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Peer-reviewedCC BYSciPost Physics Lecture Notes

Exactly solvable conformal field theories

  • Sylvain Ribault1,2
  1. 1CEA Saclay
  2. 2L'Institut de physique théorique
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Abstract

We review 2d CFT in the bootstrap approach, and sketch the known exactly solvable CFTs with no extended chiral symmetry: Liouville theory, (generalized) minimal models, limits thereof, and loop CFTs, including the O(n) O ( n ) , Potts and PSU(n) P S U ( n ) CFTs. Exact solvability relies on local conformal symmetry, and on the existence of degenerate fields. We show how these assumptions constrain the spectrum and correlation functions. We discuss how crossing symmetry equations can be solved analytically and/or numerically, leading to analytic expressions for structure constants in terms of the double Gamma function. In the case of loop CFTs, we sketch the corresponding statistical models, and derive the relation between statistical and CFT variables. We review the resulting combinatorial description of correlation functions, and discuss what remains to be done for solving the CFTs.

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Cite

BibTeXRIS
BibTeX
@article{ribault2026exactly,
  title = {{Exactly solvable conformal field theories}},
  author = {Sylvain Ribault},
  journal = {SciPost Physics Lecture Notes},
  year = {2026},
  month = jul,
  doi = {10.21468/scipostphyslectnotes.130},
  url = {https://doi.org/10.21468/scipostphyslectnotes.130},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Exactly solvable conformal field theories
AU  - Sylvain Ribault
PY  - 2026
DA  - 2026/07/22
JO  - SciPost Physics Lecture Notes
DO  - 10.21468/scipostphyslectnotes.130
UR  - https://doi.org/10.21468/scipostphyslectnotes.130
AB  - We review 2d CFT in the bootstrap approach, and sketch the known exactly solvable CFTs with no extended chiral symmetry: Liouville theory, (generalized) minimal models, limits thereof, and loop CFTs, including the O(n) O ( n ) , Potts and PSU(n) P S U ( n ) CFTs. Exact solvability relies on local conformal symmetry, and on the existence of degenerate fields. We show how these assumptions constrain the spectrum and correlation functions. We discuss how crossing symmetry equations can be solved analytically and/or numerically, leading to analytic expressions for structure constants in terms of the double Gamma function. In the case of loop CFTs, we sketch the corresponding statistical models, and derive the relation between statistical and CFT variables. We review the resulting combinatorial description of correlation functions, and discuss what remains to be done for solving the CFTs.
SN  - 2590-1990
ER  -