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.

## Contents

* 1 Motivations
* 2 Type of project
* 3 Known results and possible generalizations
* 3.1 Finite OPEs
* 3.2 Four-point functions inspired by the Potts model
* 3.3 Ashkin-Teller model
* 3.4 Brownian loop soup
* 4 Work to be done
* 5 References

## Motivations[edit | edit source]

* 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 project[edit | edit source]

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 generalizations[edit | edit source]

### Finite OPEs[edit | edit source]

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  Φ 1 2 , 1 2 × Φ 1 2 , 1 2 = Φ 0 , 0 {\displaystyle \Phi _{{\frac {1}{2}},{\frac {1}{2}}}\times \Phi _{{\frac {1}{2}},{\frac {1}{2}}}=\Phi _{0,0}} , 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.

### Four-point functions inspired by the Potts model[edit | edit source]

Cluster connectivities in the critical 2d Potts model are related to correlation functions of diagonal fields of the type  Φ 0 , 1 2 {\displaystyle \Phi _{0,{\frac {1}{2}}}} .

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  ( Φ r , s , Φ r , − s ) {\displaystyle (\Phi _{r,s},\Phi _{r,-s})} with  r ∈ 2 Z + 1 {\displaystyle r\in 2\mathbb {Z} +1} and  s ∈ Z {\displaystyle s\in \mathbb {Z} } . The spectrum in the third channel is not known. This four-point function might have an interpretation in the  O ( n ) {\displaystyle O(n)} model.

An infinite family of crossing-symmetric four-point functions of such fields was discovered[5], where the spectrum in two of the three channels is made of non-diagonal fields with indices

( r , s ) ∈ { ( r 0 , 0 ) } ∪ { ( r ≥ r 0 ∈ 2 N ∗ , s ∈ 1 r N ∗ ) } for r 0 = 2 , 4 , 6 , … {\displaystyle (r,s)\in \\{(r_{0},0)\\}\cup \left\\{\left(r\geq r_{0}\in 2\mathbb {N} ^{*},s\in {\frac {1}{r}}\mathbb {N} ^{*}\right)\right\\}\qquad {\text{for }}r_{0}=2,4,6,\dots }

Again, the spectrum in the third channel is not known.

### Ashkin-Teller model[edit | edit source]

At the central charge  c = 1 {\displaystyle c=1} , 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.[6]

Compactified free bosons exist for any central charge, although for  c ≠ 1 {\displaystyle c\neq 1} the radius is quantized.[2] It might be possible to also build orbifolds for any central charge.

### Brownian loop soup[edit | edit source]

Exact crossing-symmetric four-point functions in the Brownian loop soup were computed.[7] They depend on 5 continuous parameters: the central charge, 4 dimensions subject to a charge conservation condition, and one cross-ratio. The channel decomposition is a sum over an infinite discrete spectrum.

It is not clear whether this four-point function is just one nice function that is invariant under global conformal transformations, or whether it really belongs to an interesting CFT. The invariance of the channel spectrum under integer shifts of the left and right dimensions makes it impossible to deduce the central charge from the four-point function. Maybe there is a CFT with a symmetry algebra that is however larger than the Virasoro algebra.[8]

It might be interesting to study the factorization of four-point structure constants using the techniques of[9].

## Work to be done[edit | edit source]

* Test crossing symmetry in natural generalizations of known crossing-symmetric four-point functions. For example, the index  1 2 {\displaystyle {\tfrac {1}{2}}} 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,[10] and if possible compute them analytically.

* Look for sets of fields that close under fusion, with an associative fusion product.

## References[edit | edit source]

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. ↑ Ribault, Sylvain (2020). "Bootstrap_Virasoro v2.0". GitLab. Retrieved 2020-06-18.
6. ↑ 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.
7. ↑ Camia, Federico; Foit, Valentino F.; Gandolfi, Alberto; Kleban, Matthew (2019-12-02). "Exact Correlation Functions in the Brownian Loop Soup". arXiv.org. Retrieved 2020-07-16.
8. ↑ Ribault, Sylvain (2020-09-04). "Does this covariant function belong to some 2d CFT?". Research Practices and Tools. Retrieved 2020-09-04.
9. ↑ Ribault, Sylvain. "Diagonal fields in critical loop models". arXiv:2209.09706.
10. ↑ 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.