Sharp upper bound on the lowest non‑vacuum scaling dimension in 2D CFT (pure gravity gap problem)

Determine the sharp universal upper bound, as a function of the central charge c, on the lowest non‑vacuum scaling dimension in unitary compact two‑dimensional conformal field theories, particularly resolving whether the “pure gravity” gap Δ1−Δ0≤c/12 can be saturated or universally enforced.

Background

In AdS3/CFT2, the existence of pure gravity without matter suggests a spectral gap of order c/12 between the vacuum and the lightest non‑vacuum primary. Establishing the maximal allowed gap in general unitary compact CFTs would clarify whether such a ‘pure gravity’ dual can exist.

Hellerman initiated this program, deriving a weaker bound using modular bootstrap, and subsequent work has improved bounds numerically, but the problem of determining the exact universal bound remains unresolved.