Beauty and the Twist: The Bethe Ansatz for Twisted N=4 SYM

TL;DR

The authors develop a general framework to twist integrable spin chains dual to ${ m N}=4$ SYM, extending Lunin–Frolov-type deformations to the dilatation operator and its Bethe ansatz. They show that a phase twist determined by Cartan charges yields a deformed ${ m R}$-matrix that still satisfies the Yang–Baxter equation, and they derive twisted Bethe equations for the complete theory at one and higher loops, including a flavor- and spacetime-twisted (noncommutative) extension. The construction identifies the maximum set of integrable twists compatible with field-theory Feynman rules and discusses the implications for the ${rak su}(2|3)$ and ${rak so}(6)$ sectors, dualizations, and higher-loop generalizations. It also links these deformations to noncommutative gauge theories, offering a practical route to compute planar anomalous dimensions without explicit dilatation-operator calculations. The results illuminate how integrability survives under a broad class of marginal and non-morally deformed settings and provide a unified algebraic and field-theoretic perspective on twisted ${ m N}=4$ SYM spectra.

Abstract

It was recently shown that the string theory duals of certain deformations of the N=4 gauge theory can be obtained by a combination of T-duality transformations and coordinate shifts. Here we work out the corresponding procedure of twisting the dual integrable spin chain and its Bethe ansatz. We derive the Bethe equations for the complete twisted N=4 gauge theory at one and higher loops. These have a natural generalization which we identify as twists involving the Cartan generators of the conformal algebra. The underlying model appears to be a form of noncommutative deformation of N=4 SYM.

Figures (5)

• Figure 1: Generic phase deformation of the R-matrix
• Figure 2: The Yang-Baxter equation
• Figure 3: Dynkin diagram of $\mathfrak{su}(2|3)$. The signs in the white nodes indicate the sign of the diagonal elements of the Cartan matrix.
• Figure 4: "Beauty'' Dynkin diagram of $\mathfrak{su}(2,2|4)$.
• Figure 5: Dynkin diagram of $\mathfrak{su}(2,2|4)$ for the higher-loop Bethe equations.