– is a generalization of it. Type IIB string theory can be quantized on this background and the light-cone Hamiltonian that one obtains is $$\begin{aligned}
H_\textrm{lc} \sim E-J_1, \qquad  p^+ \sim \frac{E+J_1}{R^2}\, .\end{aligned}$$ From the condition that $H_\textrm{lc}$ and $p^+$ should stay finite in the limit, we get that $J_1=-i\partial_{\varphi_0}$ must be large. On the other hand since $\varphi_1, ..., \varphi_4$ are all fixed in the limit $R\to \infty$, we deduce from , and that $J_2$, $J_3$, $S_1$ and $S_2$ are also fixed.

We see from the above that the “no flat direction” Penrose limit corresponds to the following regime of type IIB string theory on $\mbox{AdS}_5\times S^5$ $$R \rightarrow \infty \ \mbox{with}\ E-J_1\ \mbox{fixed} , \quad \frac{E+J_1}{R^2}\ \mbox{fixed}, \quad \frac{J_1}{R^2} \ \mbox{fixed}, \quad g_s,l_s \ \mbox{fixed}$$ Translating this into ${\mathcal{N}}=4$ SYM language, it corresponds to the regime $$N \rightarrow \infty \ \mbox{with}\ E-J_1\ \mbox{fixed} , \quad \frac{E+J_1}{\sqrt{N}}\ \mbox{fixed}, \quad \frac{J_1}{\sqrt{N}} \ \mbox{fixed}, \quad {g_{\rm YM}}^2 \ \mbox{fixed}$$

The “one flat direction” Penrose limit
--------------------------------------

Now we repeat an analogous procedure and show that, by a different choice of light-cone coordinates, we obtain a generalization of the pp-wave background derived in [@Bertolini:2002nr]. We define the coordinates $\varphi_0,...,\varphi_4$ in the following 