The F-Theory QSMs

The Quadrillion F-Theory Standard Models (QSMs) were introduced in [CHLLT19] as a large class of compactifications within F-theory that exhibit promising features for particle physics model building. Due to their richness and physical relevance, these models have been extensively studied; see for example [Bie24] and the references therein.

The detailed geometric and physical properties of the QSMs are captured within FTheoryTools through the framework of Literature Models, enabling computations and explorations of their characteristics.

What are the QSM?

The QSMs correspond to a vast ensemble of F-theory compactifications constructed on elliptically fibered Calabi–Yau fourfolds with carefully chosen base geometries. Their defining feature is that they yield low-energy effective theories closely resembling the Standard Model of particle physics, including realistic gauge groups and chiral matter spectra.

The geometries arise from systematically classifying certain favorable base threefolds, often toric, and then carefully engineering elliptic fibrations that produce the Standard Model gauge group and chiral matter spectrum.

In FTheoryTools, we focus on the QSMs built over toric base spaces. It turned out that there are exactly 708 families of such QSM geometries. Each family corresponds to the fine, regular, star triangulation of one reflexive 3-dimensional polytope. Since the number of such triangulations can be astronomical — more specifically, the largest such number is a quadrillion ($10^{15}$), which gives this family of F-theory models its name — in FTheoryTools one such triangulation is chosen. In other words, we could read this as a chosen representative for each family, that can be constructed via the framework of Literature Models.

Constructing QSMs

The construction of QSMs happens via the literature model constructor. Here is an example:

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

The value assigned to the model parameter k must be altered to construct different QSMs. Recall that there are exactly 708 distinct classes of QSMs, each family corresponding to the fine, regular, star triangulations of the k-th reflexive polytope in the Kreuzer-Skarke list (cf. [KS98]).

Accordingly, k corresponds exactly to the index of the polytopes as they appear in the Kreuzer-Skarke list. The "first" QSM is associated with the 4th polytope, and the last one with index k = 4291. Between these, 706 other values correspond to QSMs. If a given k does not correspond to a QSM, the constructor raises an error.

It is worth recalling that this classifcation of base spaces is rooted in the triple intersection number of the anticanonical class of the base space. This triple intersection number can be accessed with FTheoryTools, see Base Intersection Numbers.

The Underlying Polytope

Each QSM corresponds to a family of models associated with the fine regular star triangulations of a reflexive 3-dimensional polytope. The following methods provide detailed information about the underlying polytope and its triangulations.

Vertices

Return the vertices of the underlying polytope. Vertices are normalized to rational numbers following the Polymake standard.

vertices — Method

vertices(m::AbstractFTheoryModel)

Return vertices of the F-theory model if known, otherwise throw an error.

See Underlying Polytope for more details.

Examples

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> vertices(qsm_model)
4-element Vector{Vector{QQFieldElem}}:
 [-1, -1, -1]
 [1, -1//2, -1//2]
 [-1, 2, -1]
 [-1, -1, 5]

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.