`hep-th/0511109`\
`PUTP-2182`\

**$\mathcal{N}=4$ SYM to Two Loops:\
Compact Expressions for the Non-Compact Symmetry Algebra of the ${{\mathfrak{su}}}(1,1|2)$ Sector**

<span style="font-variant:small-caps;">Benjamin I. Zwiebel</span>

*Joseph Henry Laboratories\
Princeton University\
Princeton, NJ 08544, USA*

`bzwiebel@princeton.edu`

**Abstract**

We begin a study of higher-loop corrections to the dilatation generator of $\mathcal{N}=4$ 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 ${{\mathfrak{su}}}(1,1|2)$ sector consists of nested commutators of four $\mathcal{O}(g^1)$ 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 $\mathcal{O}(g^3)$ 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 ${{\mathfrak{sl}}}(2)$ sector internal S-matrix, we confirm recent conjectures of the higher-loop Bethe ansatz of `hep-th/0412188`.

Introduction
============

Our understanding of the conjectured AdS-CFT correspondence [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj] has 