Lifshitz-like systems and AdS null deformations

Abstract

Following K. Balasubramanian and K. Narayan [J. High Energy Phys. 08 (2010) 014], we discuss certain lightlike deformations of ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{X}^{5}$ in type IIB string theory sourced by a lightlike dilaton $\ensuremath{\Phi}({x}^{+})$ dual to the $\mathcal{N}=4$ super Yang-Mills theory with a lightlike varying gauge coupling. We argue that, in the case where the ${x}^{+}$ direction is noncompact, these solutions describe anisotropic $3+1$-dim Lifshitz-like systems with a potential in the ${x}^{+}$ direction generated by the lightlike dilaton. We then describe solutions of this sort with a linear dilaton. This enables a detailed calculation of two-point correlation functions of operators dual to bulk scalars and helps illustrate the spatial structure of these theories. Following this, we discuss a nongeometric string construction involving a compactification along the ${x}^{+}$ direction of this linear dilaton system. We also point out similar IIB axionic solutions. Similar bulk arguments for ${x}^{+}$-noncompact can be carried out for deformations of ${\mathrm{AdS}}_{4}\ifmmode\times\else\texttimes\fi{}{X}^{7}$ in M theory.