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Recent work explores the candidate phases of the 4d adjoint quantumchromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjointWeyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible lowenergy 4d topological quantum field theories (TQFTs) to saturate the higher 'tHooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow fromhigh energy. In this work, we generalize the symmetry-extension method[arXiv:1705.06728] to higher symmetries, and formulate higher group cohomologyand cobordism theory approach to construct higher-symmetric TQFTs. We provethat the symmetry-extension method saturates certain anomalies, but also provethat neither $A \mathcal{P}_2(B_2)$ nor $\mathcal{P}_2(B_2)$ can be fullytrivialized, with the background 1-form field $A$, Pontryagin square$\mathcal{P}_2$ and 2-form field $B_2$. Surprisingly, this indicates anobstruction to constructing a fully 1-form center and 0-form chiral symmetry(full discrete axial symmetry) preserving 4d TQFT with confinement, a no-goscenario via symmetry-extension for specific higher anomalies. We comment onthe implications and constraints on deconfined quantum critical points (dQCP),quantum spin liquids (QSL) or quantum fermionic liquids in condensed matter,and ultraviolet-infrared (UV-IR) duality in 3+1 spacetime dimensions.

Adjoint QCD$_4$, Deconfined Critical Phenomena, Symmetry-Enriched Topological Quantum Field Theory, and Higher Symmetry-Extension