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Questions
Is a topological quantum field theory metrizable? Or else a TQFT coming from a subfactor?
For a given metric, are there always renormalization and Feynman diagrams?
Is there always a Feynman motive related to the theory?
Finally, does this Feynman motive depend on the choice of the metric?

Motivations
My motivations come from the observation that the subfactors theory and the motives theory are both an "enriched Galois theory" so that I asked myself if there is a link between these two enrichment.
The path through TQFTs and Feynman motives could be a link.
All these questions could be unified by "Is the Feynman motive a topological invariant?".

References
- Y. André; An Introduction to Motives (Pure motives, mixed motives, periods); 2004
- P. Cartier; A mad day's work: from Grothendieck to Connes and Kontsevich, the evolution of concepts of space and symmetry; 2001.
- A. Connes, D. Kreimer; Renormalization in quantum field theory and the Riemann-Hilbert problem I and II; 2000, 2001.
- A. Connes, M. Marcolli; Noncommutative Geometry, Quantum Field Theory and Motives; 2008.
- S. Henry; From toposes to non-commutative geometry through the study of internal Hilbert spaces; PhD dissertation; 2014.
- V. Kodiyalam, V.S. Sunder; Topological quantum field theories from subfactors; 2000.
- V. Kodiyalam, V. Pati, V.S. Sunder; Subfactors and 1+1-dimensional TQFTs; 2005.
- M. Marcolli; Feynman motives; 2010.

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    $\begingroup$ Please define acronyms upon first use. Here, in the subject. $\endgroup$ – garyp Oct 5 '16 at 19:14
  • $\begingroup$ @garyp: see the (original) long version on mathoverflow: mathoverflow.net/q/161641/34538 $\endgroup$ – Sebastien Palcoux Oct 19 '16 at 3:47
  • $\begingroup$ ... ok, but how does that help? $\endgroup$ – garyp Oct 19 '16 at 10:51
  • $\begingroup$ @garyp: all the background is there, and all the concepts (subfactor, planar algebra, tqft, topos, motive) have a link to their definition. $\endgroup$ – Sebastien Palcoux Oct 19 '16 at 14:34
  • $\begingroup$ Yes, but what we need here is a definition of TQFT in the subject so that people who should skip the post will know without having to read it. $\endgroup$ – garyp Oct 19 '16 at 15:41

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