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Some open intervals are denoted with the French notation, while others use the English notation. For instance, equation (3.1.61) reads [25,∞[ while Exercise 3.3 uses (0,Q).

Maybe all representation names should be put in the \mathcal style. This is currently the case only for Verma modules.

Compute structure constants of the compactified free bosons. These should be signs. However, these signs must be nontrivial in order to have the correct behaviour under permutations. Same issue with OPE coefficients. (This could be an exercise.)

The state-field correspondence is a subtle concept, which sometimes puzzles students. In Vertex Operator Algebras, the correspondence is part of the data, and subject to various axioms. If we define states and fields independently, the correspondence is an axiom. If we first define fields, the correspondence could be a definition of states. (A rather trivial definition, since the space of states is the space of fields.)

It would be nice to clarify this concept.

Consider the non-diagonal unitary minimal model (E_8,A_30), whose spectrum is for instance given in Table 3 of https://arxiv.org/pdf/0911.3242 (page 9). The fusion of the representation with Kač labels (7,1) with itself allows (1,1), (3,1), ..., (13,1), as given for instance in equation (7.70) of Di Francesco's CFT book. Not all of these representations appear in the spectrum of the (E_8,A_30) model.

Something similar happens for the E_6 minimal model I think, but not for the D-type and E_7 type ones.

I'm probably just confused by how the fusion multiplicities are defined.

• Below (1.2.14), in "necessary for building a quantum theory from the classical theory", the word "from" is misspelled as "form".
• In "We will mostly justify these further assumptions by formal considerations, which are not physically or mathematically unavoidable." (above (1.3.5)), the double negation seems to be unwanted; I would remove "not". But perhaps I misunderstood.

I'll collect here typos I find in the arXiv version

• "contolled" => "controlled" p. 36
• In eq (3.2.6) the last exponent should be Delta2-Delta3-Delta1.

The reflection coefficient in Liouville theory and other CFTs is not a fundamental quantity, as it depends on field normalizations. It could be eliminated, and only normalization-independent quantities could be kept. This would simplify the bootstrap.

To get the DOZZ formula we nevertheless need to choose specific normalizations. It would be nice to justify the choice of normalization without reference to path integral results.
The DOZZ normalization allows some degenerate OPE coefficients to be one, allows degenerate fields to be obtained in appropriate limits, and allows the two-point function to be given by the reflection coefficient.

Material on reflection could be kept as a side note and/or an exercise.

It could be interesting to include boundary conformal field theory in this review article. Some useful material can be taken from the following lecture notes:

ipht.cea.fr/Docspht//search/article.php?id=t14/141

You should add a pdf file in your release files.
"Releases" tab is In the main tab of the repository.
This functions ties release tags with codes, and we can attach products from sources,
e.g., a compiled pdf file from tex sources.
This will save trouble to compile tex and make your product easy to use.

The analytic bootstrap determination of three-point structure constants is currently done in the context of Liouville theory. This calculation could be presented as model-independent: it depends only on the assumption that degenerate fields exist.