Abstract
Recent work explores the candidate phases of the 4d adjoint quantum chromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low energy 4d topological quantum field theories (TQFTs) to saturate the higher 't Hooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow from high energy. In this work, we generalize the symmetry-extension method [arXiv:1705.06728] to higher symmetries, and formulate higher group cohomology and cobordism theory approach to construct higher-symmetric TQFTs. We prove that the symmetry-extension method saturates certain anomalies, but also prove that neither $A \mathcal{P}_2(B_2)$ nor $\mathcal{P}_2(B_2)$ can be fully trivialized, with the background 1-form field $A$, Pontryagin square $\mathcal{P}_2$ and 2-form field $B_2$. Surprisingly, this indicates an obstruction to constructing a fully 1-form center and 0-form chiral symmetry (full discrete axial symmetry) preserving 4d TQFT with confinement, a no-go scenario via symmetry-extension for specific higher anomalies. We comment on the 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.
Highlights
AND SUMMARY OF MAIN RESULTSRecent work explores the candidate phases of the adjoint quantum chromodynamics in four-dimensional (4D) spacetime (QCD4) with an SU(2) gauge group and two massless adjoint Weyl fermions [1,2,3,4].1 This adjoint QCD4 has a 1-form electric Z2 center global symmetry, which is a generalized global symmetry of a higher differential form [5]
This adjoint QCD4 has the SU(2) gauge theory coupling to the matter fields in the adjoint representation; it gains a 1-form electric Z2 center symmetry, while the usual fundamental QCD4 has the gauge theory coupling to the matter fields in the fundamental representation, which lacks the 1-form symmetry
We demonstrate the explicit construction of the 4D symmetric anomalous topological quantum field theories (TQFTs) for this 4D higher anomaly of Eq (3) in Sec
Summary
Recent work explores the candidate phases of the adjoint quantum chromodynamics in four-dimensional (4D) spacetime (QCD4) with an SU(2) gauge group and two massless adjoint Weyl fermions (equivalently, two massless adjoint Majorana fermions, or one massless adjoint Dirac fermion) [1,2,3,4].1 This adjoint QCD4 has a 1-form electric Z2 center global symmetry, which is a generalized global symmetry of a higher differential form [5]. We find that the generalized symmetryextension method can construct another symmetric 4D TQFT saturating a different higher mixed anomaly, mixing between the background gravity (or the curved spacetime geometry) and the 1-form center symmetry (denoted as Z2;1⁄21) This higher anomaly is abbreviated as the type II higher anomaly in Ref. In Appendix A, we find a potential obstruction: The type I anomaly equation (2) cannot be saturated by a symmetric anomalous finite group/higher group TQFT, at least by a symmetry-extension method. The mathematical details of our cobordism calculations can be found in a companion paper [9]
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