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Fibration structures in toric Calabi-Yau Fourfolds

Falk Rohsiepe

TL;DR

The paper develops a toric-geometric framework for identifying and analyzing fibration structures of Calabi–Yau fourfolds, focusing on elliptic and K3 fibrations via reflexive polyhedra. By introducing virtual toric prefibrations and explicit algorithms, it systematically explores when reflexive subpolyhedra yield genuine fibrations, studies obstructions, and computes perturbative heterotic gauge algebras from generic fibers and monodromy. It then applies these methods to hypersurfaces in weighted projective spaces, cataloguing a large set of virtual fibrations, their dualities, and monodromy properties, including both flat and singular cases, with extensive quantitative results. The work provides practical criteria for constructing projective triangulations that realize fibrations, and highlights the role of monodromy in shaping the resulting gauge algebras, thereby contributing to a deeper understanding of F-/heterotic dualities in four dimensions and the landscape of toric Calabi–Yau fourfolds.

Abstract

In the context of string dualities, fibration structures of Calabi-Yau manifolds play a prominent role. In particular, elliptic and K3 fibered Calabi-Yau fourfolds are important for dualities between string compactifications with four flat space-time dimensions. A natural framework for studying explicit examples of such fibrations is given by Calabi-Yau hypersurfaces in toric varieties, because this class of varieties is sufficiently large to provide examples with very different features while still allowing a large degree of explicit control. In this paper, many examples for elliptic K3 fibered Calabi-Yau fourfolds are found (not constructed) by searching for reflexive subpolyhedra of reflexive polyhedra corresponding to hypersurfaces in weighted projective spaces. Subpolyhedra not always give rise to fibrations and the obstructions are studied. In addition, perturbative gauge algebras for dual heterotic string theories are determined. In order to do so, all elliptically fibered toric K3 surfaces are determined. Then, the corresponding gauge algebras are calculated without specialization to particular polyhedra for the elliptic fibers. Finally, the perturbative gauge algebras for the fourfold fibrations are extracted from the generic fibers and monodromy.

Fibration structures in toric Calabi-Yau Fourfolds

TL;DR

The paper develops a toric-geometric framework for identifying and analyzing fibration structures of Calabi–Yau fourfolds, focusing on elliptic and K3 fibrations via reflexive polyhedra. By introducing virtual toric prefibrations and explicit algorithms, it systematically explores when reflexive subpolyhedra yield genuine fibrations, studies obstructions, and computes perturbative heterotic gauge algebras from generic fibers and monodromy. It then applies these methods to hypersurfaces in weighted projective spaces, cataloguing a large set of virtual fibrations, their dualities, and monodromy properties, including both flat and singular cases, with extensive quantitative results. The work provides practical criteria for constructing projective triangulations that realize fibrations, and highlights the role of monodromy in shaping the resulting gauge algebras, thereby contributing to a deeper understanding of F-/heterotic dualities in four dimensions and the landscape of toric Calabi–Yau fourfolds.

Abstract

In the context of string dualities, fibration structures of Calabi-Yau manifolds play a prominent role. In particular, elliptic and K3 fibered Calabi-Yau fourfolds are important for dualities between string compactifications with four flat space-time dimensions. A natural framework for studying explicit examples of such fibrations is given by Calabi-Yau hypersurfaces in toric varieties, because this class of varieties is sufficiently large to provide examples with very different features while still allowing a large degree of explicit control. In this paper, many examples for elliptic K3 fibered Calabi-Yau fourfolds are found (not constructed) by searching for reflexive subpolyhedra of reflexive polyhedra corresponding to hypersurfaces in weighted projective spaces. Subpolyhedra not always give rise to fibrations and the obstructions are studied. In addition, perturbative gauge algebras for dual heterotic string theories are determined. In order to do so, all elliptically fibered toric K3 surfaces are determined. Then, the corresponding gauge algebras are calculated without specialization to particular polyhedra for the elliptic fibers. Finally, the perturbative gauge algebras for the fourfold fibrations are extracted from the generic fibers and monodromy.

Paper Structure

This paper contains 32 sections, 9 theorems, 55 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notational conventions
  3. Calabi-Yau fibered toric hypersurfaces
  4. Calabi-Yau fibers
  5. How generic are the fibers?
  6. Obstructions
  7. Multifibrations
  8. Algorithms for finding virtual fibrations
  9. Enumerating sublattices
  10. Making use of classifications
  11. Elliptically fibered toric K3 surfaces
  12. The class of the fiber
  13. Gauge algebras
  14. Monodromy
  15. A Note on Generality
  16. ...and 17 more sections

Key Result

Lemma 3.1

Let $\sigma \subset \mathbb{Q}^N$ be a strongly convex $N$-dimensional rational cone, $\sigma' \subset \mathbb{Q}^n$ a convex rational $n$-dimensional cone and $\pi: \mathbb{Q}^N\rightarrow\mathbb{Q}^n$ a linear map with $\pi(\sigma) = \sigma'$. Let $\Sigma^{(k)} := \{\tau\, |\, \tau\,\hbox{is a fac

Figures (2)

  • Figure 1: Outer automorphisms of Lie algebras. The dashed arrows connect roots interchanged by an outer automorphism. Note, that Cartan matrix and intersection form are related by a simple switch of sign only when all roots have equal length (i.e. on the left hand side). The direction of the arrows on the right hand side may easily be deduced by using Serre relations on Chevalley generators of the invariant subalgebra. The problematic case of outer automorphisms of $A_{2n}$liecy3andf cannot occur, because a partial split toric edge diagram can only be connected when including a vertex point (which is nonsplit).
  • Figure 2: The two maximal triangulations of the basic octahedron. In both cases, one of the simplices is shaded and its edges are drawn bold. The circular marks in the centers demark the intersection point of the diagonal with the border of the lower left simplex. This point is not integer.

Theorems & Definitions (15)

  • Lemma 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Theorem 5.1
  • Definition 5.2
  • Lemma 5.3
  • Lemma 5.4
  • ...and 5 more