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A Higgs category for the cluster variety of triples of flags

Bernhard Keller, Miantao Liu

TL;DR

The paper constructs a Higgs category ${\mathcal H}$ for the cluster variety of triples of flags associated with a simply laced Dynkin diagram $Δ$, realized as a Frobenius exact dg category that is stably $2$-Calabi–Yau. It develops the relative $3$-Calabi–Yau completion framework, identifies the boundary dg category with Gorenstein projective dg modules, and proves Christ's conjecture that ${\mathcal H}$ is equivalent to the cosingularity category ${\rm cosg}(\Gamma)$. By exploiting projective domination and GP-dg module theory, the authors relate the Higgs category precisely to GP-dg modules over ${\mathcal P}_{dg}$, establishing the cyclic and braid group actions that match Goncharov–Shen cluster symmetries. The results provide a concrete categorification of the basic triangle in higher Teichmüller theory and connect the Higgs category to the cosingularity framework, yielding a robust algebraic model for the cluster geometry of triples of flags.

Abstract

The cluster variety of triples of flags (associated with a split simple Lie group of Dynkin type Delta) plays a key role in higher Teichmuller theory as developed by Fock-Goncharov, Jiarui Fei, Ian Le, ... and Goncharov-Shen. We refer to it as the basic triangle associated with Delta. In this paper, for simply laced Delta, we construct and study a Higgs category (in the sense of Yilin Wu) which we expect to categorify the basic triangle. This category is a certain exact dg category (in the sense of Xiaofa Chen) which is Frobenius and stably 2-Calabi-Yau. We show that it has indeed the expected cyclic group symmetry and that its derived category has the expected braid group symmetry. A key ingredient in our construction is a conjecture by Merlin Christ, whose proof occupies most of this paper. The proof is based on a new description of the Higgs category in terms of Gorenstein projective dg modules. Our techniques are in the spirit of Orlov in his work on triangulated categories of graded B-branes.

A Higgs category for the cluster variety of triples of flags

TL;DR

The paper constructs a Higgs category for the cluster variety of triples of flags associated with a simply laced Dynkin diagram , realized as a Frobenius exact dg category that is stably -Calabi–Yau. It develops the relative -Calabi–Yau completion framework, identifies the boundary dg category with Gorenstein projective dg modules, and proves Christ's conjecture that is equivalent to the cosingularity category . By exploiting projective domination and GP-dg module theory, the authors relate the Higgs category precisely to GP-dg modules over , establishing the cyclic and braid group actions that match Goncharov–Shen cluster symmetries. The results provide a concrete categorification of the basic triangle in higher Teichmüller theory and connect the Higgs category to the cosingularity framework, yielding a robust algebraic model for the cluster geometry of triples of flags.

Abstract

The cluster variety of triples of flags (associated with a split simple Lie group of Dynkin type Delta) plays a key role in higher Teichmuller theory as developed by Fock-Goncharov, Jiarui Fei, Ian Le, ... and Goncharov-Shen. We refer to it as the basic triangle associated with Delta. In this paper, for simply laced Delta, we construct and study a Higgs category (in the sense of Yilin Wu) which we expect to categorify the basic triangle. This category is a certain exact dg category (in the sense of Xiaofa Chen) which is Frobenius and stably 2-Calabi-Yau. We show that it has indeed the expected cyclic group symmetry and that its derived category has the expected braid group symmetry. A key ingredient in our construction is a conjecture by Merlin Christ, whose proof occupies most of this paper. The proof is based on a new description of the Higgs category in terms of Gorenstein projective dg modules. Our techniques are in the spirit of Orlov in his work on triangulated categories of graded B-branes.

Paper Structure

This paper contains 37 sections, 54 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Adjoints from Serre functors
  4. Representations up to homotopy and the dg category of triangles
  5. Relative Calabi--Yau completions and dg localizations
  6. Characterizing objects in degree
  7. The inverse dualizing module
  8. The morphism category and its relative Calabi--Yau completion
  9. The morphism category
  10. Passage to the derived categories and functors
  11. The relative Calabi--Yau completion
  12. Morphism complexes of the boundary dg category
  13. Easy isomorphisms
  14. Study of the functor
  15. The cone functor and its use
  16. ...and 22 more sections

Key Result

Lemma 2.1.1

The triangle functor $S^{-1}RS$ is left adjoint to $L$ and $SLS^{-1}$ is right adjoint to $R$.

Theorems & Definitions (111)

  • Lemma 2.1.1
  • proof
  • Proposition 2.3.1
  • proof
  • Corollary 2.3.2
  • Lemma 2.4.1
  • proof
  • Lemma 2.4.2
  • proof
  • Proposition 2.5.1
  • ...and 101 more