The Tate Form on Steroids: Resolution and Higher Codimension Fibers

TL;DR

This paper constructs and analyzes the resolution of singular Tate forms in elliptic Calabi-Yau four-folds within F-theory, elucidating fiber structures in higher codimension and their relation to gauge theories, matter, and Yukawa couplings.

Contribution

It provides a detailed resolution of singular Tate forms and characterizes higher codimension fibers, including non-minimal singularities, relevant for gauge theories in F-theory.

Findings

Fibers in higher codimension are of Kodaira type along minimal singular loci.

Irreducible fiber components correspond to weights of ADE gauge group representations.

Non-minimal singularities exhibit non-Kodaira fibers, but their splittings align with matter and Yukawa couplings.

Abstract

F-theory on singular elliptically fibered Calabi-Yau four-folds provides a setting to geometrically study four-dimensional N=1 supersymmetric gauge theories, including matter and Yukawa couplings. The gauge degrees of freedom arise from the codimension 1 singular loci, the matter and Yukawa couplings are generated at enhanced singularities in higher codimension. We construct the resolution of the singular Tate form for an elliptic Calabi-Yau four-fold with an ADE type singularity in codimension 1 and study the structure of the fibers in codimension 2 and 3. We determine the fibers in higher codimension which in general are of Kodaira type along minimal singular loci, and are thus consistent with the low energy gauge-theoretic intuition. Furthermore, we provide a complementary description of the fibers in higher codimension, which will also be applicable to non-minimal singularities. The irreducible components in the fiber in codimension 2 correspond to weights of representations of the ADE gauge group. These can split further in codimension 3 in a way that is consistent with the generation of Yukawa couplings. Applying this reasoning, we then venture out to study non-minimal singularities, which occur for A type along codimension 3, and for D and E also in codimension 2. The fibers in this case are non-Kodaira, however some insight into these singularities can be gained by considering the splitting of fiber components along higher codimension, which are shown to be consistent with matter and Yukawa couplings for the corresponding gauge groups.