Non-minimal Elliptic Threefolds at Infinite Distance II: Asymptotic Physics

TL;DR

The paper investigates infinite-distance limits in the complex structure moduli of six-dimensional F-theory via non-minimal elliptic degenerations over Hirzebruch bases, focusing on horizontal log Calabi–Yau configurations. It classifies single infinite-distance, genus-zero limits into four Kulikov-like types II.a, II.b, III.a, III.b and analyzes their physical implications through adiabatic reductions, dual heterotic descriptions, and emergent string behavior. The results support Emergent String/Weak-Coupling pictures: horizontal II.a limits realize decompactifications with defects to ten dimensions, while II.b, III.a, and III.b produce partial or global weak-coupling endpoints with corresponding KK towers, strings, and defect algebras, consistently aligning with Swampland constraints. The work also develops detailed bounds on vertical gauge ranks and brane content, and connects the F-theory degenerations to heterotic dual pictures, providing evidence for a coherent higher-dimensional decompactification framework with defect sectors in the asymptotic regime.

Abstract

We interpret infinite-distance limits in the complex structure moduli space of F-theory compactifications to six dimensions in the light of general ideas in quantum gravity. The limits we focus on arise from non-minimal singularities in the elliptic fiber over curves in a Hirzebruch surface base, which do not admit a crepant resolution. Such degenerations take place along infinite directions in the non-perturbative brane moduli space in F-theory. A blow-up procedure, detailed generally in Part I of this project, gives rise to an internal space consisting of a union of log Calabi-Yau threefolds glued together along their boundaries. We geometrically classify the resulting configurations for genus-zero single infinite-distance limits. Special emphasis is put on the structure of singular fibers in codimension zero and one. As our main result, we interpret the central fiber of these degenerations as endpoints of a decompactification limit with six-dimensional defects. The conclusions rely on an adiabatic limit to gain information on the asymptotically massless states from the structure of vanishing cycles. We also compare our analysis to the heterotic dual description where available. Our findings are in agreement with general expectations from quantum gravity and provide further evidence for the Emergent String Conjecture.

Figures (9)

• Figure 1: Infinite-distance complex structure degenerations for F-theory on an elliptic K3 surface and their associated physics. Figure adapted from Lee:2021usk.
• Figure 2: The open-chain resolutions of horizontal models constructed over the Hirzebruch surfaces $\hat{B} = \mathbb{F}_{n}$, with $5 \leq n \leq 12$, present a non-Higgsable cluster over the $(-n)$-curve of the $Y^{P}$ component of the central fiber $Y^{0}$. It corresponds to an exceptional algebra $\mathfrak{e}_{m}$, with $m=6$, $7$ or $8$ depending on the value of $n$, see \ref{['tab:non-Higgsable-clusters']}. Forcing said component to be at weak coupling enhances the non-Higgsable cluster to be non-minimal. The resolution process demands then, possibly after a base change, at least a further base blow-up. In other words, the geometry prevents global weak coupling limits by shedding a new component at strong coupling.
• Figure 3: We schematically represent the base of the central fiber of a resolved horizontal model and some generic representatives of a subset of the global divisor classes discussed in \ref{['sec:horizontal-global-divisors']}. The depiction is based on models constructed over $\hat{B} = \mathbb{F}_{1}$.
• Figure 4: Tuning vertical algebras over representatives of the global divisor $\mathcal{F}$ leads to forced enhancements over the $S_{P}$ curve in the end-component. Eventually, the gauge factor supported on $S_{P}$ corresponds to $\mathrm{E}_{8}$, meaning that further forced factorizations of $S_{P}$ in $\Delta_{P}^{\prime}$ will make it non-minimal. If we try to exceed the vertical gauge rank by a higher tuning over the representatives of $\mathcal{F}$, the model sheds a new component in which the bound is still satisfied, rendering the tuning a local enhancement. As a consequence, the bound is still respected from the global point of view that is used to assign the gauge algebras.
• Figure 5: We schematically represent the base of the central fiber of a resolved horizontal Type II.a model constructed over $\hat{B} = \mathbb{F}_{n}$, with $n \geq 1$, on the left, and various vertical slices of it on the right. A reduced number of global 7-branes are depicted: two in the divisor class $\mathcal{H}_{\infty}^{0}$, one in $\mathcal{H}_{\infty}^{1}$, one in $\mathcal{H}_{0}^{1}$ and one in $\mathcal{F}$. The first, fourth and fifth vertical slices are generic and lead to Kulikov Type II.a models. This is not true for the second vertical slice, since it overlaps with the global 7-brane in the class $\mathcal{F}$ in both components, and for the third vertical slice, since it overlaps with the global 7-brane in the class $\mathcal{H}_{\infty}^{1}$ in the $B^{0}$ component.

Theorems & Definitions (1)

• Theorem E.1