48 Crepant Paths to $\text{SU}(2)\!\times\!\text{SU}(3)$

TL;DR

This work geometrically engineers the non-Abelian sector of the Standard Model via elliptic fibrations with gauge group $\text{SU}(2)\times\text{SU}(3)$, identifying six collision types that admit crepant resolutions and assembling a network of 48 resolved fibrations connected by deformations and flops. It provides explicit topological data including Euler characteristics, Hodge numbers, and triple-intersection polynomials for each resolution, and establishes a deep link between the geometry (via hyperplane arrangements and flop graphs) and the 5d Coulomb-branch physics as well as 6d anomaly cancellation in F-theory. The paper then 6d–5d consistency is checked through 5d prepotentials, with a unique, anomaly-free matter content fixed by the Green–Schwarz mechanism and global anomaly constraints, and the results consistently uplift to a 6d theory. Overall, the work delivers a comprehensive geometric and physical accounting of SU(2)×SU(3) models, including non-Kodaira fibers and a precise mapping between crepant resolutions, hyperplane chambers, and Coulomb phases, with implications for stringy realizations of SM-like gauge sectors.

Abstract

We study crepant resolutions of Weierstrass models of $\text{SU}(2)\!\times\!\text{SU}(3)$-models, whose gauge group describes the non-abelian sector of the Standard Model. The $\text{SU}(2)\!\times\!\text{SU}(3)$-models are elliptic fibrations characterized by the collision of two Kodaira fibers with dual graphs that are affine Dynkin diagrams of type $\widetilde{\text{A}}_1$ and $\widetilde{\text{A}}_2$. Once we eliminate those collisions that do not have crepant resolutions, we are left with six distinct collisions that are related to each other by deformations. Each of these six collisions has eight distinct crepant resolutions whose flop diagram is a hexagon with two legs attached to two adjacent nodes. Hence, we consider 48 distinct resolutions that are connected to each other by deformations and flops. We determine topological invariants---such as Euler characteristics, Hodge numbers, and triple intersections of fibral divisors---for each of the crepant resolutions. We analyze the physics of these fibrations when used as compactifications of M-theory and F-theory on Calabi--Yau threefolds yielding 5d ${\mathcal N}=1$ and 6d ${\mathcal N}=(1,0)$ supergravity theories respectively. We study the 5d prepotential in the Coulomb branch of the theory and check that the six-dimensional theory is anomaly-free and compatible with a 6d uplift from a 5d theory.

Figures (24)

• Figure 1: The six collisions of singularities that give Weierstrass models whose crepant resolutions are smooth $\text{SU}(2)\!\times\!\text{SU}(3)$-models. In this graph, each arrow indicates a specialization of a Weierstrass model to another. The only collision that correspond to a non-Higgsable model is $\text{III}+\text{IV}^{\text{s}}$.
• Figure 2: The chamber structure of I$(\text{E}_3=\text{A}_1\oplus\text{A}_2, (\mathbf{1},\mathbf{3})\oplus (\mathbf{2},\mathbf{3}))$ and its adjacency graph. This also represents the structure of the extended Kähler cone of an $\text{SU}(2)\!\times\!\text{SU}(3)$-model. See Figure \ref{['2DChambersMatch']} for more details on the walls separating the chambers.
• Figure 3: Left: the complete structure of the resolutions of $\text{SU}(2)\!\times\!\text{SU}(3)$. This is a two-dimensional patch of the entire three-dimensional cones. Hence, every point (resp. line) on this picture represents a line (resp. surface). Accordingly, these eight triangles are the three-dimensional triangular cones. The point in the top of the triangle is the resolution that resolves the SU(2), and the bottom-middle point of the triangle is the first resolution that resolves SU(3) only. The point in the middle of the triangle is the point that describes the blow-up that mixes both SU($2$) and SU($3$). All these three points are connected as expected and it works as the plane of the mirrors. Right: the adjacent graph of the eight chambers of $\text{I}(\mathfrak{g},\bold{R})$ with $\mathfrak{g}=\text{A}_1\oplus \text{A}_2$ and $\bold{R}=(\bold{1},\bold{3})\oplus (\bold{2},\bold{3})$.
• Figure 4: Fiber over the generic point of the locus $S\cap T$ in Resolution I of the I$_2^{\text{s}}$+I$_3^{\text{s}}$-model.
• Figure 5: Fiber over the generic point of the locus $S\cap T\cap V(a_1)$ in Resolution I of the I$_2^{\text{s}}$+I$_3^{\text{s}}$-model.

Theorems & Definitions (19)

• Remark 2.1
• Theorem 3.1: Aluffi, Aluffi_CBU
• Theorem 3.2: Esole--Jefferson--Kang, see Euler
• Theorem 3.3: Esole--Jefferson--Kang, see Euler and AE1AE2EKYFullwood:SVW
• Remark 3.4
• Theorem 3.5
• proof
• Lemma 3.6
• Lemma 3.7
• Lemma 3.8