Small resolutions of SU(5)-models in F-theory

TL;DR

The authors construct an explicit crepant, small resolution of a Weierstrass model with a split $\tilde{A}_4$ singularity to realize an $SU(5)$ GUT in F-theory, yielding a smooth elliptic fourfold with a split $I_5$ fiber over the SU(5) divisor. The remaining higher-codimension singularities are captured by a fourfold affine binomial geometry, which admits six distinct small resolutions connected by flop transitions forming a dihedral group, and these resolutions yield nuanced codimension-three fiber structures, including an exotic $E_6^{-}$ type and an ordinary $E_6$ type depending on the resolution chosen. The study elucidates that fiber enhancements in codimension three can occur without increasing the fiber rank, and some fibers may not belong to Kodaira or extended Dynkin diagrams, challenging naive extrapolations from Tate's algorithm and supporting a rich birational geometry landscape for F-theory SU(5) models. A no-go result, leveraging Batyrev’s invariance of Betti numbers, argues against certain conjectured fiber configurations within the same birational class, highlighting deep interplays between geometry and physics in these compactifications.

Abstract

We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered fourfolds defined by Weierstrass models admitting a split A_4 singularity over a divisor of the discriminant locus. Such varieties are used to geometrically engineer SU(5) Grand Unified Theories in F-theory. The desingularization is given by a small resolution of singularities. The I_5 fiber naturally appears after resolving the singularities in codimension-one in the base. The remaining higher codimension singularities are then beautifully described by a four dimensional affine binomial variety which leads to six different small resolutions of the the elliptically fibered fourfold. These six small resolutions define distinct fourfolds connected to each other by a network of flop transitions forming a dihedral group. The location of these exotic fibers in the base is mapped to conifold points of the threefolds that defines the type IIB orientifold limit of the F-theory. The full resolution have interesting properties, specially for fibers in codimension three: the rank of the singular fiber does not necessary increase and the fibers are not necessary in the list of Kodaira and some are not even (extended) Dynkin diagram.

Figures (7)

• Figure 1: Fiber of type $T_{p,q,r}$ and $T^-_{p,q,r}$ with $1\leq p\leq q\leq r$. The fiber of type $T^-_{p,q,r}$ is obtained from $T_{p,q,r}$ by replacing the central node by a point common to the three branches. A fiber of type $T_{p,q,r}$ has rank $p+q+r-2$, while a fiber of type $T^-_{p,q,r}$ has rank $p+q+r-3$.
• Figure 2: Conjectured singular fiber enhancements of a $\mathop{\rm SU}(5)$ GUT. Starting with codimension one in the base, the codimension increases from left to right. Thinking in terms of dual graphs, the rank of the associated Dynkin diagrams increases as we move in codimension.
• Figure 3: Degeneration of singular fibers of a small resolution of the $\mathop{\rm SU}(5)$ model. Starting with codimension-one in the base, the codimension increases from left to right. In comparaison with the conjecture fiber structure, there are no fibers of type $\tilde{A}_6$ and the fibers of type $\tilde{E}_6$ are replaced by (non-Kodaira) fibers of type $E_6$ or $\tilde{E}^-_6(:=T^-_{3,3,3})$ according to the choice of the resolution. We obtain six different resolutions, four of which have fibers of type $E_6$ in codimension-three while two have fibers of type $\tilde{E}^-_6$.
• Figure 4: Fiber degeneration of a small resolution of the $\mathop{\rm SU}(5)$ model. The node $C_0$ is the one that touches the section. The nodes $C_{1\pm}$ and $C_{2\pm}$ are coming from the resolution of the singularities over a generic point of the divisor $D_{\mathfrak{su}(5)}$. The remaining nodes $C_{x}$, $C_{w}$ and $C_t$ are obtained from the resolution of the higher codimensional singularities. We have 6 possible resolutions $\mathscr{E}_{xw}$, $\mathscr{E}_{wx}$, $\mathscr{E}_{xt}$, $\mathscr{E}_{tw}$ and $\mathscr{E}_{wt}$.
• Figure 5: A torus seen as the quotient $\mathbb{C}/ (\mathbb{Z}+\tau \mathbb{Z})$.

Theorems & Definitions (7)

• Proposition 1.1: Resolution in codimension-one
• Proposition 1.2: The binomial geometry
• Proposition 1.3: A network of small resolutions
• Proposition 1.4: Fiber enhancements in codimension-two
• Proposition 1.5: Fiber enhancements in codimension-three
• Proposition 1.6: Flop transitions
• Proposition 1.7: Birational invariance and a no-go theorem