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Anomaly Cancelation in Field Theory and F-theory on a Circle

Thomas W. Grimm, Andreas Kapfer

TL;DR

The paper develops a unified framework linking four- and six-dimensional gauge anomalies to circle-reduced Kaluza-Klein theories, using higher-dimensional large gauge transformations to identify symmetry maps on the Coulomb branch that reproduce anomaly cancellation via one-loop Chern-Simons terms. It shows that in 4D Abelian and non-Abelian cases these maps form a $\mathbb{Z}^{\text{rank}}$ group, while in 6D the analogous structure yields the full anomaly constraints through weight-sum identities and Green-Schwarz data. By leveraging M-/F-theory duality, the authors assign geometric meanings to these transformations as zero-section and zero-node choices in elliptically fibered Calabi–Yau manifolds, proving the general absence of Abelian and non-Abelian anomalies in the considered F-theory geometries. Overall, the work provides a general, geometry-linked mechanism by which higher-dimensional anomaly cancellation is reflected in lower-dimensional CS terms and in the symmetry structure of F-theory compactifications.

Abstract

We study the manifestation of local gauge anomalies of four- and six-dimensional field theories in the lower-dimensional Kaluza-Klein theory obtained after circle compactification. We identify a convenient set of transformations acting on the whole tower of massless and massive states and investigate their action on the low-energy effective theories in the Coulomb branch. The maps employ higher-dimensional large gauge transformations and precisely yield the anomaly cancelation conditions when acting on the one-loop induced Chern-Simons terms in the three- and five-dimensional effective theory. The arising symmetries are argued to play a key role in the study of the M-theory to F-theory limit on Calabi-Yau manifolds. For example, using the fact that all fully resolved F-theory geometries inducing multiple Abelian gauge groups or non-Abelian groups admit a certain set of symmetries, we are able to generally show the cancelation of pure Abelian or pure non-Abelian anomalies in these models.

Anomaly Cancelation in Field Theory and F-theory on a Circle

TL;DR

The paper develops a unified framework linking four- and six-dimensional gauge anomalies to circle-reduced Kaluza-Klein theories, using higher-dimensional large gauge transformations to identify symmetry maps on the Coulomb branch that reproduce anomaly cancellation via one-loop Chern-Simons terms. It shows that in 4D Abelian and non-Abelian cases these maps form a group, while in 6D the analogous structure yields the full anomaly constraints through weight-sum identities and Green-Schwarz data. By leveraging M-/F-theory duality, the authors assign geometric meanings to these transformations as zero-section and zero-node choices in elliptically fibered Calabi–Yau manifolds, proving the general absence of Abelian and non-Abelian anomalies in the considered F-theory geometries. Overall, the work provides a general, geometry-linked mechanism by which higher-dimensional anomaly cancellation is reflected in lower-dimensional CS terms and in the symmetry structure of F-theory compactifications.

Abstract

We study the manifestation of local gauge anomalies of four- and six-dimensional field theories in the lower-dimensional Kaluza-Klein theory obtained after circle compactification. We identify a convenient set of transformations acting on the whole tower of massless and massive states and investigate their action on the low-energy effective theories in the Coulomb branch. The maps employ higher-dimensional large gauge transformations and precisely yield the anomaly cancelation conditions when acting on the one-loop induced Chern-Simons terms in the three- and five-dimensional effective theory. The arising symmetries are argued to play a key role in the study of the M-theory to F-theory limit on Calabi-Yau manifolds. For example, using the fact that all fully resolved F-theory geometries inducing multiple Abelian gauge groups or non-Abelian groups admit a certain set of symmetries, we are able to generally show the cancelation of pure Abelian or pure non-Abelian anomalies in these models.

Paper Structure

This paper contains 19 sections, 103 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Anomalies of four-dimensional theories
  3. General setup and anomaly conditions
  4. Circle compactification and Chern-Simons terms
  5. Non-Abelian models
  6. Abelian models
  7. Anomalies of six-dimensional theories
  8. General setup and anomaly conditions
  9. Circle compactification and Chern-Simons terms
  10. Non-Abelian models
  11. Abelian models
  12. Anomalies on the circle in the M-/F-theory duality
  13. F-theory on elliptically fibered Calabi-Yau manifolds
  14. Abelian anomalies from zero-section changes
  15. Non-Abelian anomalies from zero-node changes
  16. ...and 4 more sections

Figures (2)

  • Figure 1: The transformations \ref{['e:non_abelian_trafo_vectors_4d']} generate a $\mathbb{Z}^{\mathop{\mathrm{rank}}\nolimits G}$ group relating theories with different Coulomb branch parameters but the same effective theory, as long as anomalies are canceled in the four-dimensional theory. We display the lattice of related theories for $\mathop{\mathrm{rank}}\nolimits G =2$. The transformation associated to the horizontal arrow corresponds to the choice $A^1 \equiv A^{\tilde{0}}$, the other one to $A^2 \equiv A^{\tilde{0}}$.
  • Figure 2: Two different choices for zero-nodes in a theory with $A_4$ gauge algebra are depicted. The zero-nodes are colored in red, the remaining ones, corresponding to the Cartan generators, in green. Note that it is not necessary that the zero-node gets intersected by the zero-section.