The Tate Form on Steroids: Resolution and Higher Codimension Fibers

TL;DR

This work delivers a comprehensive crepant resolution of singular Tate models for ADE-type elliptic Calabi–Yau fourfolds and maps the resulting higher-codimension fiber structures to physical data in F-theory. By resolving codimension-1 singularities and tracking fiber splitting through codimension-2 and codimension-3 loci, the authors identify matter representations as weights of ADE groups and extract the corresponding Yukawa couplings, while carefully distinguishing minimal (Kodaira-type) from non-minimal, non-flat fibers. The analysis encompasses $A_n$, $D_n$, and $E_n$ series, including non-minimal loci where surface components appear, offering insight into how extra light states (e.g., M5-brane modes) may arise beyond a pure gauge-theoretic description. The resolved geometries provide a solid foundation for constructing $G_4$-fluxes and for exploring Higgsing patterns and matter couplings in four-dimensional F-theory compactifications, with direct relevance to GUT-like models and beyond. Overall, the paper clarifies how higher-codimension geometry encodes gauge, matter, and Yukawa structures in a rigorous, geometry-driven framework.

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.

Figures (22)

• Figure 1: Intersection graph of the fibers in codimension 1, resulting in the affine $A_{2k}$ Dynkin diagram. The labels denote the sections which give rise to the Cartan divisors $D_{-\alpha_i}$ as is spelled out in (\ref{['SUCartanDivs']}).
• Figure 2: Splitting of the Cartan divisors $D_{-\alpha_i}$ along the codimension 2 locus $b_1=0$ for $SU(2k+1)$. $\color{green}\bullet$ are the simple roots of $A_{2k}$, green lines indicate how they split along $b_1=0$. $\color{blue}\bullet$ are the irreducible components of the fiber over the codimension 2 locus, the blue lines give their intersection graph, which reproduces the affine $D_{2k+1}$. Curves that remain irreducible are bicolored.
• Figure 3: Codimension 2 splitting for $SU(2k+1)$ along $P=0$. Bicolored nodes correspond to Cartan divisors that remain irreducible when passing to $P=0$. The only one that splits is $\delta_k$, which splits according to the green lines. The irreducible components of the fibers are blue and their intersections are given by the blue lines. Bicolored nodes remain irreducible.
• Figure 4: Codimension 3 splitting along $b_1=b_3=0$ of the $b_1=0$ codimension 2 locus for $SU(2k+1)$. The only node that splits is $\alpha_{k+1}$, which has three components $u_{2k}^{(1,2)}$ and $v_k$. The irreducible components in codimension 3 are $\color{red}\bullet$ or bi-colored, and the red lines indiate the intersections of these. Bicolored nodes remain irreducible.
• Figure 5: Summary graph: splitting along $b_1=b_3=0$ for $SU(2k+1)$. $\color{green}\bullet$ are initial Cartan divisors, green lines indicate how they split in codimesion 2 along $b_1=0$ into $\color{blue}\bullet$. Blue lines indicate the splitting of these when passing to $b_3=0$. $\color{red}\bullet$ are the irreducible components in codimension 3, and red lines indicate their intersections.