A) + k - 3 = k,$ since $\dim_\R L = 3$, $X$0, and the genus-zero (disk) topology imposes the $X$1 term (Iacovino, 2011).

The moduli space $X$2 parameterizes stable maps $X$3 of class $X$4 with $X$5 cyclically ordered marked points on the boundary, modulo disk automorphisms.

Moduli spaces admit Kuranishi structures: each local chart is modeled on a finite-dimensional domain $X$6, obstruction bundle $X$7, and a Kuranishi map $X$8. To assemble a virtual fundamental chain, compatible multi-sections $X$9 transverse to zero are chosen, resulting in an oriented “virtual fundamental chain”: $\omega$0 of real dimension $\omega$1.

2. Algebraic Definition and Obstruction Theory

Given cohomology classes $\omega$2, the open Gromov–Witten invariant with insertions is defined by: $\omega$3 where $\omega$4 is the evaluation at the $\omega$5-th marked boundary point.

A distinctive aspect of the construction is the obstruction–bounding chain recursion. For $\omega$6, the boundary of the virtual chain may not be closed; the obstruction is encoded in a chain $\omega$7 (associated to a rooted tree with one external leg). When $\omega$8 in $\omega$9, a bounding chain $\Omega$0 with $\Omega$1 is constructed. This bounding chain is used to homotope away boundary strata and thus define the closed 0-chain: $\Omega$2 with $\Omega$3 the trivial tree, which is the OGW invariant in class $\Omega$4 with no insertions (Iacovino, 2011).

3. Recursion, Gluing Laws, and Deformation Properties

The framework is intrinsically recursive and relies on the vanishing of obstruction classes in smaller area classes:

• For each $\Omega$5, the obstruction class $\Omega$6 and bounding chain $\Omega$7 (when $\Omega$8) are defined recursively.
• The invariant $\Omega$9 exists if obstructions for all smaller areas vanish.
• The system of chains $c_1(TX)=0$0 attached to trees satisfies the recursively coherent gluing/composition law: $c_1(TX)=0$1 is obtained as a sum over splittings $c_1(TX)=0$2 with combinatorial weights.
• The invariants are invariant under deformations of the tuple $c_1(TX)=0$3, including small Hamiltonian isotopies of $c_1(TX)=0$4.
• If $c_1(TX)=0$5, the invariant is undefined in class $c_1(TX)=0$6.
• In the presence of an anti-symplectic involution $c_1(TX)=0$7 with $c_1(TX)=0$8 as fixed locus, all obstructions can be arranged to vanish, resulting in invariants matching real curve counts as in Solomon’s theory [math/0606429, (Iacovino, 2011)].

4. Explicit Computations and Examples

Elementary computation applies when the obstruction chain is trivial. For a minimal area, nontrivial class $c_1(TX)=0$9, $L\subset X$0 is compact, boundaryless, and $L\subset X$1 automatically, so

$L\subset X$2

is a naive count of Maslov-zero disks with one marked point. For higher classes, such as $L\subset X$3, the construction of $L\subset X$4 is required. The value $L\subset X$5 depends on $L\subset X$6, with ambiguity by multiples of $L\subset X$7, matching the anomaly–obstruction phenomenon in Lagrangian Floer theory.

When anti-symplectic involution techniques are available, all obstructions vanish and the resulting numbers match real disk counts for real loci of quintic or toric threefolds.

5. Relations to Legendrian Contact Homology and Knot Theory

Mahowald’s approach for $L\subset X$8 connects genus-zero open Gromov–Witten invariants with knot conormal Lagrangians $L\subset X$9 to the augmentation polynomial of Legendrian contact homology (Mahowald, 2016). For a knot $A\in H_2(X,L)$0, its conormal lifts to a Lagrangian $A\in H_2(X,L)$1 with topology $A\in H_2(X,L)$2. Genus-zero OGW invariants $A\in H_2(X,L)$3 (with degree $A\in H_2(X,L)$4 and winding $A\in H_2(X,L)$5) are encoded in the generating function: $A\in H_2(X,L)$6 The augmentation polynomial $A\in H_2(X,L)$7 specifies the mirror geometry and, conjecturally (Aganagic–Vafa), fully determines the OGW generating function via period integrals: $A\in H_2(X,L)$8 with $A\in H_2(X,L)$9 solved as $k$0.

Explicit computations show that for toric conormal Lagrangians, Atiyah–Bott localization yields closed formulas for $k$1. This data allows the inversion process to reconstruct $k$2. For the unknot

$k$3

and similar explicit polynomials exist for torus knots and non-toric cases like the figure-eight and three-twist knots.

6. Integrality, Obstruction Vanishing, and Mirror Symmetry

The “LMOV/OV integrality” conjecture posits integer BPS numbers $k$4 such that

$k$5

and the generating function decomposes accordingly. Explicit computations confirm this integrality up to high $k$6 for knots treated in the literature (Mahowald, 2016).

Anti-symplectic involutions play a critical technical role: when such an involution fixes $k$7, all obstruction chains vanish simultaneously, enabling the computation of OGW invariants without the need for bounding chains. The resulting numbers coincide with those emerging from real enumerative invariants (Iacovino, 2011).

The connection to mirror symmetry is exemplified by the Aganagic–Vafa conjecture, positing that the mirror geometry ($k$8) and the OGW generating function are controlled by the augmentation polynomial from knot contact homology, enabling concrete calculations through period integrals and formal expansions of $k$9. For knots with non-toric Lagrangians, the process relies on series expansions and inversion algorithms, yielding OGW invariants aligned with known mirror data.

7. Computational Examples and Limitations

Explicit calculations have been realized for the unknot, various torus knots, and non-toric knots such as the figure-eight and the three-twist. Localization reproduces known augmentation polynomials in the toric setting, and the inversion process matches disk invariants to the augmentation polynomial for Legendrian knot contact homology.

For non-toric cases, explicit tables of $\dim^{\vir}_\R \Mbar_{0,k}(X,L;A) = \dim_\R L + \mu(A) + k - 3 = k,$0 and $\dim^{\vir}_\R \Mbar_{0,k}(X,L;A) = \dim_\R L + \mu(A) + k - 3 = k,$1 confirm the integrality predictions. However, computational complexity grows rapidly with knot complexity due to the high degrees of $\dim^{\vir}_\R \Mbar_{0,k}(X,L;A) = \dim_\R L + \mu(A) + k - 3 = k,$2 and $\dim^{\vir}_\R \Mbar_{0,k}(X,L;A) = \dim_\R L + \mu(A) + k - 3 = k,$3 in the augmentation polynomials, making large-scale calculations challenging. A mathematically rigorous foundation for OGW invariants in general non-toric settings, beyond the toric or anti-symplectic involution cases, remains open (Mahowald, 2016).

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For fundamental constructions, obstruction theory, and the recursive definition in the closed and open case, see (Iacovino, 2011). For the relation to knot theory, the augmentation polynomial, and broad classes of examples, see (Mahowald, 2016).