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

TL;DR

This paper investigates the relationship between 4d pure Yang-Mills theory and 2d $ ext{CP}^{N-1}$ models through 't Hooft anomalies, confirming their connection at $ heta=\pi$ and supporting confinement mechanisms.

Contribution

It non-perturbatively verifies the consistency of the relation between 4d Yang-Mills and 2d $ ext{CP}^{N-1}$ models at $ heta=\pi$ using anomaly matching.

Findings

Confirmed the mixed 't Hooft anomalies match at $ heta=\pi$

Supported the confinement approach via 2d models

Established non-perturbative consistency of the relation

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 $\theta=\pi$, 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.