Mordell-Weil Torsion, Anomalies, and Phase Transitions

Abstract

We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefolds with a I${}_2$+I${}_4$ collision of singularities. The resulting gauge theory has a semi-simple Lie algebra $\mathfrak{su}(2)\oplus \mathfrak{sp}(4)$ or $\mathfrak{su}(2)\oplus \mathfrak{su}(4)$. We compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. We determine the matter representation geometrically by computing weights via intersection of curves and fibral divisors. We fix the number of charged hypermultiplets transforming in each representations by comparing the triple intersection numbers and the one-loop prepotential. This condition is enough to fix the number of representation when the Mordell-Weil group is ${\mathbb{Z}}_2$ but not when it is trivial. The vanishing of the fourth power of the curvature forms in the anomaly polynomial is enough to fix the number of representations. We discuss anomaly cancellations of the six-dimensional uplifted. In particular, the gravitational anomaly is also considered as the Hodge numbers are computed explicitly without counting the degrees of freedom of the Weierstrass equation.

arXiv comment57 pages+ references, 13 figures, 15 tables

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• The Geometry of SO(3), SO(5), and SO(6) models
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• Infinitely many 4d N=1 SCFTs with a=c
• Unbroken $E_7\times E_7$ nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals
• Characteristic numbers of elliptic fibrations with non-trivial Mordell-Weil groups
• 48 Crepant Paths to $\text{SU}(2)\times \text{SU}(3)$
• F-theory models with 3 to 8 U(1) factors on K3 surfaces
• Characteristic numbers of crepant resolutions of Weierstrass models
• 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