The SYM integrable super spin chain

Abstract

Recently it was established that the one-loop planar dilatation generator of N = 4 Super Yang-Mills theory may be identified, in some restricted cases, with the Hamiltonians of various integrable quantum spin chains. In particular Minahan and Zarembo established that the restriction to scalar operators leads to an integrable vector so(6) chain, while recent work in QCD suggested that restricting to twist operators, containing mostly covariant derivatives, yields certain integrable Heisenberg XXX chains with non-compact spin symmetry sl(2). Here we unify and generalize these insights and argue that the complete one-loop planar dilatation generator of N = 4 is described by an integrable su(2, 2|4) super spin chain. We also write down various forms of the associated Bethe ansatz equations, whose solutions are in one-to-one correspondence with the complete set of all one-loop planar anomalous dimensions in the N = 4 gauge theory. We finally speculate on the non-perturbative extension of these integrable structures, which appears to involve non-local deformations of the conserved charges.

The complete one-loop dilatation operator of super-Yang–Mills theory

“…Here, we also see some signs of integrability of the complete planar dilatation generator. The issue of integrability will, however, not be discussed in detail in this work; this is discussed in [35].…”

Finite size effects in ferromagnetic spin chains and quantum corrections to classical strings

“…The energy of these semiclassical strings admits an expansion in the effective coupling λ/J 2 , with λ the 't Hooft coupling of the gauge theory, suggesting the possibility of a precise comparison between string energies and anomalous dimensions of large N = 4 Yang-Mills operators. However operator mixing turned the computation of anomalous dimensions for large N = 4 operators into a formidable task, until the illuminating identification of the planar dilatation operator [5,6] with the Hamiltonian of an integrable quantum spin chain [7,8]. The spectrum of anomalous dimensions for large operators became then computable using the powerful technique of the Bethe ansatz, and complete agreement with the energies of semiclassical string states was found at the first two leading orders in λ [9]- [25].…”

Bethe ansatz equations for general orbifolds of 𝒩 = 4 SYM

“…More details regarding these root systems are given in Appendix A.2. The whole N = 4 theory was proved to be integrable [14], [15], [16]. Note that the spin chains with the different representations of su 2,2|4 correspond to different subclasses of operators.…”

“…Recall that in the su 2 case there is a single root γ 1 = α 12 , and the weight q 1 = s W − s Z ≡ s 2 − s 1 . Analogously, for an excitation associated with some simple root γ = α ij the corresponding weight q γ = s j − s i is 16 Conventions regarding the choice of the root system as well as the Cartan matrix are summarized in Appendix A.2.…”

“…Following [16] we will call it the beast root system. However, as it was emphasized in [53], there exists a more convenient choice from the point of view of the supersymmetry, the so-called beauty root system.…”