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The Geometry of SO(3), SO(5), and SO(6) models

Mboyo Esole, Patrick Jefferson

TL;DR

This work develops a comprehensive geometric engineering of SO(3), SO(5), and SO(6) gauge theories via elliptic fibrations with Mordell–Weil torsion $Z/2Z$, distinguishing these SO-models from Spin$(n)$. By constructing crepant resolutions of the corresponding Weierstrass models, it derives fiber structures, topological invariants, and the Coulomb-branch data for 5D $\mathcal{N}=1$ supergravity, as well as anomaly-cancellation conditions for 6D $(1,0)$ theories arising from F-theory on the same Calabi–Yau threefolds. The authors show that these SO$(n)$ models generally require adjoint matter and, in certain cases, vector representations, with explicit matches between geometric triple-intersection data and 5D prepotentials (IMS) across $\mathcal{F}$ and $\mathcal{F}_{IMS}$. The results confirm the consistency of the 5D/6D spectra with M-/F-theory compactifications and illuminate the distinct global structures of gauge groups compared to Spin$(n)$ cousins. Overall, the paper provides canonical Weierstrass realizations, detailed crepant-resolutions, and a robust geometric framework connecting elliptic-fibration geometry to higher-dimensional supergravity content.

Abstract

SO(3), SO(5), and SO(6)-models are singular elliptic fibrations with Mordell--Weil torsion Z/2Z and singular fibers whose dual fibers correspond to affine Dynkin diagrams of type A1, C2, and A3 respectively, where we emphasize the distinction between SO(n) and its universal cover Spin(n). While the SO(3)-model has been studied before, the SO(5) and SO(6)-models are studied here for the first time. By computing crepant resolutions of their Weierstrass models, we study their fiber structures and topological invariants. In the special case that the SO(n)-model is an elliptically fibered Calabi-Yau threefold, we compute the Chern-Simons couplings and matter content of a 5D N=1 supergravity theory with gauge group SO(n), which is related to M-theory compactified on this Calabi-Yau threefold. We also verify the 6D lift of the 5D matter content is necessary and sufficient for anomaly cancellation in 6D (1,0) supergravity theories geometrically engineered by F-theory compactified on the same threefold. We find that the associated 5D and 6D supergravity theories with SO(n) gauge symmetry indeed differ from their Spin(n) cousins, with one striking consequence of this distinction being that all such theories must include adjoint matter.

The Geometry of SO(3), SO(5), and SO(6) models

TL;DR

This work develops a comprehensive geometric engineering of SO(3), SO(5), and SO(6) gauge theories via elliptic fibrations with Mordell–Weil torsion , distinguishing these SO-models from Spin. By constructing crepant resolutions of the corresponding Weierstrass models, it derives fiber structures, topological invariants, and the Coulomb-branch data for 5D supergravity, as well as anomaly-cancellation conditions for 6D theories arising from F-theory on the same Calabi–Yau threefolds. The authors show that these SO models generally require adjoint matter and, in certain cases, vector representations, with explicit matches between geometric triple-intersection data and 5D prepotentials (IMS) across and . The results confirm the consistency of the 5D/6D spectra with M-/F-theory compactifications and illuminate the distinct global structures of gauge groups compared to Spin cousins. Overall, the paper provides canonical Weierstrass realizations, detailed crepant-resolutions, and a robust geometric framework connecting elliptic-fibration geometry to higher-dimensional supergravity content.

Abstract

SO(3), SO(5), and SO(6)-models are singular elliptic fibrations with Mordell--Weil torsion Z/2Z and singular fibers whose dual fibers correspond to affine Dynkin diagrams of type A1, C2, and A3 respectively, where we emphasize the distinction between SO(n) and its universal cover Spin(n). While the SO(3)-model has been studied before, the SO(5) and SO(6)-models are studied here for the first time. By computing crepant resolutions of their Weierstrass models, we study their fiber structures and topological invariants. In the special case that the SO(n)-model is an elliptically fibered Calabi-Yau threefold, we compute the Chern-Simons couplings and matter content of a 5D N=1 supergravity theory with gauge group SO(n), which is related to M-theory compactified on this Calabi-Yau threefold. We also verify the 6D lift of the 5D matter content is necessary and sufficient for anomaly cancellation in 6D (1,0) supergravity theories geometrically engineered by F-theory compactified on the same threefold. We find that the associated 5D and 6D supergravity theories with SO(n) gauge symmetry indeed differ from their Spin(n) cousins, with one striking consequence of this distinction being that all such theories must include adjoint matter.

Paper Structure

This paper contains 15 sections, 7 theorems, 37 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and summary of results
  3. Weierstrass models
  4. Intersection theory and blowups
  5. Hyperplane arrangement
  6. $G$-models and crepant resolutions of Weierstrass models
  7. Understanding $G$-models for orthogonal groups of small rank
  8. Spin($n$)-models versus SO($n$)-models
  9. Weierstrass models with Mordell--Weil group $\mathbb{Z}/2\mathbb{Z}$
  10. Canonical forms for SO($3$), SO($4$), SO($5$), and SO($6$)-models
  11. Summary of results
  12. The SO(3)-model
  13. Singularities of the Weierstrass model
  14. Crepant resolution
  15. Fiber geometry

Key Result

Theorem 2.2

Let $Z\subset X$ be the complete intersection of $d$ nonsingular hypersurfaces $Z_1$, …, $Z_d$ meeting transversally in $X$. Let $f: \widetilde{X}\longrightarrow X$ be the blowup of $X$ centered at $Z$. We denote the exceptional divisor of $f$ by $E$. The total Chern class of $\widetilde{X}$ is then

Figures (2)

  • Figure 1: Affine Dynkin diagrams corresponding to the respective Lie groups SO(3), SO(5), and SO(6). In the above graphs, the black node represents the affine node; deleting this node produces the corresponding finite Dynkin diagrams of type A$_1 \cong$ B$_1$, C$_2 \cong$ B$_2$ and A$_3 \cong$ D$_3$.
  • Figure 2:

Theorems & Definitions (12)

  • Definition 2.1: Weierstrass model
  • Theorem 2.2: Aluffi, Aluffi_CBU
  • Theorem 2.3: Esole--Jefferson--Kang, see Euler
  • Theorem 2.4: Esole--Jefferson--Kang, see Euler
  • Definition 2.6: Resolution of singularities
  • Definition 2.7: Crepant birational map
  • Definition 2.8: Hyperplane arrangement I$(\mathfrak{g},\mathbf{R})$
  • Theorem 2.9
  • Theorem 3.1
  • proof
  • ...and 2 more