Origin of Abelian Gauge Symmetries in Heterotic/F-theory Duality

TL;DR

The work addresses how Abelian gauge symmetries originate in heterotic/F-theory duality by studying F-theory models with a rank-one Mordell-Weil group and performing toric stable degeneration to the heterotic side. The authors derive both the Calabi-Yau geometry and the heterotic spectral covers, then analyze the spectral data using the elliptic curve group law to classify dual heterotic structures that produce U(1) factors. They identify three main classes—split spectral covers with S(U(m) × U(1)) structure, torsional spectral covers implying SU(m) × Zk structure, and cases with purely non-Abelian bundles whose E8 centralizers contain a U(1)—and show how the Stückelberg mechanism and gluing conditions control the massless U(1) content. The toric construction and stable degeneration procedure provide a concrete bridge between F-theory MW data and heterotic vector-bundle structure, offering a framework for building Abelian sectors in dual compactifications with potential phenomenological applications.

Abstract

We study aspects of heterotic/F-theory duality for compactifications with Abelian gauge symmetries. We consider F-theory on general Calabi-Yau manifolds with a rank one Mordell-Weil group of rational sections. By rigorously performing the stable degeneration limit in a class of toric models, we derive both the Calabi-Yau geometry as well as the spectral cover describing the vector bundle in the heterotic dual theory. We carefully investigate the spectral cover employing the group law on the elliptic curve in the heterotic theory. We find in explicit examples that there are three different classes of heterotic duals that have U(1) factors in their low energy effective theories: split spectral covers describing bundles with S(U(m) x U(1)) structure group, spectral covers containing torsional sections that seem to give rise to bundles with SU(m) x Z_k structure group and bundles with purely non-Abelian structure groups having a centralizer in E_8 containing a U(1) factor. In the former two cases, it is required that the elliptic fibration on the heterotic side has a non-trivial Mordell-Weil group. While the number of geometrically massless U(1)'s is determined entirely by geometry on the F-theory side, on the heterotic side the correct number of U(1)'s is found by taking into account a Stuckelberg mechanism in the lower-dimensional effective theory. In geometry, this corresponds to the condition that sections in the two half K3 surfaces that arise in the stable degeneration limit of F-theory can be glued together globally.

Figures (13)

• Figure 1: Computing the Weierstrass normal form (horizontal arrows) and taking the stable degeneration limit (vertical arrows) does not commute.
• Figure 2: On the left we show the reflexive polytope $\Delta_3^\circ$, while its dual $\Delta_3$ is shown on the right. In this example, the ambient space for the elliptic fiber, specified by $\Delta_2^\circ$, is given by $\text{Bl}_1 \mathbb{P}^{(1,1,2)}$.
• Figure 3: The toric morphism $f_2$.
• Figure 4: The dual polytope $\Delta_{dP_2}$ and the associated monomials.
• Figure 5: This figure illustrates a specialization of the coefficients of the hypersurface $\chi=0$ such that the resulting gauge group is enhanced to E$_7$$\times$ E$_6$$\times$ U(1), see also the discussion in Section \ref{['E7E6U1']}. In the left picture, the non-vanishing coefficients are marked by a circle in the polytope $\Delta_3$. In the right figure the new polytope, i.e. the Newton polytope of the specialized constraint $\chi=0$, is shown.

Theorems & Definitions (2)

• Lemma C.1
• proof