Two-dimensional Conformal Field Theory (2D CFT) is a branch of theoretical physics that studies two-dimensional quantum field theories that are invariant under conformal transformations. These transformations include translations, rotations, dilations (scaling), and special conformal transformations, which preserve angles but not necessarily lengths. ### Key Features of 2D CFT: 1. **Conformal Symmetry**: In two dimensions, the conformal group is infinite-dimensional.
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A two-dimensional conformal field theory is a quantum field theory with local conformal transformations. In Euclidean complex coordinates, infinitesimal transformations separate into holomorphic and antiholomorphic sectors, organized by two copies of the Virasoro algebra.
This symmetry strongly constrains correlation functions. A primary field's conformal weights fix its two-point dependence, while three-point functions are fixed up to constants; higher-point functions retain dependence on conformal cross-ratios. The operator product expansion combines local fields and records the remaining dynamical data. A free boson conformal field theory provides an explicit example. In string theory, this framework describes gauge-fixed worldsheet dynamics and connects the central charge to quantum consistency.