We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short-/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at $\theta =\pi$ [$\mathrm{SU}(2{)}_{\theta =\pi }$ YM], by turning on the background fields for both the time reversal (i.e., on unorientable manifolds) and the one-form center global symmetry. We find four siblings of ${\mathrm{SU}(2)}_{\theta =\pi }$ YM with distinct couplings to background fields, labeled by $(K_1,K_2)$: $K_1=0$, 1 specifies Kramers singlet/doublet Wilson line and new mixed higher ’t Hooft anomalies; $K_2=0$, 1 specifies the boson/fermionic Wilson line and a new Wess-Zumino-Witten–like counterterm. Higher anomalies indicate that to realize all higher $n$-global symmetries locally on $n$ simplices, the 4d theory becomes a boundary of a 5d higher-symmetry-protected topological state (SPTs, as an invertible topological quantum field theory (iTQFT) or a cobordism invariant in math, or as a 5d higher-symmetric interacting topological superconductor in condensed matter). Via Weyl’s gauge principle, by dynamically gauging the one-form symmetry, we transform 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d ${\mathrm{SO}(3)}_{\theta =\pi }$ YM-5d LRE-higher-SETs coupled system with dynamical higher-form gauge fields. We further derive new exotic anyonic statistics of extended objects such as two world sheets of strings and three world volumes of branes, physically characterizing the 5d SETs. We discover triple and quadruple link invariants potentially associated with the underlying 5d higher-gauge topological quantum field theories, hinting at a new intrinsic relation between nonsupersymmetric 4d pure YM and topological links in 5d. We provide 4d-5d lattice simplicial complex regularizations and bridge to 4d SU(2)- and SO(3)-gauged quantum spin liquids as ($3+1$)-dimensional realizations.

• Received 14 May 2019
• Revised 24 August 2019
• Corrected 31 October 2019

DOI:https://doi.org/10.1103/PhysRevD.100.085012

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Published by the American Physical Society