Geodesic
A Framework for Geometric Field Theories and their Classification in Dimension One
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: field theory, vector bundles
Mots-clés : bordism. 
@article{SIGMA_2021_17_a71,
     author = {Matthias Ludewig and Augusto Stoffel},
     title = {A {Framework} for {Geometric} {Field} {Theories} and their {Classification} in {Dimension} {One}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2021},
     volume = {17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a71/}
}
TY  - JOUR
AU  - Matthias Ludewig
AU  - Augusto Stoffel
TI  - A Framework for Geometric Field Theories and their Classification in Dimension One
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2021
VL  - 17
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a71/
LA  - en
ID  - SIGMA_2021_17_a71
ER  - 
%0 Journal Article
%A Matthias Ludewig
%A Augusto Stoffel
%T A Framework for Geometric Field Theories and their Classification in Dimension One
%J Symmetry, integrability and geometry: methods and applications
%D 2021
%V 17
%U http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a71/
%G en
%F SIGMA_2021_17_a71
Matthias Ludewig; Augusto Stoffel. A Framework for Geometric Field Theories and their Classification in Dimension One. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a71/

[1] Atiyah M., “Topological quantum field theories”, Inst. Hautes Études Sci. Publ. Math., 68, 1988, 175–186 | DOI | MR | Zbl

[2] Berwick-Evans D., Pavlov D., Smooth one-dimensional topological field theories are vector bundles with connection, arXiv: 1501.00967

[3] Bunke U., Turner P., Willerton S., “Gerbes and homotopy quantum field theories”, Algebr. Geom. Topol., 4 (2004), 407–437, arXiv: math.AT/0201116 | DOI | MR | Zbl

[4] Freed D. S., “Classical Chern–Simons theory. I”, Adv. Math., 113 (1995), 237–303, arXiv: hep-th/9206021 | DOI | MR | Zbl

[5] Houzel C., “Espaces analytiques relatifs et théorème de finitude”, Math. Ann., 205 (1973), 13–54 | DOI | MR | Zbl

[6] Metzler D., Topological and smooth stacks, arXiv: math.DG/0306176

[7] Meyer R., “Bornological versus topological analysis in metrizable spaces”, Banach Algebras and their Applications, Contemp. Math., 363, Amer. Math. Soc., Providence, RI, 2004, 249–278, arXiv: math.FA/0310225 | DOI | MR | Zbl

[8] Meyer R., Local and analytic cyclic homology, EMS Tracts in Mathematics, 3, European Mathematical Society (EMS), Zürich, 2007 | DOI | MR | Zbl

[9] Rezk C., “A model for the homotopy theory of homotopy theory”, Trans. Amer. Math. Soc., 353 (2001), 973–1007, arXiv: math.AT/9811037 | DOI | MR | Zbl

[10] Schreiber U., Waldorf K., “Parallel transport and functors”, J. Homotopy Relat. Struct., 4 (2009), 187–244, arXiv: 0705.0452 | MR | Zbl

[11] Segal G., “Classifying spaces and spectral sequences”, Inst. Hautes Études Sci. Publ. Math., 34, 1968, 105–112 | DOI | MR | Zbl

[12] Segal G., “The definition of conformal field theory”, Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., 308, Cambridge University Press, Cambridge, 2004, 421–577 | MR | Zbl

[13] Stolz S., Teichner P., “Supersymmetric field theories and generalized cohomology”, Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Proc. Sympos. Pure Math., 83, Amer. Math. Soc., Providence, RI, 2011, 279–340, arXiv: 1108.0189 | DOI | MR | Zbl

[14] Trèves F., Topological vector spaces, distributions and kernels, Academic Press, New York – London, 1967 | MR | Zbl

[15] Vistoli A., “Grothendieck topologies, fibered categories and descent theory”, Fundamental Algebraic Geometry, Math. Surveys Monogr., 123, Amer. Math. Soc., Providence, RI, 2005, 1–104, arXiv: math.AG/0412512 | DOI | MR

[16] Witten E., “Topological quantum field theory”, Comm. Math. Phys., 117 (1988), 353–386 | DOI | MR | Zbl