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

TL;DR

This paper explores the geometry of specific elliptic fibrations used in string theory to engineer G2, Spin(7), and Spin(8) gauge theories, detailing resolutions, intersection numbers, and physical implications.

Contribution

It provides a complete classification of crepant resolutions for these fibrations, links geometric flops to gauge theory Coulomb branches, and computes physical data like Chern-Simons levels and matter content.

Findings

Classified all crepant resolutions and their flop networks.

Connected geometric flops to Coulomb branch structures.

Calculated intersection numbers corresponding to physical parameters.

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.