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Koszul dual $\mathcal{A}_{\infty}$ algebras for star-shaped diagrams -- Part 1

Isabella Khan

TL;DR

This work develops a Koszul dual framework for star-shaped diagrams in bordered Heegaard Floer theory by constructing weighted $\mathcal{A}_{\infty}$-algebras $\mathcal{A}$ and $\mathcal{B}$ from horizontal star-shaped slices and defining dualizing $AA$- and $DD$-bimodules. It introduces a graphical calculus for $\mathcal{A}_{\infty}$-operations, builds diagonals to define tensor-product $\mathcal{A}_{\infty}$-structures, and computes explicit structures for the $\alpha$-bordered algebra $\mathcal{A}$ and the $\beta$-bordered algebra $\mathcal{B}$, including gradings and higher multiplications. The paper proves that these algebras admit valid higher operations and dualizing bimodules, laying the groundwork for a Koszul duality result that will be completed in Part 2. Together, these constructions enable the computation of Heegaard Floer invariants via tensor products of bordered pieces, connecting to broader frameworks such as the fully wrapped Fukaya category. The results provide concrete, combinatorial models for weighted $\mathcal{A}_{\infty}$-structures with both operationally bounded and unbounded features, and establish a robust algebraic path toward dual decompositions of bordered Heegaard Floer data.

Abstract

By slicing the Heegaard diagram for a given $3$-manifold in a particular way, it is possible to construct $\mathcal{A}_{\infty}$-bimodules, the tensor product of which retrieves the Heegaard Floer homology of the original 3-manifold. The first step in this is to construct algebras corresponding to the individual slices. Here, we use the graphical calculus for $\mathcal{A}_{\infty}$-structures introduced by Lipshitz, Ozsváth, and Thurston, to construct Koszul dual weighted $\mathcal{A}_{\infty}$-algebras $\mathcal{A}$ and $\mathcal{B}$, and dualizing bimodules for a particular star-shaped class of slice. The duality result is then proved in the sequel.

Koszul dual $\mathcal{A}_{\infty}$ algebras for star-shaped diagrams -- Part 1

TL;DR

This work develops a Koszul dual framework for star-shaped diagrams in bordered Heegaard Floer theory by constructing weighted -algebras and from horizontal star-shaped slices and defining dualizing - and -bimodules. It introduces a graphical calculus for -operations, builds diagonals to define tensor-product -structures, and computes explicit structures for the -bordered algebra and the -bordered algebra , including gradings and higher multiplications. The paper proves that these algebras admit valid higher operations and dualizing bimodules, laying the groundwork for a Koszul duality result that will be completed in Part 2. Together, these constructions enable the computation of Heegaard Floer invariants via tensor products of bordered pieces, connecting to broader frameworks such as the fully wrapped Fukaya category. The results provide concrete, combinatorial models for weighted -structures with both operationally bounded and unbounded features, and establish a robust algebraic path toward dual decompositions of bordered Heegaard Floer data.

Abstract

By slicing the Heegaard diagram for a given -manifold in a particular way, it is possible to construct -bimodules, the tensor product of which retrieves the Heegaard Floer homology of the original 3-manifold. The first step in this is to construct algebras corresponding to the individual slices. Here, we use the graphical calculus for -structures introduced by Lipshitz, Ozsváth, and Thurston, to construct Koszul dual weighted -algebras and , and dualizing bimodules for a particular star-shaped class of slice. The duality result is then proved in the sequel.
Paper Structure (20 sections, 18 theorems, 80 equations, 24 figures)

This paper contains 20 sections, 18 theorems, 80 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. $\mathcal{A}_{\infty}$ algebras, and operations as trees
  4. AA bimodules
  5. DD Bimodules
  6. Koszul Duality
  7. Diagonals and tensor products
  8. Diagonals
  9. $\mathcal{A}_{\infty}$-structures for tensor products of algebras
  10. The $\alpha$-bordered algebra $\mathcal{A}$
  11. Algebra elements and multiplication
  12. Higher operations
  13. Maslov and Alexander gradings for $\mathcal{A}$
  14. $\mathcal{A}_{\infty}$-relations
  15. The $\beta$-bordered algebra $\mathcal{B}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathcal{A}$ and $\mathcal{B}$ be the weighted $\mathcal{A}_{\infty}$-algebras defined in Sections alph and bb1, respectively. Then there exists a DD-bimodule $\:^{\mathcal{A}} X^{\mathcal{B}}$, constructed in Section DDbim, and a weighted AA-bimodule $\:_{\mathcal{B}} Y_{\mathcal{A}}$ construc and where $\:^{\mathcal{A}} \mathrm{id}_{\mathcal{A}}$ and $\:^{\mathcal{B}} \mathrm{id}_{\mathcal

Figures (24)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 19 more figures

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 2.1
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Theorem 4.3
  • proof
  • Lemma 4.4
  • Remark 4.5
  • Remark 4.6
  • ...and 27 more