F-理论中的离散对称与挠同调
On Discrete Symmetries and Torsion Homology in F-Theory
摘要：
本文研究了F-理论紧化中离散规范对称性与相应的Calabi-Yau流形上的挠同调之间的关系。集中在一个最简单的例子$\mathbb Z_2$对称性，我们表明，有两个物理上不同的方式，这样一个离散的规范对称性可以出现。首先，我们知道卡-丘三重层上的M理论的紧化是六维F理论真空的对偶，它支持一个双截面的亏格一纤维化，并且具有$\mathbb Z_2$规范对称性。我们表明，由此产生的五维理论不具有$\mathbb Z_2$对称性，但后者仅在F理论解压缩极限中出现。因此，亏格为1的卡-丘流形不具有离散挠。与双截面纤维化相关联的是不支持截面的雅可比纤维化。紧化这些相关的，但不同的品种，导致$\mathbb Z_2$对称性在五个维度，因此，我们发现明确相关的离散扭转。我们确定了预期的粒子和膜系统的离散对称的包装M2和M5膜，并提出了一个场论描述的物理两种情况下，在六维理论的圆形减少。我们的结果和方法直接推广到更大的离散对称性和四维紧化。
英文摘要：
We study the relation between discrete gauge symmetries in F-theory compactifications and torsion homology on the associated Calabi-Yau manifold. Focusing on the simplest example of a $\mathbb Z_2$ symmetry, we show that there are two physically distinct ways that such a discrete gauge symmetry can arise. First, compactifications of M-Theory on Calabi-Yau threefolds which support a genus-one fibration with a bi-section are known to be dual to six-dimensional F-theory vacua with a $\mathbb Z_2$ gauge symmetry. We show that the resulting five-dimensional theories do not have a $\mathbb Z_2$ symmetry but that the latter emerges only in the F-theory decompactification limit. Accordingly the genus-one fibred Calabi-Yau manifolds do not exhibit discrete torsion. Associated to the bi-section fibration is a Jacobian fibration which does support a section. Compactifying on these related but distinct varieties does lead to a $\mathbb Z_2$ symmetry in five dimensions and, accordingly, we find explicitly an associated discrete torsion. We identify the expected particle and membrane system of the discrete symmetry in terms of wrapped M2 and M5 branes and present a field-theory description of the physics for both cases in terms of circle reductions of six-dimensional theories. Our results and methods generalise straightforwardly to larger discrete symmetries and to four-dimensional compactifications.
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