Root bundles: Applications to F-theory Standard Models

Abstract

The study of vector-like spectra in 4-dimensional F-theory compactifications involves root bundles, which are important for understanding the Quadrillion F-theory Standard Models (F-theory QSMs) and their potential implications in physics. Recent studies focused on a superset of physical root bundles whose cohomologies encode the vector-like spectra for certain matter representations. It was found that more than 99.995% of the roots in this superset for the family B 3 (∆ • 4 ) of O(10 11 ) different F-theory QSM geometries had no vector-like exotics, indicating that this scenario is highly likely.To study the vector-like spectra, the matter curves in the F-theory QSMs were analyzed. It was found that each of them can be deformed to nodal curve that is identical across all spaces in B 3 (∆ • ). Therefore, from studying a few nodal curves, one can probe the vector-like spectra of a large fraction of F-theory QSMs. To this end, the cohomologies of all limit roots were determined, with line bundle cohomology on rational nodal curves playing a major role. A computer algorithm was used to enumerate all limit roots and analyze the global sections of all tree-like limit roots. For the remaining circuit-like limit roots, the global sections were manually determined. These results were organized into tables, which represent -to the best knowledge of the authorthe first arithmetic steps towards Brill-Noether theory of limit roots.

Towards natural and realistic E7 GUTs in F-theory

“…If the Higgs wavefunction is nearly uniform on C 56 , the Higgs will interact with all three generations of chiral matter in the same way, thus the Yukawa couplings will be undesiredly close to an identity matrix. 14 Nevertheless, especially with the presence of bulk fluxes, we expect the Higgs wavefunction to be non-uniform and peak in some smaller region. A simple scenario would be that the region intersects with C 56 in a connected small but finite range.…”

“…The other two copies of singlets have nontrivial multiplicities but belong to inert matter. 14 We thank Jonathan Heckman for pointing this out.…”

“…Among the above SM features, string theory techniques for constructing the gauge group and the chiral spectrum are well-established. While there is some recent progress on the Higgs sector [11][12][13][14][15] and the Yukawa couplings [16] in a large class of F-theory models, so far no fully precise statement on the realization of these features in a way that matches observed physics has been made in this context. On the other hand, incorporating these established features with e.g.…”

“…One of the main reasons for this is that in a realistic model we need to realize the Higgs sector, while avoiding dangerous vector-like exotics. While the techniques for computing the chiral spectrum are already at hand as above, computing the vector-like spectrum in general requires not only the G 4 flux, but the full information of C 3 in terms of line bundles on Σ and C. 5 In many cases, these things are hard to compute and some of the relevant technology has only been developed fairly recently [11][12][13][14]. Fortunately, our models admit several important simplifications such that the G 4 flux itself already determines the vector-like spectrum.…”

“…(2.30), which are in general very hard to compute since they are moduli-dependent quantities. More exotic tools like root bundles [11][12][13][14][15] may also be needed in the construction. In many cases, there is no Higgs in the low-energy theory unless some further tuning is done.…”