Construction of Calabi-Yau geometries#
We are mostly concerned with Calabi-Yau threefolds, which we abbreviate by CY3s in what follows. JAXVacua currently supports two classes of CY3 geometries: hypersurfaces in toric varieties from the Kreuzer-Skarke database and complete intersection Calabi-Yau threefolds (CICYs). In the future, support for F-theory flux compactifications on Calabi-Yau fourfolds is desirable.
CY hypersurfaces from reflexive polytopes#
Smooth Calabi-Yau threefold hypersurfaces can be obtained from triangulations of reflexive polytopes in four dimensions. These were enumerated by Kreuzer and Skarke [hep-th/0002240], finding 473,800,776 such polytopes. With the development of CYTools [2303.00757], efficient and targeted construction of such CY geometries became feasible on large scales, opening a new window to study string theory compactifications. Our implementation supports CYTools capabilities for the construction of the EFT from compactifications on such Calabi-Yau threefold hypersurfaces.
Batyrev construction#
We begin by setting notation and terminology for Calabi-Yau threefold hypersurfaces in toric varieties, obtained from triangulations of four-dimensional reflexive polytopes from the Kreuzer-Skarke list. More details on this construction can be found in [2008.01730].
Suppose that \(\Delta \subset \mathbb{Z}^4\) is a four-dimensional reflexive polytope and denote by \(\Delta^{\circ}\) its polar dual.
Let \(\mathcal{T}\) be a regular, star triangulation of \(\Delta^{\circ}\) in which every point of \(\Delta^{\circ}\) that is not interior to a facet is a vertex of a simplex of \(\mathcal{T}\). Such a triangulation is not in general a fine, regular, star triangulation (FRST), because of the omission of points interior to facets, but the associated subvarieties of \(V\) do not intersect a generic hypersurface \(X\), and so are immaterial for our analysis. With this understanding we refer to such triangulations \(\mathcal{T}\) as FRSTs. The toric fan associated to \(\mathcal{T}\) defines a four-dimensional toric variety \(V\) in which the generic anticanonical hypersurface \(X\) is a smooth Calabi-Yau threefold [alg-geom/9310003].
Regular triangulations of polytopes can be represented by a vector of heights which lifts the point collection of \(\Delta^\circ\) to one higher dimension. As is well known, there is a massive redundancy when going from polytope triangulations to Calabi-Yau hypersurfaces. Wall’s theorem asserts that Hodge numbers, triple intersection numbers, and second Chern classes completely determine the homotopy type of a compact, simply connected Calabi-Yau threefold with torsion-free homology. For Calabi-Yau hypersurfaces, Hodge numbers are fixed by polytope data, while triple intersection numbers and second Chern classes are determined purely by the induced triangulations of two-faces. Therefore, FRSTs of \(\Delta^{\circ}\) with identical restrictions to two-faces give rise to topologically equivalent Calabi-Yau threefolds.
Orientifolds#
We say that a polytope \(\Delta^{\circ}\) is trilayer if the points of \(\Delta^{\circ}\) lie in exactly three distinct affine sub-lattices of codimension one [2106.05084]. Calabi-Yau threefold hypersurfaces in toric varieties \(V\) resulting from triangulations of trilayer polytopes admit very convenient orientifold involutions. In each case there exists a toric coordinate \(x_1\) such that the involution defined by
yields, when restricted to the generic invariant hypersurface \(X \subset V\), an orientifold with \(h^{1,1}_-=h^{2,1}_+=0\), which we refer to as a trilayer orientifold. All orientifolds considered in this work are of this type.
Given a Calabi-Yau orientifold \(X/\mathcal{I}\), the D3-brane tadpole charge is
A larger tadpole \(Q_O\) admits more flux configurations and thus a richer landscape of vacua (see Type IIB Three-Form Fluxes). In practice, we restrict to geometries with \(3\leq h^{2,1}\leq 5\) and \(Q_{\text{O}}\ge 100\), or \(6\leq h^{2,1}\leq 8\) and \(Q_{\text{O}}\ge 150\).
Prime toric divisors#
A consequence of \(\Delta^{\circ}\)-favorability is that \(h^{1,1}(V) = h^{1,1}(X)\), with \(h^{1,1}(V)+4\) toric coordinates \(x_I\) generating the Cox ring. We define the prime toric divisors of \(V\) as \(\hat{D}_I = \{x_I = 0\}\), and we refer to \(D_I = \hat{D}_I \cap X\) as the prime toric divisors of \(X\). The \(D_I\) are all effective divisors, and (again using \(\Delta^{\circ}\)-favorability) they generate \(H_4(X,\mathbb{Z})\). In general there also exist effective divisors \(D\) that are non-positive integer linear combinations of the \(D_I\), which are relevant for non-perturbative contributions to the superpotential (see Moduli Stabilisation).
Complete Intersection CYs#
The JAXVacua package also supports Complete Intersection Calabi-Yau threefolds (CICYs), defined as complete intersections of hypersurfaces in products of projective spaces. The 7,890 topologically distinct CICY threefolds were classified in [hep-th/8802033]. The package provides topological data (Hodge numbers, triple intersection numbers, second Chern class) for all CICYs, enabling flux compactification studies on this class of geometries.
Background on CICYs#
A CICY threefold \(X\) is specified by a configuration matrix \([n_r \,|\, k_{ra}]\), where \(\mathbb{P}^{n_r}\) are the ambient projective factors and \(k_{ra}\) are the degrees of the defining polynomials. The Hodge numbers \(h^{1,1}\) and \(h^{2,1}\), the triple intersection numbers \(\kappa_{ijk}\), and the second Chern class \(c_2(X)\) — which enter the Kähler potential and the D3-brane tadpole — are stored in the package data. Kähler cone data are also included, enabling the construction of the prepotential.
For details on loading CICY models — both via the local model_ID parameter and via the HuggingFace database hosted on aschachner/cy-database — see the database documentation in the stringforge umbrella package.
Calabi-Yau fourfolds#
In the future, we hope to include a similar implementation for F-theory flux compactifications on Calabi-Yau fourfolds.