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

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${}_0^{\ast }$. 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.

arXiv commentV2:Typos corrected, references added, 52 pages + references, 7 tables, 13 figures

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Cited by in corpus (13)

• Infinitely many 4d $\mathcal{N}=2$ SCFTs with $a=c$ and beyond
• Flopping and Slicing: SO(4) and Spin(4)-models
• Mordell-Weil Torsion, Anomalies, and Phase Transitions
• Symmetry enhancement in 4d Spin(n) gauge theories and compactification from 6d
• Singular Geometry and Higgs Bundles in String Theory
• The Geometry of SO(3), SO(5), and SO(6) models
• Infinitely many 4d N=1 SCFTs with a=c
• Characteristic numbers of elliptic fibrations with non-trivial Mordell-Weil groups
• Characteristic numbers of crepant resolutions of Weierstrass models
• 48 Crepant Paths to $\text{SU}(2)\times \text{SU}(3)$
• The suspended pinch point and SU($2$)$\times$U($1$) gauge theories
• Matter representations from geometry: under the spell of Dynkin
• Flops and Fibral Geometry of E${}_7$-models