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From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

Aaron D. Lauda, Anthony M. Licata, Andrew Manion

Abstract

We give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement, using the algebras defined in arXiv:0905.1335 from the equivariant cohomology of toric varieties. We prove this conjecture for cyclic arrangements by showing that these algebras are isomorphic to algebras appearing in work of Ozsvath-Szabo arXiv:1603.06559 in bordered Heegaard Floer homology arXiv:0810.0687. The proof of our conjecture in the cyclic case extends work of Karp-Williams arXiv:1608.08288 on sign variation and the combinatorics of the m=1 amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of gl(1|1).

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

Abstract

We give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement, using the algebras defined in arXiv:0905.1335 from the equivariant cohomology of toric varieties. We prove this conjecture for cyclic arrangements by showing that these algebras are isomorphic to algebras appearing in work of Ozsvath-Szabo arXiv:1603.06559 in bordered Heegaard Floer homology arXiv:0810.0687. The proof of our conjecture in the cyclic case extends work of Karp-Williams arXiv:1608.08288 on sign variation and the combinatorics of the m=1 amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of gl(1|1).

Paper Structure

This paper contains 61 sections, 38 theorems, 72 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Generalities
  3. Cyclic arrangements
  4. Bordered Floer homology and the representation theory of $\mathfrak{gl}(1|1)$
  5. Positive geometries, hypertoric varieties, and the amplituhedron
  6. Outline of paper
  7. Hypertoric varieties and algebras
  8. Hyperplane arrangements and hypertoric varieties
  9. Hyperplane arrangements
  10. Hypertoric varieties
  11. Additional structure
  12. Polarizations
  13. Definitions
  14. Polarizations and hypertoric varieties
  15. Dualities
  16. ...and 46 more sections

Key Result

Theorem 1.2

[cf. Theorem thm:LeftCyclicIsom, Corollary cor:MainUnpolarizedThm, Theorem thm:RightCyclicIsom] The universal deformation $\widetilde{B}(\mathcal{V})$ associated to a cyclic hyperplane arrangement is isomorphic to the Ozsváth--Szabó algebra associated to a symmetric product of a multiply-punctured d

Figures (14)

  • Figure 1: Left: $n$ lines and $n+1$ regions between them. Right: a set of $3$ dots in the regions $\{0,1,4\} \subset \{0,\ldots,6\}$.
  • Figure 2: A cyclic arrangement with left cyclic and right cyclic polarizations for $n=4, k=1$.
  • Figure 3: A cyclic arrangement with left cyclic and right cyclic polarizations for $n=4, k=2$.
  • Figure 4: The decorated surface $(F,Z,\underline{\alpha})$.
  • Figure 5: From left to right: arc diagrams $\mathcal{Z}_l$, $\mathcal{Z}_r$, $\mathcal{Z}_{\mathop{\mathrm{full}}\nolimits}$, and $\mathcal{Z}'$ for $n=3$.
  • ...and 9 more figures

Theorems & Definitions (92)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5: Definition 3.1 and Remark 3.1 of Gale
  • Definition 2.6
  • Remark 2.7
  • ...and 82 more