Extra-Dimensional \eta-Invariants and Anomaly Theories

Anomalies of a quantum field theory (QFT) constitute fundamental non-perturbatively robust data. In this paper we extract anomalies of 5D superconformal field theories (SCFTs) directly from the underlying extra-dimensional geometry. We show that all of this information can be efficiently extracted from extra-dimensional $\eta$-invariants, bypassing previously established approaches based on computationally cumbersome blowup / resolution techniques. We illustrate these considerations for 5D SCFTs engineered in M-theory by non-compact geometries $X=\mathbb{C}^3/\Gamma$ with finite subgroup $\Gamma\subset SU(3)$, where the anomalies are determined by the $\eta$-invariants of the asymptotic boundary $\partial X=S^5/\Gamma$. Our results apply equally to Abelian and non-Abelian $\Gamma$, as well as isolated and non-isolated singularities. In the setting of non-isolated singularities we further analyze the interplay of anomaly structures across different strata of the singular locus. Our considerations extend readily to backgrounds which are not global orbifolds, as well as those which do not preserve supersymmetry.