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

CFT description of some particular systems

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 $N\ne 1,2$ and $2<Q<4$?

Space of CFTs

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 $c\ge 25$? See Virasoro CFTs with a large central charge.

Technical problems

Algebraic structures and conformal blocks

• Virasoro conformal blocks at rational central charge.

• Prove Zamolodchikov's recursive representation of w:Virasoro conformal blocks. (See Section 4.3 of ref.) 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.)

• Compute the fusion product of representations of the affine Lie algebra ${\widehat{𝔰\mathfrak{\ell }}}_2$, including the representations that appear in the $S{L}_2(ℝ)$ w:Wess-Zumino-Witten model, and the degenerate representations needed to bootstrap that model. (See Section 4.4.3 of ref. and ref.)

Solving specific CFTs

• Boundary Liouville theory with $c\le 1$.

• Conformal Toda theory.

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

• Solving the SL(2,R) WZW model.

• 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.

Various issues

• Computing entanglement entropy in 2d CFTs.