Wrapping interactions and the genus expansion of the 2-point function of composite operators

Abstract

We perform a systematic analysis of wrapping interactions for a general class of theories with color degrees of freedom, including N=4 SYM. Wrapping interactions arise in the genus expansion of the 2-point function of composite operators as finite size effects that start to appear at a certain order in the coupling constant at which the range of the interaction is equal to the length of the operators. We analyze in detail the relevant genus expansions, and introduce a strategy to single out the wrapping contributions, based on adding spectator fields. We use a toy model to demonstrate our procedure, performing all computations explicitly. Although completely general, our treatment should be particularly useful for applications to the recent problem of wrapping contributions in some checks of the AdS/CFT correspondence.

Figures (12)

• Figure 1: The composite operators $\mathcal{O}^{}_{R}$\ref{['corep']} and the Green function $V_{2R}$\ref{['Vrep']} are the building blocks of the $2$-point function $(\mathcal{O}^{1}_{R},V_{2R},\mathcal{O}^{2}_{R})$
• Figure 2: On a compact Riemann surface, the external lines of the Green function $V_{2R}$ are connected to the vertex at $\infty$. These connections separate the Riemann suface into two parts. Lines that cross these connections require adding of the wrapping handle, depicted in gray.
• Figure 3: The cyclic permutations of the external legs have been moved to the vertex at $\infty$. The resolution of the cyclic permutation requires adding one handle \ref{['rcyclVinfty']}. An exception is the case \ref{['rhandelcyclVinfty']} in which the crossing leg contains a non-planar self energy correction. In this case the genus does not change. Two cyclic permutations in the same \ref{['rlcyclVinfty1']} or in opposite directions \ref{['rlcyclVinfty2']} can be resolved with a single handle.
• Figure 4: On a compact Riemann surface, the two operators $\mathcal{O}^{e}_{R}$ in the $2$-point function have a finite separation. This allows one to occupy the wrapping path, depicted in gray.
• Figure 5: Decomposition of the $2$-point function $(\mathcal{O}^{1}_{R},V_{2R},\mathcal{O}^{2}_{R})$ into interaction part $\frac{V_{2R}}{S_R}$ without any permutations of the external legs and the permutations $\frac{S_R}{C_R}$ without cyclic permutations $C_R$. $V_{2R}$ contains only one diagram $D^{K(h)}_{2R}$ of each equivalence class with minimal genus $h$. The cyclic permutations are taken into account by using all cyclic permutations of the operators $\mathcal{O}^{e}_{R}$, denoted by $C_R(\mathcal{O}^{e}_{R})$.