Extra-Dimensional η-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 $η$-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 $X=\mathbb{C}^3/Γ$ with finite subgroup $Γ\subset SU(3)$, where the anomalies are determined by the $η$-invariants of the asymptotic boundary $\partial X=S^5/Γ$. Our results apply equally to Abelian and non-Abelian $Γ$, 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.

💡 Research Summary

The paper introduces a novel geometric framework for extracting ’t Hooft anomalies of five‑dimensional superconformal field theories (5D SCFTs) engineered in M‑theory by non‑compact orbifolds (X=\mathbb{C}^{3}/\Gamma). Traditionally, one resolves the singularity, computes triple intersection numbers of non‑compact divisors, and reduces the eleven‑dimensional supergravity Chern‑Simons terms to obtain a Symmetry Topological Field Theory (SymTFT) that encodes the anomalies. This blow‑up/resolution procedure is technically demanding, often ambiguous, and fails for non‑isolated or non‑abelian singularities where a smooth resolution may not exist.

The authors propose to bypass all of these steps by focusing exclusively on the asymptotic boundary (\partial X=S^{5}/\Gamma). They argue that the relevant anomaly data are captured entirely by appropriate η‑invariants defined on this (possibly singular) five‑manifold. The reasoning follows from the Atiyah‑Patodi‑Singer index theorem on manifolds with boundary: the bulk index receives a contribution from the η‑invariant of the boundary, and in the M‑theory context the bulk index is precisely the coefficient of the eleven‑dimensional Chern‑Simons term. Consequently, the η‑invariant directly yields the mixed 1‑form/0‑form anomalies of the 5D SCFT.

The paper proceeds in several stages. Section 2 reviews the dimensional reduction picture (\partial X\to\mathrm{SymTFT}(T_X)) and the role of defect groups and BF‑type sectors. Section 3 collects the mathematical background on η‑invariants, emphasizing their modular properties and the fact that they live in (\mathbb{Q}/\mathbb{Z}). Section 4 illustrates the method in two well‑understood settings: (i) 7D Super‑Yang‑Mills obtained from (\mathbb{C}^{2}/\Gamma_{\text{ADE}}) and (ii) non‑supersymmetric 6D string backgrounds. In both cases the η‑invariant reproduces known mixed anomalies, confirming the approach.

The core results are presented in Section 5. For isolated cyclic orbifolds (\Gamma\cong\mathbb{Z}N) the boundary is a Lens space (S^{5}/\mathbb{Z}N); the η‑invariant can be expressed as a simple rational function of (N) and matches the triple‑intersection computation. For non‑isolated singularities—either because (\Gamma) is non‑cyclic or because the singular locus has higher‑dimensional strata—the boundary becomes a stratified space. The authors define η‑invariants on each stratum (\Sigma_i) and introduce the notion of “anomaly refinement”: the sum (\sum_i \eta_i) gives the basic anomaly, while additional integer‑valued corrections arise from the way strata intersect. Explicit calculations for families such as (\mathbb{C}^{3}/\mathbb{Z}{2n+1}(1,1,2n-1)) and (\mathbb{C}^{3}/\mathbb{Z}{2n+2}(1,1,2n)) are carried out in Appendix A, showing perfect agreement with the traditional blow‑up results.

A key advantage of the η‑invariant method is its universality. It requires only the topology of (\partial X) and the representation theory of (\Gamma); the interior geometry of (X) never enters. Hence the technique applies equally to non‑supersymmetric backgrounds, to orbifolds that are not global quotients, and to situations where the supergravity approximation breaks down near the singularity. Moreover, because the η‑invariant directly determines the coefficient of the BF‑type term in the SymTFT, the approach provides an immediate bridge between the geometric data and the full symmetry structure (including higher‑form symmetries) of the 5D SCFT.

In the concluding section the authors outline several future directions: extending the framework to 4D and 3D theories, exploring connections with anomaly inflow in holographic AdS/CFT setups, and investigating possible refinements when torsional cohomology is present. Overall, the paper establishes η‑invariants as a powerful, resolution‑independent tool for computing anomalies of quantum field theories engineered from singular extra‑dimensional geometries.