Classifying bases for 6D F-theory models

TL;DR

This work provides a geometric classification of six-dimensional F-theory vacua by analyzing the divisor structure of the base surface. By cataloging non-Higgsable clusters (NHCs) formed from irreducible divisors with negative self-intersection and determining their minimal gauge algebras and matter via monodromy analysis, it shows that any maximally Higgsed 6D theory decomposes into a product of a fixed set of gauge-summand types. It further derives bounds on the number of tensor multiplets $T$ in terms of the NHC content and the base geometry, and demonstrates how linear chains of curves constrain the allowed bases; notably achieving a maximal example with $T=193$. While the results strongly constrain the landscape of 6D F-theory models, they leave open whether all potential Higgsings are realizable in every base, and whether non-F-theory 6D models could realize spectra beyond this geometric classification. The framework advances systematic classification and provides tools for enumerating bases and their allowed gauge content in 6D F-theory compactifications.

Abstract

We classify six-dimensional F-theory compactifications in terms of simple features of the divisor structure of the base surface of the elliptic fibration. This structure controls the minimal spectrum of the theory. We determine all irreducible configurations of divisors ("clusters") that are required to carry nonabelian gauge group factors based on the intersections of the divisors with one another and with the canonical class of the base. All 6D F-theory models are built from combinations of these irreducible configurations. Physically, this geometric structure characterizes the gauge algebra and matter that can remain in a 6D theory after maximal Higgsing. These results suggest that all 6D supergravity theories realized in F-theory have a maximally Higgsed phase in which the gauge algebra is built out of summands of the types su(3), so(8), f_4, e_6, e_8, e_7, (g_2 + su(2)), and su(2) + so(7) + su(2), with minimal matter content charged only under the last three types of summands, corresponding to the non-Higgsable cluster types identified through F-theory geometry. Although we have identified all such geometric clusters, we have not proven that there cannot be an obstruction to Higgsing to the minimal gauge and matter configuration for any possible F-theory model. We also identify bounds on the number of tensor fields allowed in a theory with any fixed gauge algebra; we use this to bound the size of the gauge group (or algebra) in a simple class of F-theory bases.

Figures (2)

• Figure 1: All possible clusters of intersecting curves with self-intersection of each curve $-2$ or below. For each cluster the corresponding gauge algebra is noted and the gauge algebra and matter content are listed in Table \ref{['f:primitives']}
• Figure 2: Periodic linear chains of divisors with simple gauge algebras. Chains shown are all those that satisfy a local maximality condition on the self-intersection numbers of the divisors. Bounds on the number of tensors given the gauge group place a limit on the size of such chains.