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.

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 $$N\neq 1,2$$ and $$21$$, 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 $$c\geq 25$$?