PP-Waves and Holography

TL;DR

This work investigates holography in the pp-wave limit of $AdS_5\times S^5$ (BMN), clarifying how normalizable bulk modes relate to states and non-normalizable modes to boundary sources. It argues that the holographic direction is the radial coordinate $r$ of the first $\mathbb{R}^4$, with two $\mathbb{R}^4$ factors leading to extra non-normalizable modes in the second $\mathbb{R}^4$ that may lack a direct $SYM$ interpretation in the strict limit. The mode analysis, based on the 10D scalar Laplacian, yields a radial-oscillator structure with $E = \alpha(\ell+2)$ and asymptotics $\phi \sim e^{-\alpha r^2/4} r^{\ell} Y_{\ell}$, underscoring a positive-$p_-$ selection for physical states. Finally, the authors outline a holographic procedure for computing correlation functions via a bulk-to-boundary propagator and on-shell action, while acknowledging subtleties from global pp-wave coordinates and indicating directions for a fuller dictionary.

Abstract

We consider aspects of holography in the $pp$-wave limit of $AdS_5\times S5$. This geometry contains two $\RR4$'s, one obtained from $AdS_5$ directions, and the other from the $S 5$. We argue that the holographic direction in the $pp$-wave background can be taken to be $r$, the radial direction in the first $\RR4 $. Normalizable modes correspond to states, and non-normalizable modes correspond to deformations of the boundary theory. In the strict $pp$-wave limit, there are additional non-normalizable modes in the second $\RR 4$, which have no apparent super-Yang-Mills interpretation. We outline the procedure for calculating correlation functions holographically.