A class of prime fusion categories of dimension $2^N$

TL;DR

This paper introduces and analyzes a new class of fusion categories called N-Ising categories, detailing their structure, properties, and the existence of braided and degenerate variants, expanding understanding of fusion category classifications.

Contribution

It defines N-Ising fusion categories, proves their primeness in the braided case, and characterizes their role as extensions of rank 2 pointed categories.

Findings

Every braided N-Ising fusion category is prime.

Existence of slightly degenerate N-Ising braided categories for all N > 2.

Braided extensions of rank 2 pointed categories are structured by N-Ising categories.

Abstract

We study a class of strictly weakly integral fusion categories $\mathfrak{I}_{N, \zeta}$, where $N \geq 1$ is a natural number and $\zeta$ is a $2^N$th root of unity, that we call $N$-Ising fusion categories. An $N$-Ising fusion category has Frobenius-Perron dimension $2^{N+1}$ and is a graded extension of a pointed fusion category of rank 2 by the cyclic group of order $\mathbb Z_{2^N}$. We show that every braided $N$-Ising fusion category is prime and also that there exists a slightly degenerate $N$-Ising braided fusion category for all $N > 2$. We also prove a structure result for braided extensions of a rank 2 pointed fusion category in terms of braided $N$-Ising fusion categories.