Constraining the spectral gap of 2d CFTs just above c=1

Abstract

We numerically study the space of compact unitary two-dimensional conformal field theories (2d CFTs) with central charge $c>1$ using the modular bootstrap. We focus on bounding the spectral gap $Δ_{\rm gap}$ of such theories, namely the smallest scaling dimension among the non-vacuum primary operators. We use both a primal method that directly searches for consistent spectra and a dual method that yields rigorous exclusions. A key feature of our analysis is requiring the degeneracies of primary operators to be positive integers. At $c$ very near $1$, the integrality constraint greatly reduces the allowed spectral gap in both methods. For example, at $c-1 = 3\times 10^{-3}$ integrality strengthens the rigorous bound from $Δ_{\rm gap} \lesssim 0.5$ to $Δ_{\rm gap} < 0.041$. Our results suggest that the spectral gap might vanish as $c$ approaches $1$ from above.