Compute braided G-crossed extensions of Vect_Γ^Q under isometric G-actions

Determine the braided G-crossed extension C of the pointed braided fusion category Vect_Γ^Q, where Γ is a finite abelian group equipped with a quadratic form Q and G is a group acting faithfully by isometries of (Γ, Q), up to modification by a class in H^3(G, C^×). Concretely construct C (with identity component C_1 = Vect_Γ^Q) and specify its grading and coherence data so that the extension is fully explicit up to the cohomological 3-cocycle freedom.

Background

In the paper, B = Vect_Γ^Q denotes the pointed braided fusion category of Γ-graded vector spaces with associator and braiding determined by a quadratic form Q on the finite abelian group Γ. If a group G acts by isometries on the discriminant form (Γ, Q), the seminal results of Etingof–Nikshych–Ostrik and Davydov–Nikshych guarantee existence and uniqueness of braided G-crossed extensions up to a 3-cocycle in H^3(G, C^×) when the obstruction vanishes. However, computing these extensions explicitly (beyond existence and abstract classification) remains challenging.

The authors illustrate explicit solutions in special cases (e.g., G = Z_2 and certain parity conditions on Γ) and connect these computations to orbifold categories of lattice vertex operator algebras. The open problem asks for explicit determination of the unique braided G-crossed extension of Vect_Γ^Q for general isometric G-actions, clarifying the full categorical structure up to the cohomological freedom.