Editing Conformal field theory (section)

== Features ==

=== Unitarity ===

A conformal field theory is '''unitary''' if its space of states has a positive definite [[scalar product]] such that the dilation operator is self-adjoint. Then the scalar product endows the space of states with the structure of a [[Hilbert space]].

In Euclidean conformal field theories, unitarity is equivalent to '''reflection positivity''' of correlation functions: one of the [[Osterwalder-Schrader axioms]].<ref name='prv18'/>

Unitarity implies that the conformal dimensions of primary fields are real and bounded from below. The lower bound depends on the spacetime dimension {{tmath|1= d }}, and on the representation of the rotation or Lorentz group in which the primary field transforms. For scalar fields, the unitarity bound is<ref name='prv18'/>
: <math> 
\Delta \geq \frac12(d-2).
</math>

In a unitary theory, three-point structure constants must be real, which in turn implies that four-point functions obey certain inequalities. Powerful numerical bootstrap methods are based on exploiting these inequalities.

=== Compactness ===

A conformal field theory is '''compact''' if it obeys three conditions:<ref name="br19"/>
* All conformal dimensions are real.
* For any <math>\Delta\in\mathbb{R}</math> there are finitely many states whose dimensions are less than {{tmath|1= \Delta }}.
* There is a unique state with the dimension {{tmath|1= \Delta =0 }}, and it is the '''vacuum state''', i.e. the corresponding field is the '''identity field'''.

(The identity field is the field whose insertion into correlation functions does not modify them, i.e. {{tmath|1= \left\langle I(x)\cdots \right\rangle = \left\langle \cdots \right\rangle }}.) The name comes from the fact that if a 2D conformal field theory is also a [[sigma model]], it will satisfy these conditions if and only if its target space is compact.

It is believed that all unitary conformal field theories are compact in dimension {{tmath|1= d>2 }}. Without unitarity, on the other hand, it is possible to find CFTs in dimension four<ref>{{cite journal|last1=Levy|first1=T.|last2=Oz|first2=Y.|year=2018|title=Liouville Conformal Field Theories in Higher Dimensions|journal=JHEP|volume=1806|issue=6|pages=119|arxiv=1804.02283|doi=10.1007/JHEP06(2018)119|bibcode=2018JHEP...06..119L|s2cid=119441506}}</ref> and in dimension <math>4 - \epsilon</math><ref name="jm18" /> that have a continuous spectrum. And in dimension two, [[Liouville field theory|Liouville theory]] is unitary but not compact.

=== Extra symmetries ===

A conformal field theory may have extra symmetries in addition to conformal symmetry. For example, the Ising model has a <math>\mathbb{Z}_2</math> symmetry, and superconformal field theories have supersymmetry.