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Relations among $\mathbb{P}$-Twists

Andreas Hochenegger, Andreas Krug

TL;DR

This work analyzes when autoequivalences given by $\mathsf{P}$-twists of $\mathbb{P}^n[k]$-objects satisfy relations. The authors build a framework of spherification functors that transport $\mathbb{P}$-twists to spherical twists, allowing them to apply Volkov’s results on spherical twists to deduce that, under technical hypotheses, two non-isomorphic $\mathbb{P}^n[k]$-objects either commute precisely when they are orthogonal or generate a free group $F_2$; in particular, no further relations occur. They provide constructive criteria for the existence of spherification functors (geometric via $j_*$ and dg-algebraic via cones) and establish key compatibilities between $\mathsf{P}$-twists and $\mathsf{T}$-twists, with applications to hyperkähler geometries and related symplectic contexts. The paper also discusses open questions about extending these results beyond pairs of objects and exploring generalizations to $\mathbb{P}$-functors and higher-order relations.

Abstract

Given two $\mathbb{P}$-objects in some algebraic triangulated category, we investigate the possible relations among the associated $\mathbb{P}$-twists. The main result is that, under certain technical assumptions, the $\mathbb{P}$-twists commute if and only if the $\mathbb{P}$-objects are orthogonal. Otherwise, there are no relations at all. In particular, this applies to most of the known pairs of $\mathbb{P}$-objects on hyperkähler varieties. In order to show this, we relate $\mathbb{P}$-twists to spherical twists and apply known results about the absence of relations between pairs of spherical twists.

Relations among $\mathbb{P}$-Twists

TL;DR

This work analyzes when autoequivalences given by -twists of -objects satisfy relations. The authors build a framework of spherification functors that transport -twists to spherical twists, allowing them to apply Volkov’s results on spherical twists to deduce that, under technical hypotheses, two non-isomorphic -objects either commute precisely when they are orthogonal or generate a free group ; in particular, no further relations occur. They provide constructive criteria for the existence of spherification functors (geometric via and dg-algebraic via cones) and establish key compatibilities between -twists and -twists, with applications to hyperkähler geometries and related symplectic contexts. The paper also discusses open questions about extending these results beyond pairs of objects and exploring generalizations to -functors and higher-order relations.

Abstract

Given two -objects in some algebraic triangulated category, we investigate the possible relations among the associated -twists. The main result is that, under certain technical assumptions, the -twists commute if and only if the -objects are orthogonal. Otherwise, there are no relations at all. In particular, this applies to most of the known pairs of -objects on hyperkähler varieties. In order to show this, we relate -twists to spherical twists and apply known results about the absence of relations between pairs of spherical twists.
Paper Structure (10 sections, 16 theorems, 50 equations)

This paper contains 10 sections, 16 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Enter spherical and $\mathbb{P}$-objects
  3. From $\mathbb{P}$-objects to spherical objects
  4. Introducing spherification functors
  5. The guiding example
  6. Existence of spherification functors
  7. Properties of spherification functors
  8. Returning from spherical objects to $\mathbb{P}$-objects
  9. Twists along orthogonal $\mathbb{P}$-objects
  10. Open questions and speculation

Key Result

Theorem A

Let $P_1,\ldots,P_m$ be $\mathbb{P}^n[k]$-objects in some algebraic triangulated category, such that $k\ge 2$ is even, $n\ge 2$, $\mathop{\mathrm{\mathsf{gcd}}}\nolimits(k,nk/2)>1$ and $\mathop{\mathrm{\mathsf{Hom}}}\nolimits^*(P_i,P_j)$ is concentrated in degree $nk/2$ for $i\neq j$ (or zero). Then

Theorems & Definitions (51)

  • Theorem A: \ref{['cor:formal:spherification']}
  • Theorem B: \ref{['thm:main']}
  • Definition 1
  • Proposition 1: ST
  • Proposition 2
  • Remark 1
  • Remark 2
  • Definition 2: ALdg, HM
  • Proposition 3: ALdg
  • Example 1
  • ...and 41 more