  We study the partition function of type IIA string theory on 10-manifolds of
the form T^2 x X where X is 8-dimensional, compact, and spin. We pay particular
attention to the effects of the topological phases in the supergravity action
implied by the K-theoretic formulation of RR fields, and we use these to check
the T-duality invariance of the partition function. We find that the partition
function is only T-duality invariant when we take into account the T-duality
anomalies in the RR sector, the fermionic path integral (including 4-fermi
interaction terms), and 1-loop corrections including worldsheet instantons. We
comment on applications of our computation to speculations about the role of
the Romans mass in M-theory. We also discuss some issues which arise when one
attempts to extend these considerations to checking the full U-duality
invariance of the theory.
