Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Anomaly Equations and Intersection Theory

Daniel S. Park

TL;DR

The paper analyzes anomaly cancellation in six-dimensional ${ m N}=(1,0)$ supergravities and shows that the associated anomaly equations can be translated into three intersection-theory identities on a smooth resolution of an elliptically fibered Calabi–Yau threefold. By employing M/F-theory duality, it identifies the geometric counterparts of abelian anomaly coefficients via Shioda-type cycles and establishes a concrete link between low-energy data (spectra and anomalies) and the geometry of the base and fiber. It provides a detailed mapping of non-abelian sectors through Cartan data and reconstructs the abelian sector by introducing type-S cycles, proving that $b_{ij}=-\pi(S_i\cdot S_j)$ and showing how charged multiplets arise from M2-branes on shrinking curves. The results offer a geometric framework for anomaly constraints in 6D, discuss implications for string universality, and suggest directions for extending these insights to 4D F-theory models and generalized Kodaira-type constraints for abelian charges.

Abstract

Six-dimensional supergravity theories with N=(1,0) supersymmetry must satisfy anomaly equations. These equations come from demanding the cancellation of gravitational, gauge and mixed anomalies. The anomaly equations have implications for the geometrical data of Calabi-Yau threefolds, since F-theory compactified on an elliptically fibered Calabi-Yau threefold with a section generates a consistent six-dimensional N=(1,0) supergravity theory. In this paper, we show that the anomaly equations can be summarized by three intersection theory identities. In the process we also identify the geometric counterpart of the anomaly coefficients---in particular, those of the abelian gauge groups---that govern the low-energy dynamics of the theory. We discuss the results in the context of investigating string universality in six dimensions.

Anomaly Equations and Intersection Theory

TL;DR

The paper analyzes anomaly cancellation in six-dimensional supergravities and shows that the associated anomaly equations can be translated into three intersection-theory identities on a smooth resolution of an elliptically fibered Calabi–Yau threefold. By employing M/F-theory duality, it identifies the geometric counterparts of abelian anomaly coefficients via Shioda-type cycles and establishes a concrete link between low-energy data (spectra and anomalies) and the geometry of the base and fiber. It provides a detailed mapping of non-abelian sectors through Cartan data and reconstructs the abelian sector by introducing type-S cycles, proving that and showing how charged multiplets arise from M2-branes on shrinking curves. The results offer a geometric framework for anomaly constraints in 6D, discuss implications for string universality, and suggest directions for extending these insights to 4D F-theory models and generalized Kodaira-type constraints for abelian charges.

Abstract

Six-dimensional supergravity theories with N=(1,0) supersymmetry must satisfy anomaly equations. These equations come from demanding the cancellation of gravitational, gauge and mixed anomalies. The anomaly equations have implications for the geometrical data of Calabi-Yau threefolds, since F-theory compactified on an elliptically fibered Calabi-Yau threefold with a section generates a consistent six-dimensional N=(1,0) supergravity theory. In this paper, we show that the anomaly equations can be summarized by three intersection theory identities. In the process we also identify the geometric counterpart of the anomaly coefficients---in particular, those of the abelian gauge groups---that govern the low-energy dynamics of the theory. We discuss the results in the context of investigating string universality in six dimensions.

Paper Structure

This paper contains 23 sections, 129 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. 6D $(1,0)$ Theories and Anomaly Cancellation
  3. The Massless Spectrum
  4. The Anomaly Equations and Anomaly Coefficients
  5. Summary
  6. Six-dimensional F-theory Backgrounds
  7. A Very Brief Review of the Non-Abelian Sector
  8. M/F-theory Duality and Abelian Anomaly Coefficients
  9. M/F-theory Duality
  10. The Non-Abelian Sector
  11. The Abelian Sector
  12. Summary
  13. Anomaly Equations and Intersection Theory
  14. Summary and Discussion
  15. Lie Algebra and Intersection Theory
  16. ...and 8 more sections

Figures (8)

  • Figure 1: Resolved fiber for $A_n$. The curves $\alpha$ corresponding to root vectors are in solid lines while the monodromy invariant fibers $\gamma$ corresponding to coroots are in dotted lines.
  • Figure 2: Resolved fiber for $D_n$.
  • Figure 3: Resolved fiber for $E_n$.
  • Figure 4: Resolved fiber for $A_1$.
  • Figure 5: Resolved fiber for $B_n$. The curves $\alpha$ corresponding to root vectors are in solid lines while the monodromy invariant fibers $\gamma$ corresponding to coroots are in dotted lines.
  • ...and 3 more figures