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Effective action of 6D F-Theory with U(1) factors: Rational sections make Chern-Simons terms jump

Thomas W. Grimm, Andreas Kapfer, Jan Keitel

TL;DR

This work develops a comprehensive framework to obtain the $6D$ $(1,0)$ F-theory action with Abelian and non-Abelian factors by leveraging M-theory on a resolved CY$_3$ and a circle-reduced 6D theory. One-loop Chern–Simons terms arising from integrating out massive Coulomb-branch and KK modes reveal the charged spectrum and anomaly structure, with the extended relative Mori cone and non-holomorphic zero sections playing crucial roles in the F-theory limit. The authors demonstrate precise matches between 5D CS data and 6D anomalies for several explicit CY$_3$ examples, thereby determining the full charged matter content and confirming anomaly cancellation. Their approach links geometry—via the Mordell–Weil group, Shioda maps, and Mori cones—to low-energy couplings, providing a powerful diagnostic for consistent F-theory compactifications with $U(1)$ factors and highlighting how KK states can affect CS levels when the zero section is non-holomorphic.

Abstract

We derive the six-dimensional (1,0) effective action arising from F-theory on an elliptically fibered Calabi-Yau threefold with multiple sections. The considered theories admit both non-Abelian and Abelian gauge symmetries. Our derivation employs the M-theory to F-theory duality in five-dimensions after circle reduction. Five-dimensional gauge and gravitational Chern-Simons terms are shown to arise at one-loop by integrating out massive Coulomb branch and Kaluza-Klein modes. In the presence of a non-holomorphic zero section, we find an improved systematic for performing the F-theory limit by using the concept of the extended relative Mori cone. In this situation Kaluza-Klein modes can become lighter than Coulomb branch modes and a jump in the Chern-Simons levels occurs. By determining Chern-Simons terms for various threefold examples we are able to compute the complete six-dimensional charged matter spectrum and show consistency with six-dimensional anomalies.

Effective action of 6D F-Theory with U(1) factors: Rational sections make Chern-Simons terms jump

TL;DR

This work develops a comprehensive framework to obtain the F-theory action with Abelian and non-Abelian factors by leveraging M-theory on a resolved CY and a circle-reduced 6D theory. One-loop Chern–Simons terms arising from integrating out massive Coulomb-branch and KK modes reveal the charged spectrum and anomaly structure, with the extended relative Mori cone and non-holomorphic zero sections playing crucial roles in the F-theory limit. The authors demonstrate precise matches between 5D CS data and 6D anomalies for several explicit CY examples, thereby determining the full charged matter content and confirming anomaly cancellation. Their approach links geometry—via the Mordell–Weil group, Shioda maps, and Mori cones—to low-energy couplings, providing a powerful diagnostic for consistent F-theory compactifications with factors and highlighting how KK states can affect CS levels when the zero section is non-holomorphic.

Abstract

We derive the six-dimensional (1,0) effective action arising from F-theory on an elliptically fibered Calabi-Yau threefold with multiple sections. The considered theories admit both non-Abelian and Abelian gauge symmetries. Our derivation employs the M-theory to F-theory duality in five-dimensions after circle reduction. Five-dimensional gauge and gravitational Chern-Simons terms are shown to arise at one-loop by integrating out massive Coulomb branch and Kaluza-Klein modes. In the presence of a non-holomorphic zero section, we find an improved systematic for performing the F-theory limit by using the concept of the extended relative Mori cone. In this situation Kaluza-Klein modes can become lighter than Coulomb branch modes and a jump in the Chern-Simons levels occurs. By determining Chern-Simons terms for various threefold examples we are able to compute the complete six-dimensional charged matter spectrum and show consistency with six-dimensional anomalies.

Paper Structure

This paper contains 28 sections, 230 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. On the geometry of Calabi-Yau threefolds with Abelian gauge factors
  3. General remarks on elliptic fibrations
  4. Topology of elliptically fibered Calabi-Yau threefolds
  5. Effective action via M-theory
  6. Reducing $(1,0)$ supergravity on a circle
  7. M-theory on a Calabi-Yau threefold with $U(1)$s
  8. Classical matching of F-theory and M-theory
  9. One-loop Chern-Simons terms and anomaly cancelation
  10. Review of 6D anomalies with $U(1)$s
  11. One-loop Chern-Simons terms in circle reduced theories
  12. Gravitational and mixed anomalies via one-loop Chern-Simons terms
  13. Pure gauge anomalies and one-loop Chern-Simons terms
  14. Creating new Chern-Simons terms for $m_{CB} > m_{KK}$
  15. Systematics and concrete examples of Calabi-Yau threefolds with Abelian gauge factors
  16. ...and 13 more sections

Figures (7)

  • Figure 1: Pure gravitational anomaly
  • Figure 2: Triangle diagram inducing $k_{000}$
  • Figure 3: Triangle diagram inducing $k_0$
  • Figure 4: Abelian-gravitational anomaly
  • Figure 5: Triangle diagram inducing $k_{0mn}$
  • ...and 2 more figures