[Paper Review] Orbifolding Frobenius Algebras

Ralph M. Kaufmann|ArXiv.org|Jul 23, 2001

Algebraic structures and combinatorial models13 references4 citations

TL;DR

This paper introduces a complete axiomatic framework for Frobenius algebras with finite group actions, termed orbifold Frobenius algebras, and establishes their geometric realization via cobordism categories. It demonstrates that twisted sectors—previously treated as auxiliary—are essential for mirror symmetry, showing that orbifolding $A_n$ by $bZ/(n+1)bZ$ yields self-dual mirror pairs and provides the first mathematical derivation of the $A_{2n+3}/bZ/2bZ \to D_n$ transition using Frobenius algebra duality.

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.

Motivation & Objective

• To develop a rigorous axiomatic theory of Frobenius algebras with finite group actions, modeling physical orbifold theories in string and topological field theory.
• To provide a geometric realization of these algebras through cobordism categories, extending Atiyah-Dubrovin-Dijkgraaf-Segal formalism to the orbifold setting.
• To clarify the role of twisted sectors as fundamental structures, not mere technicalities, by showing their necessity for mirror symmetry in singularity examples.
• To introduce and study special $G$-Frobenius algebras, particularly Jacobian $G$-Frobenius algebras, to capture structures from quasi-homogeneous singularities and their symmetries.
• To establish a duality transformation that identifies orbifolding as a mechanism for mirror symmetry, exemplified by the duality between $(A_n, A_1)$ and $(A_1, A_n)$ via $bZ/(n+1)bZ$ action.

Proposed method

• Axiomatize Frobenius algebras with two distinct $G$-actions differing by a character twist, corresponding to natural multiplication and scalar product structures.
• Define a non-commutative multiplication on the direct sum of all twisted sectors *before* taking $G$-invariants, a novel construction not previously considered.
• Construct geometric cobordism categories whose functors to vector spaces are parameterized by these orbifold Frobenius algebras.
• Introduce the class of special $G$-Frobenius algebras, including Jacobian $G$-Frobenius algebras, to model Landau-Ginzburg and singularity theories.
• Use graded and super-graded versions to incorporate parity choices for twisted sectors, enabling a full treatment of Ramond and Neveu-Schwarz sectors.
• Apply the duality transformation to show that orbifolding $A_n$ by $bZ/(n+1)bZ$ yields a self-dual mirror pair, and derive the $A_{2n+3}/\bbZ/2\bbZ \to D_n$ correspondence mathematically without path integrals.

Experimental results

Research questions

• RQ1 How can Frobenius algebras with finite group actions be axiomatized to model orbifold theories in physics and geometry?
• RQ2 What is the geometric significance of twisted sectors, and why are they essential rather than auxiliary in orbifold constructions?
• RQ3 Can the process of orbifolding be interpreted as a form of mirror symmetry, and if so, how does it act on Frobenius algebra structures?
• RQ4 How do the Ramond and Neveu-Schwarz sectors in Landau-Ginzburg models relate to the cohomology of singularities under group actions?
• RQ5 What is the mathematical structure underlying the transition from $A_{2n+3}$ to $D_n$ via $\bbZ/2\bbZ$ quotient, and how can it be derived without path integrals?

Key findings

• Orbifolding the $A_n$ singularity with $\bbZ/(n+1)\bbZ$ action produces a self-dual mirror pair, where the sum of twisted sectors is dual to the untwisted sector.
• The $A_{2n+3}/\bbZ/2\bbZ$ orbifold yields the $D_n$ singularity, and this correspondence is derived mathematically via Frobenius algebra duality, avoiding path integrals.
• The $A_n$ Frobenius algebra with trivial grading and trivial $G$-action on even sectors is isomorphic to the $r$-spin curve algebra, confirming its appearance in the A-model of mirror symmetry.
• For $G \subset O(n,\bbC)$ with $s_g^- = 0$, the $G$-twisted algebra becomes a twisted group algebra, with structure determined by $H^2(G, k^*)$.
• The $G$-invariant part of the Ramond sector in the $A_{2n-3}$ model with $\bbZ/2\bbZ$ action yields $A_{n-2}$, matching expectations from Arnold's singularity theory.
• The duality transformation on Jacobian $G$-Frobenius algebras shows that $(A_n, A_1)$ is mirror dual to $(A_1, A_n)$ under $\bbZ/(n+1)\bbZ$ orbifolding, providing a new mechanism for mirror symmetry.