A framework for geometric field theories and their classification in dimension one
Abstract
In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with a particular kind of geometric structure. We take particular care to establish a notion of smooth variation of geometric structures, so that it makes sense to require the output of our field theory functors 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.05721
 Bibcode:
 2020arXiv200105721L
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Algebraic Topology
 EPrint:
 51 pages, 4 figures