Editing Conformal field theory (section)

=== Global vs local conformal symmetry in two dimensions ===

The global conformal group of the [[Riemann sphere]] is the group of [[Möbius transformation]]s {{tmath|1= \mathrm{PSL}_2(\mathbb{C}) }}, which is finite-dimensional. 
On the other hand, infinitesimal conformal transformations form the infinite-dimensional [[Witt algebra]]: the [[conformal Killing equation]]s in two dimensions, <math>\partial_\mu \xi_\nu + \partial_\nu \xi_\mu = \partial \cdot\xi \eta_{\mu \nu},~</math> reduce to just the Cauchy-Riemann equations, {{tmath|1= \partial_{\bar{z} } \xi(z) = 0 = \partial_z \xi (\bar{z}) }}, the infinity of modes of arbitrary analytic coordinate transformations <math>\xi(z)</math> yield the infinity of [[Killing vector field]]s {{tmath|1= z^n\partial_z }}.

Strictly speaking, it is possible for a two-dimensional conformal field theory to be local (in the sense of possessing a stress-tensor) while still only exhibiting invariance under the global {{tmath|1= \mathrm{PSL}_2(\mathbb{C}) }}. This turns out to be unique to non-unitary theories; an example is the biharmonic scalar.<ref name="raj11"/> This property should be viewed as even more special than scale without conformal invariance as it requires <math>T_\mu{}^\mu</math> to be a total second derivative.

Global conformal symmetry in two dimensions is a special case of conformal symmetry in higher dimensions, and is studied with the same techniques. This is done not only in theories that have global but not local conformal symmetry, but also in theories that do have local conformal symmetry, for the purpose of testing techniques or ideas from higher-dimensional CFT. In particular, numerical bootstrap techniques can be tested by applying them to [[Minimal model (physics)|minimal models]], and comparing the results with the known analytic results that follow from local conformal symmetry.