# Chern-Simons and framing dependence$.$

I posted this question to physics.SE last week (cf. here), but it got not attention. I hope it is not too trivial to post it here.

According to ref.1, the correlation functions of a Chern-Simons theory are topological invariants, up to the so-called framing, that is, the trivialisation of $$TM\oplus TM$$. The origin of this framing dependence is the UV divergences, which require a regularisation. The standard choice is to use point-splitting, where links are fattened up.

1. How exactly does framing the link introduce a dependence on the trivialisation of $$TM\oplus TM$$? What role does this bundle play here?

2. How can an observable possibly depend on a trivialisation? To me, that's like saying that, for example, the temperature of a black hole depends on the system of coordinates you use. Nonsense!

References.

1. Chern-Simons Theory and Topological Strings, M. Mariño, arXiv:hep-th/0406005.
• Can you give a more precise reference in your document? I am surprised that "framing" here means trivializing $TM \oplus TM$ as opposed to $TM$ itself. Regarding (2), do you also object to the notion of spin structures? Sometimes you need extra structure to pin down an invariant that might otherwise have some discrete ambiguity. – Mike Miller Nov 30 '18 at 16:23
• @MikeMiller 1) See e.g., the last paragraph of the first column on page 3. 2) I am mostly fine with a dependence on a spin stricture (not every CS theory depends on it though). My main objection is not a dependence on any extra structure, but a dependence on a trivialisation. To me, it seems that nothing should really depend on a trivialisation. – AccidentalFourierTransform Nov 30 '18 at 16:28
• But it's the same story: you're using it to pin down an invariant that might have discrete ambiguity. Note that in particular you only care about the homotopy class of trivialization, of which if $E$ is an $n$-dimensional trivializable vector bundle, there are $[M, SO(n)]$ many of these. – Mike Miller Nov 30 '18 at 16:32
• Homotopy classes of maps; there are countably many, sometimes finitely many. (This still leaves open precisely what the framing does and how things change when you modify it.) – Mike Miller Nov 30 '18 at 17:13
• I think by "framing" the question means Atiyah 2-framing. – Noah Snyder Dec 1 '18 at 3:28