Anomalies in the Duality Groupoid Projective Functoriality and Inflow in Four-Dimensional Abelian S-Duality

Electromagnetic duality in 4D abelian gauge theory admits a natural description in terms ofa duality groupoid, whose objects encode choices of charge lattice, line-operator spectrum, andtangential structure. Quantum mechanically, each duality transformation acts projectively onthe one-dimensional state space, with phases governed by a 2-cocycle α ∈ Z2SL(2,Z);U(1) .We prove three main results: (i) by lifting to the metaplectic cover and using the Dedekind–ηmultiplier, we give an explicit cocycle representative on the standard generators S,T ∈ SL(2,Z);(ii) the class α ∈ H2SL(2,Z);U(1) ∼ = µ12 obstructs any strict implementation of SL(2,Z),yielding a no-go criterion for functoriality; (iii) the same cocycle is identified with the phase ofa 5D invertible bulk partition function, expressed via a signature η-invariant together with aquadratic-refinement (Arf) contribution determined by the duality bundle.