Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symplectic higher Auslander correspondence for type A

Ilaria Di Dedda

Abstract

We prove that the Fukaya-Seidel categories of a certain family of singularities on $\mathbb{C}^d$ are equivalent to the perfect derived categories of higher Auslander algebras of Dynkin type A. We relate these to the Fukaya-Seidel categories of Brieskorn-Pham singularities and to the partially wrapped Fukaya categories considered by Dyckerhoff-Jasso-Lekili. We provide a symplectic interpretation to higher Auslander correspondence of type A in terms of Fukaya-Seidel categories of Lefschetz fibrations.

Symplectic higher Auslander correspondence for type A

Abstract

We prove that the Fukaya-Seidel categories of a certain family of singularities on are equivalent to the perfect derived categories of higher Auslander algebras of Dynkin type A. We relate these to the Fukaya-Seidel categories of Brieskorn-Pham singularities and to the partially wrapped Fukaya categories considered by Dyckerhoff-Jasso-Lekili. We provide a symplectic interpretation to higher Auslander correspondence of type A in terms of Fukaya-Seidel categories of Lefschetz fibrations.
Paper Structure (17 sections, 53 theorems, 130 equations, 22 figures)

This paper contains 17 sections, 53 theorems, 130 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Higher Auslander– Reiten theory
  5. Fukaya categories of symmetric products of surfaces
  6. Fukaya– Seidel categories
  7. Partially wrapped Fukaya categories
  8. The derived equivalence FSf perfA
  9. An invariant description of Ffhat
  10. A distinguished basis for decoupled singularities
  11. A distinguished basis for Brieskorn– Pham
  12. The Fukaya--Seidel category Ffnd
  13. The derived equivalence Ffnd Wnd
  14. A tilting object for And
  15. Type A symplectic higher Auslander correspondence
  16. ...and 2 more sections

Key Result

Proposition 1.1

The Fukaya– Seidel category of the Brieskorn– Pham singularity $\hat{f}_{n,d}$ defined by is quasi-equivalent to the perfect derived category $\textnormal{perf}(\hat{B}_{n,d})$ of the algebra $\hat{B}_{n,d}:=(\hat{B}_{n})^{\otimes d}$, where $\hat{B}_{n}$ is the path algebra of the $A_{n}$ quiver over a field $\mathbf{k}$ with alternating orientation of the arrows, as in Figure fig:Bhat

Figures (22)

  • Figure 1: The quivers underlying $\hat{B}_{n}$ (left), $\hat{B}_{n,d}$ (centre) and $B_{n,d}$ (right), with all possible commutativity relations, for $n=4$ and $d=2$.
  • Figure 2: Left-to-right composition of strands diagrams, in the strands algebra with $2$ strands in $4$ places.
  • Figure 3: Differential in the strands algebra with $3$ strands in $5$ places.
  • Figure 4: Base of a Lefschetz fibration, with regular value on $\partial \mathbb{D}$.
  • Figure 5: Vanishing paths for $P$ (left), $Q$ (centre), and $P+Q$ (right), for $\mu=2$, $\nu=3$.
  • ...and 17 more figures

Theorems & Definitions (135)

  • Proposition 1.1: Theorem \ref{['thm:mainthm1']}
  • Theorem 1.2: Theorem \ref{['thm:mainthm2']}
  • Theorem 1.3: Theorem \ref{['thm:mainthm3']}
  • Corollary 1.4
  • Theorem 1.5: Theorem \ref{['prop:uniquetopfibr']}, Corollary \ref{['mainthmsympAus']}, Definition \ref{['def:Sigma']}
  • Definition 2.1
  • Definition 2.2: OT12
  • Proposition 2.3: OT12
  • Theorem 2.4: Auroux1
  • Proposition 2.6: Auroux1
  • ...and 125 more