$7$维规范理论
Gauge theory in dimension $7$
摘要：
我们首先回顾了$G_2$ -流形的概念，定义在主$G_2$（“规范”）丛在$7$维流形，然后讨论它们与超引力的关系。在第二个线程中，我们专注于结合子流形，并提出他们的变形理论。特别地，我们详细讨论了一个具有协结合边界条件的变形问题。它的无穷小变形空间可以等同于一个椭圆型方程的解空间，其指数由拓扑公式给出。
英文摘要：
We first review the notion of a $G_2$-manifold, defined in terms of a principal $G_2$ ("gauge") bundle over a $7$-dimensional manifold, before discussing their relation to supergravity. In a second thread, we focus on associative submanifolds and present their deformation theory. In particular, we elaborate on a deformation problem with coassociative boundary condition. Its space of infinitesimal deformations can be identified with the solution space of an elliptic equation whose index is given by a topological formula.
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