In recent years, there have been two independent but related developments in the study of irrelevant deformations in two dimensional quantum field theories (QFTs). The first development is the deformation of a two dimensional QFT by the determinant of the energy momentum stress tensor, commonly referred to as $T{\bar T}$ deformation. The second development is in two dimensional holographic field theories which are dual to string theory in asymptotically Anti-de Sitter (AdS) spacetimes. In this latter development, the deformation is commonly referred to as single-trace $T{\bar T}$ deformation. The single-trace $T{\bar T}$ deformation corresponds in the bulk to a string background that interpolates between AdS spacetime in the infrared (IR) and a linear dilaton spacetime (vacuum of little string theory (LST)) in the ultraviolet (UV). It serves as a useful tool and guide to better understand and explore holography in asymptotically AdS and non-AdS spacetimes in a controlled setting. In particular, it is useful to gain insights into holography in flat spacetimes. The dissertation is devoted to the study of single-trace $T{\bar T}$ deformation and its single-trace generalizations in theories with $U(1)$ currents, namely $J\bar T$ and $T\bar J$ deformations, in the context of gauge/gravity duality. In the dissertation I present new results in the study of holography in asymptotically non-AdS spacetimes. I discuss two point correlation functions in single-trace $T{\bar T}$ deformation, and entanglement entropy and entropic $c$-function in single-trace $T{\bar T}$, $J\bar T$ and $T\bar J$ deformations. I show that two point functions in position space have both real parts and imaginary parts. I also show that the imaginary parts are non-perturbative. The imaginary parts correspond in momentum space to branch cuts, which signal non-locality. I obtain exact result for entanglement entropy associated with a spatial region of finite size. I also show that in the UV for a particular combination of the deformation couplings the leading order dependence of the entanglement entropy on the size is given by a square root but not logarithmic function. Such power law dependence of the entanglement entropy on the size is quite distinct and interesting. I also give exact result for the entropic $c$-function and show that it is regularization schemes independent, positive and monotonic, which are similar to the behaviors observed in conventional local QFTs. I also discuss its distinctive features in the UV.




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