T T ¯ deformations and the pp -wave correspondence

In this paper we consider TT¯ deformations in the context of pp waves obtained from gravity duals. We propose a deformation of AdS5×S5 similar to the deformation of the single trace TT¯ deformation of AdS3×S3×T4 with NS-NS flux, and study it through the Penrose limit, concluding that it must correspond to some dipole theory, probably noncommutative. We TT¯ deform the world sheet string for the AdS5×S5 pp wave, and find a corresponding spin chain Hamiltonian. Finally, directly TT¯ deforming the spin chain Hamiltonian obtained from the pp wave, we find that it corresponds to an equivalent Berenstein-Maldacena-Nastase (BMN) sector of the N=4 super Yang-Mills.