====== List of CFTs ======

Here is a list of CFTs. They are organized by dimension. Note that some CFTs, like the gaussian theories and [[percolation|percolation]], can be defined in any number of dimensions.

Some special classes of CFTs:

  * [[tag:holographic]]
  * [[tag:superconformal]]
  * [[tag:logarithmic]]
  * [[tag:topological]]
  * [[tag:non-unitary]]
  * [[tag:nonrelativistic]]

===== One dimension =====

In 1d, the conformal Ward identity $T_\mu{}^\mu = 0$ implies $T_{00} = H = 0$. Therefore quantum systems which truly respect conformal invariance are non-local (see [[https://arxiv.org/pdf/1105.1772.pdf|here]] for another argument). Nonetheless, there are a number of interesting models exhibiting conformal symmetry or near conformal symmetry in one dimension.

{{topic>1d}}

===== Two dimensions =====

[[2d CFTs]] are the best understood class of CFTs, due to their larger [[Virasoro symmetries]]. 2d CFTs differ based on their central charge; those with $c<1$ comprise considerably simpler class of [[minimal models]]. 

{{topic>minimal models}}

Those with $c>1$ are more complicated and in general are not fully classified yet. 

{{topic>2d}}
===== Three dimensions =====

{{topic>3d}}
===== Four dimensions =====

[[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.31.851]]

[[https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.122.211601]]

{{topic>4d}}

===== Five dimensions =====
{{topic>5d}}
===== Six dimensions =====

[[https://arxiv.org/abs/hep-th/0608014]]

[[https://arxiv.org/abs/1805.06467]]

{{topic>6d}}
===== Higher dimensions =====

There are no known unitary interacting CFTs in $d>6$. 

There are constructions of non-unitary theories in higher dimensions (see [[tag:non-unitary|non-unitary CFTs]]).

There are no interacting [[tag:superconformal|superconformal field theories]] in $d>6$ because there are no superconformal algebras (satisfying certain assumptions). This follows from the classification of superconformal algebras due to [[https://inspirehep.net/literature/120988|Nahm]]. See also the proof of [[https://arxiv.org/abs/hep-th/9712074|Minwalla]] for a more recent discussion.


===== Fractional dimensions =====

These apparently exist, even nonperturbatively. What's the conformal symmetry group? Possibly relevant: [[http://mr.crossref.org/iPage?doi=10.1070%2FRM1988v043n02ABEH001720]]

Here's an example of the Ising model on the Sierpinski carpet: [[https://arxiv.org/abs/cond-mat/9802018]].

Here's some bootstrap work: [[https://arxiv.org/abs/1309.5089]].