The relative Deligne tensor product over pointed braided fusion categories (Q2685347)

Let \(A\) be a finite abelian group and write \(\boldsymbol{Vect}_{A}\) for the fusion category of -graded finite dimensional vector spaces. Working over an algebraically closed field of characteristic zero, a formula for the relative Deligne tensor product of all finite semisimple module categories over \ with the trivial symmetric braiding was given in [\textit{P. Etingof} et al., Quantum Topol. 1, No. 3, 209--273 (2010; Zbl 1214.18007)]. The author [Cah. Topol. Géom. Différ. Catég. 63, No. 1, 3--24 (2022; Zbl 1481.18021)] extended their computations to obtain a formula for the relative Deligne tensor product over \(\boldsymbol{Vect}_{A}^{\beta}\), where \(\beta\) is any braiding on \(\boldsymbol{Vect}_{A}\). This paper generalizes the result in two directions. \begin{itemize} \item[1.] The base field can be any algebraically closed field. \item[2.] The underlying fusion category carries any associator \(\omega\), and the relative Deligne tensor product of all finite semisimple module categories \(\boldsymbol{Vect}_{A}^{(\alpha,\beta)}\) with a compatible braiding \(\beta\) is computed, so that a formula for the fusion rule of \(\boldsymbol{Mod}(\boldsymbol{Vect}_{A}^{(\alpha,\beta)})\) is obtained. \end{itemize} The author gives tables of the relative Deligne tensor product over \(\boldsymbol{Vect}_{A}^{(\alpha,\beta)}\), where \(A\) is either \(\mathbb{Z}/p\), \(\mathbb{Z}/p^{2}\), \(\mathbb{Z}/2\oplus\mathbb{Z}/2\), or \(\mathbb{Z}/4\oplus\mathbb{Z}/2\) with various choices of associator and compatible braiding. The Picard groups of \(\mathbb{Z}/p\oplus\mathbb{Z} /p\) and \(\mathbb{Z}/2\oplus\mathbb{Z}/2\oplus\mathbb{Z}/2\) with specific choices of associators and compatible braidings are also computed. It should be noted that, for some specific groups and braidings, surprisingly similar formulas have been obtained in [\textit{K. Roumpedakis} et al., ``Higher gauging and non-invertible condensation defects'', Preprint, \url{arXiv:2204.02407}] through high energy physics considerations, the precise relation of which with this work remains to be clarified.