  We study the effects of having multiple Spin structures on the partition
function of the spacetime fields in M-theory. This leads to a potential anomaly
which appears in the eta-invariants upon variation of the Spin structure. The
main source of such spaces are manifolds with nontrivial fundamental group,
which are also important in realistic models. We extend the discussion to the
Spin^c case and find the phase of the partition function, and revisit the
quantization condition for the C-field in this case. In type IIA string theory
in ten dimensions, the mod 2 index of the Dirac operator is the obstruction to
having a well-defined partition function. We geometrically characterize
manifolds with and without such an anomaly and extend to the case of nontrivial
fundamental group. The lift to KO-theory gives the alpha-invariant, which in
general depends on the Spin structure. This reveals many interesting connection
to positive scalar curvature manifolds and constructions related to the
Gromov-Lawson-Rosenberg conjecture. In the twelve-dimensional theory bounding
M-theory, we study similar geometric questions, including choices of metrics
and obtaining elements of K-theory in ten dimensions by pushforward in K-theory
on the disk fiber. We interpret the latter in terms of the families index
theorem for Dirac operators on the M-theory circle and disk. This involves
superconnections, eta-forms, and infinite-dimensional bundles, and gives
elements in Deligne cohomology in lower dimensions. We illustrate our
discussion with many examples throughout.
