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Bordered algebras and the wrapped Fukaya category

Isabella Khan

TL;DR

The paper proves an \(\mathcal{A}_{\infty}\)-isomorphism between endomorphism algebras in the wrapped Fukaya category of a star diagram and the star algebras from bordered knot Floer homology, by modeling the star algebras as unweighted deformations of associative algebras and computing Hochschild cohomology to guarantee uniqueness. It then uses explicit model calculations within the wrapped Fukaya category to realize these deformations as endomorphism algebras, establishing the correspondences \(\mathcal{A} \cong \mathrm{End}_{\mathcal{W}(M)}(\alpha_1\oplus \cdots \oplus \alpha_N)\) and \(\mathcal{B} \cong \mathrm{End}_{\mathcal{W}(M)}(\beta_1\oplus \cdots \oplus \beta_N)\). The approach relies on cobar duality between the underlying associative algebras and a careful Hochschild cohomology analysis to prove both existence and uniqueness of the deformations with prescribed higher operations. Model calculations demonstrate that the endomorphism algebras carry the same higher products and generator structure, enabling an explicit isomorphism. This work lays groundwork for extending these ideas to weighted Koszul-dual A-infinity algebras and provides a concrete bridge between bordered knot Floer data and wrapped Fukaya categories, with potential for broad computational applications.

Abstract

This paper establishes an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and the class of A-infinity algebras related to bordered knot Floer homology, called star algebras, which the author first constructed in her previous work. By viewing the star algebras as A-infinity deformations of underlying associative algebras and making several calculations with Hochschild cohomology, we verify that the star algebras are unique with a given set of generators and basic A-infinity relations. We then make model calculations in order to establish that the endomorphism algebras have these generators and basic operations, so that the desired isomorphism follows.

Bordered algebras and the wrapped Fukaya category

TL;DR

The paper proves an -isomorphism between endomorphism algebras in the wrapped Fukaya category of a star diagram and the star algebras from bordered knot Floer homology, by modeling the star algebras as unweighted deformations of associative algebras and computing Hochschild cohomology to guarantee uniqueness. It then uses explicit model calculations within the wrapped Fukaya category to realize these deformations as endomorphism algebras, establishing the correspondences \(\mathcal{A} \cong \mathrm{End}_{\mathcal{W}(M)}(\alpha_1\oplus \cdots \oplus \alpha_N)\) and \(\mathcal{B} \cong \mathrm{End}_{\mathcal{W}(M)}(\beta_1\oplus \cdots \oplus \beta_N)\). The approach relies on cobar duality between the underlying associative algebras and a careful Hochschild cohomology analysis to prove both existence and uniqueness of the deformations with prescribed higher operations. Model calculations demonstrate that the endomorphism algebras carry the same higher products and generator structure, enabling an explicit isomorphism. This work lays groundwork for extending these ideas to weighted Koszul-dual A-infinity algebras and provides a concrete bridge between bordered knot Floer data and wrapped Fukaya categories, with potential for broad computational applications.

Abstract

This paper establishes an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and the class of A-infinity algebras related to bordered knot Floer homology, called star algebras, which the author first constructed in her previous work. By viewing the star algebras as A-infinity deformations of underlying associative algebras and making several calculations with Hochschild cohomology, we verify that the star algebras are unique with a given set of generators and basic A-infinity relations. We then make model calculations in order to establish that the endomorphism algebras have these generators and basic operations, so that the desired isomorphism follows.
Paper Structure (15 sections, 21 theorems, 86 equations, 14 figures)

This paper contains 15 sections, 21 theorems, 86 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Algebraic Definitions
  3. $\mathcal{A}_{\infty}$-algebras
  4. The cobar complex
  5. Hochschild cohomology and $\mathcal{A}_{\infty}$-deformations
  6. The Wrapped Fukaya category
  7. Definitions
  8. Gradings
  9. Constructions with $\mathcal{A}$ and $\mathcal{B}$
  10. The $\mathcal{A}_{\infty}$-algebra $\mathcal{A}$
  11. The $\mathcal{A}_{\infty}$-algebra $\mathcal{B}$
  12. Uniqueness of $\mathcal{A}$ and $\mathcal{B}$
  13. Computing the cobar algebras of $\mathcal{A}_0$ and $\mathcal{B}_0$
  14. Computing the Hochschild cohomology
  15. Proof of Theorem \ref{['fukaya']}

Key Result

Theorem 1.1

Let $\alpha_1, \ldots, \alpha_N$, $\beta_1, \ldots \beta_N$ denote the $\alpha$- and $\beta$-arcs, respectively. Let $\mathcal{W}(M)$ denote the Fukaya category of the star diagram with $N$ boundary circles, as defined in Section fukayaDef. Let $\mathcal{A}$ and $\mathcal{B}$ be the $\mathcal{A}_{\i and

Figures (14)

  • Figure 1:
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  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 9 more figures

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Remark 2.8
  • Proposition 2.9
  • ...and 25 more