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Sigma Models on Flags

Kantaro Ohmori, Nathan Seiberg, Shu-Heng Shao

TL;DR

The general flag model exhibits several new elements that are not present in the special case of the well-known jats:inline-formula, which depends on more parameters, its global symmetry can be larger, and its ’t Hooft anomalies can be more subtle.

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.

Sigma Models on Flags

TL;DR

The general flag model exhibits several new elements that are not present in the special case of the well-known jats:inline-formula, which depends on more parameters, its global symmetry can be larger, and its ’t Hooft anomalies can be more subtle.

Abstract

We study (1+1)-dimensional non-linear sigma models whose target space is the flag manifold , with a specific focus on the special case . These generalize the well-known model. The general flag model exhibits several new elements that are not present in the special case of the 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 and for specific values of the parameters the model is gapless in the IR and is described by an WZW model. Some of the techniques we present can also be applied to other cases.

Paper Structure

This paper contains 39 sections, 185 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Lagrangian and the Parameters
  3. The Lagrangian
  4. The Lagrangian
  5. Counterterms
  6. Discrete Global Symmetry and 't Hooft Anomaly
  7. $\mathbb{S}_N$ Symmetry
  8. $\mathbb{Z}_N$ Symmetry
  9. $\mathbb{Z}_N$ Invariant Deformations
  10. $\mathbb{Z}_2$ Symmetries
  11. Deformation of the WZW Model
  12. Global Symmetry
  13. Deformation to the Flag Sigma Model
  14. The WZW Action
  15. Symmetry-Preserving Relevant Deformations
  16. ...and 24 more sections

Figures (3)

  • Figure 1: The UV WZW model flows in the IR to the WZW CFT. This flow preserves the full $G_{WZW}$ global symmetry. Here we deform this UV theory by a potential that restricts the field to take values in the flag ${\cal M}$ and breaks the WZW symmetry to $G_{UV}\subset G_{WZW}$. Then we explore whether the sigma model can flow to the WZW CFT (dashed line in the diagram). The global symmetry of each theory in the figure is written below it.
  • Figure 2: The gauged linear sigma model description of the flag manifold $SU(3)/U(1)^2$. Each node represents a $U(1)$ vector multiplet and the square represents the $SU(3)$ flavor symmetry. $A_i,B^i$ and $C$ denote the chiral superfields in the fundamental representation of the two gauge groups connected by the corresponding arrow. $i=1,2,3$ is the flavor $SU(3)$ index. There is a superpotential term $W= \sum_{i=1}^3A_iB^iC$.
  • Figure 3: The non-Abelian gauged linear sigma model description of the flag manifold $SU(3)/U(1)^2$. It is related to the Abelian quiver in Figure \ref{['fig:SU3Ab']} by a duality move on the middle node.