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Anomalies on ALE spaces and phases of gauge theory

Mohamed M. Anber

TL;DR

This work shows that ALE spaces, especially the Eguchi–Hanson geometry, reveal new ’t Hooft anomalies invisible on traditional backgrounds by exploiting the nontrivial second cohomology and the RP^3 boundary. The anomaly is computed via a 5D mapping torus and the η-invariant, combining bulk characteristic classes with boundary data, and it yields strengthened IR constraints on vector-like and chiral gauge theories. By comparing UV (semiclassical) and IR (composite) descriptions, the authors demonstrate that EH anomalies can rule out infrared realizations that would otherwise match on standard manifolds, and in some cases necessitate additional degrees of freedom or TQFTs. The analysis also clarifies the Hilbert-space interpretation of the EH anomaly and its RG-flow behavior as the bolt size a varies relative to the strong scale Λ. Overall, EH anomalies provide a new, geometry-driven probe of infrared dynamics with potential to refine dualities and confinement mechanisms.

Abstract

We show that certain 't~Hooft anomalies that evade detection on commonly used closed four-dimensional manifolds become visible when a quantum field theory is placed on asymptotically locally Euclidean (ALE) spaces. As a concrete example, we use the Eguchi-Hanson (EH) space, whose defining features are its nontrivial second cohomology generated by the self-intersecting two-sphere and its asymptotic boundary $\mathbb{RP}^3$, which carries torsion and thus furnishes additional cohomological data absent on conventional backgrounds. For a theory with symmetry $G_1\times G_2$, we turn on background flux for $G_1$ and probe potential anomalies by performing a global $G_2$ transformation; the resulting anomaly is captured by a five-dimensional mapping torus. The anomaly receives contributions from the four-dimensional characteristic classes on EH space as well as from the $η$-invariant associated with the $\mathbb{RP}^3$ boundary. The anomaly uncovered in this way leads to new constraints on asymptotically free gauge theories. In particular, infrared composite spectra that successfully match anomalies on standard manifolds may nevertheless fail to reproduce the EH anomaly, and can therefore be excluded as the complete infrared realization of the symmetries.

Anomalies on ALE spaces and phases of gauge theory

TL;DR

This work shows that ALE spaces, especially the Eguchi–Hanson geometry, reveal new ’t Hooft anomalies invisible on traditional backgrounds by exploiting the nontrivial second cohomology and the RP^3 boundary. The anomaly is computed via a 5D mapping torus and the η-invariant, combining bulk characteristic classes with boundary data, and it yields strengthened IR constraints on vector-like and chiral gauge theories. By comparing UV (semiclassical) and IR (composite) descriptions, the authors demonstrate that EH anomalies can rule out infrared realizations that would otherwise match on standard manifolds, and in some cases necessitate additional degrees of freedom or TQFTs. The analysis also clarifies the Hilbert-space interpretation of the EH anomaly and its RG-flow behavior as the bolt size a varies relative to the strong scale Λ. Overall, EH anomalies provide a new, geometry-driven probe of infrared dynamics with potential to refine dualities and confinement mechanisms.

Abstract

We show that certain 't~Hooft anomalies that evade detection on commonly used closed four-dimensional manifolds become visible when a quantum field theory is placed on asymptotically locally Euclidean (ALE) spaces. As a concrete example, we use the Eguchi-Hanson (EH) space, whose defining features are its nontrivial second cohomology generated by the self-intersecting two-sphere and its asymptotic boundary , which carries torsion and thus furnishes additional cohomological data absent on conventional backgrounds. For a theory with symmetry , we turn on background flux for and probe potential anomalies by performing a global transformation; the resulting anomaly is captured by a five-dimensional mapping torus. The anomaly receives contributions from the four-dimensional characteristic classes on EH space as well as from the -invariant associated with the boundary. The anomaly uncovered in this way leads to new constraints on asymptotically free gauge theories. In particular, infrared composite spectra that successfully match anomalies on standard manifolds may nevertheless fail to reproduce the EH anomaly, and can therefore be excluded as the complete infrared realization of the symmetries.

Paper Structure

This paper contains 26 sections, 196 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Eguchi--Hanson space, background fluxes, and fermions
  3. Eguchi--Hanson space: a crash course
  4. General background fluxes
  5. Fermions, global consistency, and zero modes
  6. The Atiyah--Patodi--Singer (APS) index theorem
  7. Gluing EH spaces and the $\eta$-invariant
  8. The EH space as a probe of anomalies
  9. Diagnosis of anomalies
  10. The new anomaly on EH space
  11. RG flow and varying the bolt size
  12. Regime $a\Lambda \ll 1$ (bolt scale in the UV).
  13. Regime $a\Lambda \gg 1$ (bolt scale in the IR).
  14. No double counting of anomaly-carrying states.
  15. Are there boundary degrees of freedom on $\mathbb{RP}^3$?
  16. ...and 11 more sections

Figures (1)

  • Figure 1: Sewing EH (on the right) and $\hbox{EH}'$ (an orientation reversal of EH but with a different $U(1)$ flux) via their $\mathbb{RP}^3$ boundary, to give ${\cal M}=\hbox{EH}\cup\hbox{EH}'$ (on the left). The sewing must happen such that the gauge bundles over both spaces match smoothly via $\mathbb{RP}^3$, so that we avoid a $\mathbb {Z} _2$ jump in the holonomy seen by the fermions. The sewing must also not change the number of fermion zero modes (represented by the arrows) on the original EH space.