Higher-Loop Integrability in N=4 Gauge Theory

TL;DR

Beisert surveys the construction of the dilatation operator in N=4 SYM, showing how planar integrability yields new long-range spin-chain models and allows Bethe-ansatz techniques to compute higher-loop anomalous dimensions. The work demonstrates integrability at and beyond one loop, including a long-range Bethe ansatz compatible with SU(2) subsectors and partial agreement with semiclassical string theory, but identifies a persistent discrepancy starting at three loops that may be due to wrapping effects or order-of-limits issues. It argues for an asymptotic, all-loop description valid for long operators and outlines major open problems, including nonperturbative formulations and wrapping corrections.

Abstract

The dilatation operator measures scaling dimensions of local operator in a conformal field theory. Algebraic methods of constructing the dilatation operator in four-dimensional N=4 gauge theory are reviewed. These led to the discovery of novel integrable spin chain models in the planar limit. Making use of Bethe ansaetze, a superficial discrepancy in the AdS/CFT correspondence was found, we discuss this issue and give a possible resolution.

Figures (11)

• Figure 1: Duality between gauge theory local operators and spin chains and similarity to a closed string.
• Figure 2: Generic Spin Chain.
• Figure 3: Dynkin diagrams and Dynkin labels for subsectors.
• Figure 4: Action of the dilatation operator.
• Figure 5: One-loop contributions to the dilatation operator.