The plane-wave/super Yang-Mills duality

This article reviews the plane-wave/super Yang-Mills duality, which states that strings on a plane-wave background are dual to a particular large $R$-charge sector of $\mathcal{N}=4$, $D=4$ superconformal $\mathrm{U}\left(N\right)$ gauge theory. This duality is a specification of the usual anti–de Sitter/conformed field theory (AdS/CFT) correspondence in the “Penrose limit.” The Penrose limit of ${\mathrm{AdS}}_5\times S^5$ leads to the maximally supersymmetric ten-dimensional plane wave (henceforth “the” plane wave) and corresponds to restricting to the large $R$-charge sector, the Berenstein-Maldacena-Nastase (BMN) sector, of the dual superconformal field theory. After reviewing the necessary background, the authors state the duality and review some of its supporting evidence. They discuss the suggestion by ’t Hooft that Yang-Mills theories with gauge groups of large rank might be dual to string theories and the realization of this conjecture in the form of the AdS/CFT duality. Plane waves as exact solutions of supergravity and their appearance as Penrose limits of other backgrounds are considered, followed by an overview of string theory on the plane-wave background, discussing the symmetries and spectrum. The article then makes precise the statement of the proposed duality and classifies the BMN operators. It examines the gauge theory side of the duality, studying both quantum and nonplanar corrections to correlation functions of BMN operators and their operator-product expansions. The important issue of operator mixing and the resultant need for rediagonalization is stressed. Finally, the article studies strings on the plane wave via light-cone string field theory and demonstrates agreement between the one-loop correction to the string mass spectrum and the corresponding quantity in the gauge theory. A new presentation of the relevant superalgebra is given.

Figure 1

Typical Feynman rules for adjoint fields and sample planar and nonplanar diagrams.

Figure 2

Feynman diagrams containing a single insertion of the interaction Hamiltonian. The two different impurities are labeled $\varphi$ and $\psi$.

Figure 3

Diagrams contributing to the scalar wave-function renormalization at one loop. The first is a scalar tadpole, the second a gauge-boson loop, and the third a fermion loop.

Figure 4

Irreducible toroidal diagrams contributing to $⟨\mathrm{Tr}{Z}^J\mathrm{Tr}{\overline{Z}}^J⟩$. The arrows indicate the direction in which traces are taken.

Figure 5

Diagram depicting the phase shift in a torus diagram with no interactions. The solid lines represent an arbitrary number of $Z$ fields, and the dashed lines represent the contraction between two $\varphi$’s or $\psi$’s.

Figure 6

Three-string interaction vertex in the light-cone gauge. Note that, due to closed string boundary conditions, $\sigma =0$, $\sigma =2\pi {\alpha }_1$, and $\sigma =2\pi ({\alpha }_1+{\alpha }_2)$ are identified and $I$ is the interaction point.