Configuration Spaces of Finite Representation Type Algebras
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908741508530176 |
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| author | Arkani-Hamed, Nima Frost, Hadleigh Plamondon, Pierre-Guy Salvatori, Giulio Thomas, Hugh |
| author_facet | Arkani-Hamed, Nima Frost, Hadleigh Plamondon, Pierre-Guy Salvatori, Giulio Thomas, Hugh |
| contents | To every finite-dimensional $\mathbb C$-algebra $Λ$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first introduced in the context of open string theory and moduli space of ordered points on the real projective line by Koba and Nielsen, rediscovered by Brown as "dihedral co-ordinates", and recently generalized to any finite type hereditary algebras. We show that each such variety is irreducible and admits a rational parametrization. The assignment is functorial: algebra quotients correspond to monomial maps among the varieties. The non-negative real part of each variety has boundary strata that are controlled by Jasso reduction. These non-negative parts naturally define a generalization of open string integrals in physics, exhibiting factorization and splitting properties that do not come from a worldsheet picture. We further establish a family of Rogers dilogarithm identities extending results of Chapoton beyond the Dynkin case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24870 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Configuration Spaces of Finite Representation Type Algebras Arkani-Hamed, Nima Frost, Hadleigh Plamondon, Pierre-Guy Salvatori, Giulio Thomas, Hugh Representation Theory High Energy Physics - Theory 16G20 (Primary) 81R99 (Secondary) To every finite-dimensional $\mathbb C$-algebra $Λ$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first introduced in the context of open string theory and moduli space of ordered points on the real projective line by Koba and Nielsen, rediscovered by Brown as "dihedral co-ordinates", and recently generalized to any finite type hereditary algebras. We show that each such variety is irreducible and admits a rational parametrization. The assignment is functorial: algebra quotients correspond to monomial maps among the varieties. The non-negative real part of each variety has boundary strata that are controlled by Jasso reduction. These non-negative parts naturally define a generalization of open string integrals in physics, exhibiting factorization and splitting properties that do not come from a worldsheet picture. We further establish a family of Rogers dilogarithm identities extending results of Chapoton beyond the Dynkin case. |
| title | Configuration Spaces of Finite Representation Type Algebras |
| topic | Representation Theory High Energy Physics - Theory 16G20 (Primary) 81R99 (Secondary) |
| url | https://arxiv.org/abs/2512.24870 |