Fusion 3-Categories for Duality Defects

We study the fusion 3-categorical symmetries for quantum theories in (3+1) d with self-duality defects. Such defects have been realized physically by half-space gauging in theories with 1-form symmetries A1 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). 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 the construction of duality defects either directly in 4d, or from the 5d Symmetry Topological Field Theory (SymTFT). 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 3Vect (A1). Firstly, we use invertible bimodule 3-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the Brauer-Picard 4-groupoid of 3Vect (A1) can be identified with the Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of 3Vect (A1), 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. We perform explicit computations for Z/2 and Z/4 graded 3TY categories.