Configuration Spaces of Finite Representation Type Algebras

TL;DR

This paper introduces affine varieties associated with finite-dimensional algebras of finite representation type, revealing their geometric properties and connections to physics, algebra, and dilogarithm identities.

Contribution

It generalizes the construction of configuration varieties for finite type hereditary algebras and explores their geometric, algebraic, and physical properties.

Findings

Each variety is irreducible and admits a rational parametrization.

The varieties' non-negative parts have boundary strata controlled by Jasso reduction.

The work extends Rogers dilogarithm identities beyond the Dynkin case.

Abstract

To every finite-dimensional $\mathbb C$-algebra $\Lambda$ 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.