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# Advances in Theoretical and Mathematical Physics

## Volume 23 (2019)

### Number 4

### Flopping and slicing: $\operatorname{SO}(4)$ and $\operatorname{Spin}(4)$-models

Pages: 1003 – 1066

DOI: https://dx.doi.org/10.4310/ATMP.2019.v23.n4.a2

#### Authors

#### Abstract

We study the geometric engineering of gauge theories with gauge group $\operatorname{Spin}(4)$ and $\operatorname{SO}(4)$ using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graphs A1. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell–Weil group of the elliptic fibration is required to be trivial for $\operatorname{Spin}(4)$ and $\mathbb{Z} / 2 \mathbb{Z}$ for $\operatorname{SO}(4)$.

Each of these models has two possible crepant resolutions connected by a flop. We also compute a generating function for the Euler characteristic of such elliptic fibrations over a base of arbitrary dimensions. In the case of a threefold, we also compute the triple intersection numbers of the fibral divisors. In the case of Calabi–Yau threefolds, we also compute their Hodge numbers and check the cancellations of anomalies in a six-dimensional supergravity theory.

M.E. is supported in part by the National Science Foundation (NSF) grant DMS-1701635 “Elliptic Fibrations and String Theory”. M.J.K. would like to acknowledge a partial support from NSF grant PHY-1352084.

Published 16 January 2020