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