Exploring non-invertible symmetries in free theories

Symmetries corresponding to local transformations of the fundamental fields that leave action invariant give rise (invertible) topological defects, which obey group-like fusion rules. One can construct more general (codimension-one) defects by specifying a map between gauge-invariant operators from one side defect and such on other side. In this work, we apply construction Maxwell theory in four dimensions free compact scalar two dimensions. case theory, show mixes field strength $F$ its Hodge dual $\star F$ be at most an $SO(2)$ rotation. For rational values bulk coupling $\theta$-angle find explicit Lagrangian realizes angle $\varphi$ $\cos \varphi$ is also rational. We further determine Wilson 't Hooft lines they are non-invertible. repeat analysis for $\phi$ only discrete maps: trivial one, $Z_2$ $d\phi \rightarrow -d\phi$, $\mathcal{T}$-duality-like i \star d\phi$, product last two.