  We incorporate the microscopic assumptions that lead to a certain
generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional
spin chains into the conformal bootstrap. Our approach accounts for the "LSM
anomaly" possessed by these spin chains through a combination of modular
bootstrap and correlator bootstrap of symmetry defect operators. We thus obtain
universal bounds on the local operator content of (1+1)$d$ conformal field
theories (CFTs) that could describe translationally invariant lattice
Hamiltonians with a $\mathbb Z_N\times \mathbb Z_N$ symmetry realized
projectively at each site. We present bounds on local operators both with and
without refinement by their global symmetry representations. Interestingly, we
can obtain non-trivial bounds on charged operators when $N$ is odd, which turns
out to be impossible with modular bootstrap alone. Our bounds exhibit
distinctive kinks, some of which are approximately saturated by known theories
and others that are unexplained. We discuss additional scenarios with the
properties necessary for our bounds to apply, including certain multicritical
points between (1+1)$d$ symmetry protected topological phases, where we argue
that the anomaly studied in our bootstrap calculations should emerge.
