Orbifolding Frobenius Algebras

TL;DR

This paper develops a comprehensive theory of Frobenius algebras with group actions, connecting algebraic structures to geometric and physical theories like orbifolds, and extends the framework to graded cases with applications to singularity quotients.

Contribution

It introduces and axiomatizes Frobenius algebras with group actions, linking them to geometric cobordism categories and extending the theory to graded and super-graded cases.

Findings

Defined a geometric cobordism category parameterized by these algebras

Extended the theory to graded and super-graded Frobenius algebras

Applied the framework to Frobenius algebras from quasi-homogeneous singularities

Abstract

We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define a geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super-graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi-homogeneous singularities and their symmetries.