According to the definition given by [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)], non-degenerate braided fusion categories \(\mathcal{C}\) and \(\mathcal{D}\) are Witt equivalent when there exist fusion categories \(\mathcal{A}\) and \(\mathcal{B}\), and a braided equivalence \(\mathcal{C} \boxtimes\mathcal {Z A} \simeq \mathcal {D}\boxtimes \mathcal {Z B}\), where \(\mathcal {Z A}\) and \(\mathcal {Z A} \) denote the monoidal centers (also called Drinfeld centers) of the fusion categories \(\mathcal{A}\) and \(\mathcal{B}\). The Witt equivalence class containing a category \(C\) is denoted by \([C]\). The set of Witt equivalence classes under the Deligne tensor product \(\boxtimes\) is an abelian group. The definition of the Witt group \(W\) depends a priori on the choice of the ground field, but in this paper, like in [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)], the ground field \(\boldsymbol{k}\) is taken as an algebraically closed field of characteristic zero and it can be shown that different choices for \(\boldsymbol{k}\) lead to isomorphic Witt groups. The unit is \([\mathrm{Vec}]\), where \(\mathrm{Vec}\) denotes denote the fusion category of finite dimensional vector spaces over \(\boldsymbol{k}\).
For any simple Lie algebra \(\mathfrak{g}\) over \(\mathbb{C}\) and a positive integer \(k\) called the level, one gets a modular category, i.e., a non-degenerated braided fusion category with a ribbon structure, by taking the semisimplification of the tilting module category of the quantum group \(U_q(\mathfrak{g})\) specialized at a root of unity \(q\) determined by \(\mathfrak{g}\) and \(k\), see [B. Bakalov and A. Kirillov jun., Lectures on tensor categories and modular functors. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0965.18002)]. This category is denoted \(\mathfrak{g}_k\).
The authors study the structure of the Witt subgroup generated by the categories \(\mathcal{C}_r = \mathfrak{so}(2r)_{2r}\), i.e., they choose \(\mathfrak{g}\) of type \(\mathfrak{so}(2r)\), the rank being \(r\), and consider the particular case where the level \(k\) is chosen as twice the rank. If \(r \geq 3\) their general discussion depends on the parity of \(r\), whereas the case \(r=2\) (where \(\mathfrak{g}\) is not simple) requires a separate study.
The notion of a connected étale algebra object \(A\) in a braided fusion category \(\mathcal{C}\) is formally defined and studied in [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)], although the existence and usefulness of such algebra objects was already recognized and discussed in the previous literature, for instance in [A. Kirillov jun. and V. Ostrik, Adv. Math. 171, No. 2, 183–227 (2002; Zbl 1024.17013)]. When \(\mathcal{C}\) is the fusion category \(Rep(G)\) for some finite group \(G\), the regular algebra \(A=Fun(G)\) is such a connected étale algebra in \(\mathcal{C}\). The choice of a connected étale algebra object \(A\) in a given braided fusion category \(\mathcal{C}\) plays an important role in the present analysis. Indeed, it can be shown [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)] that if \(A\) is a connected étale algebra in \(\mathcal{C}\), the category \(C_A\) of right \(A\)-modules, is a fusion category; moreover, some particular \(A\)-modules, called local, or dyslectic, in this paper (they are also called dyslectic in [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)], but ambichiral in [J. Böckenhauer et al., Commun. Math. Phys. 210, No. 3, 733–784 (2000; Zbl 0988.46047)] or in several older conformal field theory articles and in the theory of bimodules [A. Ocneanu, “Paths on Coxeter diagrams: from Platonic solids and singularities to minimal models and subfactors”, in: Lectures on Operator Theory. Providence: AMS Publications. 243–323 (1999)], are such that \(C_A^0\), the full subcategory of local modules of \(C_A\), is a braided fusion category like \(\mathcal{C}\) itself, and it is shown (also in [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)]) that if \(A\) in \(\mathcal{C}\) is a connected étale algebra, then the Witt classes \([C_A^0 ]\) and \([\mathcal{C}]\) are equal. For this reason, in order to study the Witt class of a braided fusion category \(\mathcal{C}\), it suffices to study the smaller category \(C_A^0\).
The authors follow the above path to study the Witt subgroup generated by the braided fusion categories \(\mathcal{C}_r = \mathfrak{so}(2r)_{2r}\).
In a first step they describe connected étale algebras \(A\) in \(\mathcal{C}_r\) and show that, if \(r\) is even, \((\mathcal{C}_r)_A^0\) is an unpointed modular category \(\mathcal{D}_r\) (a fusion category is unpointed if it has only one invertible object), whereas, if \(r\) is odd, it is the tensor product of an unpointed modular category \(\mathcal{D}_r\) by a well-defined pointed modular category (that they call \(\mathrm{SEM}\)) characterized by the pair \((\mathbb{Z}/2\mathbb{Z},Q)\) where \(Q\) is a particular quadratic form on the abelian group \(\mathbb{Z}/2\mathbb{Z}\). In both cases (assuming \(r \geq 3\)) they show for the Witt classes the equality \([\mathcal{D}_r]^2 = [\mathrm{Vec}]\).
In the next step they study the Witt subgroup generated by the classes \([\mathcal{D}_r]\), or, rather, the subgroups generated by families of such \([\mathcal{D}_r]\)’s since one has to separate cases according to the parity of \(r\). In the even case the discussion actually depends on \(r\bmod 8\), but in all cases the authors show that the obtained subgroups are isomorphic with direct products \(\mathbb{Z}/2\mathbb{Z}\) factors, the key point being to prove that there are no nontrivial relations (no finite product of distinct \([\mathcal{D}_r]\) yields \( [\mathrm{Vec}]\)).
In the last section the authors consider non-degenerate braided fusion category \(\mathcal{C}\) that are simultaneously completely anisotropic (the only connected étale algebra \(A \in \mathcal{C}\) is \(A = 1\)), simple (it has no non-trivial fusion subcategories) and unpointed (\(1\) is the only one invertible object). The subgroup of \(W\) generated by such categories is called \(W_S\). The following question was asked in [A. Davydov et al., J. Reine Angew. Math. 677, 135–177 (2013; Zbl 1271.18008)]: Is there a class of order \(2\) in \(W_S\)? The present authors give a positive answer by showing that the class \([D_4]\) indeed obeys the above conditions. It is already known (as a by-product of their analysis) that it has order \(2\) in \(W\), it remains to show that it belongs to \(W_S\), this is done at the end of the article.
The methods used in the various sections are often number-theoretical and illustrate the usefulness of several invariants for distinguishing Witt classes, in particular the signature homomorphism (defined, using Galois theory, in terms of the Perron-Frobenius dimension of the studied fusion category), as well as other quantities like the so-called higher central charges. Independently of the advocated purpose of the article, the several sections contain useful general information about many aspects of fusion categories and about the various numerical quantities that one can attach to their study.