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Deformations of N=4 SYM and integrable spin chain models

David Berenstein, Sergey A. Cherkis

TL;DR

The paper investigates the fate of integrability when N=4 SYM is marginally deformed, showing that planar one-loop spectra reduce to twisted XXZ spin chains for orbifold-like deformations, while generic deformations break full integrability though integrable subsectors can appear. It provides detailed Bethe-ansatz analyses for twisted chains, constructs an exact SO_q(6) XXZ Hamiltonian via quantum-group methods, and demonstrates that such an SO_q(6) chain does not typically arise from a simple Lagrangian deformation preserving conformality. A broader matrix-model perspective is developed, arguing that a multi-matrix quantum mechanics framework is a natural arena for realizing general spin-chain dynamics and potentially connecting to holographic duals of plane-wave geometries. Overall, the work delineates sharp no-go results for full integrability beyond orbifold twists and highlights the central role of quantum-group symmetries in understanding the landscape of integrable deformations of N=4 SYM.

Abstract

Beginning with the planar limit of N=4 SYM theory, we study planar diagrams for field theory deformations of N=4 which are marginal at the free field theory level. We show that the requirement of integrability of the full one loop dilatation operator in the scalar sector, places very strong constraints on the field theory, so that the only soluble models correspond essentially to orbifolds of N=4 SYM. For these, the associated spin chain model gets twisted boundary conditions that depend on the length of the chain, but which are still integrable. We also show that theories with integrable subsectors appear quite generically, and it is possible to engineer integrable subsectors to have some specific symmetry, however these do not generally lead to full integrability. We also try to construct a theory whose spin chain has quantum group symmetry SO_q(6) as a deformation of the SO(6) R-symmetry structure of N=4 SYM. We show that it is not possible to obtain a spin chain with that symmetry from deformations of the scalar potential of N=4 SYM. We also show that the natural context for these questions can be better phrased in terms of multi-matrix quantum mechanics rather than in four dimensional field theories.

Deformations of N=4 SYM and integrable spin chain models

TL;DR

The paper investigates the fate of integrability when N=4 SYM is marginally deformed, showing that planar one-loop spectra reduce to twisted XXZ spin chains for orbifold-like deformations, while generic deformations break full integrability though integrable subsectors can appear. It provides detailed Bethe-ansatz analyses for twisted chains, constructs an exact SO_q(6) XXZ Hamiltonian via quantum-group methods, and demonstrates that such an SO_q(6) chain does not typically arise from a simple Lagrangian deformation preserving conformality. A broader matrix-model perspective is developed, arguing that a multi-matrix quantum mechanics framework is a natural arena for realizing general spin-chain dynamics and potentially connecting to holographic duals of plane-wave geometries. Overall, the work delineates sharp no-go results for full integrability beyond orbifold twists and highlights the central role of quantum-group symmetries in understanding the landscape of integrable deformations of N=4 SYM.

Abstract

Beginning with the planar limit of N=4 SYM theory, we study planar diagrams for field theory deformations of N=4 which are marginal at the free field theory level. We show that the requirement of integrability of the full one loop dilatation operator in the scalar sector, places very strong constraints on the field theory, so that the only soluble models correspond essentially to orbifolds of N=4 SYM. For these, the associated spin chain model gets twisted boundary conditions that depend on the length of the chain, but which are still integrable. We also show that theories with integrable subsectors appear quite generically, and it is possible to engineer integrable subsectors to have some specific symmetry, however these do not generally lead to full integrability. We also try to construct a theory whose spin chain has quantum group symmetry SO_q(6) as a deformation of the SO(6) R-symmetry structure of N=4 SYM. We show that it is not possible to obtain a spin chain with that symmetry from deformations of the scalar potential of N=4 SYM. We also show that the natural context for these questions can be better phrased in terms of multi-matrix quantum mechanics rather than in four dimensional field theories.

Paper Structure

This paper contains 14 sections, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Supersymmetric marginal deformations of ${\cal N}=4$ SYM
  3. One loop anomalous dimensions as a spin chain Hamiltonian
  4. Bethe ansatz for the twisted spin chain
  5. The three state chain Hamiltonian
  6. Towards non-supersymmetric integrable deformations
  7. A brief introduction to the quantum groups $SU_q(M)$
  8. Spin chains with quantum group symmetry
  9. $SO(6)$ integrable spin chains
  10. Symmetry considerations
  11. Exact $SO(6)$ XXZ Hamiltonian
  12. Corresponding Lagrangian deformation
  13. Correspondence between integrable spin chains and matrix quantum mechanics
  14. Conclusion

Figures (3)

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