[Paper Review] Asymptotic shifting numbers in triangulated categories

Yuwei Fan, Simion Filip|arXiv (Cornell University)|Aug 14, 2020

Mathematical Dynamics and Fractals34 references4 citations

TL;DR

This paper introduces shifting numbers as asymptotic invariants that measure the translation behavior of autoequivalences in triangulated categories, analogous to Poincaré rotation numbers in dynamics. It establishes that these invariants form quasimorphisms on the group of autoequivalences in key examples, such as derived categories of elliptic and abelian surfaces, and links them to entropy and Bridgeland stability conditions.

ABSTRACT

We introduce invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangulated category translates inside the category. The invariants are analogous to Poincare translation numbers that are widely used in dynamical systems. We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences. Additionally, we relate our shifting numbers to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich.

Motivation & Objective

• To define and study asymptotic invariants—shifting numbers—that measure the translation of autoequivalences in triangulated categories.
• To establish that these invariants satisfy properties analogous to Poincaré translation numbers, including conjugacy invariance and homogeneity.
• To show that shifting numbers yield quasimorphisms on the group of autoequivalences in specific geometric settings, such as derived categories of elliptic and abelian surfaces.
• To connect shifting numbers to existing invariants in categorified dynamics, including entropy functions and Bridgeland stability conditions.
• To provide explicit constructions of quasimorphisms on groups like PSL_2(Z) using categorical data, particularly in Calabi–Yau categories of A_2 quivers.

Proposed method

• Define upper and lower shifting numbers τ⁺(F) and τ⁻(F) as asymptotic limits of Ext-distance functions ε⁺ and ε⁻ between a split generator G and its iterates FⁿG.
• Use Ext-distance functions ε⁺(E₁,E₂) = max{k ∈ ℤ : Hom(E₁, E₂[−k]) ≠ 0} and ε⁻(E₁,E₂) = min{k ∈ ℤ : Hom(E₁, E₂[−k]) ≠ 0} to measure categorical displacement.
• Prove that τ⁺(F) and τ⁻(F) are independent of the choice of split generator and satisfy key properties: shift invariance, conjugacy invariance, and homogeneity.
• Establish a link between shifting numbers and the entropy function of Dimitrov, Haiden, Katzarkov, and Kontsevich via Legendre duality.
• Construct explicit quasimorphisms on PSL₂(ℤ) using the Rademacher function and its homogenization, showing they arise from categorical autoequivalences.
• Demonstrate that in derived categories of elliptic and abelian surfaces, the shifting number gives a quasimorphism on the group of autoequivalences, with explicit formulas in terms of intersection forms and braid group actions.

Experimental results

Research questions

• RQ1 How can one define asymptotic invariants that measure the translation of an autoequivalence in a triangulated category, analogous to Poincaré rotation numbers in dynamics?
• RQ2 In which categories do the shifting numbers τ⁺ and τ⁻ yield quasimorphisms on the group of autoequivalences?
• RQ3 How are shifting numbers related to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich?
• RQ4 Can shifting numbers be computed explicitly in geometric examples such as derived categories of elliptic curves or abelian surfaces?
• RQ5 What is the role of Bridgeland stability conditions in interpreting the shifting numbers as phases or asymptotic translation data?

Key findings

• The upper and lower shifting numbers τ⁺(F) and τ⁻(F) are well-defined, finite, and independent of the choice of split generator in a triangulated category.
• The shifting number τ(F) = ½(τ⁺(F) + τ⁻(F)) satisfies τ(Fⁿ) = n·τ(F) for all n ∈ ℤ when F is an autoequivalence and the category admits a Serre functor.
• In the derived category of an elliptic curve, the shifting number defines a quasimorphism on the group of autoequivalences, with explicit values determined by intersection theory.
• For the derived category of an abelian surface, the shifting number gives a quasimorphism that factors through the action on cohomology and is related to the distinguished component of the Künneth decomposition.
• In the Calabi–Yau category of the A₂ quiver, the shifting number construction yields a quasimorphism on the braid group via a central extension, with values determined by the homogenized Rademacher function on PSL₂(ℤ).
• The shifting numbers are related to the Legendre transform of the entropy function, providing a dual perspective on categorical dynamics.