String Interactions in PP-wave from N = 4 Super Yang Mills

TL;DR

The paper develops a framework to compute non-planar, genus-g corrections to BMN operator correlators in free N=4 SYM by recasting string interactions in the pp-wave background as cubic vertices built from free planar three-point functions. It demonstrates precise agreements between field-theory torus amplitudes and corresponding pp-wave string theory loop diagrams at one and two loops, including explicit x- and m-dependent results and symmetry-factor relations. The results bolster the pp-wave/CFT correspondence and provide a practical diagrammatic scheme for higher-genus correlators, while outlining challenges and directions for extending to interacting Yang-Mills theory and deeper holographic interpretation.

Abstract

We consider non-planar contributions to the correlation functions of BMN operators in free N = 4 super Yang Mills theory. We recalculate these non-planar contributions from a different kind of diagram and find some exact agreements. The vertices of these diagrams are represented by free planar three point functions, thus our calculations provide some interesting identities for correlation functions of BMN operators in N = 4 super Yang Mills theory. These diagrams look very much like loop diagrams in a second quantized string field theory, thus these identities could possibly be interpreted as natural consequences of the pp-wave/CFT correspondence.

Figures (10)

• Figure 1: It is recently pointed out in chu2 that higher point string interactions in pp-wave can be reduced to cubic interactions under some double pinching limits. For example, the skeleton diagram with $s$, $t$ and $u$ channels appear in the computation of a planar four point function $\langle\bar{O}_1\bar{O}_2O_3O_4\rangle$ as we take some specific double pinching limits. We will only need to consider cubic interactions in our calculations of the string theory diagram.
• Figure 2: There are 2 diagrams contributing the one loop string propagation. The BMN string $O^J_{n,-n}$ can split into two strings $O^{J_1}_{l,-l}$, $O^{J_2}$ or $O^{J_1}_0$, $O^{J_2}_0$ and joining back into another string $O^J_{m,-m}$. We denote contributions to these two diagrams $P_1$ and $P_2$.
• Figure 3: Feymann diagram of torus contraction of large $N$ gauge indices semeharvmit. We are contracting non-planarly by dividing the string into 4 segments.
• Figure 4: There are three diagrams contribute to the torus three point function. We denote their contributions by $Q_1$, $Q_2$ and $Q_3$. Here we use single line notation. One would check these diagrams indeed have a power of $1/N^3$ in double line notation. The top line and the bottom lines represent the long string and short strings. In the three diagrams we have divided the long string into 6 groups and the short strings into (5,1), (4,2) and (3,3) groups. Each group is represented by a single line here.
• Figure 5: String theory diagrams contribute to $\langle\bar{O}^JO^{J_1}O^{J_2}\rangle_{torus}$ organized in 3 groups. Diagrams in $R_1$ and $R_2$ are corrections to one particle propagator while diagrams $R_3$ are amputated. The two diagrams in $R_3$ are symmetric by exchange of the two decayed operators.