五参数Hulek-Verrill流形的镜像对称
Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds
AI总结：
本研究构造五参数Hulek-Verrill流形的镜像，验证镜像对称预测，计算亏格0和1的瞬子数，发现其背后的$S_6$对称性及更大的无限双曲Coxeter群结构，揭示瞬子数的正网特性。
摘要：
我们研究了五参数Calabi-Yau三维流形的镜像，这类流形最初由Hulek和Verrill在研究Calabi-Yau流形ζ函数的观测模行为的背景下提出。借助环面几何，我们可以简单显式地构造这些镜像，结果表明它们是已知的流形，且具有多种椭圆纤维化方式。通过研究奇异纤维，我们能够确定镜像流形上的低次有理曲线，这验证了通过在大复结构点附近研究镜像映射得到的镜像对称预测。我们还对Hulek-Verrill流形的周期及其单值群进行了深入研究。在镜像流形上，我们计算了亏格为0和1的瞬子数，由于$h^{1,1}=5$，这些瞬子数由5个指标标记。这些指标上存在明显的置换对称性，但除此之外还存在令人惊讶的数值重复现象。我们将这一现象追溯到一个$S_{6}$对称性，该对称性通过Hulek-Verrill流形复结构模空间的某些构造得以显现。作为诸多推论之一，我们由此发现该模空间存在六个大复结构极限。正是围绕这些点中的任意一点展开预势的自由度，导致了瞬子数的这种对称性。一个有趣的事实是，作用于瞬子数的群比$S_6$更大，实际上是一个无限双曲Coxeter群，我们对该群进行了研究。群轨道具有“网”结构，在特定限定条件下，瞬子数仅当属于我们所称的“正网”时才非零。这种结构对所有亏格的瞬子数都有影响。
英文摘要：
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.
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