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Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds

by Philip Candelas, Xenia de la Ossa, Pyry Kuusela, Joseph McGovern

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Submission summary

Authors (as registered SciPost users): Philip Candelas · Pyry Kuusela · Joseph McGovern · Xenia de la Ossa
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-04-13
Date submitted: 2023-02-17 10:56
Submitted by: McGovern, Joseph
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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.

List of changes

We have fixed the typos mentioned and also added clarification on the different
triangulations in section 2.

Also, we have added a new section on outlook, which discusses the questions
raised in the first report.

Bottom of p.23
We have clarified that, although we do not prove an isomorphism between the manifolds studied by Hulek and Verrill and those that we study, they are certainly birational. This is sufficient for investigating the mirror-symmetry problems that we consider.

We have deleted a sentence below the Hodge diamond to avoid possible confusion about isomorphisms or lack thereof between manifolds.

p.32, above the start of section 3.3
We clarify that although the GKZ approach would provide the periods, we prefer to use the method outlined which yields the same periods owing to the fact that the fundamental period is already determined by the original equations (1.2) acting on the algebraic torus where no coordinate X is zero. The other periods follow by employing a variation on the method of Frobenius.

Published as SciPost Phys. 15, 144 (2023)

Reports on this Submission

Anonymous Report 1 on 2023-2-19 (Invited Report)


I think that the points addressed in the previous report have been clarified, and this paper can be published in the present form.

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Anonymous Report 2 on 2023-2-19 (Invited Report)


All the suggestions have been implemented in a satisfactory manner. The acceptance criteria of the journal are easily met. The article is of a very high quality, publication in its current form is recommended.

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