How do open WDVV equations compute open Gromov-Witten invariants for Lagrangians?

Even dimensions block multiple boundary constraints

Open Gromov-Witten invariants count pseudoholomorphic disks whose boundary lies on a Lagrangian submanifold L inside a closed symplectic manifold X. Solomon and Tukachinsky packaged these counts as coefficients of a superpotential, with a formal variable s recording boundary point constraints. The obstacle the paper addresses appears when dim L is even: s then has odd degree, and graded commutativity forces s squared to equal zero. A superpotential built that way cannot record counts with more than one boundary point constraint, such as disks through three points of L.

The paper's route around this obstruction is to let the coefficients be noncommutative. It establishes the open WDVV equations for A-infinity algebras over noncommutative differential graded algebras, extending the Solomon-Tukachinsky equations. The resulting recursive formulas are what make open invariants with several boundary point constraints computable for even-dimensional Lagrangians.

Bounding cochains both mark points and cancel bubbling

The main difficulty in defining open invariants is disk bubbling: the relevant moduli spaces have codimension-one boundary strata, so disk counts change as the auxiliary data vary in one-parameter families. Fukaya-Oh-Ohta-Ono introduced bounding cochains as solutions of a Maurer-Cartan equation in the resulting A-infinity algebra, and Solomon and Tukachinsky built the superpotential from point-like bounding cochains, families b parameterized by formal variables of degree 1 minus dim L.

The family b plays two roles at once. It serves as the boundary point constraint, and it cancels bubbling contributions. The paper's point-like condition is the identity of the pairing of the unit with b against minus s, so the boundary variable is tied directly to the cochain rather than added by hand. Point-like bounding cochains over noncommutative coefficients come from joint work of the author with Solomon, and the present paper supplies the WDVV structure that operates on them.

A recursion seeded by one open count

The open WDVV equations are formal partial differential equations satisfied jointly by the closed potential and the open superpotential. Their value is computational: they relate invariants with more boundary or interior insertions to invariants with fewer, so a small amount of initial data propagates outward. The paper writes Omega_k(t) for the generating function of open invariants with k boundary point constraints, where the bulk coordinates t record interior constraints.

For the real locus X(R) of an even-dimensional complex quadric of dimension n greater than two, the recursion determines all open Gromov-Witten invariants from the genus-zero closed invariants of X and one open disk count in degree one. The paper also states the converse direction: given two independent open counts in degree one, the recursion recovers all genus-zero closed Gromov-Witten invariants of X. The closed inputs are the invariants computed in the cited literature, not new measurements made here.

Quadrics show multi-point invariants can be nontrivial

Applying the equations to the real loci of even-dimensional complex quadrics of dimension greater than two yields all open Gromov-Witten invariants for those pairs. The computations show that open invariants with multiple boundary point constraints can be nontrivial in arbitrary dimensions. This is the concrete payoff of the noncommutative extension: the invariants that a graded-commutative coefficient algebra cannot record are exactly the ones that turn out to carry information here.

For the quadric, the one-boundary-point count identifies the Fukaya-category summand containing the real locus, while counts with several boundary point constraints reveal finer structure in the corresponding summand of quantum cohomology. The paper reports that there are no Maslov-two disks for this case, so the character evaluated on c1(TX) at zero bulk vanishes.

Open counts define a character on a quantum subalgebra

The paper also builds a map from a natural subalgebra of big quantum cohomology, using open Gromov-Witten invariants, for a Lagrangian endowed with a point-like bounding cochain. For a bulk class u, the derivative of Omega_1 counts disks with one more interior constraint, through a cycle Poincare dual to u. Under the stated hypothesis, the character is L(u) equal to that derivative, and the open WDVV equations are what prove multiplicativity.

The construction is described as being in the spirit of the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono, with the point-like bounding cochain playing the role of the critical point and L of the Euler element playing the role of the critical value. At zero bulk the Euler element is c1(TX), and L of it is the Novikov-weighted count of Maslov-two disks through a point of L. The paper notes that the character construction does not require a prior computation of a Floer group.

Three boundary points separate the two extensions

For the even quadric, the boundary character has two extensions to the full big quantum cohomology algebra. They agree on the Euler element, and at zero bulk that common value on c1(TX) is zero, so an eigenvalue does not tell them apart. They differ on the Poincare dual p of the Lagrangian, where they take opposite values, and that difference is determined by L(p star p) equal to 2 Omega_3(t).

The invariants on the right count disks with three boundary point constraints, precisely the invariants a graded-commutative coefficient algebra cannot record. This is where the paper locates the refinement: point-like bounding cochains carry more information about closed quantum cohomology than weak bounding cochains of Floer theory do, and the three-boundary-point invariants are the measurement that distinguishes the two extensions.

Scope of the recursion and its coefficient extensions

The geometric setting is a closed symplectic manifold of dimension 2n with n greater than one, a compact connected oriented Lagrangian with a fixed relative spin structure, and a tame almost complex structure. The existence of a bounding pair is an additional hypothesis rather than a theorem of the paper; the point-like families for the even-dimensional Lagrangian spheres considered are supplied by the joint work with Solomon. The quadric computation is carried out for n greater than two and even.

Coefficient extensions are handled in stages rather than assumed throughout. The construction is formulated axiomatically, separating structure maps and their identities from the coefficient extensions used in applications, and for the quadric the square root is adjoined only after the relevant invariants have been computed. The character theorem itself requires the condition that a specific Maurer-Cartan coefficient vanish, so the character is not automatic for every Lagrangian.

The quadric computation settles which invariants matter

The paper's most decision-relevant comparison is internal to the even quadric. The one-boundary-point count and the character value on c1(TX) at zero bulk both land on the same coarse information: the summand of quantum cohomology where multiplication by c1(TX) acts nilpotently, which the real locus split-generates. Neither distinguishes the two extensions of the character.

The three-boundary-point invariants do. Because the two extensions agree on the Euler element but take opposite values on the Poincare dual of the Lagrangian, with the difference fixed by 2 Omega_3(t), a reader who wants to know which extension is in play should look at the three-boundary-point count rather than at an eigenvalue. That is the paper's specific priority: multi-boundary-point open invariants, not the single-point count, are what refine the closed quantum-cohomological information for even-dimensional Lagrangians.

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