Penrose limit, spontaneous symmetry breaking, and holography in app-wave background

We argue that the gauge theory dual to the type IIB string theory in a ten-dimensional $\mathrm{pp}$-wave background resides on a Euclidean subspace spanning four of the eight transverse coordinates. We then show that the evolution of the string along one of the light cone directions in the bulk is identifiable as the RG flow of the gauge theory, a relation facilitating the “holography” of the $\mathrm{pp}$-wave background. The “holography” reorganizes the dual gauge theory into theories defined over Hilbert subspaces of fixed R charge. The reorganization breaks the $\mathrm{SO}\mathrm{}(4,2)\mathrm{}\times \mathrm{SO}\mathrm{}\left(6\right)\mathrm{}$ symmetry to a maximal subgroup $\mathrm{SO}\mathrm{}\left(4\right)\mathrm{}\times \mathrm{SO}\mathrm{}\left(4\right)\mathrm{}$ spontaneously. We argue that the low-energy string modes may be regarded as Goldstone modes resulting from such a symmetry breaking pattern.