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Integrable Spin Chain and Operator Mixing in N=1,2 Supersymmetric Theories

Xiao-Jun Wang, Yong-Shi Wu

TL;DR

This work shows that planar one-loop operator mixing for holomorphic scalars in certain ${\cal N}=1,2$ gauge theories and Wess-Zumino models maps to integrable spin chains (e.g., ${\rm SU}(3)$ or ${\rm SU}(2)$), even away from conformal points. The integrable structure is driven by the superpotential and persists across orbifolded descendants of ${\cal N}=4$ SYM, with conformal points aligning with known ${\cal N}=4$ chains but yielding distinct spectra when deformations are present. Bethe Ansatz analyses illustrate how operator mixing differs from the ${\cal N}=4$ case due to discrete gauge-indices, while still enabling exact anomalous dimensions in the holomorphic sector. The authors conjecture a universal integrable framework for all ${\cal N}=1,2$ orbifolded daughters with non-abelian global symmetry, extending the reach of spin-chain methods to a broader class of gauge theories.

Abstract

We study operator mixing, due to planar one-loop corrections, for composite operators in D=4 supersymmetric theories. We present some N=1,2 Yang-Mills and Wess-Zumino models, in which the planar one-loop anomalous dimension matrix in the sector of holomorphic scalars is identified with the Hamiltonian of an integrable quantum spin chain of SU(3) or SU(2) symmetry, even if the theory is away from the conformal points. This points to a more universal origin of the integrable structure beyond superconformal symmetry. We also emphasize the role of the superpotential in the appearance of the integrable structure. The computations of operator mixing in our examples by solving Bethe Ansatz equations show some new features absent in N=4 SYM.

Integrable Spin Chain and Operator Mixing in N=1,2 Supersymmetric Theories

TL;DR

This work shows that planar one-loop operator mixing for holomorphic scalars in certain gauge theories and Wess-Zumino models maps to integrable spin chains (e.g., or ), even away from conformal points. The integrable structure is driven by the superpotential and persists across orbifolded descendants of SYM, with conformal points aligning with known chains but yielding distinct spectra when deformations are present. Bethe Ansatz analyses illustrate how operator mixing differs from the case due to discrete gauge-indices, while still enabling exact anomalous dimensions in the holomorphic sector. The authors conjecture a universal integrable framework for all orbifolded daughters with non-abelian global symmetry, extending the reach of spin-chain methods to a broader class of gauge theories.

Abstract

We study operator mixing, due to planar one-loop corrections, for composite operators in D=4 supersymmetric theories. We present some N=1,2 Yang-Mills and Wess-Zumino models, in which the planar one-loop anomalous dimension matrix in the sector of holomorphic scalars is identified with the Hamiltonian of an integrable quantum spin chain of SU(3) or SU(2) symmetry, even if the theory is away from the conformal points. This points to a more universal origin of the integrable structure beyond superconformal symmetry. We also emphasize the role of the superpotential in the appearance of the integrable structure. The computations of operator mixing in our examples by solving Bethe Ansatz equations show some new features absent in N=4 SYM.

Paper Structure

This paper contains 16 sections, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The ${\cal N}=1$ model
  3. Matrix of Renormalization Factors
  4. Operator mixing and spin chain
  5. Generalization
  6. Lift to the ${\cal N}=4$ SYM
  7. Other ${\cal N}=1$ SYM and Wess-Zumino model
  8. An ${\cal N}=2$ SYM
  9. A conjecture
  10. Summary
  11. Generalized Heisenberg (anti-)ferromagnet
  12. Anomalous dimensions from Bethe Ansatz
  13. Physics
  14. One impurity
  15. Two impurities
  16. ...and 1 more sections

Figures (3)

  • Figure 1: A quiver diagram for D3-branes on a $\mathbb{C}^3/\mathbb{Z}_3$ orbifold. At the nodes we have vectormultiplets in the gauge group indicated, while the arrows connecting each pair of nodes correspond to the bi-fundamental fields.
  • Figure 2: One-loop planar diagrams on correlation functions (\ref{['9']}).
  • Figure 3: The cascade of integrable structure in SYM.