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# Advances in Theoretical and Mathematical Physics

## Volume 24 (2020)

### Number 2

### $SU(n) \times \mathbb{Z}_2$ in F-theory on K3 surfaces without section as double covers of Halphen surfaces

Pages: 459 – 490

DOI: https://dx.doi.org/10.4310/ATMP.2020.v24.n2.a5

#### Author

#### Abstract

We investigate F-theory models with a discrete $\mathbb{Z}_2$ gauge symmetry and $SU(n)$ gauge symmetries. We utilize a class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index $2$, to yield genus-one fibered K3 surfaces with a bisection, but lacking a global section. We consider F‑theory compactifications on these K3 surfaces times a K3 surface to build such models. We construct Halphen surfaces of index $2$ with type $I_n$ fibers, and we take double covers of these surfaces to obtain K3 surfaces without a section with two type $I_n$ fibers, and K3 surfaces without a section with a type $I_{2n}$ fiber. We study these models to advance the understanding of gauge groups that form in F‑theory compactifications on the moduli of bisection geometries.

Our results also show that the Halphen surfaces of index $2$ can have type $I_n$ fibers up to $I_9$. We construct an example of such a surface and determine the complex structure of the Jacobian of this surface. This allows us to precisely determine the non-Abelian gauge groups that arise in F‑theory compactifications on genusone fibered K3 surfaces obtained as double covers of this Halphen surface of index $2$, with a type $I_9$ fiber times a K3 surface. We also determine the $U(1)$ gauge symmetries for compactifications when K3 surfaces as double covers of Halphen surfaces with type $I_9$ fiber are ramified over a smooth fiber.

This work is partially supported by Grant-in-Aid for Scientific Research #16K05337 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

Published 10 July 2020