Relating 't Hooft Anomalies of 4d Pure Yang-Mills and 2d $\mathbb{CP}^{N-1}$ Model

TL;DR

The paper investigates a proposed 4d to 2d correspondence between pure SU(N) Yang-Mills theory on a center-twisted three-torus and the 2d CP^{N-1}-model on a flavor-twisted circle. By analyzing the mixed t'Hooft anomaly between center symmetry and time-reversal at theta = π in 4d and performing a twisted compactification, the authors show that the anomaly precisely reduces to the corresponding 2d anomaly in the CP^{N-1}-model. This non-perturbative consistency check strengthens the program of deriving confinement and mass-gap features of 4d YM from the 2d CP^{N-1}-model. The results also imply nontrivial vacuum structure and metastable vacua in the higher-dimensional theory, with potential extensions to theories including matter fields.

Abstract

It has recently been shown that a center-twisted compactification of the four-dimensional pure $SU(N)$ Yang-Mills theory on a three-torus gives rise to the two-dimensional $\mathbb{CP}^{N-1}$-model on a circle with a flavor-twisted boundary condition. We verify the consistency of this statement non-perturbatively at theta angle $θ=π$, in terms of the mixed 't Hooft anomalies for flavor symmetries and the time-reversal symmetry. This provides further support for the approach to the confinement of four-dimensional Yang-Mills theory from the two-dimensional $\mathbb{CP}^{N-1}$-model.