We offer a streamlined and computationally powerful characterization of higher representations (higher charges) for defect operators under generalized symmetries, employing the powerful framework of Symmetry TFT \( \mathcal{Z}\left(\mathcal{C}\right) \) . For a defect \( \mathcal{D} \) of codimension p, these representations (charges) are in one-to-one correspondence with gapped boundary conditions for the SymTFT \( \mathcal{Z}\left(\mathcal{C}\right) \) on a manifold Y = Σd−p+1 × Sp−1, and can be efficiently described through dimensional reduction. We explore numerous applications of our construction, including scenarios where an anomalous bulk theory can host a symmetric defect. This generalizes the connection between ’t Hooft anomalies and the absence of symmetric boundary conditions to defects of any codimension. Finally we describe some properties of surface charges for 3 + 1d duality symmetries, which should be relevant to the study of Gukov-Witten operators in gauge theories.
