$G_4$-Fluxes

In F-theory compactifications on Calabi–Yau 4-folds, $G_4$-fluxes are essential ingredients for defining consistent and physically meaningful vacua. They affect various physical aspects, such as the chiral spectrum in the 4D effective theory, tadpole cancellation, and moduli stabilization. This page provides an accessible introduction to working with $G_4$-fluxes in FTheoryTools, including background, construction methods, and available functionality.

Background: What Is a $G_4$-Flux?

Let $\widehat{Y}_4$ be a smooth Calabi–Yau 4-fold obtained via crepant resolution of a singular elliptically fibered 4-fold $Y_4 \to B_3$. A $G_4$-flux is a cohomology class:

\[G_4 \in H^{2, 2}(\widehat{Y}_4, \mathbb{R}) \equiv H^{2, 2}(\widehat{Y}_4, \mathbb{C}) \cap H^4(\widehat{Y}_4, \mathbb{R})\]

subject to quantization, transversality, and several additional consistency conditions:

Quantization (cf. Witten 1997): $G_4 + \frac{1}{2} c_2(\widehat{Y}_4) \in H^4(\widehat{Y}_4, \mathbb{Z})$
Transversality: The flux must be orthogonal to both the base and the fiber in a precise cohomological sense.
Additional constraints, such as D3-tadpole cancellation, self-duality, and primitivity, are also typically imposed. One often requires that the $G_4$-flux preserves the non-abelian gauge symmetry encoded in the fibration.

These conditions ensure compatibility with fundamental physical principles like anomaly cancellation and D3-brane charge quantization.

For further theoretical background, see the exposition in Weigand 2018.

Modeling $G_4$-Fluxes

In practice, FTheoryTools focuses on candidate $G_4$-fluxes: elements of the rational cohomology group $H^{2,2}(\widehat{Y}_4, \mathbb{Q})$, i.e.

\[G_4 \in H^{2, 2}(\widehat{Y}_4, \mathbb{Q}) \equiv H^{2, 2}(\widehat{Y}_4, \mathbb{C}) \cap H^4(\widehat{Y}_4, \mathbb{Q})\,.\]

We work with rational coefficients due to both the quantization condition–-which implies that valid fluxes must have rational coefficients–-and the practical advantages this offers in computation. Currently, FTheoryTools cannot fully verify the quantization condition, so we adopt the term candidate flux. We elaborate on our approach to the quantization condition below.

The space $H^{2,2}(\widehat{Y}_4, \mathbb{C})$ admits an orthogonal decomposition:

\[H^{2, 2}(\widehat{Y}_4, \mathbb{C}) = H^{2, 2}_\text{hor}(\widehat{Y}_4, \mathbb{C}) \oplus H^{2, 2}_\text{vert}(\widehat{Y}_4, \mathbb{C}) \oplus H^{2, 2}_\text{rem}(\widehat{Y}_4, \mathbb{C})\,.\]

Horizontal fluxes arise from variations of the complex structure,
Vertical fluxes are built from wedge products of (1,1)-forms,
Remainder fluxes account for the rest.

FTheoryTools focuses on the vertical part.

Standing Assumption

FTheoryTools assumes that $\widehat{Y}_4$ is defined as a hypersurface in a complete, smooth toric variety $X_\Sigma$. (In principle, it would suffice for the hypersurface to be smooth and the ambient space to be simplicial. However, we compute characteristic classes of the ambient space and restrict them to the hypersurface to obtain $c_2(\widehat{Y}_4)$, and this requires the ambient space to be smooth.)

Workflow

Under this assumption, we can access a subspace of the vertical cohomology:

\[\left. H^{2,2}(X_\Sigma, \mathbb{Q}) \right|_{\widehat{Y}_4} \subseteq H^{2,2}_{\text{vert}}(\widehat{Y}_4, \mathbb{Q}) \,.\]

We refer to elements in this subspace as ambient vertical fluxes.

Theorem 9.3.2 of Cox, Little, Schenck 2011 states that for complete, simplicial toric varieties, $H^{p, q}(X_\Sigma, \mathbb{Q}) = 0$ whenever $p \ne q$. Consequently, $H^{2, 2}(X_\Sigma, \mathbb{Q}) = H^4(X_\Sigma, \mathbb{Q})$. Furthermore, Theorem 12.4.1 of Cox, Little, Schenck 2011 establishes that $H^4(X_\Sigma, \mathbb{Q})$ is isomorphic to $R_{\mathbb{Q}}(\Sigma)_2$, the space of degree-2 monomials (under the standard grading) in the cohomology ring $R_{\mathbb{Q}}(\Sigma)$ of $X_\Sigma$. This cohomology ring is the quotient of $\mathbb{Q}[x_1, \dots, x_r]$, where $r$ is the number of rays in the polyhedral fan $\Sigma$ of the toric ambient space $X_\Sigma$, by the sum of the Stanley–Reisner ideal and the ideal of linear relations.

To construct a generating set of ambient candidate $G_4$-fluxes, we proceed as follows:

Construct the cohomology ring $R_{\mathbb{Q}}(\Sigma)$.
Identify a basis of degree-2 monomials in $R_{\mathbb{Q}}(\Sigma)$.
Restrict these monomials to $\widehat{Y}_4$.

While performing the restriction to $\widehat{Y}_4$, dependencies may arise:

Some monomials vanish identically,
Others may become linearly dependent.

FTheoryTools currently detects only obviously trivial restrictions (e.g., cycles whose intersection with $\widehat{Y}_4$ is empty). It does not eliminate all possible dependencies–-hence the result is a generating set, not a minimal basis. This pragmatic choice enables efficient and scalable construction of vertical flux candidates across a wide range of F-theory geometries.

FTheoryTools includes specialized algorithms to perform steps 1 through 3. These algorithms—described in detail in the appendix of BMT25—are especially effective for large geometries such as those studied in Taylor, Wang 2015, where traditional Gröbner basis techniques are computationally impractical. Even for more manageable QSM geometries (cf. Cvetič, Halverson, Ling, Liu, Tian 2019), we observe a significant performance boost.

Necessary Conditions for Flux Quantization

Verifying the flux quantization condition (cf. Witten 1997) is generally difficult. In practice, FTheoryTools performs necessary consistency checks to gather evidence for (or against) proper quantization.

Let $H$ denote the hypersurface divisor class defining $\widehat{Y}_4$ inside the ambient toric variety $X_\Sigma$. Let $\{D_i\}$ be a basis for the toric divisor classes of $X_\Sigma$, and let $\hat{c}_2 \in H^{2,2}(X_\Sigma, \mathbb{Q})$ be a class whose restriction gives the second Chern class of the Calabi–Yau hypersurface, i.e., $\left. \hat{c}_2 \right|_{\widehat{Y}_4} = c_2(\widehat{Y}_4)$.

As elaborated above, in FTheoryTools, a $G_4$-flux candidate is modeled by a class $g \in H^{2,2}(X_\Sigma, \mathbb{Q})$, and its restriction to $\widehat{Y}_4$ defines the actual flux candidate. A necessary condition for quantization of this flux $G_4$—modelled by $g \in H^{2,2}(X_\Sigma, \mathbb{Q})$—is that, for all pairs of indices $i$, $j$, the following integral evaluates to an integer:

\[\int_{X_\Sigma}{\left(g + \frac{1}{2} \hat{c}_2 \right) \wedge [H] \wedge [D_i] \wedge [D_j]}\,,\]

where $[D]$ indicates the Poincaré dual cohomology class to the divisor $D$. It is these conditions that we check in FTheoryTools.

Even these elementary checks can be computationally expensive on large geometries. Therefore, FTheoryTools allows users to skip quantization checks during flux construction using the keyword argument check = false. This can be useful in early-stage explorations or for performance optimization.

Working with $G_4$-Fluxes

Constructing $G_4$-Fluxes

In FTheoryTools, $G_4$-fluxes are created from cohomology classes associated with a given resolved F-theory model.

g4_flux — Method

g4_flux(model::AbstractFTheoryModel, class::CohomologyClass; check::Bool = true)

Constructs a candidate $G_4$-flux for a resolved F-theory model from a given cohomology class on the toric ambient space.

By default, check = true enables basic consistency and quantization checks. Set check = false to skip these checks, which can improve performance or allow for exploratory computations.

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

julia> cohomology_ring(ambient_space(qsm_model), check = false);

julia> g4_class = cohomology_class(anticanonical_divisor_class(ambient_space(qsm_model)), quick = true)^2;

julia> g4f = g4_flux(qsm_model, g4_class, check = false)
G4-flux candidate
  - Elementary quantization checks: not executed
  - Transversality checks: not executed
  - Non-abelian gauge group: breaking pattern not analyzed
  - Tadpole cancellation check: not computed

Experimental

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