The Geometry of G$_2$, Spin(7), and Spin(8)-models

TL;DR

The paper develops a rigorous geometric framework for Weierstrass models arising from Step 6 of Tate's algorithm with an $I_0^*$-fiber over a divisor $S$, yielding $G_2$, Spin$(7)$, and Spin$(8)$ gauge theories in F-theory/M-theory. It provides canonical forms, proves the existence (where possible) of crepant resolutions, and maps the network of flops to the Coulomb branches via a hyperplane arrangement $I(rak g,f R)$, while computing triple intersection numbers to read off Chern–Simons terms and matter content. The authors determine representations from fibral divisors, count 5D hypermultiplets through the IMS prepotential, and verify 6D anomaly cancellations (including global$G_2$ anomalies), establishing a consistent geometry–gauge correspondence. The results deliver explicit resolutions and a detailed dictionary linking the geometry of elliptic fibrations to the physics of 5D and 6D theories for these exceptional groups, enabling precise gauge-theory engineering in string theory contexts.

Abstract

We study the geometry of elliptic fibrations given by Weierstrass models resulting from Step 6 of Tate's algorithm. Such elliptic fibrations have a discriminant locus containing an irreducible component $S$, over which the generic fiber is of Kodaira type I$^*_0$. In string geometry, these geometries are used to geometrically engineer G$_2$, Spin($7$), and Spin($8$) gauge theories. We give sufficient conditions for the existence of crepant resolutions. When they exist, we give a complete description of all crepant resolutions and show explicitly how the network of flops matches the Coulomb branch of the associated gauge theories. We also compute the triple intersection numbers in each chamber. Physically, they correspond to the Chern-Simons levels of the gauge theory and depend on the choice of a Coulomb branch. We determine the representations associated with these elliptic fibrations by computing intersection numbers with fibral divisors and then interpreting them as weights of a representation. For a five-dimensional gauge theory, we compute the number of hypermultiplets in each representation by matching the triple intersection numbers with the superpotential of the theory. We also discuss anomaly cancellations of a six-dimensional supergravity theory obtained by a compactification of F-theory on an elliptically fibered Calabi--Yau threefold corresponding to a G$_2$, Spin($7$), or Spin($8$) gauge theory.

Figures (7)

• Figure 1: Non-Kodaira fibers appearing in the fiber structures of G$_2$, Spin($7$), and Spin($8$)-models
• Figure 2: Chambers of the hyperplane arrangement I$(\text{G$_2$},\mathbf{7})$ or equivalently, the Coulomb phases of a G$_2$ gauge theory with matter in the representation $\mathbf{7}$. There is a unique chamber since the non-zero weights of $\mathbf{7}$ are the short roots of G$_2$.
• Figure 5: Geometric fiber degeneration of $\widetilde{G}_2^{S_3}$-model with valuation $(2,3,6)$.
• Figure 8: Geometric fiber degeneration of a Spin($7$)-model with $v_{S}(a_2)=1$ and $v_{S}(a_6)=4$. When $v_{S}(a_2)>1$, the degeneration graph contracts to the middle row. When there are multiple fibers, the one on the left corresponds to the monomial resolution and the one on the right to its flop.
• Figure 9: Geometric fiber degeneration of a Spin($7$)-model with $v_{S}(a_2)=1$ and $v_{S}(a_6)\geq 5$.

Theorems & Definitions (51)

• Definition 2.1: Fiber type
• Definition 2.2: Dual graph
• Definition 2.3: $\mathcal{K}$-model
• Definition 2.4: $G$-model
• Definition 2.5: Saturated set of weights
• Definition 2.6: Saturation of a subset
• Proposition 2.7
• proof
• Theorem 2.8
• proof