Categorical continuous symmetry

We define the symmetry category in 1+1d for continuous 0-form G-symmetry to be Sky^τ (G), the category of skyscraper sheaves of finite dimensional vector spaces with finite support on the group manifold of G, where τ ∈ H⁴(BG, ℤ) is the anomaly. We propose that the corresponding 2+1d SymTFT is described by the Drinfeld center of Sky^τ (G). We show explicitly the way that τ twists the convolution tensor product of the objects of Sky^τ (G). As a concrete example, we present the S and T-matrices for the simple anyons of the resulting Z(Sky^τ (G)) category for G = U(1), both for the cases without or with anomaly, and discuss the topological boundary conditions as Lagrangian algebra of Z(Sky^τ (U (1))). We also present the definition of Sky^τ (G) and Z(Sky^τ (G)) for the non-abelian case of G = SU(2), as well as the speculated modular data. We point out that in order to have a physically relevant center and Lagrangian algebras, it is necessary to generalize Sky^τ (G) to a larger category, which we argue to be closely related to the category of quasi-coherent sheaves on G_ℂ with convolution tensor product twisted by τ.