Tohoku Mathematical Journal
2020

September
SECOND SERIES VOL. 72, NO. 3

Tohoku Math. J.
72 (2020), 411-424

Title METAPLECTIC CATEGORIES, GAUGING AND PROPERTY $F$

Author Paul Gustafson, Eric C. Rowell and Yuze Ruan

(Received August 28, 2018, revised February 14, 2019)
Abstract. $N$-Metaplectic categories, unitary modular categories with the same fusion rules as $SO(N)_2$, are prototypical examples of weakly integral modular categories generalizing the model for the Ising anyons, i.e. metaplectic anyons. A conjecture of the second author would imply that images of the braid group representations associated with metaplectic categories are finite groups, i.e. have property $F$. While it was recently shown that $SO(N)_2$ itself has property $F$, proving property $F$ for the more general class of metaplectic modular categories is an open problem. We verify this conjecture for $N$-metaplectic modular categories when $N$ is odd, exploiting their recent enumeration together with a characterization in terms of Galois conjugation and twisting. In another direction, we prove that when $N$ is divisible by 8 the $N$-metaplectic categories have 3 non-trivial bosons, and the boson condensation procedure applied to 2 of these bosons yields $\frac{N}{4}$-metaplectic categories. Otherwise stated: any $8k$-metaplectic category is a $\mathbb{Z}_2$-gauging of a $2k$-metaplectic category, so that the $N$ even metaplectic categories lie towers of $\mathbb{Z}_2$-gaugings commencing with $2k$- or $4k$-metaplectic categories with $k$ odd.

Mathematics Subject Classification. Primary 18D10; Secondary 20F36.

Key words and phrases. Modular tensor categories, symmetry gauging, Property $F$ conjecture.

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