Anomaly Inflow and the $η$-Invariant

TL;DR

The authors present a nonperturbative framework for fermion anomalies via the $\eta$-invariant and APS boundary conditions, unifying perturbative and global anomalies as a bulk–boundary inflow problem. By deriving a precise inflow formula, $Z(Y,\mathsf{L}) = |\Pf(D_W^+)|\,\exp(-i\pi\eta/2)$ (or $|\Det D_W^+|\exp(-i\pi\eta_D)$ in appropriate cases), they embed a boundary theory in a bulk gapped system and connect the phase of the boundary path integral to cobordism invariants. The work grounds anomaly cancellation in the Dai–Freed framework and cobordism theory, illustrating with explicit examples in dimensions $d=1,2,3,4$ including topological insulators and the Standard Model, thereby providing a robust, general language for both perturbative and nonperturbative anomalies. This has broad implications for SPT phases and the long-distance behavior of gapped boundary theories compared to their anomaly-free bulk completions.

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 $η$-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.