Global Anomalies and Bordism Invariants in One Dimension

We consider massless Majorana fermion systems with $G=\mathbb{Z}_N$, $SO(N)$, and $O(N)$ symmetry in one-dimensional spacetime. In these theories, phase ambiguities of the partition functions are given as the exponential of the $\eta$-invariant of the Dirac operators in two dimensions, which is a bordism invariant. We construct sufficient numbers of bordism invariants to detect all bordism classes. Then, we classify global anomalies by calculating the $\eta$-invariant of these bordism classes.