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

Here is a list of CFTs. They are organized by dimension. Note that some CFTs, like the [[gaussian|gaussian theories]] and [[percolation|percolation]], can be defined in any number of dimensions. This allows their conformal data to be analytically continued between dimensions, connecting otherwise apparently different CFTs.

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 [[list_of_cfts#nonlocal_cfts|nonlocal]] (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/continuous dimension =====

Non-chiral CFTs can often be defined in continuous dimension $d$, though they are usually explicilty non-unitary for noninteger $d$ due to evanescent operators that have negative norm [[https://arxiv.org/abs/1512.00013 | 1]]. In the limit as $d$ approaches an integer, these evanescent operators decouple from the CFT.

These CFTs 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]] [[https://arxiv.org/abs/1811.07751 | 2]] [[https://arxiv.org/abs/2207.10118 | 3]] [[https://arxiv.org/pdf/2210.03051 | 4]].

There is a possible connection to non-conformal D$p$-brane holography [[http://arxiv.org/abs/2503.18770]].

===== Nonlocal CFTs =====

It is possible to define quantum field theories with full SO$(d+1,1)$ symmetry that do not have a local action (i.e. $S= \int d^d x O(x)$ for some local operator $x$). These are called nonlocal CFTs, long-range CFTs, or just Conformal Theories (CTs).

The simplest example of these are the [[gaussian|generalized free fields]] (GFFs, also called mean field theory), which are Gaussian: we declare one field to have a conformal two-point function, and then determine all other correlators by the usual Wick contractions.

By perturbing one of these GFFs $\phi$ of arbitrary scaling dimension $\Delta$ by $\phi^4$ and tuning to the critical point, we can construct the the long-range Ising CFT in arbitrary dimension $d$.
The long-range Ising CFT becomes the standard Ising CFT (+ a decoupled free field) in the limit $\Delta \to \Delta_{\phi, \text{Ising}}$ (i.e. $1/8$ in $d=2$, $0.518$ in $d=3$) [[https://arxiv.org/abs/1703.05325 | 1]]. 

From the perspective of conformal data, the signal that these theories are nonlocal is that they do not possess a local stress tensor. That is, in the OPE of these theories there is no conserved spin-2 operator. This is reasonable because the stress tensor is an operator measuring the response of the theory to a change of the geometry. If the theory is nonlocal, then the response will not be local, so $T^{\mu\nu}$ cannot be a local operator.