Add boundary conformal field theory

[ribault]

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

[blefloch]

Bearing in mind that I have limited experience, it seems to me that the most illuminating point of view on the classification of branes is the annulus bootstrap giving Cardy's conditions.

1. Compute the S-matrix. That's simply a Gaussian integral plus modularity of η.

2. Require the existence of an identity brane such that an open string stretching between a pair of identity branes has as its single state the identity. The annulus partition function constrains the wavefunction Ψ_1(α) to obey Ψ_1(α)Ψ_1(Q-α) = S_1^α. The reflection coefficient relating Vα and V{Q-α} is known and gives the ratio Ψ_1(α) / Ψ_1(Q-α). That gives Ψ_1(α) up to a sign. By continuity in α that sign is constant and almost certainly a matter of convention fixed by the semiclassical limit.

3. Cardy's condition for any other brane is that the open string spectrum between the identity and that brane has integer multiplicities. A basis of solutions is given by branes B such that the spectrum of 1—B strings is a single Virasoro representation. The annulus partition function tells us Ψ_μ(α) = Ψ_1(α) S_μ^α / S_1^α.

4. Comment that that ratio of S-matrix elements is the effect of a Verlinde loop operator.

Of course it's still valuable to also discuss bootstrap. Presumably there should also be a part about boundary-changing operators. Sorry for the long-winded comment.

[blefloch]

Chatting with Joaquín uncovered a way that avoids the worry about why Ψ_1(α) / Ψ_1(Q-α) should give the reflection coefficient R(P). Let α=Q/2+iP. Restrict all integrals to P∈[0,∞). Then instead of Ψ_1(P)Ψ1(-P)=S{1P} we get Ψ_1(P) Ψ1(P) = S{1P} 〈PP〉 = S_{1P} R(P) and we're done.

[ribault]

Thank you @blefloch for the ideas on how to present boundary CFT. Let me use this opportunity for proposing a different approach.

Boundary CFT is a vast subject and may deserve a text of comparable length as the bulk story. Keeping the orginal idea to only deal with topologically trivial surfaces, the title could be "Conformal field theory on the disc".

Then I have to explain why the annulus and torus may not be fundamental. The reason is that the modular S-matrix is hard to define in theories with W algebras, because characters depend only on one variable and need not be linearly independent when representations have several parameters. Without the S-matrix, we however seem to lose interesting results such as the Verlinde formula. To avoid losing these results, we may replace the S-matrix with the disc one-point function (i.e. @blefloch's wavefunction). This quantity is not problematic in theories with W algebras. This one-point function could be the fundamental quantity in terms of which the S-matrix is expressed, rather than the other way around.

A drawback of this approach is that its motivation relies on W algebras, and such algebras are absent from the present text. However, it might make sense to introduce W algebras in boundary CFT only, given how poorly we understand conformal Toda theory in the bulk.