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Topological algebra of symplectic geometry of symmetric powers

Vivek Shende, Peng Zhou

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.

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.

Paper Structure

This paper contains 14 sections, 45 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Clustering rules
  3. Kähler potentials
  4. Recollections on smoothing plurisubharmonic functions
  5. Factorizable Kähler potentials on symmetric powers
  6. Sectorial covers for symmetric products
  7. Box decomposition of $\mathop{\mathrm{Sym}}\nolimits^n \mathbb{R}$
  8. Recollections on Liouville structures
  9. Sectorial covers
  10. Weinstein-ness
  11. Proof of Theorems \ref{['thm: functor']} and \ref{['thm: intro cover']}
  12. Calculations
  13. Three easy pieces
  14. Cutting corners

Key Result

Theorem 1.5

To each $W \in \mathop{\mathrm{Sect}}\nolimits_2$, we may associate a symmetric power $\mathop{\mathrm{Sym}}\nolimits^{(n)}(W) \in \mathop{\mathrm{Sect}}\nolimits_{2n}$, agreeing with the set-theoretic symmetric power for surfaces without boundary. This assignment extends to a functor $\mathop{\math

Theorems & Definitions (109)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • proof
  • Remark 1.9
  • ...and 99 more