CFT Zoo

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list_of_cfts [2026/03/05 14:52]
Ludo Fraser-Taliente [Fractional dimensions] added nonunitarity, more bootstrap evidence. 		list_of_cfts [2026/03/05 16:00] (current)
Ludo Fraser-Taliente [One dimension]
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 ====== List of CFTs ====== ====== 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.+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: Some special classes of CFTs:
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 ===== One dimension ===== ===== 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.+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}} {{topic>1d}}
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 ===== Fractional/continuous dimension ===== ===== 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 [[https://arxiv.org/abs/1512.00013 | 1]].+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]] These CFTs apparently exist, even nonperturbatively. What's the conformal symmetry group? Possibly relevant: [[http://mr.crossref.org/iPage?doi=10.1070%2FRM1988v043n02ABEH001720]]
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 There is a possible connection to non-conformal D$p$-brane holography [[http://arxiv.org/abs/2503.18770]]. 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.
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