Fusion 3-Categories for Duality Defects

We study the fusion 3-categorical symmetries for quantum theories in (3+1)-dimensions with self-duality defects. Such defects have been realized physically by half-space gauging in theories with one-form symmetries A[1] for an abelian group A, and have found applications in the continuum and the lattice. These fusion 3-categories will be called (generalized) Tambara-Yamagami fusion 3-categories (3TY), in analogy to the TY fusion 1-categories. We consider the Brauer-Picard and Picard 4-groupoids to construct these categories using a 3-categorical version of the extension theory introduced by Etingof, Nikshych and Ostrik. These two 4-groupoids correspond to looking at the construction of duality defects either directly from the 4d point of view, or from the point of view of the 5d Symmetry Topological Field Theory (SymTFT). At this categorical level, the Witt group of non-degenerate braided fusion 1-categories naturally appears in the aforementioned 4-groupoids and represents enrichments of standard duality defects by (2+1)d TFTs. Our main objective is to study graded extensions of the fusion 3-category $3\mathrm{Vect}\left(A\right[1\left]\right)$ for some finite abelian group A, which is the symmetry category associated to a (3+1)d theory with 1-form symmetry A. Firstly, we do so explicitly using invertible bimodule 3-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the Brauer-Picard 4-groupoid of $3\mathrm{Vect}\left(A\right[1\left]\right)$ can be identified with the Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of $3\mathrm{Vect}\left(A\right[1\left]\right)$ , which represents topological defects of the SymTFT, is completely described by a sylleptic strongly fusion 2-category formed by topological surface defects of the SymTFT. These are classified by a finite abelian group equipped with an alternating 2-form. We relate the Picard 4-groupoid of the corresponding braided fusion 3-categories with a generalized Witt group constructed from certain graded braided fusion 1-categories using a twisted Deligne tensor product. In tractable examples, we are able to carry out explicit computations so as to understand the categorical structure of the $Z/2$ and $Z/4$ graded 3TY categories.