{\mathbb{C}}P^3$ background of type IIA supergravity with one explicit space-like isometry.

Penrose limits and pp-waves with explicit isometries {#sec:stringtheory}
====================================================

In this section we derive a Penrose limit of $\mbox{AdS}_5 \times S^5$ which results in a new pp-wave background with two space-like isometries. We then show how to obtain a general pp-wave background which, for appropriate choices of the parameters of the background, reproduces all the known pp-wave backgrounds which are obtained through a Penrose limit procedure on $\mbox{AdS}_5 \times S^5$. We begin the section by writing down a slightly generalized version of the previously found Penrose limits of $\mbox{AdS}_5 \times S^5$ with zero and one explicit space-like isometries [@Blau:2002dy; @Berenstein:2002jq; @Bertolini:2002nr].

In $\mbox{AdS}_5 \times S^5$, the Penrose limit consists in considering a particle in the center of $\mbox{AdS}_5 $ that is moving very rapidly on a geodesic of $S^5$. This means that the angular momentum along the direction in which the particle is moving is very large ($J \to
\infty$). Then by taking the limit $R \to \infty$, where $R$ is the radius of $\mbox{AdS}_5$ and $S^5$, but such that the ratio $J/R^2$ remains fixed, the geometry of $\mbox{AdS}_5 \times S^5$ reduces to a plane-wave geometry.

An important point 