Geometric constraints in dual F-theory and heterotic string compactifications

TL;DR

This work develops a comprehensive framework for classifying and analyzing 4D heterotic/F-theory dual pairs where the heterotic side uses a smooth elliptically fibered Calabi–Yau threefold with a single section and smooth vector bundles, and the F-theory side uses a dual base ${f B}_3$ that is a ${f P}^1$-bundle over a base ${f B}_2$ with an elliptically fibered Calabi–Yau fourfold. By deriving and matching geometric constraints on ${f B}_3$ (and the twists defining the bundle) to stability and topological data on the heterotic side, the authors show that the set of such dual pairs is finite; they then explicitly enumerate all toric bases, obtaining 4962 dual bases over 16 toric ${f B}_2$'s. A central result is the close correspondence between F-theory geometric constraints and heterotic bundle existence conditions, including the base-point-free condition and limits on gauge enhancement, which in turn influence the presence of chiral matter in the dual pair. The study emphasizes the role of G-flux in potentially lifting moduli and altering symmetry patterns beyond purely geometric expectations, and it provides a concrete, toric dataset to guide future construction of heterotic bundles with exceptional gauge groups. Overall, the paper lays groundwork for a systematic 4D heterotic/F-theory classification and highlights rich avenues for extending duality beyond the smooth, sectioned, toric regime.

Abstract

We systematically analyze a broad class of dual heterotic and F-theory models that give four-dimensional supergravity theories, and compare the geometric constraints on the two sides of the duality. Specifically, we give a complete classification of models where the heterotic theory is compactified on a smooth Calabi-Yau threefold that is elliptically fibered with a single section and carries smooth irreducible vector bundles, and the dual F-theory model has a corresponding threefold base that has the form of a P^1 bundle. We formulate simple conditions for the geometry on the F-theory side to support an elliptically fibered Calabi-Yau fourfold. We match these conditions with conditions for the existence of stable vector bundles on the heterotic side, and show that F-theory gives new insight into the conditions under which such bundles can be constructed. In particular, we find that many allowed F-theory models correspond to vector bundles on the heterotic side with exceptional structure groups, and determine a topological condition that is only satisfied for bundles of this type. We show that in many cases the F-theory geometry imposes a constraint on the extent to which the gauge group can be enhanced, corresponding to limits on the way in which the heterotic bundle can decompose. We explicitly construct all (4962) F-theory threefold bases for dual F-theory/heterotic constructions in the subset of models where the common twofold base surface is toric, and give both toric and non-toric examples of the general results.

Figures (2)

• Figure 1: The possible Higgsing/Enhancement chains for smooth heterotic/F-theory dual pairs; modified from Bershadsky:1996nh. Figure depicts Higgsing possibilities based on heterotic bundles with structure group $H \subset E_8$, which match with dual F-theory models. F-theory gauge groups from Kodaira singularities with $f, g$ having nonzero degrees of vanishing lie to the right of the vertical red dashed line, such gauge groups can be forced from the geometry (geometrically "non-Higgsable") and cannot be unHiggsed to anything left of the line. The $SU(3)$'s and $SU(2)$'s connected near the bottom by horizontal dashed lines correspond to transitions between different Kodaira types in F-theory from type $IV, III$ to type $I_3, I_2$. The top row above the horizontal blue dashed line corresponds to an alternative Higgsing sequence from $E_8$ to $SU(3), SU(2)$ with non-standard commutants ( e.g. $H =SU(3) \times G_2$ for upper $SU(3)$), generically associated with matter in the adjoint representation, which on the F-theory side involves wrapping on higher genus curves for 6D models. Note that in F-theory models that do not have heterotic duals, further unHiggsing ( e.g. to $SU(N > 6)$) can occur. Note also that in the heterotic theory some Higgsing chains lead to product gauge groups, as discussed further in text.
• Figure 2: Constraints imposed on dual monomials in the Weierstrass function $g$ parameterized by $a, b, c$ in the toric description of a twist $\tilde{t}_i = 1$ over a curve of self-intersection $-n = -2$; the depicted constraints on $b, c$ correspond to weakest conditions, which hold at $a = -B= -6$. For a smooth F-theory geometry, at least one monomial in the shaded region (not including the boundary at $b > c + 6$, or the corresponding region for $f (B = 4)$ must be nonzero. The circled point $(5, -2)$ corresponds to the monomial $yz^4$ in coordinates where $z = 0$ corresponds to $\Sigma_-$ and $y = 0$ corresponds to $D_i$. This point is relevant in ruling out gauge algebra factors ${\mathfrak{e}}_7,{\mathfrak{e}}_8$ on $\Sigma_-$ under these conditions (§\ref{['sec:bpf']}).)