[Paper Review] A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations

Mboyo Esole, Monica Jinwoo Kang+1 more|arXiv (Cornell University)|Sep 30, 2014

Algebraic Geometry and Number Theory39 references43 citations

TL;DR

This paper introduces the Q₇(ℒ,𝒮) model, a new smooth elliptic fibration with a rank-one Mordell-Weil group, generalizing Weierstrass and quartic models. It provides a complete classification of singular fibers, derives a generalized Sethi-Vafa-Witten formula for Euler characteristics, and constructs a semi-stable weak coupling limit in F-theory, proving topological tadpole matching that ensures D3-brane charge conservation between F-theory and its orientifold limit.

ABSTRACT

We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q$_7(\mathscr{L},\mathscr{S})$ model. It naturally generalizes several previous models of elliptic fibrations popular in the F-theory literature. The model is also explicitly smooth, thus relevant physical quantities can be computed in terms of topological invariants in straight manner. Since the general fiber is defined by a cubic curve, basic arithmetic operations on the curve can be done using the chord-tangent group law. We will use this model to determine the spectrum of singular fibers of an elliptic fibration of rank one and compute a generating function for its Euler characteristic. With a view toward string theory, we determine a semi-stable degeneration which is understood as a weak coupling limit in F-theory. We show that it satisfies a non-trivial topological relation at the level of homological Chern classes. This relation ensures that the D3 charge in F-theory is the same as the one in the weak coupling limit.

Motivation & Objective

• To construct a new, smooth, geometrically explicit model for elliptic fibrations with a Mordell-Weil group of rank one, generalizing existing models in F-theory.
• To classify the spectrum of singular fibers in such fibrations using the Q₇(ℒ,𝒮) model.
• To derive a generating function for the Euler characteristic of the fibration over a base of arbitrary dimension.
• To construct a semi-stable degeneration of the Q₇(ℒ,𝒮) model as a weak coupling limit in F-theory.
• To prove a topological relation in the Chow ring linking Chern classes of the fibration and its degeneration, ensuring D3-brane charge conservation between F-theory and the orientifold limit.

Proposed method

• The Q₇(ℒ,𝒮) model is defined as a smooth hypersurface in a projective bundle over a base variety B, parameterized by two line bundles ℒ and 𝒮.
• The general fiber is a cubic curve with Newton polygon a reflexive quadrilateral with seven boundary lattice points, corresponding to a genus-one curve with non-trivial Mordell-Weil group.
• The model is shown to be birational to a Jacobi quartic, enabling explicit computation of rational sections and Mordell-Weil group structure.
• Singular fibers are classified via degeneration types, including irreducible and reducible fibers, using the geometry of the fibration and discriminant loci.
• A weak coupling limit is constructed fiberwise, mapping the elliptic fibration to an orientifold theory, with brane spectrum and fluxes explicitly computed.
• A topological identity in the Chow ring is derived, relating the total Chern class of the fibration to those of subvarieties in the degeneration, proving tadpole matching.

Experimental results

Research questions

• RQ1 What is the complete classification of singular fibers in an elliptic fibration with a Mordell-Weil group of rank one?
• RQ2 How can a smooth, geometrically explicit model be constructed for such fibrations, generalizing existing Weierstrass and quartic models?
• RQ3 What is the generating function for the Euler characteristic of such fibrations over a base of arbitrary dimension?
• RQ4 How does the semi-stable degeneration of the Q₇(ℒ,𝒮) model correspond to the weak coupling limit in F-theory?
• RQ5 Does the D3-brane charge computed in F-theory match that in the weak coupling orientifold limit, and if so, why?

Key findings

• The Q₇(ℒ,𝒮) model is a smooth, geometrically explicit hypersurface in a projective bundle, providing a natural generalization of the Weierstrass and Jacobi quartic models.
• The model admits two rational sections, corresponding to a Mordell-Weil group of rank one, with explicit rational points derived from the cubic form.
• A complete classification of singular fibers is obtained, including both irreducible and reducible types, based on the degeneration of the cubic curve.
• A generalized Sethi-Vafa-Witten formula is derived as a generating function for the Euler characteristic of the fibration over a base of arbitrary dimension.
• The semi-stable degeneration of the Q₇(ℒ,𝒮) model is constructed and shown to correspond to the weak coupling limit in F-theory, with consistent brane and flux spectrum.
• A non-trivial topological relation in the Chow ring is proven, ensuring that the D3-brane charge in F-theory matches that in the weak coupling orientifold limit.