Exact Spectrum of Anomalous Dimensions of Planar $\mathit{N}=4$ Supersymmetric Yang-Mills Theory

We present a set of functional equations defining the anomalous dimensions of arbitrary local single trace operators in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It takes the form of a $Y$ system based on the integrability of the dual superstring $\sigma$ model on the five-dimensional anti–de Sitter space (${\mathrm{AdS}}_5\times {\mathrm{S}}^5$) background. This $Y$ system passes some very important tests: it incorporates the full asymptotic Bethe ansatz at large length of operator $L$, including the dressing factor, and it confirms all recently found wrapping corrections. The recently proposed ${\mathrm{AdS}}_4/\mathrm{\text{three-dimensional}}$ conformal field theory duality is also treated in a similar fashion.

Figure 1

$\mathbf{T}$-shaped fat hook for the $Y$ and $T$ systems [19]. The middle double line separates the two subgroups with extended $SU(2|2{)}_L$ and $SU(2|2{)}_R$ symmetries.

Figure 2

Fat hook for ${\mathrm{AdS}}_4/{\mathrm{CFT}}_3$. The $OSp(2,2|6)$ symmetry of the ABJM theory, with two momentum carrying nodes, and the $SU(2|2)$ subgroup is manifest in the diagram.