Classifying bases for 6D F-theory models (1201.1943v3)

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.

• The paper presents a comprehensive classification framework for 6D F-theory models by examining the divisor structures underlying elliptic Calabi–Yau bases.
• It employs algebraic geometry to analyze intersecting divisors, identifying non-Higgsable clusters that determine minimal gauge groups and matter content.
• The findings establish tensor multiplet bounds that constrain supergravity model complexity, guiding future theoretical and AI-driven research.

Insights into 6D F-theory Model Classification

The paper, "Classifying bases for 6D F-theory models," by Morrison and Taylor, offers a structured classification framework for six-dimensional (6D) supergravity theories constructed via F-theory. This research focuses on elucidating the geometric features of the bases supporting elliptically fibered Calabi-Yau threefolds, which identify non-Higgsable clusters (NHCs) of gauge symmetries within these theories. The goal is to categorize viable 6D supergravity models by understanding the possible configurations of these foundational bases and the implications of their curvature and intersection properties on gauge symmetries and anomaly cancellations.

Overview of Methodology

Key to this classification process is the analysis of the divisor structure on the base surfaces. The authors employ algebraic geometry to investigate configurations of intersecting divisors, analyzing the intersection pairing and canonical class behavior. Through such analyses, various clusters of divisors, termed NHCs, are identified as essential elements of F-theory constructions. These clusters are characterized by constraints that ensure specific gauge algebras manifest in the theory's physical degrees of freedom. They explore configurations that support non-abelian gauge groups and pinpoint those not subject to further Higgsing in maximally Higgsed phases.

Detailed Findings

The research identifies several key clusters justified by F-theory bases:

• Single irreducible divisors provide minimal gauge groups like su(3), so(8), and more complex exceptions like e6, e7, and e8. These predominantly arise for divisors with negative self-intersections.
• More complex clusters, such as intersecting (−3) and (−2) curves, lead to gauge algebras encompassing g2 ⊕ su(2), su(2) ⊕ so(7) ⊕ su(2), and present specific matter content due to the non-trivial intersection properties.

These divisor classes lead to implications for the bound on tensor multiplets T in the associated supergravity models. For each identified NHC, there is a derived upper bound on T, establishing a finite limit to the overall complexity (as represented by tensor multiplets) of potential F-theory models.

Implications for Future AI and Theoretical Developments

On the theoretical front, this classification paves the way for a systematic exploration and potential enumeration of the entire space of 6D F-theory models, suggesting constraints on the configurations that can give rise to consistent, anomaly-free six-dimensional supergravity theories. The classification also informs potential mathematical advancements in understanding the geometry of elliptically fibered Calabi-Yau manifolds and their divisor structures.

From a practical standpoint, this framework aids researchers in identifying viable models more efficiently, cutting through the vast landscape of possible theories. Looking forward, these insights could deepen our understanding of how these models might manifest in lower-dimensional physical theories and pave the way toward formulating analogous frameworks for four-dimensional theories.

Conclusion

Morrison and Taylor’s systematic approach to classifying 6D F-theory models imparts a robust understanding of the underlying geometric constraints and their connection to gauge symmetries in these theories. By coupling algebraic geometry with quantum field theory principles, this work offers substantial strides towards a comprehensive classification of the space of 6D supergravity constructions, providing a blueprint for future explorations within the theoretical high-energy physics community.