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. As such, 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 classification and enumeration to relate them to SO(N )2. In another direction, we prove that when N is divisible by 8 the N -metaplectic categories have 3 nontrivial bosons, and the boson condensation procedure applied to 2 of these bosons yields N 4 -metaplectic categories. Otherwise stated: any 8k-metaplectic category is a Z2-gauging of a 2k-metaplectic category, so that the N even metaplectic categories lie towers of Z2-gaugings commencing with 2k-or 4k-metaplectic categories with k odd.
We construct projective (unitary) representations of Hecke groups from the vector spaces associated with Witten-Reshetikhin-Turaev topological quantum field theory of higher genus surfaces. In particular, we generalize the modular data of Temperley-Lieb-Jones modular categories. We also study some properties of the representation. We show the image group of the representation is infinite at low levels in genus 2 by explicit computations. We also show the representation is reducible with at least three irreducible summands when the level equals 4l + 2 for l ≥ 1.
We study the Witt classes of the modular categories SO(2r) 2r associated with quantum groups of type Dr at (4r − 2)-th roots of unity. From these classes we derive infinitely many Witt classes of order 2 that are linearly independent modulo the subgroup generated by the pointed modular categories. In particular we produce an example of a simple, completely anisotropic modular category that is not pointed whose Witt class has order 2, answering a question of Davydov, Müger, Nikshych and Ostrik. Our resultsshow that the trivial Witt class [Vec] has infinitely many square roots modulo the pointed classes, in analogy with the recent construction of infinitely many square roots of the Ising Witt classes modulo the pointed classes constructed in a similar way from certain type Br modular categories. We compare the subgroups generated by the Ising square roots and [Vec] square roots and provide evidence that they also generate linearly independent subgroups.
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