Low Dimensional Topology

August 25, 2015

Heisenberg-picture TQFT

Filed under: Quantum topology — dmoskovich @ 8:06 am

This interesting-looking preprint has just appeared on ArXiv:

Theo Johnson-Freyd, Heisenberg-picture quantum field theory, arXiv:1508.05908

It argues for a different category-theoretical formalism for TQFT than the `Schroedinger-picture‘ Atiyah-Segal-type axiomatization that we are used to. The `Heisenberg-picture‘ functor it proposes has as its target a category whose top level is pointed vector spaces instead of numbers, and whose second to top level is associative algebras instead of vector spaces. The preprint argues that this formalism is better physically motivated, and one might dream that it is better-suited to analyze “semiclassical limit” conjectures such as the AJ conjecture and its variants.

I’m very happy to see this sort of playing-around with the foundations of TQFT, which I am happy to believe are too rigid. I expect there should be a useful Dirac picture also, and that there are other alternative axiomatizations also. Let’s see where this all leads!

1 Comment »

  1. Skein algebras associated to surfaces (especially the torus) are interesting algebras representation-theoretically (although the more elementary definition of “links in the thickened surface modulo skein relations” goes back to the early 90’s). The definition in the last section of the paper you mentioned (C-labeled graphs in a 3-manifold M) seems somewhat similar to other constructions, e.g. the degree 0 part of the Blob complex of Morrison-Walker (although I think the basic idea has been around for a while). The recent Brochier, Ben-Zvi, Jordan paper has a very nice description of related(?) algebras for punctured surfaces and C = Rep U_q(g).

    I’d be curious to see if more abstract definitions can be used to prove theorems about more complicated manifolds, e.g. knot complements. For classical skein algebras, the only theorems I’ve heard for more general 3-manifolds only hold for q=1. I believe (some version of) the AJ conjecture can be stated using (classical) skein modules, so there are interesting theorems that are possibilities.

    Comment by Peter Samuelson — October 6, 2015 @ 8:14 pm | Reply

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