The Witt classes of $SO(2r)_{2r}$

We study the Witt classes of the modular categories $SO(2r)_{2r}$ associated with quantum groups of type $D_r$ 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 results show 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 $B_r$ 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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