Critical flavor number for chiral symmetry breaking in QED3
In plain words
Electrodynamics in two space dimensions with N species of massless electrons is scale invariant at long distances when N is large. Below which N this fails is unknown.
Precise statement
3D QED with $N_{f}$ two-component Dirac fermions of unit charge ($N_{f}$ even, parity invariant) flows to an interacting CFT for large $N_{f}$. Determine $N_{f}^{c}$ such that for $N_{f} < N_{f}^{c}$ the IR breaks the $\mathrm{SU}(N_{f})$ flavor symmetry or is otherwise not conformal; in particular decide $N_{f} = 2$ and $N_{f} = 4$ (the latter describes the Dirac spin liquid).
What would settle it
Conformal bootstrap islands or continuum-extrapolated lattice simulations that bracket $N_{f}^{c}$ and determine the fate of $N_{f}=2\text{ and }4$.
Status in the literature
Estimates from Schwinger-Dyson equations, $\epsilon$ expansion, the F-theorem, lattice simulations and bootstrap disagree; recent lattice and bootstrap evidence favors conformality at $N_f = 4$.