The SL(2,Z) S-duality groupoid of 4D abelian gauge theory acts only projectively on one-dimensionalstate spaces. The obstruction classα ∈H2 SL(2,Z);U(1)(1)measures the failure of strict duality composition. Quantum S-duality depends on global data(charge lattices, allowed line operators) and on tangential (spin/pin) structure of the backgroundfour-manifold X, so duality is more naturally modeled as a groupoid of frames than as an actionon a single presentation. From the anomaly-inflow viewpoint, duality defects carry an ’t Hooftanomaly measured by α, which must be canceled by a 3D invertible theory on the defect network.Concretely, this perspective predicts an invertible inflow term whose partition function supplies therequired multiplier phases in duality compositions; in Maxwell realizations with duality activated alongnontrivial loops/defects, the relevant bulk phase is governed by the combination −ηsignature/4+Arf(q)as in Eq.(7) of 2.
SIKX HILTON (Fri,) studied this question.