My talk will aim to be a friendly introduction for condensed matter friends, mathematicians, and QFT theorists alike --- I shall quickly review and warm up the use of higher symmetries and anomalies of gauge theories and condensed matter systems. Then I will present the results of recent work [arXiv:1904.00994].

We explore 4d time-reversal symmetric pure Yang-Mills (YM) gauge theories of an SU(2) gauge group with a second-Chern-class topological term at *θ*=*π*(*SU*(2)*θ*=*πYM*)

living as boundary conditions of 5d topological state (5d bordism invariants of mod 2 class, or 5d higher-1-form-center-symmetry-protected interacting (SPT) "topological superconductor" in condensed matter). We find their "Fantastic Four Siblings" with four sets of new higher "anomalies" associated with the Kramers singlet/doublet and bosonic/fermionic properties of Wilson lines. Via Weyl's gauge principle, by dynamically gauging the 1-form center symmetry, we transform a 5d bulk SPTs into a symmetry-enriched topologically ordered state (SETs) as 5d higher-gauge TQFT. Apply the tool introduced in [arXiv:1612.09298], we derive new exotic anyonic statistics of extended objects such as 2-worldsheet of strings and 3-worldvolume of branes, which physically characterize the 5d SETs. We discover new triple and quadruple link invariants potentially associated with the underlying 5d TQFTs, hinting an intrinsic relation between non-supersymmetric 4d pure YM and topological links in 5d. We provide lattice simplicial complex and "condensed matter" regularizations and then comment on 4d gauge dynamics.