Sigma Models on Flags

TL;DR

This paper explores sigma models with flag manifold target spaces, revealing new features, symmetries, and anomaly structures, and identifies conditions under which these models are gapless and described by WZW models.

Contribution

It introduces a comprehensive analysis of flag sigma models, generalizing the $ ext{CP}^{N-1}$ model, and uncovers new phenomena related to parameters, symmetries, and anomalies.

Findings

Certain flag models are gapless and described by $SU(N)_1$ WZW models.

The models depend on more parameters and can have larger global symmetries.

The 't Hooft anomalies are more subtle and varied in these models.

Abstract

We study (1+1)-dimensional non-linear sigma models whose target space is the flag manifold $U(N)\over U(N_1)\times U(N_2)\cdots U(N_m)$, with a specific focus on the special case $U(N)/U(1)^{N}$. These generalize the well-known $\mathbb{CP}^{N-1}$ model. The general flag model exhibits several new elements that are not present in the special case of the $\mathbb{CP}^{N-1}$ model. It depends on more parameters, its global symmetry can be larger, and its 't Hooft anomalies can be more subtle. Our discussion based on symmetry and anomaly suggests that for certain choices of the integers $N_I$ and for specific values of the parameters the model is gapless in the IR and is described by an $SU(N)_1$ WZW model. Some of the techniques we present can also be applied to other cases.