TQFTs and Feynman motives

Question

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 it?
Finally, does this Feynman motive depend on the choice of the metric?

Clarification by the OP given in a comment here

"First, all these questions could be unified by "Is the Feynman motive a topological invariant?". My motivation comes 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."

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

This post imported from StackExchange Physics at 2014-10-06 20:59 (UTC), posted by SE-user Sébastien Palcoux

Comments

Can you expand on these questions? Maybe give some motivation or examples where you have some understanding. Since you have so many questions, maybe consider just asking one of them or writing separate questions.

---

To start at the beginning, what do you want to mean by a metric on a topological field theory? Hence what do you want to mean by a TQFT being "metrizable"?