Open problems in 2d CFT

Two-dimensional conformal field theory has been an active topic of research since the 1980s, with applications to statistical physics and quantum gravity.

Big questions

1. Is diffusion-limited aggregation in two dimensions conformally invariant? If yes, which CFT describes it?

2. Which CFT describes KPZ surface growth?

3. Which CFT describes the infrared limit of $N$ coupled $Q$ -state Potts models in 2d, coupled by the energy, with $N\neq 1,2$ and $2<Q<4$ ?

4. For compact unitary Virasoro-CFTs with $c>1$ , which values of the central charge $c$ are possible?

5. Are there compact unitary CFTs with $c\in \mathbb {Q}$ that are neither rational, nor exactly marginal deformations of rational CFTs?

6. Are there exactly solvable non-diagonal Virasoro-CFTs with $c\geq 25$ ? See Virasoro CFTs with a large central charge.

Prove Zamolodchikov's recursive representation of w:Virasoro conformal blocks. (See Section 4.3 of ref.^[1]) Related issue: prove the convergence of Virasoro conformal blocks.

Define an interchiral algebra that leads to the known interchiral representations and interchiral conformal blocks. (See Section 4.2 of ref.^[1])

Compute the fusion product of representations of the affine Lie algebra ${\hat {\mathfrak {s\ell }}}_{2}$ , including the representations that appear in the $SL_{2}(\mathbb {R} )$ w:Wess-Zumino-Witten model, and the degenerate representations needed to bootstrap that model. (See Section 4.4.3 of ref.^[2] and ref.^[3])

Generalized D-series minimal models, i.e. non-rational limits of D-series minimal models, see Solvable non-diagonal 2d CFTs.

Logarithmic minimal models are believed to exist at rational central charges, with primary fields in the extended Kac table. They might be constructed as limits of generalized minimal models.^[4]

↑ ^1.0 ^1.1 Ribault, Sylvain (2024). "Exactly solvable conformal field theories". GitHub. Retrieved 2024-08-31.
↑ Ribault, Sylvain (2014). "Conformal field theory on the plane". arXiv:1406.4290 [hep-th].
↑ Stocco, Dario (2022-09-18). "The torus one-point block of 2d CFT and null vectors in sl(2)". arXiv.org. Retrieved 2024-10-30.
↑ Ribault, Sylvain (2019-06-25). "On 2d CFTs that interpolate between minimal models". SciPost Physics 6 (6). doi:10.21468/scipostphys.6.6.075. ISSN 2542-4653. https://arxiv.org/abs/1809.03722.