Global Anomalies and Bordism Invariants in One Dimension

TL;DR

This paper investigates global anomalies in one-dimensional massless Majorana fermion systems with various symmetries, using bordism invariants and the $ a$-invariant to classify and detect all anomalies.

Contribution

It constructs a comprehensive set of bordism invariants to classify global anomalies in 1D Majorana fermion systems with specific symmetries.

Findings

All bordism classes are detected by constructed invariants.

The $ a$-invariant effectively classifies global anomalies.

Partition function ambiguities are characterized by bordism invariants.

Abstract

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.