Commutative Algebras in Fibonacci Categories

TL;DR

This work analyzes commutative ribbon algebras within Fibonacci modular categories and their tensor powers to determine maximality phenomena in rational conformal field theories. By encoding module-category data through NIM-representations of Fib$^{\boxtimes \ell}$, the authors show that Fib$^{\boxtimes \ell}$ is completely anisotropic when the factors are not mutually inverse, implying there are no nontrivial separable, ribbon, commutative algebras. Consequently, chiral algebras whose representation category is a product of Fibonacci categories are maximal, including cases corresponding to the Yang-Lee model M(2,5) and affine algebras G_{2,1} and F_{4,1} (and their tensor powers). The paper also establishes the existence of four non-equivalent Fibonacci modular categories Fib$_u$ (u a primitive 10th root of unity) and proves that any modular category containing a Fibonacci subcategory decomposes as a tensor product with an additional factor; these results constrain possible RCFT extensions and VOA constructions involving Fibonacci-type categories.

Abstract

By studying NIM-representations we show that the Fibonacci category and its tensor powers are completely anisotropic; that is, they do not have any non-trivial separable commutative ribbon algebras. As an application we deduce that a chiral algebra with the representation category equivalent to a product of Fibonacci categories is maximal; that is, it is not a proper subalgebra of another chiral algebra. In particular the chiral algebras of the Yang-Lee model, the WZW models of G2 and F4 at level 1, as well as their tensor powers, are maximal.

Theorems & Definitions (9)

• Lemma 1
• Proposition 2
• Proposition 3
• Theorem 4
• Lemma 5
• Theorem 6
• Theorem 7
• Lemma 8
• proof