  We study the topological $G_2$ and $Spin(7)$ strings at 1-loop. We define new
double complexes for supersymmetric NSNS backgrounds of string theory using
generalised geometry. The 1-loop partition function then has a target-space
interpretation as a particular alternating product of determinants of
Laplacians, which we have dubbed the analytic torsion. In the case without flux
where these backgrounds have special holonomy, we reproduce the worldsheet
calculation of the $G_2$ string and give a new prediction for the $Spin(7)$
string. We also comment on connections with topological strings on Calabi-Yau
and K3 backgrounds.
