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Configuration Spaces of Finite Representation Type Algebras

Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, Giulio Salvatori, Hugh Thomas

TL;DR

This work constructs and analyzes a broad class of affine varieties $\widetilde{\mathcal{M}}_\Lambda$ attached to every finite representation type algebra $\Lambda$, defined via $u$-equations and $\widehat{F}$-polynomials. It proves irreducibility, provides a rational parametrization, and shows functorial behavior under quotients, with boundary strata governed by Jasso reductions that realize factorization generalizing string amplitudes. The authors connect representation theory, tropical geometry, and algebraic geometry through the $g$-vector fan, toric embeddings, and a detailed study of non-negative parts, culminating in Rogers dilogarithm identities valid for all finite representation type algebras. They develop a robust framework that extends cluster-algebraic and string-theoretic insights from Dynkin types to the full finite representation type setting, and supply extensive examples and applications in small algebras. The results open pathways for applications in geometry, number theory, and mathematical physics, providing canonical parametrizations and structural decompositions via Jasso reductions and toric geometry.

Abstract

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.

Configuration Spaces of Finite Representation Type Algebras

TL;DR

This work constructs and analyzes a broad class of affine varieties attached to every finite representation type algebra , defined via -equations and -polynomials. It proves irreducibility, provides a rational parametrization, and shows functorial behavior under quotients, with boundary strata governed by Jasso reductions that realize factorization generalizing string amplitudes. The authors connect representation theory, tropical geometry, and algebraic geometry through the -vector fan, toric embeddings, and a detailed study of non-negative parts, culminating in Rogers dilogarithm identities valid for all finite representation type algebras. They develop a robust framework that extends cluster-algebraic and string-theoretic insights from Dynkin types to the full finite representation type setting, and supply extensive examples and applications in small algebras. The results open pathways for applications in geometry, number theory, and mathematical physics, providing canonical parametrizations and structural decompositions via Jasso reductions and toric geometry.

Abstract

To every finite-dimensional -algebra of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by " variables" satisfying "-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.
Paper Structure (41 sections, 60 theorems, 124 equations)

This paper contains 41 sections, 60 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Summary of main results.
  3. Generalized factorization and stringy amplitudes
  4. Introduction from a cluster algebras point of view
  5. Motivation from number theory
  6. Algebra background
  7. Almost split morphisms
  8. The category $K_\Lambda$
  9. The $H^0$ functor
  10. The $g$-vector fan
  11. Dimension vectors
  12. $F$-polynomials
  13. $\widehat{F}$-polynomials
  14. Jasso Reduction
  15. Varieties defined by $\widehat{F}$-equations and $u$-equations
  16. ...and 26 more sections

Key Result

Proposition 2.1

Suppose that $0\to \tau N \to M \to N \to 0$ is an almost split sequence, and $X$ is an indecomposable module. Then:

Theorems & Definitions (120)

  • Conjecture 1.1
  • Proposition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 110 more