$\beta$-deformation on a slanted torus and deformed $pp$ wave

We discuss the $\beta$-deformation of ${\mathrm{AdS}}_5\times S^5$ which incorporates the $SL(2,\mathbb{R})$ symmetry of the type IIB theory. The axion-dilaton is identified with a two-torus from an 11-dimensional viewpoint. We consider the null geodesic with equal component angular momenta to take the Penrose limit of the deformed ${\mathrm{AdS}}_5\times S^5$. We study the bosonic part of the string sigma model and the spectrum of the string in the $pp$-wave background.