Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.04863

A Higgs category for the cluster variety of triples of flags

Bernhard Keller·Miantao Liu
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.RTmath.ACmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bernhard Keller, Miantao Liu. 2026-06-03. A Higgs category for the cluster variety of triples of flags. https://arxiv.org/abs/2509.04863

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A local relative trace formula for F*\SL(2,F)

In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible elements in F. This is perhaps one of the simplest non-trivial analogs of the trace formula, and also a motivating example for the author's work (in preparation) on the relative trace formula.

math.RT↗
arXiv2026

Semi-infinite parabolic IC-sheaf

Let G be a connected reductive group, P its parabolic subgroup. We consider the parabolic semi-infinite category of sheaves on the affine Grassmanian of G and construct the parabolic version of the semi-infinite IC-sheaf of each orbit. We establish some of its properties and relate it to sheaves on the Drinfeld compactification of the moduli stack Bun_P of P-torsors on a curve. We also relate the parabolic semi-infinite IC-sheaf with the dual baby Verma object on the spectral side.

math.RT↗
arXiv2026

The Grothendieck group of an extriangulated category

In this paper, we investigate the split Grothendieck group $K^{\rm sp}_{0}(\mathcal{M})$ of a $d$-rigid subcategory $\mathcal{M}$ in an extriangulated category $\mathscr{C}$. As applications, we prove the following results: (1) If $\mathcal{M}$ is a silting subcategory, then the Grothendieck group $K_{0}(\mathscr{C})$ is isomorphic to $K_{0}^{\rm sp}(\mathcal{M})$; (2) If $\mathcal{M}$ is a $d$-cluster tilting subcategory, then $K_{0}(\mathscr{C})$ is isomorphic to the index Grothendieck group $K_{0}^{\rm in}(\mathcal{M})$; (3) Let $\mathcal{C}_{A_{n}}^{d}$ be the $d$-cluster category of type $A_n$. If $d$ is even, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}/(n+1)\mathbb{Z}$. If $d$ is odd, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}$ if $n$ is odd; $K_0(\mathcal{C}_{A_{n}}^{d})\cong 0$ if $n$ is even.

math.RT↗