Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 8

$\tfrac{1}{2}$Calabi–Yau $4$-folds and four-dimensional F-theory on Calabi–Yau $4$-folds with $\mathrm{U}(1)$ factors

Pages: 2119 – 2140



Yusuke Kimura (KEK Theory Center, Institute of Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki, Japan)


In this study, four dimensional $N=1$ F‑theory models with multiple $\mathrm{U}(1)$ gauge group factors are constructed. A class of rational elliptic 4‑folds, which we call as “$\tfrac{1}{2}$Calabi–Yau 4‑folds,” is introduced, and we construct the elliptically fibered 4‑folds by utilizing them. This yields a novel approach for building families of elliptically fibered Calabi–Yau 4‑folds with positive Mordell–Weil ranks. The introduced $\tfrac{1}{2}$Calabi–Yau 4‑folds possess the characteristic property wherein the sum of the ranks of the singularity type and the Mordell–Weil group is always equal to six. This interesting property enables us to construct the elliptically fibered Calabi–Yau 4‑folds of various positive Mordell–Weil ranks. From one to six $\mathrm{U}(1)$ factors form in four dimensional F‑theory on the resulting Calabi–Yau 4‑folds. We also propose the geometric condition on the base 3‑fold of the built Calabi–Yau 4‑folds that allows four dimensional F‑theory models that have heterotic duals to be distinguished from those that do not.

Published 14 September 2022