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

## TL;DR

This paper investigates the construction of 4d topological quantum field theories for adjoint QCD$_4$, revealing obstructions to fully trivializing certain higher anomalies and discussing implications for quantum critical phenomena and dualities.

## Contribution

It generalizes the symmetry-extension method to higher symmetries and proves specific anomalies cannot be fully trivialized, indicating constraints on TQFTs with certain symmetries.

## Key findings

- Certain higher anomalies cannot be trivialized, indicating obstructions to specific TQFT constructions.
- The symmetry-extension method saturates some anomalies but not others, revealing limitations.
- Implications for deconfined quantum critical points and dualities in 4d quantum field theories.

## 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.