$T\overline{T}$ Deformation: Introduction and Some Recent Advances

This review explores recent advances in the theory of $T\bar{T}$ deformation, an irrelevant yet solvable deformation of quantum field theories defined via the quadratic form of the energy-momentum tensor. It addresses classical and quantum aspects, highlighting significant developments across various fields, including field theory, holography, and string theory. Classically, $T\bar{T}$ deformation manifests through multiple geometric interpretations, notably random geometry, Jackiw-Teitelboim-like gravity, and uniform light-cone gauge frameworks. For quantum aspects, the deformation introduces notable features such as non-locality, UV-IR mixing, solvable renormalization structures, and intriguing modifications to correlation functions and entanglement properties. Furthermore, the paper examines the profound relationship between $T\bar{T}$ deformation and holography, particularly within the mixed boundary conditions/cutoff AdS holography proposal and holographic entanglement entropy. Connections to string theory through single-trace deformations and their holographic duals further reveal the deformed structure of the worldsheet. This review synthesizes recent developments and outlines potential directions for future research in the study of $T\bar{T}$-like deformation.