CFT Zoo

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list_of_cfts [2021/04/07 23:30]
Brian McPeak [One dimension] 		list_of_cfts [2021/09/14 01:58] (current)
Brian McPeak [Two dimensions]
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   * [[tag:topological]]   * [[tag:topological]]
   * [[tag:non-unitary]]   * [[tag:non-unitary]]
 +  * [[tag:nonrelativistic]]
  
 ===== One dimension ===== ===== One dimension =====
  
-Truly conformally invariant 1d theories must have $H = 0$. 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 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}} {{topic>1d}}
  
-[[Defect Conformal Field Theory]] 
 ===== Two dimensions ===== ===== Two dimensions =====
  
-[[2d CFTs]] are the best understood class of CFTs, due to their larger [[Virasoro symmetries]]+[[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]]. 
  
-[[Minimal Models]]+{{topic>minimal models}}
  
-  * [[2d Ising]] +Those with $c>1$ are more complicated and in general are not fully classified yet. 
-  * All the others +
- +
-[[Liouville Theory]] +
- +
-[[Narain Theories]] +
- +
-[[Percolation]]+
  
 +{{topic>2d}}
 ===== Three dimensions ===== ===== Three dimensions =====
  
 {{topic>3d}} {{topic>3d}}
 ===== Four dimensions ===== ===== 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}} {{topic>4d}}
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 {{topic>5d}} {{topic>5d}}
 ===== Six dimensions ===== ===== Six dimensions =====
 +
 +[[https://arxiv.org/abs/hep-th/0608014]]
 +
 +[[https://arxiv.org/abs/1805.06467]]
  
 {{topic>6d}} {{topic>6d}}
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 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. 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]].
Last modified: 2021/04/07 23:30
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