N=2 Super Yang-Mills and the XXZ spin chain

TL;DR

This study shows that in planar ${\cal N}=2$ Super Yang-Mills, the one-loop anomalous-dimension matrix for scalar composites coincides with a closed XXZ spin-chain Hamiltonian with anisotropy $\Delta>1$ (specifically $\Delta=3$), revealing an integrable structure beyond the ${\cal N}=4$ case. The analysis demonstrates that certain operators, notably those built from the complex scalar, are protected at one loop and that the gauge coupling running does not affect their RG flow at this order, with two-loop effects needed to break conformal invariance. The ground state corresponds to ${\rm Tr}(\phi^{L})$ while excitations are impurities; the nonzero anisotropy implies a mass gap in the spectrum, suggesting a string-theoretic interpretation via fluxes in the dual background. These results motivate extending the integrable description to finite-size operators and higher loops, and exploring holographic connections to understand the role of fluxes in generating the observed anisotropy.

Abstract

We analyse the renormalisation properties of composite operators of scalar fields in the N=2 Super Yang-Mills theory. We compute the matrix of anomalous dimensions in the planar limit at one-loop order in the 't Hooft coupling, and show that it corresponds to the Hamiltonian of an integrable XXZ spin chain with an anisotropy parameter Delta>1. We suggest that this parameter could be related to the presence of non-trivial two-form fluxes in the dual supergravity background. We find that the running of the gauge coupling does not affect the renormalization group equations for these composite operators at one-loop order, and argue that this is a general property of gauge theories which is not related to supersymmetry.

Figures (2)

• Figure 1: Feynman diagrams contributing at one--loop. The thick horizontal line joins the fields belonging to the composite operator.
• Figure 2: Spin chain