Higgs Category for Cluster Flags

• Higgs Category is a categorical framework that generalizes cluster algebras and flag moduli, built as an exact dg category with Frobenius and stably 2-Calabi–Yau structures.
• It employs relative Ginzburg dg algebras and quiver moduli to construct a categorification of the basic triangle in higher Teichmüller theory with vanishing higher extensions.
• A key result is Merlin Christ’s conjecture, proved via a correspondence with Gorenstein projective dg modules, establishing an equivalence with a cosingularity category.

The Higgs category is a categorical framework that arises in both representation theory and geometry, generalizing the structures underlying cluster algebras, quiver moduli, and higher Teichmüller spaces. In the context of the cluster variety of triples of flags for a split simple Lie group of Dynkin type Δ, the Higgs category provides a categorification of the so-called “basic triangle”—the moduli space of configurations of three flags—central to higher Teichmüller theory. The category is constructed as an exact dg category (in the sense of Xiaofa Chen), equipped with a Frobenius and stably 2-Calabi–Yau structure, cyclic group symmetry, and braid group symmetry on its derived category. A key technical component of the construction and its structural properties is a conjecture of Merlin Christ, proved in the paper, stating that the Higgs category is equivalent (via restriction) to a cosingularity category of dg modules, with the proof relying on a new description of the Higgs category in terms of Gorenstein projective dg modules, in the vein of Orlov’s graded B-branes framework (Keller et al., 5 Sep 2025).

1. Construction of the Higgs Category in the Triples of Flags Setting

The foundational construction of the Higgs category in this context employs a differential graded (dg) enhancement of the relative cluster category. Starting from a quiver associated to the Dynkin type Δ, one forms a relative Ginzburg dg algebra (typically denoted Π₃), reflecting the combinatorics and potential of the cluster data. The perfect derived category $\operatorname{per}(\Pi_3)$ is then considered, and its localization, the relative cluster category $\mathcal{C}$, is taken by modding out the thick subcategory of “perfect-valued objects” (those corresponding to acyclic or frozen/factorizable data).

The Higgs category is then defined as the full extension-closed subcategory of $\mathcal{C}$ comprising objects $X$ satisfying:

$$\operatorname{Ext}^i_\mathcal{H}(X,Y) = 0 \quad \text{for all}\; i>0 \;\text{and all}\; Y.$$

This selects those objects in the cluster category which are “Higgs” in the sense that they do not factor through the (projective–injective) frozen subcategory and whose higher extensions vanish, thereby corresponding to the “non-frozen” combinatorial sector.

This construction relies on a boundary dg category (arising from “relative” Calabi–Yau completions) and functorial resolutions using mapping cones and projective–injective objects associated with frozen data.

2. Structural Properties: Frobenius, Calabi–Yau, and Symmetries

The constructed Higgs category is exact and Frobenius, so that every projective object is injective and vice versa. The stable category (modulo projective–injectives) is 2-Calabi–Yau:

$$\operatorname{Ext}^1_{stable}(X,Y) \cong D\operatorname{Ext}^1_{stable}(Y,X)$$

providing a categorical reflection of the cluster exchange structure.

The derived category (dg enhancement) features symmetry properties matching structures found in higher Teichmüller theory:

• Cyclic group symmetry: Induced by the Calabi–Yau property, a “rotation” auto-equivalence operates on the Higgs category, often of order 3 or 6, depending on the Dynkin diagram’s automorphisms. This realizes cluster automorphisms as cyclic actions at the categorical level.
• Braid group symmetry: The derived category admits an action of the braid group $B_\Delta$ on Δ via spherical twists and silting mutations centered at simple objects; this group action structures the derived auto-equivalences and governs the relations among different cluster–tilting (or silting) subcategories.

These structural symmetries are crucial in matching the categorical invariants with the cluster combinatorics and the modular group actions that play a central role in higher Teichmüller theory.

3. Categorification of the Basic Triangle and Cluster Structures

In higher Teichmüller theory (à la Fock–Goncharov), the cluster variety of triples of flags—the “basic triangle”—serves as the base object in constructing more elaborate moduli spaces. The Higgs category categorifies this basic triangle by replacing coordinate algebras or cluster variables with objects, morphisms, and extension structures in the category. More specifically, the Grothendieck group $K_0(\mathcal{H})$ captures (part of) the upper cluster algebra associated with the variety.

The cluster exchange relations map to distinguished triangles (or more precisely, “exchange sequences”) in the category. Canonical bases, positivity, and quantization phenomena found in higher Teichmüller theory are expected to lift to categorical statements about exceptional objects, silting reductions, and braid group orbits in the Higgs category.

4. Merlin Christ’s Conjecture and Its Proof

A crucial result proved in the paper is Merlin Christ’s conjecture: the restriction functor $$\Phi: \mathcal{H}_\mathrm{dg} \to \operatorname{cosg}(\mathcal{H}_\mathrm{dg})$$, from the dg-enhanced Higgs category to its cosingularity category, is an equivalence of $k$-linear categories and induces isomorphisms on negative degree morphism spaces.

The cosingularity category records “singular” objects modulo projective–injectives and is central to understanding the Calabi–Yau and extensional structure. The proof proceeds by:

1. Essential surjectivity: Every object in the cosingularity category can be lifted to the Higgs category.
2. Full faithfulness: By describing objects in the Higgs category explicitly as Gorenstein projective dg modules (see below), one shows that any morphism in the cosingularity category has a preimage in the Higgs category, and all relations are respected.

This result ensures that the Higgs category captures all categorical and homological features associated with the singularity structure expected from the basic triangle and its associated cluster geometry.

5. Gorenstein Projective DG Modules and Comparison with Orlov

A novel technical ingredient is the realization of the Higgs category in terms of Gorenstein projective dg modules. The correspondence is made explicit:

$$\mathcal{H}_\mathrm{dg} \simeq \operatorname{gpr}(\mathcal{P}_{\mathrm{dg}})$$

where $\operatorname{gpr}(\mathcal{P}_{\mathrm{dg}})$ denotes the full subcategory of the dg module category over the boundary (projective–injective) dg subcategory consisting of Gorenstein projective modules (i.e., dg modules admitting a totally acyclic resolution).

This approach leverages projective domination and reflexivity to replace structures arising from gradings (as in Orlov’s triangulated B-brane categories) with intrinsic homological and dg-module-theoretic machinery. Notably, whereas Orlov’s framework employs extra gradings and matrix factorizations to model categories of singularities, the Higgs category here is constructed and analyzed via its exact dg structure and derived functors, without reference to an additional grading.

This module-theoretic description provides conceptual clarity and new homological techniques for analyzing the symmetries and extension structures, as well as for studying how automorphisms (e.g., cyclic and braid group actions) manifest on the category.

6. Symmetry Realizations and Implications in Higher Teichmüller Theory

The correspondence between the categorical symmetries (cyclic, braid group) and the combinatorial symmetries of the cluster variety is explicitly realized in the Higgs category. For example, rotation functors match the expected automorphisms of the triple-of-flags variety, and braid group actions correspond to moves between different configurations of cluster–tilting objects.

This symmetry matching is essential for realizing the modular group actions proposed in higher Teichmüller theory, relating the algebraic and topological operations on moduli of flags to categorical autoequivalences and mutations in the Higgs category.

The categorification afforded by the Higgs category provides a powerful bridge between arithmetic geometry, geometric representation theory, and modern cluster algebraic approaches to moduli spaces, suggesting deep connections to canonical bases, quantization, and positivity phenomena.

Summary Table: Main Features of the Higgs Category for Triples of Flags

Feature	Description	Reference
Construction	Exact dg category from relative Calabi–Yau (Ginzburg) algebra, subcategory of cluster cat.	(Keller et al., 5 Sep 2025)
Structural Properties	Frobenius, stably 2-Calabi–Yau, cyclic and braid group symmetries	(Keller et al., 5 Sep 2025)
Categorical Description	Via Gorenstein projective dg modules over a boundary dg subcategory	(Keller et al., 5 Sep 2025)
Relation to Cluster Structures	Categorifies cluster variety of triples of flags (“basic triangle” for Dynkin type Δ)	(Keller et al., 5 Sep 2025)
Key Theorem	Merlin Christ’s conjecture: equivalence with cosingularity category	(Keller et al., 5 Sep 2025)
Methods	Projective domination, reflexivity, comparisons with graded B-branes (à la Orlov)	(Keller et al., 5 Sep 2025)

The Higgs category thus supplies an intricate, symmetry-rich, and homologically robust categorical model for the cluster variety of triples of flags, directly reflecting the combinatorics and geometry of higher Teichmüller theory and encoding deep arithmetic, representation-theoretic, and geometric information.