A Deformation of the Compact Fukaya Category via the Relative Fukaya Category

Abstract: We give a proof of Conjecture $1.3$ of [SBEAS24].

• The paper demonstrates that the deformation of the compact Fukaya category via symplectic cohomology yields a filtered quasi-equivalence with Seidel’s relative Fukaya category.
• It employs a rigorous L∞ and A∞ framework using Maurer–Cartan elements and the closed-open map to encode divisor-induced enumerative geometry.
• The result substantiates predictions in homological mirror symmetry and opens avenues for computing deformation spaces in non-compact, log Calabi–Yau settings.

Deformations of the Compact Fukaya Category via the Relative Fukaya Category

Introduction and Main Results

The paper "A Deformation of the Compact Fukaya Category via the Relative Fukaya Category" (2607.03234) addresses Conjecture 1.3 of [strom2024maurer], providing a rigorous confirmation that the wrapped Fukaya category, deformed via classes arising from the compactification divisor, is filtered quasi-equivalent to Seidel’s relative Fukaya category. Precisely, for a monotone symplectic manifold $(M,\omega)$ with an orthogonal normal crossings symplectic divisor $D$, and $X = M \setminus D$, the author constructs a deformation of the compact Fukaya category $\mathcal{F}(X)$ (which is the Fukaya category of compact Lagrangians in $X$) and shows that its deformation via the closed-open map and Maurer–Cartan elements in symplectic cohomology matches the $A_\infty$-structure of the relative Fukaya category $\mathcal{F}(M, D)$.

The core statement is:

Theorem: The full subcategory $\mathcal{W}(M, D)$ of the deformed wrapped Fukaya category whose objects are closed Lagrangian branes is filtered quasi-equivalent to Seidel's relative Fukaya category $\mathcal{F}(M, D)$.

This result resolves a foundational conjecture in the field, substantiating that the enumerative geometry encoded by the divisor $D$ supplies the nontrivial deformation parameters of the compact Fukaya category.

Structural Ingredients: Categories and Filtrations

The deformation-theoretic framework involves the wrapped Fukaya category $D$0, considered over the Novikov ring $D$1, where $D$2 tracks the intersection multiplicities with each component of $D$3 (with grading prescribed by the formula $D$4 in terms of the coefficients in the anticanonical divisor decomposition). The deformation utilizes the $D$5-Maurer–Cartan theory applied to the symplectic cohomology $D$6, using the closed-open (CO) map as an $D$7-morphism into the Hochschild cochains of the wrapped Fukaya category. Maurer–Cartan elements in $D$8 yield, via CO, curved filtered $D$9-deformations of $X = M \setminus D$0.

The key technical constructs and results:

• For a generic choice of auxiliary data (Hamiltonians, almost complex structures, perturbations), all moduli spaces—domain curves, holomorphic curve counts, compactifications—support the machinery needed for curved, filtered $X = M \setminus D$1-structures and their deformations.
• The deformation is governed by the image of an explicit MC element $X = M \setminus D$2 under CO, which is computed via a homotopy-theoretic ODE producing a unique, $X = M \setminus D$3-adically convergent solution in the completion.
• Relative Fukaya categories $X = M \setminus D$4 are presented in the framework of $X = M \setminus D$5-pre-categories with objects the same as those of $X = M \setminus D$6, morphisms enhanced with Novikov variables, and $X = M \setminus D$7-operations encoding intersection multiplicities and relative Gromov–Witten invariants.

Moduli Spaces and Holomorphic Curve Theory

A considerable portion of the text is devoted to the sophisticated combinatorial and analytic apparatus underlying moduli spaces of pseudo-holomorphic disks, including:

• Deligne-Mumford and Fulton-MacPherson compactifications, modeling degeneration of curves and boundary strata via decorated ribbon trees and colored trees (tracking disk, sphere, and divisor insertions).
• Enhanced moduli spaces parametrizing pointed disks, spheres, flavors (sprinkles), weights (asymptotic data), and choices of strip/cylindrical ends, with relevant symmetry and forgetful operations.
• Consistent choices of perturbation/auxiliary data on these spaces, crucial for defining $X = M \setminus D$8- and $X = M \setminus D$9-operations that respect all necessary equivalences and filtrations.

This setup guarantees that curve counts are appropriately signed and graded, and that bubbling and compactness phenomena are controlled to ensure $\mathcal{F}(X)$0-relations hold up to homotopy and filtration.

The Closed-Open Map and Deformations

The closed-open (CO) map—central to the construction—realizes symplectic cohomology classes as Hochschild cochains (operations) on the Fukaya category, preserving the $\mathcal{F}(X)$1-structure up to filtration. The deformation $\mathcal{F}(X)$2 is defined as the push-forward along CO of a canonical Maurer–Cartan element $\mathcal{F}(X)$3 arising from the divisor.

Key steps include:

• The explicit gauge equivalence between the tautological Maurer–Cartan element $\mathcal{F}(X)$4 and the canonical element $\mathcal{F}(X)$5, mediated by a homotopy ODE as in the theory of filtered DGLAs.
• Analysis of the resulting deformation functors at the level of derived categories/spectral sequences—specifically, comparison of deformation classes modulo $\mathcal{F}(X)$6 (where all intersection phenomena with $\mathcal{F}(X)$7 disappear) and identification of associated gradeds.

Main Comparison and Quasi-Equivalence

The equivalence is established by constructing a zig-zag of filter-preserving $\mathcal{F}(X)$8-functors, as the deformation structures on both the deformed wrapped Fukaya category and the relative Fukaya category are shown to admit strict identifications at the chain level (under sufficiently generic and compatible choices of data). Critical ingredients:

• The moduli spaces controlling the $\mathcal{F}(X)$9-operations in both settings coincide on the nose when restricted to compact Lagrangians (by integrability maximum principles and positivity of intersection).
• The deformation parameters (the Novikov variables) count disk configurations meeting $X$0 with prescribed multiplicity and distribution, which are operationally the same in both constructions.
• Any additional choices (Hamiltonians, almost complex structures, etc.) are absorbed via the quasi-equivalence invariance of the Fukaya category and its deformations.

This analytic and algebraic control produces a strict chain-level isomorphism between the relevant $X$1-categories, and the zig-zag sequence of quasi-equivalences completes the equivalence in the homotopical derived sense.

Implications and Future Directions

This result clarifies how divisor compactifications induce nontrivial deformation theory for Fukaya categories of affine varieties, effectively encoding "relative enumerative geometry" as algebraic deformations of the open (wrapped) Fukaya category's $X$2-structure. The precise identification also supplies an explicit mechanism by which symplectic cohomology—through the closed-open map—indexes the universal deformation space of such Fukaya categories, paralleling predictions from homological mirror symmetry. Practical consequences touch on computations of objects, morphisms, and deformation spaces in non-exact, non-compact settings, with direct relevance for log Calabi–Yau and Landau-Ginzburg models as explored in recent work [borman2024l_, ganatra2024homological, pomerleano2024symplectic].

Potential future directions include:

• Extension of these deformation identifications to broader contexts, such as higher-dimensional pairs, logarithmic and bulk deformations, and the presence of more general singularities.
• Computations in explicit mirror symmetry models, analyzing the effect of the deformation on derived equivalences and physical D-brane categories.
• Study of the global deformation theory (versal/universal characteristics) of Fukaya categories for complements of normal crossings (and worse) divisors.

Conclusion

The paper provides a definitive comparison between the deformation theory of the compact Fukaya category via symplectic cohomology and the relative Fukaya category determined by a compactification divisor. The filtered quasi-equivalence is established with precise control over analytic and algebraic data, synthesizing contemporary $X$3 and $X$4-algebra techniques. This bridges enumerative symplectic invariants and categorical deformation theory, with substantial impact on mirror symmetry, symplectic topology, and the study of noncompact Calabi–Yau varieties.