Extra-Dimensional η-Invariants and Anomaly Theories

TL;DR

The authors present a practical framework to extract anomalies of higher-dimensional QFTs engineered by orbifold geometries X=C^n/Γ directly from the boundary geometry ∂X using η-invariants, avoiding blowups and resolution-dependent data. They connect anomaly data to symmetry structures via SymTFT reductions, showing that 1-form and mixed anomalies in 5D SCFTs can be computed from boundary η-invariants on spaces like S^5/Γ, with careful treatments of isolated and non-isolated singularities and stratified loci. The approach is tested in 7D and 6D contexts and extended to non-supersymmetric backgrounds, with explicit computations and refinements (e.g., twisted η-invariants) and comparisons to bulk intersection theory. The work provides a geometrically transparent, computationally efficient route to symmetry data and anomaly structures in strongly coupled QFTs, while highlighting connections to K-theory and bordism-based perspectives for future exploration.

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 $η$-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.

Figures (4)

• Figure 1: We sketch the internal dimensions of $Y_{12}$ projected onto the radial coordinates of $X_{11-d}$ and $Z_{11-d}$ denoted $r,r_\bot$ respectively. 11D supergravity is a good approximation on the red line which carries at every point, at radius $r$ in $X_{11-d}$, the fiber $(\partial X)_{10-d}$. This fiber collapses at $r=0$ forming the singularity. At fixed radius $r=r_*$ the radial slice of $X_{11-d}$ is $(\partial X)_{10-d}^{\:\!r=r_*}$. The space $Z_{11-d}^{\:\!r=r_*}$ is smooth and has a boundary given by this copy of the link and fills it in along a transverse auxiliary dimension (blue). The spaces $(\partial X)_{10-d}^{\:\!r=r_*}$ as well as $Z_{11-d}^{\:\!r=r_*}$ are topologically equivalent for all $r_*> 0$. The dashed line denotes the cut used to construct $Y_{12}^\circ$ via excision of a small ball centered on the singularity. The space $Y_{12}^\circ$ projects onto the shaded region.
• Figure 2: We sketch the geometry $X=\mathbb{C}^3/{\mathbb Z}_N(m_1,m_2,m_3)$. The picture depicts the singularities $\mathscr{S}$ which consist of up to three 2D cones cut out by coordinate planes. Each cone supports an A-type ADE singularity in codimension-4. This singularity enhances at the tip of the cone at $z_1,z_2,z_3=0$ where the singularity enhances to codimension-6 (where the 5D SCFT is supported).
• Figure 3: Triangulated and refined toric diagrams for the orbifold $\mathbb{C}^3/{\mathbb Z}_{2n+1}$ with an action carrying weights $(1,1,2n-1)$. We give the triangulated diagrams for the first four cases $n=1,2,3,4$ associated with the crepant resolution. The black dots indicate $n$ compact toric divisors, located at $(x,y)=(0,0), \dots , (0,n-1)$. The 3 brown dots indicate the non-compact divisors $z_i=0$ denoted $D_{z_i}$ located at $(-1,0),(1,2n-1),(0,-1)$ for $i=1,2,3$ respectively.
• Figure 4: Triangulated and refined toric diagrams for the orbifold $\mathbb{C}^3/{\mathbb Z}_{2n+2}$ with an action carrying weights $(1,1,2n)$. We give the triangulated diagrams for the first four cases $n=1,2,3,4$ associated with the crepant resolution. The black dots indicate $n$ compact toric divisors, located at $(x,y)=(0,0), \dots , (0,n-1)$. The 3 brown dots indicate the non-compact divisors $z_i=0$ denoted $D_{z_i}$ located at $(-1,0),(1,2n-1),(0,-1)$ for $i=1,2,3$ respectively. The single blue dot indicates the non-compact exceptional divisor at $(0,n)$