Canonical Bounding Cochains in Fukaya Algebras

• Canonical bounding cochains are formal solutions to the Maurer–Cartan equation in Fukaya algebras, uniquely defined up to gauge equivalence to cancel disk bubbling.
• Point-like bounding chains, with top-degree terms proportional to the Poincaré dual of a point, enable the construction of genus zero open Gromov–Witten invariants.
• The theory extends analytically and noncommutatively to manage deformation and obstruction, ensuring robust invariant computations in various dimensions.

A canonical family of bounding cochains is a distinguished, gauge-equivalence class (or analytic sheaf) of formal solutions to the Maurer–Cartan equation in the Fukaya $A_\infty$-algebra of a Lagrangian submanifold, parameterized so that the associated family encodes geometric constraints—such as boundary point insertions—in the construction of genus zero open Gromov–Witten (OGW) invariants. The canonical property ensures uniqueness up to gauge and often parametrizes the putative “boundary constraints” analogous to the insertion of cycles in standard Gromov–Witten theory. In particular, the point-like family, where the top-degree part is Poincaré dual to a point, enables the construction of invariants that specialize to classical enumerative invariants (e.g., Welschinger's invariants) in low dimensions and provides a mechanism to overcome analytic obstacles such as the bubbling of $J$-holomorphic disks in higher dimensions (Solomon et al., 2016, Kosloff et al., 26 Nov 2025).

1. Maurer–Cartan Theory and Bounding Cochains

Let $(X, \omega)$ be a symplectic manifold and $L \subset X$ a relatively-spin Lagrangian. The Fukaya $A_\infty$-algebra $C = A^*(L; \Lambda)$, with Novikov coefficients $\Lambda$, encodes the $J$-holomorphic disk counts via structure maps

$m_k^\gamma : C^{\otimes k} \rightarrow C$

parametrized by an (optional) closed bulk deformation class $\gamma \in A^2(X, L; \mathbb{Q})$. The $A_\infty$-relations ensure the algebraic consistency of these maps.

A bounding cochain, or weak Maurer–Cartan element, is an odd-degree element $b \in C^1$ solving the deformed Maurer–Cartan equation

$$\sum_{k=0}^\infty m_k^\gamma(b, \dots, b) = c \cdot 1$$

for some $c \in \Lambda$ of degree two. The gauge-equivalence class of $(\gamma, b)$ determines a deformation of the $A_\infty$-algebra, crucial for canceling disk bubbling phenomena and for defining open invariants with geometric constraints (Solomon et al., 2016).

2. Point-Like Bounding Chains

Point-like bounding chains arise where the top-degree piece of each $b_\beta$ in the Novikov expansion

$b = \sum_{\beta} T^\beta b_\beta$

is proportional to $\mathrm{PD}(\text{pt})$, the Poincaré dual of a point in $L$. This closely mimics boundary point insertions used in enumerative geometry. The coefficient of the top-degree term is then a formal parameter $s$, tracking boundary constraint multiplicities. The construction is canonical up to gauge equivalence, with the set of possible classes parameterized by the integral of $b$ against the volume form, $\int_L b \in \Lambda_{1-n}$, and the bulk deformation $[\gamma] \in H^2(X, L; \mathbb{Q})$ (Solomon et al., 2016, Kosloff et al., 26 Nov 2025).

3. Obstruction Theory and Canonical Families

Obstruction theory classifies and constructs bounding cochains via recursive solutions to the Maurer–Cartan equation. Two bounding pairs $(\gamma, b)$ and $(\gamma', b')$ are gauge-equivalent if and only if $[\gamma] = [\gamma']$ and $\int_L b = \int_L b'$. Given $H^*(L; R) \cong H^*(S^n; R)$ or analogous vanishing conditions, there is a bijection

$$\rho: \{\text{bounding pairs}\}/(\text{gauge}) \to H^2(X,L; \mathbb{Q}) \oplus \Lambda_{1-n}$$

which, for given $(\gamma, s)$, produces a unique canonical one-parameter family $b(s)$. This is achieved by expanding $b_0$ as a multiple of $\mathrm{PD}(\text{pt})$ and solving for higher Novikov order corrections to annihilate the obstructions $\mathfrak{o}_j$ at each stage, leveraging the vanishing of $H^{\neq 0,n}(L)$ (Solomon et al., 2016, Kosloff et al., 26 Nov 2025).

In even dimensions, where the family parameter $s$ is odd, the parameter must be non-commutative to allow for invariants with multiple boundary constraints. The obstruction theory then proceeds in the non-commutative setting with twisted cohomology, using a spectral sequence whose differential structure ensures obstruction vanishing given the point-like normalization $\int_L b \neq 0$ (Kosloff et al., 26 Nov 2025).

4. Analytic and Noncommutative Extensions

In the non-archimedean analytic framework, the Maurer–Cartan moduli space is realized as a rigid analytic space over the Novikov field $\Lambda = \mathbb{C}((T^\mathbb{R}))$, defined by the vanishing of a collection of convergent power series $Q_j$ on the formal torus associated to $H_1(L;\mathbb{Z})$. The universal bounding cochain then becomes a sheaf-valued global analytic function on this space, and proper unobstructedness, defined as $Q_j = 0$ identically, is an analytic notion preserved under deformation and wall-crossing (Yuan, 4 Jan 2024).

For even-dimensional $L$, a non-commutative extension is essential: $C$ is extended by a free algebra $S = R\langle s \rangle$ with odd generator $s$, allowing for families $b(s)$ in $(s \cdot \widehat{C})^1$. Convergence of the Maurer–Cartan series is ensured by pseudo-completeness, a two-step filtration that provides control over both the Novikov energy and non-commutative powers of $s$ (Kosloff et al., 26 Nov 2025).

Approach	Parameter Algebra	Key Feature
Commutative (odd $n$)	$\Lambda$	Simple s-family; trivial obstruction
Noncommutative (even $n$)	$R\langle s \rangle$	Noncommuting $s$; obstruction theory

5. Superpotentials and Open Gromov–Witten Invariants

Given a canonical point-like family $b(s)$ and bulk class $\gamma$, the deformed cyclic superpotential is defined as

$$\Omega(\gamma, b) = (-1)^n \left[ \sum_{k\geq 0} \frac{1}{k+1} \langle m_k^\gamma(b^{\otimes k}), b \rangle_F \right]$$

or with additional correction terms to remove “type-D” (sphere) contributions. The Taylor expansion of $\Omega(\gamma(s), b(s))$ in $s$ and $t_j$ (interior insertions) yields

$$\Omega(\gamma(s), b(s)) = \sum_{\beta, k, l} \mathrm{OGW}_{\beta, k}^L(\ldots) T^\beta s^k \prod_j t_j^{l_j}$$

evaluated at $s=0$ and $t_j=0$, which defines the genus zero, boundary and interior-constraint open Gromov–Witten invariants. These satisfy the expected enumerative axioms such as grading, symmetry, unit and divisor relations, and are symplectic deformation invariant (Solomon et al., 2016, Kosloff et al., 26 Nov 2025).

6. Special Cases and Comparisons

In dimensions $2$ and $3$, obstruction theory collapses: any $b = s \,\mathrm{PD}(\text{pt})$ is bounding. The resulting invariants recover Welschinger's signed counts of real rational curves, with explicit matching up to normalization factors:

$$\sum_{\chi_L(\beta)=d} \mathrm{OGW}_{\beta, k_{d, l}}(\mathrm{PD}(\text{pt})^l) = \pm 2^{1-l} \mathrm{Welschinger}_{d, l}$$

In higher odd dimensions, the presence of an anti-symplectic involution allows for similar constructions relating to Georgieva’s invariants (Solomon et al., 2016, Kosloff et al., 26 Nov 2025). For even dimensions and non-commutative $s$-families, point-like bounding cochains provide a mechanism to define invariants for rational cohomology spheres and for Lagrangians with sphere-like cohomology in specified dimensions (Kosloff et al., 26 Nov 2025).

7. Analytic Continuation and Wall-Crossing

The non-archimedean viewpoint introduces the analytic Maurer–Cartan moduli $MC(L)$ as a rigid analytic subvariety. Within a smooth family $L_s$ of Lagrangians, proper unobstructedness (vanishing $Q_j$) for one $s$ ensures unobstructedness for all $L_s$ in the family through analytic continuation. Wall-crossing automorphisms parameterize the transition of superpotentials and bounding cochains under Lagrangian isotopies and monodromy in the parameter space (Yuan, 4 Jan 2024).

Monodromy or Galois actions correspond to automorphisms of the parameter algebra, ensuring gluing of the local moduli spaces, and give rise to local systems of affinoid spaces encoding the family of bounding cochains.

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The theory of canonical families of bounding cochains is central to the modern construction of open Gromov–Witten invariants with boundary constraints. It unifies the geometric, analytic, and noncommutative frameworks and provides a robust mechanism for both the computation and deformation-theoretic foundation of invariants in symplectic topology (Solomon et al., 2016, Yuan, 4 Jan 2024, Kosloff et al., 26 Nov 2025).