[Paper Review] Non-Higgsable abelian gauge symmetry and F-theory on fiber products of rational elliptic surfaces

David R. Morrison, Daniel S. Park+1 more|arXiv (Cornell University)|Oct 21, 2016

Black Holes and Theoretical Physics66 references22 citations

TL;DR

This paper constructs a broad class of Calabi–Yau threefolds via fiber products of rational elliptic surfaces with section, systematically classifying combinations of Kodaira fiber types that yield at worst canonical singularities. The resulting spaces support elliptic fibrations with non-Higgsable abelian gauge groups in F-theory compactifications, providing explicit examples where abelian gauge symmetries cannot be Higgsed to nonabelian ones without introducing unresolvable physical singularities.

ABSTRACT

We construct a general class of Calabi--Yau threefolds from fiber products of rational elliptic surfaces with section, generalizing a construction of Schoen to include all Kodaira fiber types. The resulting threefolds each have two elliptic fibrations with section over rational elliptic surfaces and blowups thereof. These elliptic fibrations generally have nonzero Mordell--Weil rank. Each of the elliptic fibrations has a physical interpretation in terms of a six-dimensional F-theory model with one or more non-Higgsable abelian gauge fields. Many of the models in this class have mild singularities that do not admit a Calabi--Yau resolution; this does not seem to compromise the physical integrity of the theory and can be associated in some cases with massless hypermultiplets localized at the singular loci. In some of these constructions, however, we find examples of abelian gauge fields that cannot be "unHiggsed" to a nonabelian gauge field without producing unphysical singularities that cannot be resolved. The models studied here can also be used to exhibit T-duality for a class of little string theories.

Motivation & Objective

• To generalize Schoen's construction of Calabi–Yau threefolds via fiber products of rational elliptic surfaces to include all Kodaira fiber types.
• To classify pairs of fiber types that yield Calabi–Yau threefolds with at worst canonical singularities in the total space.
• To identify physical F-theory models with non-Higgsable abelian gauge fields arising from elliptic fibrations with nonzero Mordell–Weil rank.
• To explore the physical consistency of mild singularities that do not admit Calabi–Yau resolutions, particularly in relation to massless hypermultiplets.
• To identify cases where abelian gauge fields cannot be 'unHiggsed' to nonabelian gauge fields without introducing unphysical singularities.

Proposed method

• Construct Calabi–Yau threefolds as fiber products of two rational elliptic surfaces with section, allowing all Kodaira fiber types in the components.
• Apply Miranda and Grassi's results on partial resolutions via blowups of the base to achieve flat elliptic fibrations with trivial canonical bundle.
• Use Persson’s classification of singular fiber combinations in rational elliptic surfaces to systematically enumerate viable fiber product configurations.
• Analyze the Mordell–Weil group of the resulting fibrations to identify abelian gauge symmetries in F-theory compactifications.
• Assess the physical viability of singularities by checking whether they can be resolved while preserving the Calabi–Yau condition and physical consistency.
• Compare the constructed models with known F-theory backgrounds, including those with Mordell–Weil rank 1–8, and identify them as special cases of the general construction.

Experimental results

Research questions

• RQ1 Which combinations of Kodaira fiber types in the component rational elliptic surfaces yield Calabi–Yau threefolds with at worst canonical singularities under fiber product construction?
• RQ2 Can non-Higgsable abelian gauge symmetries in F-theory be systematically realized through fiber products of rational elliptic surfaces with section?
• RQ3 Do mild singularities in the total space of the fiber product compromise the physical consistency of the resulting F-theory model?
• RQ4 Are there abelian gauge fields in this construction that cannot be 'unHiggsed' to nonabelian gauge fields without introducing unresolvable singularities?
• RQ5 How do these constructions relate to known F-theory models with nonzero Mordell–Weil rank, particularly the 13 exceptional bases from Martini-WT?

Key findings

• The construction yields 13 specific models that match all 13 known bases with nonzero Mordell–Weil rank in F-theory compactifications, confirming their universality within this class.
• For the model with r=8, N=4, n₀=-2, n∞=-2, the Hodge numbers are h¹¹=19 and h²¹=19, indicating a highly nontrivial geometry with maximal Mordell–Weil rank.
• In the case r=6, N=4, n₀=-2, n∞=-6, the Hodge numbers are h¹¹=35 and h²¹=11, showing a complex structure with significant topological richness.
• The model with r=4, N=4, n₀=-6, n∞=-6 has h¹¹=51 and h²¹=3, demonstrating that high Mordell–Weil rank can coexist with low Hodge number for h²¹.
• For r=2, N=3, n₀=-5, n∞=-6, the Hodge numbers h¹¹=61 and h²¹=1 indicate a very high complex structure moduli space dimension relative to Kähler moduli.
• The paper identifies cases—such as r=2, N=3, n₀=-1, n∞=-2—where abelian gauge fields cannot be unHiggsed to nonabelian ones without introducing singularities that cannot be resolved while preserving the Calabi–Yau condition.