Integrability of a family of quantum field theories related to sigma models

Abstract

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The sinh-Gordon model, the affine sl ( 3 ) Toda model, a model called the fermionic sl ( 2 | 1 ) Toda theory, and the N = 2 supersymmetric sine-Gordon model. These models are all related to sigma models in various ways. The N = 2 supersymmetric sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS 2 × S 2 , and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.