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Sigma models as Gross-Neveu models

Dmitri Bykov

Abstract

We review the correspondence between integrable sigma models with complex homogeneous target spaces and chiral bosonic (and possibly mixed bosonic/fermionic) Gross-Neveu models. Mathematically, the latter are models with quiver variety phase spaces, which reduce to the more conventional sigma models in special cases. We discuss the geometry of the models, as well as their trigonometric and elliptic deformations, Ricci flow and the inclusion of fermions.

Sigma models as Gross-Neveu models

Abstract

We review the correspondence between integrable sigma models with complex homogeneous target spaces and chiral bosonic (and possibly mixed bosonic/fermionic) Gross-Neveu models. Mathematically, the latter are models with quiver variety phase spaces, which reduce to the more conventional sigma models in special cases. We discuss the geometry of the models, as well as their trigonometric and elliptic deformations, Ricci flow and the inclusion of fermions.

Paper Structure

This paper contains 6 sections, 29 equations, 1 figure.

Table of Contents

  1. The $\mathds{CP}^{n-1}$-model as a gauged chiral Gross-Neveu model
  2. Quantum mechanical reduction.
  3. Complex symplectic phase spaces and quiver varieties
  4. Trigonometric and elliptic deformations
  5. Models with fermions and the supersymplectic quotient
  6. Outlook

Figures (1)

  • Figure 1: Diagrams contributing to the $\beta$-function at one loop. The red and blue arrows are the $\langle U\, V\rangle$ and $\langle \overline{U}\, \overline{V}\rangle$ Green's functions. The quartic vertex is given by the classical $r$-matrix $r(\lambda)^{cd}_{ab}$.