N=4 SYM to Two Loops: Compact Expressions for the Non-Compact Symmetry Algebra of the su(1,1|2) Sector

TL;DR

The paper tackles higher-loop corrections to the dilatation generator in non-compact sectors of N=4 SYM, focusing on the su(1,1|2) sector where interactions proliferate with loop order.It introduces an auxiliary generator and an iterative algebraic method to derive a compact two-loop dilatation operator and the accompanying O(g^3) symmetry algebra, bypassing brute-force diagrammatics and not requiring the planar limit.The two-loop operator and symmetry corrections are shown to reproduce known field-theory anomalous dimensions and the bosonic sl(2) S-matrix, providing strong evidence for two-loop integrability in this non-compact sector and enabling non-planar extensions with wrapping interactions.Tests across planar anomalous dimensions, the sl(2) S-matrix, and the su(1|1) sector corroborate the approach, while the framework opens avenues to higher-loop and larger-sector generalizations.

Abstract

We begin a study of higher-loop corrections to the dilatation generator of N=4 SYM in non-compact sectors. In these sectors, the dilatation generator contains infinitely many interactions, and therefore one expects very complicated higher-loop corrections. Remarkably, we find a short and simple expression for the two-loop dilatation generator. Our solution for the non-compact su(1,1|2) sector consists of nested commutators of four O(g) generators and one simple auxiliary generator. Moreover, the solution does not require the planar limit; we conjecture that it is valid for any gauge group. To obtain the two-loop dilatation generator, we find the complete O(g^3) symmetry algebra for this sector, which is also given by concise expressions. We check our solution using published results of direct field theory calculations. By applying the expression for the two-loop dilatation generator to compute selected anomalous dimensions and the bosonic sl(2) sector internal S-matrix, we confirm recent conjectures of the higher-loop Bethe ansatz of hep-th/0412188.