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We explore 4d Yang-Mills gauge theories (YM) living as boundary conditions of5d gapped short/long-range entangled (SRE/LRE) topological states.Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gaugegroup with a second-Chern-class topological term at $\theta=\pi$(SU(2)$_{\theta=\pi}$ YM), by turning on background fields for both thetime-reversal (i.e., on unorientable manifolds) and 1-form center globalsymmetry. We find four classes of time-reversal and Lorentz symmetry-enrichedSU(2)$_{\theta=\pi}$ YM, labeled by $(K_1, K_2)$: $K_1=0,1$ specifies Kramerssinglet/doublet Wilson line and new mixed higher 't Hooft anomalies; $K_2=0,1$specifies boson/fermionic Wilson line and a new Wess-Zumino-Witten-likecounterterm. Higher anomalies indicate that to realize all higher $n$-globalsymmetries locally on $n$-simplices, the 4d theory becomes a boundary of a 5dhigher-symmetry-protected topological state (SPTs, as an invertible topologicalquantum field theory (TQFT) or a cobordism invariant in math, or as a 5dhigher-symmetric interacting topological superconductor in condensed matter).By dynamically gauging the 1-form symmetry, we transform a 5d bulk SRE SPTsinto an LRE symmetry-enriched topologically ordered state (SETs); thus weobtain the 4d SO(3)$_{\theta=\pi}$ YM-5d LRE-higher-SETs coupled system withhigher-form gauge fields. We further derive new exotic anyonic statistics ofextended objects such as 2-worldsheet of strings and 3-worldvolume of branes,physically characterizing the 5d SETs. We discover triple and quadruple linkinvariants associated with the 5d higher-gauge TQFTs, hinting at a relationbetween non-supersymmetric 4d pure YM and topological links in 5d. We provide4d-5d lattice simplicial complex regularizations and bridge to 4d quantum spinliquids. We constrain gauge dynamics by higher anomalies and a highersymmetry-extension method.

Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory