Enhanced gauge symmetry in 6D F‐theory models and tuned elliptic Calabi‐Yau threefolds

Abstract

We systematically analyze the local combinations of gauge groups and matter that can arise in 6D F-theory models over a fixed base. We compare the low-energy constraints of anomaly cancellation to explicit F-theory constructions using Weierstrass and Tate forms, and identify some new local structures in the "swampland" of 6D supergravity and SCFT models that appear consistent from low-energy considerations but do not have known F-theory realizations. In particular, we classify and carry out a local analysis of all enhancements of the irreducible gauge and matter contributions from "non-Higgsable clusters," and on isolated curves and pairs of intersecting rational curves of arbitrary self-intersection. Such enhancements correspond physically to unHiggsings, and mathematically to tunings of the Weierstrass model of an elliptic CY threefold. We determine the shift in Hodge numbers of the elliptic threefold associated with each enhancement. We also consider local tunings on curves that have higher genus or intersect multiple other curves, codimension two tunings that give transitions in the F-theory matter content, tunings of abelian factors in the gauge group, and generalizations of the "E 8 " rule to include tunings and curves of self-intersection zero. These tools can be combined into an algorithm that in principle enables a finite and systematic classification of all elliptic CY threefolds and corresponding 6D F-theory SUGRA models over a given compact base (modulo some technical caveats in various special circumstances), and are also relevant to the classification of 6D SCFT's. To illustrate the utility of these results, we identify some large example classes of known CY threefolds in the Kreuzer-Skarke database as Weierstrass models over complex surface bases with specific simple tunings, and we survey the range of tunings possible over one specific base.arXiv:1605.08052v2 [hep-th]

Top down approach to 6D SCFTs

“…Similarly, the theory with Dirac pairing appears to be consistent from a bottom up perspective, but so far an explicit F-theory construction has not been found [83]. The same is true for a theory with this Dirac pairing but gauge algebra su(10) ⊕ su(3) and charged hypermultiplets (10, 3) ⊕ 17(10, 1) ⊕ 2(1, 3), (3.31) In these cases, it is not yet clear whether this discrepancy represents a failure of the top down perspective or the bottom up perspective.…”

On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds

“…A direct analysis of the related structure of K3 fibrations for many of the toric hypersurface constructions in the Kreuzer-Skarke database was carried out in [16], demonstrating directly the prevalence of fibrations by smaller-dimensional Calabi-Yau fibers among known Calabi-Yau threefolds. The study of F-theory has led to a systematic methodology for constructing and classifying elliptic Calabi-Yau threefolds [17][18][19][20][21][22]. Comparing the structure of geometries constructed in this way to the Kreuzer-Skarke database shows that at large Hodge numbers, virtually all Calabi-Yau threefolds that are known are in fact elliptic.…”

Fibration structure in toric hypersurface Calabi-Yau threefolds

“…In particular, with the standard ordering of the 16 fiber polytopes (used in e.g. [27] and [23]), condition I holds for fibers 2,5,7,8,9,11,12,13,14,15,16; of the fibers where condition I fails, condition II holds for fibers 1, 3, 6, 10; and condition III holds for the sole remaining case, fiber 4. By simply checking these conditions for the S i for each polytope, we can ascertain whether or not there is a reflexive 2D subpolytope ∇ 2 of ∇.…”

“…Using a combination of methods and insights from mathematics and physics, and building on Grassi's results for minimal models of the bases that support elliptic fibrations [7], we have a good global understanding of the connected space of elliptic Calabi-Yau threefolds, the bounds on their Hodge numbers, and a systematic approach to constructing such threefolds that can be used to essentially enumerate all elliptic Calabi-Yau threefolds at large h 2,1 , though there are technical issues that make a complete enumeration at smaller values of h 2,1 currently out of reach (see e.g. [8][9][10][11][12][13]). There is a growing body of evidence that most known Calabi-Yau threefolds are in fact elliptic or genus one fibered (at least up to birational equivalence).…”