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Journal Article

A framework for geometric field theories and their classification in dimension one

MPS-Authors
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Ludewig,  Matthias
Max Planck Institute for Mathematics, Max Planck Society;

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Stoffel,  Augusto
Max Planck Institute for Mathematics, Max Planck Society;

External Resource

https://doi.org/10.3842/SIGMA.2021.072
(Publisher version)

https://doi.org/10.48550/arXiv.2001.05721
(Preprint)

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2001.05721.pdf
(Preprint), 772KB

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Citation

Ludewig, M., & Stoffel, A. (2021). A framework for geometric field theories and their classification in dimension one. Symmetry, Integrability and Geometry: Methods and Applications, 17: 072. doi:10.3842/SIGMA.2021.072.


Cite as: https://hdl.handle.net/21.11116/0000-0009-B90A-C
Abstract
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.