Penrose Limits of Orbifolds and Orientifolds

TL;DR

This work analyzes the Penrose (pp-wave) limits of various AdS_p × S^q backgrounds and their orbifolds/orientifolds, showing that the resulting pp-waves typically have singular wave fronts such as $\mathbb{C}^2/\Gamma$, $\mathbb{C}^4/\Gamma$, or $\mathbb{R}\times \mathbb{C}^4/\Gamma$. When possible, desingularization to ALpp spaces via ALE gravitational instantons (e.g., Eguchi-Hanson, Gibbons-Hawking) yields smooth or asymptotically locally pp-wave backgrounds, preserving supersymmetry. The paper details explicit limits for $AdS_3\times S^3$, $AdS_5\times S^5$, $AdS_4\times S^7$, and $AdS_7\times S^4$, and extends to orientifolds (e.g., $AdS_5\times \mathbb{R}P^5$) and F-theory contexts, including D3/D7-brane configurations. It also presents geometric embeddings of D-dimensional pp-waves into flat $M^{2,D}$ and discusses the role of ALE/Holonomy spaces in preserving partial supersymmetry. Overall, the work provides a systematic framework for constructing and desingularizing pp-wave limits of AdS/CFT-related geometries and highlights their potential holographic interpretations via BMN-type sectors.

Abstract

We study the Penrose limit of various AdS_p X S^q orbifolds. The limiting spaces are waves with parallel rays and singular wave fronts. In particular, we consider the orbifolds AdS_3 X S^3/Γ, AdS_5 X S^5/Γand AdS_{4,7} X S^{7,4}/Γwhere Γacts on the sphere and/or the AdS factor. In the pp-wave limit, the wave fronts are the orbifolds C^2/Γ, C^4/Γand R XC^4/Γ, respectively. When desingularization is possible, we get asymptotically locally pp-wave backgrounds (ALpp). The Penrose limit of orientifolds are also discussed. In the AdS_5 X RP^5 case, the limiting singularity can be resolved by an Eguchi-Hanson gravitational instanton. The pp-wave limit of D3-branes near singularities in F-theory is also presented. Finally, we give the embedding of D-dimensional pp-waves in flat M^{2,D} space.