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Integrable Deformations of Strings on Symmetric Spaces

Timothy J. Hollowood, J. Luis Miramontes, David M. Schmidtt

TL;DR

This work presents a framework to construct integrable deformations of sigma-models with symmetric-space targets by combining non-abelian T-duality with gauged F/F WZW models, yielding a Lagrangian formulation that preserves integrability while deforming the underlying symplectic structure. The deformation leads to two decoupled classical current algebras and admits a consistent Hamiltonian description, with two key limits: a non-abelian T-dual regime and a current-current gauged-WZW regime. Extending to symmetric spaces, the deformations maintain integrability and map under Pohlmeyer reduction to a relativistic symmetric-space sine-Gordon theory in the large coupling limit, thereby connecting string dynamics on deformed backgrounds to relativistic integrable field theories. In the quantum regime, the deformations correspond to quantum-group structures and IRF/RSOS representations, and the formalism naturally extends to semi-symmetric spaces relevant for AdS$_5\times S^5$ string theory, highlighting potential exact S-matrices and consistent string backgrounds.

Abstract

A general class of deformations of integrable sigma-models with symmetric space F/G target-spaces are found. These deformations involve defining the non-abelian T dual of the sigma-model and then replacing the coupling of the Lagrange multiplier imposing flatness with a gauged F/F WZW model. The original sigma-model is obtained in the limit of large level. The resulting deformed theories are shown to preserve both integrability and the equations-of-motion, but involve a deformation of the symplectic structure. It is shown that this deformed symplectic structure involves a linear combination of the original Poisson bracket and a generalization of the Faddeev-Reshetikhin Poisson bracket which we show can be re-expressed as two decoupled F current algebras. It is then shown that the deformation can be incorporated into the classical model of strings on R x F/G via a generalization of the Pohlmeyer reduction. In this case, in the limit of large sigma-model coupling it is shown that the theory becomes the relativistic symmetric space sine-Gordon theory. These results point to the existence of a deformation of this kind for the full Green-Schwarz superstring on AdS5 x S5.

Integrable Deformations of Strings on Symmetric Spaces

TL;DR

This work presents a framework to construct integrable deformations of sigma-models with symmetric-space targets by combining non-abelian T-duality with gauged F/F WZW models, yielding a Lagrangian formulation that preserves integrability while deforming the underlying symplectic structure. The deformation leads to two decoupled classical current algebras and admits a consistent Hamiltonian description, with two key limits: a non-abelian T-dual regime and a current-current gauged-WZW regime. Extending to symmetric spaces, the deformations maintain integrability and map under Pohlmeyer reduction to a relativistic symmetric-space sine-Gordon theory in the large coupling limit, thereby connecting string dynamics on deformed backgrounds to relativistic integrable field theories. In the quantum regime, the deformations correspond to quantum-group structures and IRF/RSOS representations, and the formalism naturally extends to semi-symmetric spaces relevant for AdS string theory, highlighting potential exact S-matrices and consistent string backgrounds.

Abstract

A general class of deformations of integrable sigma-models with symmetric space F/G target-spaces are found. These deformations involve defining the non-abelian T dual of the sigma-model and then replacing the coupling of the Lagrange multiplier imposing flatness with a gauged F/F WZW model. The original sigma-model is obtained in the limit of large level. The resulting deformed theories are shown to preserve both integrability and the equations-of-motion, but involve a deformation of the symplectic structure. It is shown that this deformed symplectic structure involves a linear combination of the original Poisson bracket and a generalization of the Faddeev-Reshetikhin Poisson bracket which we show can be re-expressed as two decoupled F current algebras. It is then shown that the deformation can be incorporated into the classical model of strings on R x F/G via a generalization of the Pohlmeyer reduction. In this case, in the limit of large sigma-model coupling it is shown that the theory becomes the relativistic symmetric space sine-Gordon theory. These results point to the existence of a deformation of this kind for the full Green-Schwarz superstring on AdS5 x S5.

Paper Structure

This paper contains 19 sections, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Deformed Principal Chiral Model
  3. Extending beyond $\text{SU}(2)$
  4. Non-abelian T-duality and bosonized Thirring model
  5. Integrability
  6. Hamiltonian Structure of the Deformed Principal Chiral Model
  7. The Deformed Symmetric Space Theories
  8. The deformed theory
  9. Equations-of-motion and integrability
  10. The Hamiltonian Structure of the Deformed Theories
  11. The Hamiltonian
  12. Relation to other work
  13. Strings on Deformed Symmetric Spaces: the Pohlmeyer Reduction
  14. A parameterization
  15. The Symmetric Space Sine-Gordon Theories
  16. ...and 4 more sections

Figures (1)

  • Figure 1: The vertex and IRF labels for the 2-body S-matrix elements. For the case of $\text{SU}(2)$ in the vertex picture, the particles are labelled by $i,j,k,l\in\{\pm\frac{1}{2}\}$ the weights of the spin $\frac{1}{2}$ representation of $\text{SU}(2)$. In the IRF or SOS picture on the right, one labels the vacua $a,b,c,d\in\{0,\frac{1}{2},1,\frac{3}{2},\ldots\}$ which are spins of arbitrary irreducible representations of $\text{SU}(2)$ with $|a-b|=|b-c|=|a-d|=|d-c|=\frac{1}{2}$. The particles are then interpretated as kinks $K_{ab}+K_{bc}\to K_{ad}+K_{dc}$ interpolating between adjacent vacua. In the restricted case the spins are restricted to the finite set $\{0,\frac{1}{2},1,\ldots,\frac{k}{2}-1\}$, where $q=e^{i\pi/k}$.