Extra-Dimensional \eta-Invariants and Anomaly Theories

TL;DR

This paper introduces a method to extract anomalies of 5D SCFTs directly from extra-dimensional geometry using eta-invariants, simplifying previous complex computational approaches.

Contribution

It demonstrates that eta-invariants can efficiently determine anomalies in 5D SCFTs engineered via M-theory geometries, applicable to various singularity types.

Findings

Anomalies are determined by eta-invariants of the boundary geometry.

Method applies to both Abelian and non-Abelian orbifold groups.

Analysis extends to non-supersymmetric backgrounds and non-global orbifolds.

Abstract

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.