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

TL;DR

This paper derives a compact, two-loop dilatation generator for the non-compact su(1,1|2) sector of N=4 SYM, providing explicit algebraic expressions and confirming conjectures related to the higher-loop Bethe ansatz.

Contribution

It presents a novel, simplified expression for the two-loop dilatation generator in a non-compact sector, valid for any gauge group, and derives the complete symmetry algebra at this order.

Findings

Derived concise two-loop dilatation generator expressions.

Confirmed higher-loop Bethe ansatz conjectures.

Validated results with existing field theory calculations.

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.