Bordism for the 2-group symmetries of the heterotic and CHL strings

In the presence of a nonzero B-field, the symmetries of the ${\mathrm{E}}_8\times {\mathrm{E}}_8$ heterotic string form a 2-group, or a categorified group, as do the symmetries of the CHL string. We express the bordism groups of the corresponding tangential structures as twisted string bordism groups, then compute them through dimension 11 modulo a few unresolved ambiguities. Then, we use these bordism groups to study anomalies and defects for these two string theories.