Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds

Abstract

We study the mirrors of five-parameter Calabi-Yau threefolds first studied by Hulek and Verrill in the context of observed modular behaviour of the zeta functions for Calabi-Yau manifolds. Toric geometry allows for a simple explicit construction of these mirrors, which turn out to be familiar manifolds. These are elliptically fibred in multiple ways. By studying the singular fibres, we are able to identify the rational curves of low degree on the mirror manifolds. This verifies the mirror symmetry prediction obtained by studying the mirror map near large complex structure points. We undertake also an extensive study of the periods of the Hulek-Verrill manifolds and their monodromies. On the mirror, we compute the genus-zero and -one instanton numbers, which are labelled by 5 indices, as $h^{1,1}=5$. There is an obvious permutation symmetry on these indices, but in addition there is a surprising repetition of values. We trace this back to an $S_6$ symmetry made manifest by certain constructions of the complex structure moduli space of the Hulek-Verrill manifold. Among other consequences, we see in this way that the moduli space has six large complex structure limits. It is the freedom to expand the prepotential about any one of these points that leads to this symmetry in the instanton numbers. An intriguing fact is that the group that acts on the instanton numbers is larger than $S_6$ and is in fact an infinite hyperbolic Coxeter group, that we study. The group orbits have a 'web' structure, and with certain qualifications the instanton numbers are only nonzero if they belong to what we term 'positive webs'. This structure has consequences for instanton numbers at all genera.

arXiv commentv3: a conclusions section added, clarified the role of periods and birationality

References in corpus (21)

• Phases of $N=2$ Theories In Two Dimensions
• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Black Hole Entropy in M-Theory
• Holomorphic Anomalies in Topological Field Theories
• Mirror Symmetry, Mirror Map and Applications to Complete Intersection Calabi-Yau Spaces
• Elliptic integral evaluations of Bessel moments
• Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties
• New Calabi-Yau Manifolds with Small Hodge Numbers
• Analytic Structure of all Loop Banana Amplitudes
• Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties
• M-Theory and Topological Strings--II
• Perturbative Corrections to Kahler Moduli Spaces
• Worldsheet Instantons and Torsion Curves, Part B: Mirror Symmetry
• A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two
• PALP - a User Manual
• Hodge Numbers for CICYs with Symmetries of Order Divisible by 4
• Toric geometry and local Calabi-Yau varieties: An introduction to toric geometry (for physicists)
• Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations
• Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds
• Attractors with Large Complex Structure for One-Parameter Families of Calabi-Yau Manifolds

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