Topological algebra of symplectic geometry of symmetric powers

Abstract: To a noncompact orientable surface with no closed boundary, we associate the sum of Fukaya categories of (Liouville sectors associated to) its symmetric powers. We construct sectorial covers with the combinatorics of the bar resolution to show this association extends to an open 2d topological field theory -- without naming a Lagrangian, let alone a holomorphic disk. In particular, we recover results of Rouquier and Manion on extending Heegaard-Floer theory down to an interval.

• The paper constructs an open 2d TFT by mapping noncompact surfaces to Fukaya categories via symmetric Liouville sectors.
• It introduces sectorial covers with bar resolution combinatorics to rigorously establish categorical gluing and descent in symplectic geometry.
• The approach unifies symplectic and representation-theoretic methods, impacting bordered Floer theory and categorified quantum groups.

Topological Algebra and Symplectic Geometry of Symmetric Powers: An Expert Synthesis

Overview and Context

The paper "Topological algebra of symplectic geometry of symmetric powers" (2604.01208) by Shende and Zhou provides an open-2d-TFT construction from symplectic geometry of Liouville sectors associated to symmetric powers of noncompact orientable surfaces. The work develops new topological and algebraic tools for analyzing the Fukaya categories of these symmetric products, extending sectorial descent and gluing formalism, and connecting to powerful representation-theoretic frameworks for Heegaard-Floer homology and its categorified enhancements.

2D Topological Field Theory and Gluing Constructions

The authors formalize a symmetric monoidal category $\mathscr{O}$ of noncompact, oriented surfaces-with-boundary (with all boundary components noncompact) and proper inclusions as morphisms, equipped with the symmetric monoidal structure of disjoint union. They demonstrate that the assignment taking such a surface to the sum of the Fukaya categories of symmetric powers of associated Liouville sectors extends to an open 2d TFT. This TFT is constructed without explicit choice of Lagrangians or holomorphic curve analysis, instead relying on sectorial covers whose combinatorics mirror the bar resolution.

The notion of gluing is axiomatized as requiring that for a gluing configuration $\Sigma = \Sigma_L \cup_A \Sigma_R$, the category associated to $\Sigma$ can be reconstructed (up to equivalence) as a tensor product of $\mathcal{F}(\Sigma_L)$ and $\mathcal{F}(\Sigma_R)$ over $\mathcal{F}(A)$. This is realized via descent along sectorial open covers modeled on combinatorics of the bar construction in algebra.

Symmetric Powers as Liouville Sectors

A pivotal technical contribution is the functorial construction of symmetric powers of surfaces within the setting of Liouville sectors, resolving the problem that symmetric powers of manifolds-with-boundary are in general not themselves manifolds-with-boundary, let alone Liouville sectors. The authors establish a systematic method to endow these symmetric products with sectorial structures, compatible with the open cover descent formalism.

For a given surface $\Sigma \in \mathscr{O}$, the $n$-th symmetric product $\Sym^{(n)}(\Sigma)$ is made Weinstein, and these assignments collectively produce a functor

$\Sym : (\mathscr{O}, \sqcup) \to (\Sect^N, \times)$

mapping surfaces to sequences of Liouville sectors of all symmetric degrees. The construction hinges on intricate regularized products and center of mass coordinates on symmetric products, with careful patching of Kähler potentials via cluster decomposition rules and Varouchas-Richberg smoothing.

Sectorial Covers, Mayer-Vietoris, and Categorical Descent

The authors engineer sectorial open covers of symmetric powers, constructed via pre-image pullbacks of covers of $\Sigma = \Sigma_L \cup_A \Sigma_R$0 under the real part projection. These covers are highly structured: the charts and their intersections are organized so that the corresponding diagrams coincide, up to deformation equivalence, with the bar construction for gluing configurations. Regularity (in the sense of forming sectorial collections) is verified using explicit control over boundary defining functions, and transversality is established at the level of center-of-mass coordinates.

By functoriality and the sectorial descent theorem of Ganatra-Pardon-Shende [GPS2], the Fukaya category assignment satisfies the gluing property, establishing that the composed functor

$\Sigma = \Sigma_L \cup_A \Sigma_R$1

is an open 2d TFT in stable $\Sigma = \Sigma_L \cup_A \Sigma_R$2-linear categories.

Algebraic Ramifications and Categorical Complexes

The authors connect the resulting categorical structures directly to representation-theoretic constructions in cornered Heegaard-Floer theory, recovering and strengthening main theorems of Rouquier and Manion. Specifically, they identify the structure and monoidal properties of $\Sigma = \Sigma_L \cup_A \Sigma_R$3 (where $\Sigma = \Sigma_L \cup_A \Sigma_R$4) with the categorified half of $\Sigma = \Sigma_L \cup_A \Sigma_R$5 [rouquier-manion], showing the module categories are precisely categorical complexes as in Christ-Dyckerhoff-Walde.

Moreover, the explicit gluing holds at the monoidal level, with the monoidal structure corresponding to coproduct totalization in the sense of [Christ-Dyckerhoff-Walde]. The authors provide a comprehensive pushout formula for the result of gluing via the pair-of-pants, and show associativity at the categorical level emerges formally from the TFT structure.

Numerical and Contrasting Results

• The vanishing results for symmetric powers of the half disk and interval are established geometrically, producing that the only nontrivial summands of $\Sigma = \Sigma_L \cup_A \Sigma_R$6 occur for $\Sigma = \Sigma_L \cup_A \Sigma_R$7 (both $\Sigma = \Sigma_L \cup_A \Sigma_R$8-mod), with higher degrees vanishing.
• The explicit algebraic structures on the Fukaya categories $\Sigma = \Sigma_L \cup_A \Sigma_R$9 (for the pair-of-pants $\Sigma$0) recover and clarify the triangular and self-adjoint constructions present in algebraic approaches to categorial $\Sigma$1-modules.

Theoretical Implications

The work sets out a paradigm by which topological field theories valued in categories of symplectic origin can be constructed without resorting to holomorphic curve techniques, with descent and gluing controlled purely topologically and combinatorially. This reframing unifies symplectic and representation-theoretic viewpoints (notably, the connection to categorified quantum groups), empowering computations and generalizations in bordered Floer theory and related invariants.

By providing a fully functorial and coefficient-independent approach, the methodology offers a robust foundation for developing categorifications in symplectic geometry and for systematic constructions of higher representation theoretic structures (e.g., generalized Khovanov-Rouquier-Tensor categories, Heegaard-Floer-theoretic link invariants).

Prospects and Future Directions

The authors indicate that their framework adapts to the symplectic geometry underlying generalizations of Heegaard-Floer, including higher-dimensional variants and categorifications of quantum enveloping algebras for larger groups. There is clear relevance for the symplectic-geometric approaches to knot homology and for the construction of invariants via microlocal sheaf theory [GPS3, shende-microlocal].

Additionally, the alignment between categorical descent formulas and representation-theoretic tensor products suggests avenues for enhancing the bridge between algebraic and geometric categorification, potentially impacting higher representation theory, TQFT, and symplectic topology.

Conclusion

The article delivers a comprehensive and technically sophisticated framework for the topological algebra of Fukaya categories of symmetric powers, fashioning a geometric and categorical infrastructure to support the construction and computation of open 2d TFTs and their algebraic avatars. The techniques introduced have strong applicability for future work in symplectic categorification, the study of link invariants, and the development of higher symplectic representation theory.