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}U(N)/U(1)N. These generalize the well-known \mathbb{CP}^{N-1}ℂℙN−1 model. The general flag model exhibits several new elements that are not present in the special case of the \mathbb{CP}^{N-1}ℂℙN−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_INI and for specific values of the parameters the model is gapless in the IR and is described by an SU(N)_1SU(N)1 WZW model. Some of the techniques we present can also be applied to other cases.
Highlights
One possibility is that the N ⊂ GUV is unbroken and acts trivially in the IR, and there is another emergent N symmetry that combines with PSU(N ) to form GW ZW
Based on the above considerations, we argue that the flag sigma model with special parameters flows to the SU(N )1 WZW conformal field theory (CFT) in the IR
In particular we argue that the U(r M )/U(M )r sigma model with special parameters and with r, M sufficiently large flows to the SU(r M )1 WZW CFT
Summary
It is significant that the only GUV -invariant relevant operator in the WZW CFT is the identity operator This means that for a range of parameters the flow from the sigma model can hit this fixed point. One possibility is that the N ⊂ GUV is unbroken and acts trivially in the IR, and there is another emergent N symmetry that combines with PSU(N ) to form GW ZW In this case we can no longer use the N ⊂ N to restrict the relevant deformations in the SU(N ) WZW model. An identical argument carries over as long as the flag sigma model on M does not have the global symmetry PSU(N ) × N with the same anomaly as in the SU(N ) WZW model Another possibility is that the entire GUV symmetry decouples in the IR..
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