$4d$ $SU(2)$ Yang-Mills gauge theory, $2d$ $\mathbb C\mathbb P^1$ sigma model, anomalies and cobordisms

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 $SU(2)$-Yang-Mills (YM) gauge theory at $\theta = \pi$, via $5d$ cobordism invariants. Second, we propose a set of ’t Hooft anomalies of $2d$ $\mathbb C\mathbb P^1$-sigma model at $\theta = \pi$, by enlisting all possible $3d$ cobordism invariants and selecting the matched terms. We derive a correspondence between $5d$ and $3d$ new invariants. Thus we broadly prove a potentially complete anomaly-matching between $4d$ $SU(2)$ YM and $2d$ $\mathbb C\mathbb P^1$ model. This is my joint work with Juven Wang and Yunqin Zheng.