Reality of a primal/dual gap for low-c spectral gaps

Determine whether there exist modular-invariant torus partition functions with integer-valued primary degeneracies (consistent with Virasoro symmetry) for central charges c in (1.00, 1.06) and spectral gaps Δ_gap in (0.3, 0.5), or prove that such partition functions cannot exist in this region; equivalently, ascertain whether a genuine discrepancy (“primal/dual gap”) exists between primal constructions with integrality and the dual modular bootstrap bound Δ_gap ≤ c/6 + 1/3 in this regime.

Background

The study uses a primal modular bootstrap approach with integer-valued degeneracies to construct candidate partition functions at low central charge. While many candidates are found, the authors consistently fail to obtain solutions for c approximately in (1.00, 1.06) with Δ_gap around (0.3, 0.5), despite these values satisfying the known dual bound Δ_gap ≤ c/6 + 1/3.

This persistent failure suggests a possible inherent obstruction—termed a primal/dual gap—potentially reflecting stronger constraints arising from integrality of degeneracies and higher-spin contributions than those captured by the dual bound. Establishing whether this gap is real would clarify the true landscape of allowed spectra near c≈1.