Anomaly Inflow and the $\eta$-Invariant

Edward Witten; Kazuya Yonekura

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arXiv · 1909.08775

Anomaly Inflow and the $\eta$-Invariant

Edward Witten·Kazuya Yonekura

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Abstract

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 $\eta$-invariant of Atiyah, Patodi, and Singer instead of the Chern-Simons function. Here we give a nonperturbative description of anomaly inflow, involving the $\eta$-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 $\eta$-invariant in $D$ dimensions. This $\eta$-invariant is a cobordism invariant whenever perturbative anomalies cancel.

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Edward Witten, Kazuya Yonekura. 2019-09-19. Anomaly Inflow and the $\eta$-Invariant. https://arxiv.org/abs/1909.08775

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