We study higher symmetries and anomalies of 4d $$ \mathfrak{so} $$ so (2nc) gauge theory with 2nf flavors. We find that they depend on the parity of nc and nf, the global form of the gauge group, and the discrete theta angle. The contribution from the fermions plays a central role in our analysis. Furthermore, our conclusion applies to $$ \mathcal{N} $$ N = 1 supersymmetric cases as well, and we see that higher symmetries and anomalies match across the Intriligator-Seiberg duality between $$ \mathfrak{so} $$ so (2nc) ↔$$ \mathfrak{so} $$ so (2nf − 2nc + 4).

We perform a study of time reversal symmetry of abelian anyons A in 2+1 dimensions, in the spin structure independent cases. We will find the importance of the group C of time-reversalsymmetric anyons modulo anyons composed from an anyon and its time reversal. Possible choices of local Kramers degeneracy are given by quadratic refinements of the braiding phases of C, and the anomaly is then given by the Arf invariant of the chosen quadratic refinement. We also give a concrete study of the cases when |A| is odd or A = (Z 2 ) N .

We revisit various topological issues concerning four-dimensional ungauged and gauged Wess-Zumino-Witten (WZW) terms for SU and SO quantum chromodynamics (QCD), from the modern bordism point of view. We explain, for example, why the definition of the 4d WZW terms requires the spin structure. We also discuss how the mixed anomaly involving the 1-form symmetry of SO QCD is reproduced in the low-energy sigma model.