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 x S^2, and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.

66 pages; V2: minor corrections, references added

References in corpus (6)

• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Pohlmeyer reduction of AdS_5 x S^5 superstring sigma model
• Baxter's Q-operators for supersymmetric spin chains
• Quantization of soliton systems and Langlands duality
• Quantization of the N=2 Supersymmetric KdV Hierarchy
• Serre presentations of Lie superalgebras

Cited by in corpus (13)

• Irregular singularities in Liouville theory
• Yangians, S-matrices and AdS/CFT
• Non-Linear Integral Equations for the SL(2,R)/U(1) black hole sigma model
• Free Fermions, vertex Hamiltonians, and lower-dimensional AdS/CFT
• Asymptotic representations and q-oscillator solutions of the graded Yang-Baxter equation related to Baxter Q-operators
• Integrable Lambda Models And Chern-Simons Theories
• On Form Factors of the conjugated field in the non-linear Schröodinger model
• Lambda Models From Chern-Simons Theories
• Massless $Ad{S}_2$ scattering and Bethe ansatz
• On $Ad{S}_2/CF{T}_1$ transfer matrices, Bethe ansatz and scale invariance
• Pohlmeyer reduction for superstrings in AdS space
• Geometry of Massless Scattering in Integrable Superstring
• A study of integrable form factors in massless relativistic $Ad{S}_2$