论文标题
几何场理论的框架及其在维度上的分类
A Framework for Geometric Field Theories and their Classification in Dimension One
论文作者
论文摘要
在本文中，我们开发了几何函数场理论的一般框架，这意味着所有所讨论的边界均具有几何结构。我们特别注意建立这种几何结构平滑变化的概念，因此要求我们的田间理论的输出顺利地依赖输入是有意义的。然后，我们在歧管$ M $上的$ 1 $维领域理论（有或没有方向）的情况下测试我们的框架。这里的期望是，这种字段理论等于与连接上$ m $的矢量捆绑包的数据，并且在非方向的情况下，是非排定双线性配对的其他数据；我们证明在我们的框架中确实是这种情况。
In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with geometric structures. We take particular care to establish a notion of smooth variation of such geometric structures, so that it makes sense to require the output of our field theory to depend smoothly on the input. We then test our framework on the case of $1$-dimensional field theories (with or without orientation) over a manifold $M$. Here the expectation is that such a field theory is equivalent to the data of a vector bundle over $M$ with connection and, in the nonoriented case, the additional data of a nondegenerate bilinear pairing; we prove that this is indeed the case in our framework.
