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Exactly solvable conformal field theories
by Sylvain Ribault
Submission summary
| Authors (as registered SciPost users): | Sylvain Ribault |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2411.17262v3 (pdf) |
| Date submitted: | Nov. 20, 2025, 3:59 p.m. |
| Submitted by: | Sylvain Ribault |
| Submitted to: | SciPost Physics Lecture Notes |
| for consideration in Collection: |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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 $O(n)$, Potts and $PSU(n)$ 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.
Author comments upon resubmission
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Reply to Referee 1.
Weakness #1: Hopefully this is addressed to some extent by performing the requested changes, plus some more.
Weakness #2: I gave a generic answer to this concern by adding a paragraph in the introduction, starting from "While a CFT can be defined as a solution". A specific answer for each CFT under consideration is outside the scope of this text. The status of minimal models is not clear to me, it probably depends on the series.
1- Corrected as suggested, see C01.
2- No change: My notation is more compact and it is standard although less popular than the suggested notation.
3- No change: For spectrum, momentum, etc, regular plurals are correct and more logical than Latin nominative plurals (why nominative?), although less popular. I am sorry that they make the text less readable to some readers, but I believe they make things easier for an international audience.
4- Removed the confusion, see C03.
5- Clarified this point, see C04.
6- Clarified this point, see C05.
7- Added suggested comment, see C08.
8- Added a clause about normalization before what is now Eq. (1.23).
9- There was indeed a mistake in the formulas, which is now corrected by flipping the sign of $\epsilon$ in its definition.
10- Removed the offending statement, see C09. While probably true for generic central charge, it is not an essential statement.
11- Removed the offending statement, see C11.
12- No change. Fields are not defined as operators on a space, they are defined as vectors in representations of the Virasoro algebra. So there is no ambiguity, except for readers who have other formalisms in mind. Referring to these other formalisms would risk confusing less knowledgeable readers.
13- Removed $\mathcal{S}$.
14- Yes indeed, see C03.
15- Only added a statement after (2.33) about the identity field in minimal models. Not more, for fear of confusing readers with this subtle point. In a minimal model, there is no field $V^d_{\langle 1,1\rangle}$ distinct from $V^f_{\langle 1,1\rangle}$. In other words, the second singular vector of $V^d_{\langle 1,1\rangle}$ vanishes in all of the model's correlation functions. This would not be true if there existed fields outside the Kac table.
16- Clarified the definition of fusion multiplicity, see C12. I think the good definition is not the number of independent intertwiners, rather it should refer to a decomposition into indecomposables.
17- No remark with that number.
18- and 19- See C13 and C14.
20- When integrating over a continuous spectrum such as $P\in\mathbb{R}$, it does not matter what happens on a set of measure zero such as the set of degenerate momentums $P_{(r,s)}$. The question would matter if we were specifically interested in taking the limit $P\to P_{(1,1)}$ as in (3.42) in a correlation function, but in GDMMs it is not known what happens in this limit.
21- Right, see C18.
22- Fixed.
23- Fixed.
24- Fixed.
25- Mentioned the SciPost reviewers in the acknowledgements. These days, in this field, journal data are not needed for finding papers, and do not give much useful information in terms of quality or else. A journal can be a venue for improving a paper as we are doing right now, but this does not imply that journal data should be displayed to readers in the bibliography. This said, I can add the data if the editors so wish.
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Reply to Referee 2.
1- This text is about the bootstrap approach. More intuition about the stress-energy tensor is not needed, and would require introducing the Lagrangian approach, so it is outside the scope. --- Basic calculations can be found in exercises. Where to find exercises is indicated in the introduction. --- I have added the origin of the A-D-E terminology for minimal models, see C19. --- Torus and modular invariance are outside the scope of this text, as announced in the introduction. The study of modular invariant partition functions was historically relevant, but I am rederiving the same spectrums in a simpler way.
2- For lattice loop models, I have added reference [36]. As suggested in the introduction, the Coulomb gas approach is only of historical interest, and irrelevant to current research. Results from that approach are rederived in a simpler way in this text.
3- Fixed terminology for representation theory, see C06. Fixed terminology on null and singular vectors: the correct term singular vector is now used throughout the text, while null vectors are only mentioned in the context of the Shapovalov form.
4- Other approaches to Liouville theory are outside the scope. See the Wikipedia article on Liouville theory (most of which I wrote) and references therein.
5- Cited in the introduction.
6- For indecomposable representations, see C10. --- It was already said explicitly in Section 2.4.2 that logarithmic representations appear in loop CFTs. I welcome the suggestion to strengthen this aspect. See changes C07 and C17.
List of changes
C02. In the introduction, mentioned Teschner's seminal work on the analytic bootstrap in Liouville theory.
C03. In the Virasoro algebra's commutation relations (1.5), the central generator is made implicit, and the surrounding explanations modified accordingly.
C04. Added "(whether or not $L_0$ is diagonalizable in that representation)" when discussing eigenvalues of $L_0$.
C05. After (1.8), added the precision "where the unit operator $1\in \mathcal{L}$ is obtained in the case $k=0$".
C06. In Section 1.1.2, clarified the definition of a highest-weight representation.
C07. At the end of Section 1.1.2, defined logarithmic representations.
C08. Added Eqs. (1.17) and (1.18) on the behaviour of singular vectors under $\beta\to -\beta$ and $\beta\to \beta^{-1}$.
C09. After (1.26), removed the statement about the basis of the quotient representation.
C10. Before (1.27), added statement on reducibility and non-vanishing singular vectors.
C11. At the beginning of Section 1.1.5, the involution is no longer called an automorphism.
C12. Rewritten the second half of Section 1.2.4, in order to clarify the properties of the fusion product, and better define fusion multiplicities.
C13. Added precision on limit of fields in (2.37).
C14. Rewritten (2.42) more correctly.
C15. After (2.60), added comment on singular vectors.
C16. At the beginning of Section 2.4, made the distinction between critical loop models and loop CFTs.
C17. At the end of Section 2.4.2, added a reference on logarithmic representations, and a reference to the section on logarithmic blocks.
C18. Around Eq. (2.72), introduced the distinction between the group and the category of representations.
C19. In Section 2.5.3, mentioned the A-D-E classification of minimal models.
C20. Rewritten Section 3.3.3, removing the concept of reference structure constants.
C21. At the beginning of Section 4.2, added a second paragraph on interchiral symmetry, with a comparison to supersymmetry.
C22. Rewritten the introduction of Section 5.
C23. Revised Section 5.2.1, which now include the formulas for $n$ and $w(P)$.
C24. Rewritten Section 5.3.3.
C25. Simplified Section 5.5.1 in light of more recent developments.
C26. Terminology: replaced 'relative deviation' with 'deviation'.
C27. Around Eq. (1.23), added assumption on singular vector normalization, and flipped the sign of $\epsilon$ in its definition.