Non-pseudounitary fusion (Q2065617)

Fusion categories are both a generalization of the categories of representations of finite groups and an algebraic axiomatization of quantum symmetry. From a technical standpoint, fusion categories \(\mathcal{C}\) over \(\mathbb{C}\) are \(\mathbb{C}\)-linear semisimple tensor categories with finitely many isomorphism classes of simple objects, the set of which is denoted \(\mathcal{O}(\mathcal{C})\) and a tensor unit \(\boldsymbol{1}\in\mathcal{O}(\mathcal{C})\). The skeleton of a fusion category \(\mathcal{C}\) is its Grothendieck ring \(K(\mathcal{C})\), which is a fusion ring [\textit{P. Etingof} et al., Tensor categories. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1365.18001), Definition 3.1.7]. There are infinitely many fusion rings \(R\) which do not arise in this fashion. Therefore, it is said that a fusion category \(\mathcal{C}\) is a \textit{categorification} of \(R\) if \(R=K(\mathcal{C})\), and reflexively that \(R\) is \textit{categorifiable} if there exists a fusion category \(\mathcal{C}\) with \(R=K(\mathcal{C})\). Every fusion category \(\mathcal{C}\) possesses the dimension function [\textit{P. Etingof} et al., Tensor categories. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1365.18001), Proposition 3.3.6 (i)] \[ \operatorname{FPdim}:K(\mathcal{C})\rightarrow\mathbb{C} \] which for each \(X\in\mathcal{O}(\mathcal{C})\) is computed as the Frobenius-Perron eigenvalue of the matrix of tensoring with \(X\). Fusion categories \(\mathcal{C}\) which possess a spherical structure [\textit{P. Etingof} et al., Tensor categories. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1365.18001), \S 4.7] are of another dimension \[ \operatorname{dim}:K(\mathcal{C})\rightarrow\mathbb{C} \] often referred to as \textit{categorical dimension}. When \[ \operatorname{dim}(\mathcal{C})=\operatorname{FPdim}(\mathcal{C} ) \] it is said that \(\mathcal{C}\) is \textit{pseudounitary} and that \(K(\mathcal{C})\) admits a \textit{pseudounitary categorification}. Pseudounitary fusion categories are privy to a vast range of tools and results which generic fusion categories are not. The principal objective in this paper is to show that the set of fusion rings which admit a pseudounitary categorification is a strict subset of the set of all categorifiable fusion rings. Exactly speaking, the main result is the following theorem. Theorem. There exists a fusion category of rank \(6\) whose Grothendieck ring does not admit any pseudounitary categorifications. The proof of the above theorem is outlined in \S 2 while the specific details are relegated to \S 3--6. To this end, only examples from the representation theory of quantum groups at roots of unity are needed. The rank \(6\) fusion category the author provides is \(\mathcal{C}(\mathfrak{so}_{5},3/2)_{\mathrm{ad}}\) or \(\mathcal{C}(\mathfrak{so}_{5},9,q)_{\mathrm{ad}}\) in the notation of [\textit{A. Schopieray}, Contemp. Math. 747, 1--26 (2020; Zbl 1436.18018)] with \(q=\exp(\pi i/9)\). It has the structure of a \textit{modular tensor category}. Modular tensor categories of rank strictly less than \(6\) have been classified completely [\textit{P. Bruillard} et al., Int. Math. Res. Not. 2016, No. 24, 7546--7588 (2016; Zbl 1404.18016)], where all necessary fusion rings are of pseudounitary categorifications. Therefore the classification of modular tensor categories has entered a new and inherently non-pseudounitary kingdom.