A Higgs category for the cluster variety of triples of flags
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912628991852544 |
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| author | Keller, Bernhard Liu, Miantao |
| author_facet | Keller, Bernhard Liu, Miantao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04863 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Higgs category for the cluster variety of triples of flags Keller, Bernhard Liu, Miantao Representation Theory Commutative Algebra Combinatorics 16E45, 13F60 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. |
| title | A Higgs category for the cluster variety of triples of flags |
| topic | Representation Theory Commutative Algebra Combinatorics 16E45, 13F60 |
| url | https://arxiv.org/abs/2509.04863 |