Crossing-symmetric four-point functions in 2d CFT

An important step towards solving CFTs in the conformal bootstrap approach is to find crossing-symmetric four-point functions. Crossing symmetry of some four-point functions is a non-trivial contstraint on the spectrum and structure constants. For the existence of a consistent CFT, we need additional constraints: that four-point structure constants factorize into three-point structure constants, and that all possible four-point functions are crossing-symmetric, not just some of them.

In 2d CFT, there are a number of known exact solutions of crossing symmetry that do not belong to any known consistent CFT. This project is about studying these solutions, generalizing them in order to find new solutions, and looking for consistent CFTs based on these solutions - for generic values of the Virasoro algebra's central charge.


  • Getting a better idea of the space of consistent CFTs, by investigating some of the rare exactly solvable examples.
  • Potential applications to statistical physics, quantum gravity, string theory.

Type of projectEdit

Tools: A good understanding of the bootstrap approach to 2d CFT is required. Then the project involves analytic calculations and numerical checks of crossing symmetry, possibly based on existing Python code at GitLab. Moreover, the project involves scanning the literature for known solutions of crossing symmetry.

Chances of success: It is almost certainly possible to find new examples of crossing-symmetric four-point functions by generalizing known examples. Building a consistent CFT is considerably more involved, as this may require large classes of four-point functions, some of which might have an exotic behaviour.[1]

Length and difficulty: This is a very much open-ended project, with the possibility of some quick and easy results, and the scope for larger undertakings.

Known results and possible generalizationsEdit

Finite OPEsEdit

By definition, a finite OPE is an OPE of two primary fields that can be written as a linear combination of finitely many primary fields. In particular, such OPEs are manifestly convergent.

Examples of crossing-symmetric four-point functions that involve finite OPEs:

  • Four-point functions with at least one degenerate field. (In particular, four-point functions in generalized minimal models.[2])
  • Four-point functions that can be computed as Coulomb gas integrals.
  • Four-point functions based on the OPE  , where the fields are labelled by Kac table indices.[3] This OPE might be related to the analytic continuation of the OPE of Liouville theory.

A four-point function inspired by the Potts modelEdit

Cluster connectivities in the critical 2d Potts model are related to correlation functions of diagonal fields of the type  . A crossing-symmetric four-point function of such fields was discovered[4], where the spectrum in two of the three channels is made of non-diagonal fields of type   with   and  . The spectrum in the third channel is not known.

Ashkin-Teller modelEdit

At the central charge  , the critical Ashkin-Teller model can be exactly solved thanks to its description as a compactified free boson orbifold, and it depends on a continuous parameter called the radius. This leads to a wealth of crossing-symmetric four-point function, especially if one forgets the affine symmetry and looks at all Virasoro primary fields.[5]

Compactified free bosons exist for any central charge, although for   the radius is quantized.[2] It might be possible to also build orbifolds for any central charge.

Work to be doneEdit

  • Test crossing symmetry in natural generalizations of known crossing-symmetric four-point functions. For example, the index   can be generalized to half-integers, and diagonal fields can be generalized to non-diagonal fields. Given an ansatz for the exact spectrum, crossing symmetry can be numericaly tested using the method of reference[4].
  • Determine whether the structure constants obey the analytic bootstrap equations,[6] and if possible compute them analytically.
  • Look for sets of fields that close under fusion, with an associative fusion product.


  1. Ribault, Sylvain (2019). "The non-rational limit of D-series minimal models". arXiv:1909.10784 [hep-th]. 
  2. 2.0 2.1 Ribault, Sylvain (2014). "Conformal field theory on the plane". arXiv:1406.4290 [hep-th]. 
  3. Esterlis, Ilya; Fitzpatrick, A. Liam; Ramirez, David M. (2016). "Closure of the operator product expansion in the non-unitary bootstrap". Journal of High Energy Physics 2016 (11). doi:10.1007/jhep11(2016)030. ISSN 1029-8479. 
  4. 4.0 4.1 Picco, Marco; Ribault, Sylvain; Santachiara, Raoul (2016-10-27). "A conformal bootstrap approach to critical percolation in two dimensions". SciPost Physics 1 (1). doi:10.21468/scipostphys.1.1.009. ISSN 2542-4653. 
  5. Zamolodchikov, Al.B. (1987). "Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions". Nuclear Physics B (Elsevier BV) 285: 481–503. doi:10.1016/0550-3213(87)90350-6. ISSN 0550-3213. 
  6. Migliaccio, Santiago; Ribault, Sylvain (2018). "The analytic bootstrap equations of non-diagonal two-dimensional CFT". Journal of High Energy Physics 2018 (5). doi:10.1007/jhep05(2018)169. ISSN 1029-8479.