Mixed anomalies and the modular bootstrap

Lieb-Shultz-Mattis (LSM) theorems constrain the low-energy physics of translationally invariant 1D lattice models with projective symmetry representations in each unit cell by preventing them from being trivially gapped. This leaves spontaneous symmetry breaking or gaplessness as the only possibilities. Furthermore, in the low energy effective field theory of such models there is a mixed ’t Hooft anomaly between the translation symmetry and the internal symmetry. We use the modular bootstrap to put quantitative bounds on aspects of (1+1)D conformal field theories (CFTs) that saturate these LSM-type anomalies. In particular, we rigorously constrain the allowed values of the central charge for certain CFTs with an LSM-type anomaly that can describe fixed points of symmetry-enforced gapless phases.