Boundary Liouville theory

Liouville theory is an interesting 2d conformal field theory, which has been solved in the sense that 3pt structure constants are exactly known.

The theory is also quite well understood in the presence of a boundary, with boundary conditions described by ZZ branes or FZZT branes. These boundary conditions correspond to representations of the w:Virasoro algebra: degenerate representations and Verma modules respectively. However, this picture is only valid for central charges in $ℂ-(-\mathrm{\infty },1]$. For $c\le 1$, boundary Liouville theory is less well understood.

Motivations

Non-perturbative effects in the Virasoro minimal string.

Existing works

An article by Bautista and Bawane tries to construct the $c\le 1$ counterparts of FZZT branes in the bootstrap approach.

The article on the Virasoro minimal string conjectures two families of boundary conditions with one discrete parameter, which they call half-ZZ branes.

Challenges

Divergence of annulus partition functions.

Possible approaches

Limit from A-series minimal models

Liouville theory with $c\le 1$ can be obtained as a limit of diagonal minimal models, whose central charges are dense in the half-line $c\le 1$. In practice, it could be good to do it in two steps:

• Obtain boundary conditions in generalized minimal models.
• Obtain Liouville theory from generalized minimal models as a limit at fixed $c$.

In generalized minimal models, we expect to find boundary conditions labelled by two integers $m,n\in {\mathbb{N}}^{*}$. The only subtlety is that annulus partition functions probably diverge, just like the torus partition function. Then the Liouville limit could be taken at fixed $m,n$, leading to ZZ-like boundary conditions. The further limit $m\beta -n{\beta }^{-1}\to p$ with $m,n\to \mathrm{\infty }$ and $c=13-6{\beta }^2-6{\beta }^{-2}$ would yield FZZT-like boundary conditions. It might be possible to cook up other limits, and obtain boundary conditions that are neither FZZT-like nor ZZ-like.

An advantage of this approach is that the conformal bootstrap constraints are automatically satisfied. Another advantage is that non-perturbative effects in the Virasoro minimal string might follow from the same effects in Minimal Liouville gravity, where they are understood to some extent. A weakness of the approach is that there is some freedom of having prefactors when taking the limit, making it unclear which quantities are supposed to be finite or to diverge.

Construction from chiral CFT

In a simple diagonal CFT like Liouville theory, the structure constants are closely related to the fusion and modular kernels for Virasoro conformal blocks. This might still hold for $c\le 1$. However, the kernels in question become less regular than for generic central charge, and less well understood.