Symmetry-protected phases of matter have been at the forefront of condensed matter physics in recent years. Bosonic symmetry-protected phases have been interpreted in terms of anomalies and group cohomology. The present article aims to develop an analogous description of fermionic symmetry-protected phases, such as the topological insulators that have been seen experimentally in 2 or 3 space dimensions. The relevant mathematical concepts include the Atiyah-Singer index theorem and the Atiyah-Patodi-Singer eta invariant.
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Edward Witten (2016) studied this question.
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