[Paper Review] 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces

Gabriella Martini, Washington Taylor|arXiv (Cornell University)|Apr 25, 2014

Black Holes and Theoretical Physics28 references12 citations

TL;DR

This paper systematically classifies 162,404 smooth $\mathbb{C}^*$-surfaces—generalizations of toric surfaces with a single $\mathbb{C}^*$ action—that serve as bases for elliptically fibered Calabi-Yau threefolds in 6D F-theory compactifications. The method extends toric geometry to include non-toric but $\mathbb{C}^*$-equivariant surfaces, revealing new Calabi-Yau geometries with previously unknown Hodge numbers and non-Higgsable $U(1)$ gauge factors due to Mordell-Weil group rank greater than zero.

ABSTRACT

We carry out a systematic study of a class of 6D F-theory models and associated Calabi-Yau threefolds that are constructed using base surfaces with a generalization of toric structure. In particular, we determine all smooth surfaces with a structure invariant under a single C^* action (sometimes called "T-varieties" in the mathematical literature) that can act as bases for an elliptic fibration with section of a Calabi-Yau threefold. We identify 162,404 distinct bases, which include as a subset the previously studied set of strictly toric bases. Calabi-Yau threefolds constructed in this fashion include examples with previously unknown Hodge numbers. There are also bases over which the generic elliptic fibration has a Mordell-Weil group of sections with nonzero rank, corresponding to non-Higgsable U(1) factors in the 6D supergravity model; this type of structure does not arise for generic elliptic fibrations in the purely toric context.

Motivation & Objective

• To extend the classification of 6D F-theory compactifications beyond strictly toric base surfaces by including a broader class of smooth surfaces with a $\mathbb{C}^*$ action.
• To identify and enumerate all smooth $\mathbb{C}^*$-surfaces that can serve as bases for elliptically fibered Calabi-Yau threefolds with a section.
• To explore the resulting Calabi-Yau geometries, particularly those with new Hodge numbers and non-Abelian or Abelian gauge group structures not realizable in the purely toric framework.
• To investigate the emergence of non-Higgsable $U(1)$ gauge factors in F-theory models arising from Mordell-Weil group rank greater than zero in the generic elliptic fibration.

Proposed method

• The authors classify smooth surfaces with a single $\mathbb{C}^*$ action (i.e., $\mathbb{C}^*$-surfaces) that can serve as bases for elliptically fibered Calabi-Yau threefolds with a section.
• They use the framework of T-varieties, focusing on surfaces admitting a $\mathbb{C}^*$ action but not necessarily a $(\mathbb{C}^*)^2$ action, generalizing the standard toric approach.
• The classification is based on the intersection structure of effective irreducible divisors with self-intersection $\leq -2$, which directly determines the non-Abelian gauge group in the maximally Higgsed 6D supergravity theory.
• The Weierstrass model $y^2 = x^3 + fx + g$ is used to describe the elliptic fibration, with $f$ and $g$ sections of $\mathcal{O}(-4K)$ and $\mathcal{O}(-6K)$, respectively, where $K$ is the canonical class of the base.
• The Hodge numbers $h^{1,1}$ and $h^{2,1}$ of the resulting Calabi-Yau threefolds are computed from the base geometry and fibration data.
• The presence of non-Higgsable $U(1)$ gauge factors is detected via the Mordell-Weil group of rational sections, which can have rank greater than zero in non-toric $\mathbb{C}^*$-surfaces.

Experimental results

Research questions

• RQ1 What is the complete set of smooth $\mathbb{C}^*$-surfaces that can serve as bases for elliptically fibered Calabi-Yau threefolds with a section in 6D F-theory compactifications?
• RQ2 Which of these bases lead to Calabi-Yau threefolds with Hodge numbers not realizable in the purely toric classification?
• RQ3 Can non-Higgsable $U(1)$ gauge factors arise in F-theory compactifications over non-toric $\mathbb{C}^*$-surfaces, and if so, how are they encoded in the geometry?
• RQ4 How does the Mordell-Weil group of sections of the elliptic fibration relate to the gauge group structure in the resulting 6D supergravity theory for these generalized bases?
• RQ5 What is the structure of the chain of $-2$-curves and their intersections in these $\mathbb{C}^*$-surfaces, and how does it determine the gauge group and matter content?

Key findings

• The authors identify a total of 162,404 distinct smooth $\mathbb{C}^*$-surfaces that can serve as bases for elliptically fibered Calabi-Yau threefolds with a section.
• Among the resulting Calabi-Yau threefolds, 14 examples have Hodge numbers not previously found in the toric classification, including $h^{1,1}=62, h^{2,1}=2$ and $h^{1,1}=61, h^{2,1}=1$, indicating new geometries.
• Several models exhibit a Mordell-Weil group of sections with rank greater than zero, corresponding to non-Higgsable $U(1)$ gauge factors, a feature that does not occur in generic toric fibrations.
• The model with $r=8$, $N=4$, $n_0=-2$, $n_∞=-2$ yields $T=9$, $h^{1,1}=19$, $h^{2,1}=19$, and is a special limit of $dP_9$ with two $I_0^*(\hat{D}_4)$ $-2$-clusters.
• The chain structure of $-2$-curves in the base determines the gauge group and matter content, with examples like $(-1,-3,-1,-3,-1)$ and $(-2,-1,-2)$ appearing across multiple models.
• The method successfully generalizes the toric classification, revealing that non-toric $\mathbb{C}^*$-surfaces can support physically distinct F-theory vacua with richer gauge and matter content than previously known.