Virasoro CFTs with a large central charge

From the w:AdS/CFT correspondence, we expect the existence of two-dimensional CFTs with a large central charge. In order to be dual to three-dimensional gravity, such CFTs are also expected to have only Virasoro symmetry, a vacuum state with a large gap above it, and to be non-diagonal and unitary.

Motivations from AdS/CFT and three-dimensional gravity

Summarize main CFT implications of the works of Fitzpatrick, Kaplan, Walters.

$c=\frac{3{\ell }_{\text{AdS}}}{2G_{\text{Newton}}}$

Role of chaos.

Role of the light, heavy limits.

Matter vs pure gravity.

Known two-dimensional CFTs

Here we review some known two-dimensional CFTs whose chiral symmetry algebras are the Virasoro algebra. None of them has all the desired properties of a CFT dual to three-dimensional gravity.

Solvable non-diagonal CFTs

In the $Q$-state Potts model and the $O(n)$ model are non-diagonal CFTs with discrete spectrums. Their non-diagonal fields have dimensions $(\mathrm{\Delta },\overline{\mathrm{\Delta }})=({\mathrm{\Delta }}_{(r,s)},{\mathrm{\Delta }}_{(r,-s)})$ with $r,s\in \mathbb{Q}$. Such dimensions, with fractional Kac indices, are natural because the condition for the spin to be integer boils down to $rs\in \mathbb{Z}$, and because integer Kac indices play an important role in the solvability of minimal models, w:Liouville theory, etc. But the total conformal dimension is then

${\mathrm{\Delta }}_{(r,s)}+{\mathrm{\Delta }}_{(r,-s)}=\frac{c-1}{12}-\frac{1}{2}{r}^2{b}^2-\frac{1}{2}{s}^2{b}^{-2}\text{with}c=13+6b^2+6b^{-2}$

This is bounded from below for $r,s\to \mathrm{\infty }$ provided

$\mathrm{\Re }{b}^2<0⟺\mathrm{\Re }c<13$

So CFTs of this type cannot exist for $c\to +\mathrm{\infty }$. They are also non-unitary, and the identity may or may not be the ground state, depending on the central charge.

There is an analogous obstruction in integrable approaches to conformal blocks. The relation between w:Virasoro conformal blocks at $c=1$ and solutions of the Painlevé VI equation involves an infinite sum over blocks. The natural generalization of this sum at generic central charge would diverge for $\mathrm{\Re }c>13$.

Liouville theory

w:Liouville theory is unitary and exists for any $c\in ℂ$, but it is diagonal, does not have a vacuum state in the spectrum, and there is no gap above the ground state.

Generalized minimal models

Generalized minimal models have discrete spectrums made of degenerate fields, and exist for any $c\in ℂ$. However, the spectrum is not bounded from below if $\mathrm{\Re }c>13$. This does not prevent GMMs from existing, thanks to their fusion rules.

Possible approaches

Understanding analytic solutions of the crossing symmetry equations

In order to better understand the analytic properties of the fusing matrix, and of the crossing symmetry equations themselves, it would be interesting to study the known analytic solutions. In quite a few cases, analytic solutions are known, but it has not been proved analytically that they are in fact solutions.

In simple diagonal CFTs such as w:Liouville theory and diagonal minimal models, the 3pt structure constant is a special case of the fusing matrix, and crossing symmetry follows from the Pentagon equation. There is a similar story about D-series minimal models. Other analytic solutions are probably less well understood.

Solving the crossing symmetry equations at c=25

At $c=1$, the crossing symmetry equations are more explicit and easier to solve than at generic $c$. This is due to the relation of conformal blocks with solutions of the Painlevé VI equation and isomonodromic connections. This leads to an explicit formula for the fusing matrix that is much simpler than the general formula. As a result, one can try to solve crossing symmetry equations analytically.

The cases $c=25$ and $c=1$ are related by $b\to ib$ and are formally very similar. This raises the hope that simple formulas for the fusing matrix can also be found at $c=25$. Actually, the known general Ponsot-Teschner formula for the fusing matrix is valid for $c\in ℂ-(-\mathrm{\infty },1)$, the general formula for $c\le 1$ is still missing, and finding it raises similar issues. Knowing that Liouville theory is crossing-symmetric is a big hint.

Analyticity in spin

To find manageable solutions, we should use the right variables. At small central charge, the right variables seem to be Kac indices and conformal blocks, which reflect the existence of degenerate fields. At large central charge, degenerate fields might not play an important role. But if we are not using Kac indices, how do we efficiently encode the structure of a non-diagonal spectrum, and the integer values of the conformal spin?

Maybe we should use the analyticity in spin properties that are known to occur in CFTs with global symmetry. Functions of the integer spin would be traded for analytic functions of a continuous variable. In particular, conformal blocks would be traded for Regge blocks. This might allow us to write non-diagonal spectrums that are bounded from below, which cannot easily be done with Kac indices if $\mathrm{\Re }c>13$.