Gauge symmetries and matter fields in $\mathrm{F}$-theory models without section — compactifications on double cover and Fermat quartic $\mathrm{K}3$ constructions times $\mathrm{K}3$

We investigate gauge theories and matter fields in $\mathrm{F}$-theory compactifications on genus-one fibered Calabi–Yau $4$-folds without a global section. In this study, genus-one fibered Calabi–Yau $4$-folds are built as direct products of a genus-one fibered $\mathrm{K}3$ surface that lacks a section times a $\mathrm{K}3$ surface. We consider i) double covers of $\mathbb{P}^1 \times \mathbb{P}^1$ ramified along a bidegree $(4,4)$ curve, and ii) complete intersections of two bidegree $(1,2)$ hypersurfaces in $\mathbb{P}^1 \times \mathbb{P}^3$ to construct genus-one fibered $\mathrm{K}3$ surfaces without a section. $E_7$ gauge group arises in some $\mathrm{F}$-theory compactifications on double covers times $\mathrm{K}3$. We show that the tadpole can be cancelled for an $\mathrm{F}$-theory compactification on complete intersection $\mathrm{K}3$ times $\mathrm{K}3$, when complete intersection $\mathrm{K}3$ is isomorphic to the Fermat quartic, and the complex structure of the other $\mathrm{K}3$ surface in the direct product is appropriately chosen.

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This work is supported by Grant-in-Aid for JSPS Fellows No. 26•2616.

Published 29 March 2018