F-理论中的半超多重态和不完全/完全归结
Half-hypermultiplets and incomplete/complete resolutions in F-theory
摘要：
我们认为决议的余维2增强奇异性从$SO(12)$到$E_7$和从$E_7$到$E_8$在六维F-理论，其中半超多重产生的通用复杂结构实现它们。增强点处的特殊纤维表现出不同的结构，这取决于碰撞7-膜如何接近规范7-膜的堆叠，正如莫里森和泰勒先前在从$SU(6)$增强到$E_6$的情况下所观察到的那样。当碰撞膜以$O(s)$（其中$s$是基空间沿规范7-膜的坐标）的形式接近它们时，解析过程最终得到的例外纤维比小平分类天真地预期的要少，并且具有包括半整数交叉数的非Dynkin交叉矩阵。我们确认，在增强点的例外纤维形式的极值射线的锥的正权重的相关的伪实表示，解释了为什么半超多重峰出现在那里。通过改变在这个过程中爆炸的奇点的顺序，我们得到，对于$SO(12)\rightarrow E_7$和$E_7\rightarrow E_8$，每隔一行的相应的盒图的相交图。我们提出了详细的推导例外纤维的交点图在奇异性增强点，通过检查如何一个例外的曲线是解除了由于随后的爆破过程中产生的图表。当碰撞膜以$O(s^2)$的形式接近膜的堆叠时，我们在增强点处获得了额外的锥状奇点，这完成了先前发现的增强群的完整Dynkin图。
英文摘要：
We consider resolutions of codimension-two enhanced singularities from $SO(12)$ to $E_7$ and from $E_7$ to $E_8$ in six-dimensional F-theory, where a half-hypermultiplet arises for generic complex structures achieving them. The exceptional fibers at the enhanced point exhibit different structures depending on how the colliding 7-brane approaches the stack of gauge 7-branes, as previously observed by Morrison and Taylor in the case of the enhancement from $SU(6)$ to $E_6$. When the colliding brane approaches them as $O(s)$, where $s$ is the coordinate of the base space along the gauge 7-branes, the resolution process ends up with fewer exceptional fibers than naively expected from the Kodaira classification, with a non-Dynkin intersection matrix including half-integral intersection numbers. We confirm that the exceptional fibers at the enhanced point form extremal rays of the cone of the positive weights of the relevant pseudo-real representation, explaining why a half-hypermultiplet arises there. By altering the ordering of the singularities blown up in the process, we obtain, for both $SO(12)\rightarrow E_7$ and $E_7\rightarrow E_8$, the intersection diagram on every other row of the corresponding box graphs. We present detailed derivations of the intersection diagrams of the exceptional fibers at the singularity enhanced points by examining how an exceptional curve is lifted up on the chart arising due to the subsequent blowing-up process. When the colliding brane approaches the stack of branes as $O(s^2)$, we obtain additional conifold singularity at the enhanced point, which completes the full Dynkin diagram of the enhanced group as was found previously.
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