Anomaly Inflow and the η-Invariant

Edward Witten, Kazuya Yonekura

Research output: Chapter in Book/Report/Conference proceedingChapter


Perturbative fermion anomalies in spacetime dimension d have a well-known relation to Chern-Simons functions in dimension D = d + 1. This relationship is manifested in a beautiful way in “anomaly inflow” from the bulk of a system to its boundary. Along with perturbative anomalies, fermions also have global or nonperturbative anomalies, which can be incorporated by using the η-invariant of Atiyah, Patodi, and Singer instead of the Chern-Simons function. Here we give a nonperturbative description of anomaly inflow, involving the η-invariant. This formula has been expected in the past based on the Dai-Freed theorem, but has not been fully justified. It leads to a general description of perturbative and nonperturbative fermion anomalies in d dimensions in terms of an η-invariant in D dimensions. This η-invariant is a cobordism invariant whenever perturbative anomalies cancel.

Original languageEnglish
Title of host publicationMemorial Volume for
Subtitle of host publicationShoucheng Zhang
PublisherWorld Scientific Publishing Co.
Number of pages70
ISBN (Electronic)9789811231711
ISBN (Print)9789811231704
Publication statusPublished - 2021 Jan 1

ASJC Scopus subject areas

  • Engineering(all)
  • Materials Science(all)
  • Physics and Astronomy(all)


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