New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and $\mathbb{CP}^{\mathrm{N}-1}$ Sigma Model

We hypothesize a new and more complete set of anomalies of certain quantum field theories (QFTs) and then give an eclectic verification. First, we propose a set of 't Hooft higher anomalies of 4d time-reversal symmetric pure SU(N)-Yang-Mills (YM) gauge theory with a second-Chern-class topological term at $\theta=\pi$, via 5d cobordism invariants (higher symmetry-protected topological states), with N = $2, 3, 4$ and others. Second, we propose a set of 't Hooft anomalies of 2d $\mathbb{CP}^{\mathrm{N}-1}$-sigma models with a first-Chern-class topological term at $\theta=\pi$, by enlisting all possible 3d cobordism invariants and selecting the matched terms. Based on algebraic/geometric topology, QFT analysis, manifold generator correspondence, condensed matter inputs such as stacking PSU(N)-generalized Haldane quantum spin chains, and additional physics criteria, we derive a correspondence between 5d and 3d new invariants. Thus we broadly prove a potentially complete anomaly-matching between 4d SU(N) YM and 2d $\mathbb{CP}^{\mathrm{N}-1}$ models at N = 2, and suggest new (but maybe incomplete) anomalies at N = 4. We formulate a higher-symmetry analog of "Lieb-Schultz-Mattis theorem" to constrain the low-energy dynamics.