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!


  1. Chern-Simons Theory and Topological Strings, M. Mariño, arXiv:hep-th/0406005.

In his Jones polynomial paper, Witten introduced two types of framings: $3$-manifold framing and knot framing. The first one is connected to the homotopy classes of trivialization of the tangent bundle of the $3$-manifold, the second is connected to the homotopy classes of the trivializations of the normal bundle to the knot. He explains the second point here (on the bottom of page 71).

Witten remarks on the analogy between these two framings in the footnote of page 364 in the Witten-Jones paper.

The basic reason that he had to introduce the framing dependences is that he sought to obtain topological invariants; when he did not, he found that he can trade the metric dependence by framing $3$-manifold framing dependence in the first case and the knot diffeomorphism dependence by knot framing dependence in the second case.

For the first case, when he computed the partition function in the semiclassical approximation of the path integration around a background gauge field, he obtained an expression dependent on the $\eta$ invariant, which appears whenever a partition function of a first order (Dirac-like) operator (because the fluctuation term is linear in the derivatives). The $\eta$ invariant is metric dependent and Witten got rid of this metric dependence by the following trick, first he wrote: $$\eta(A) = \eta_{\mathrm{grav}} + (\eta(A) -\eta_{\mathrm{grav}} ) $$ The second term is metric invariant, where $\eta_{\mathrm{grav}}$ is the gravitational $\eta$ invariant computed from a spin connection $\omega$. Then he added by hand a gravitational Chern-Simons $I(\omega)$ to get a pre-factor: $$\eta_{\mathrm{grav}} + \frac{1}{12\pi} I(\omega)$$ The above pre-factor is metric independent but framing dependent.

In the knot case, the correlation function depends on the knots cross linking numbers which are benign but also on the self-linking numbers which are ambiguous. There are regularization schemes with finite results, but they are not diffeomorphism invariants. Here Witten regularized the self-linking integral by choosing a slightly displaced knot. The displaced knot is not unique; the result depends on the homotopy trivialization class of the knot's normal bundle. The result is diffeomorphism invariant but knot framing dependent.

Now, for your question about the physical significance of the dependence on the trivializations. A trivialization of a bundle is defined by its transition functions between charts. These transition functions are not unique; they are Čech cocycles which one can modify by adding coboundaries; these are just gauge transformations. I gave examples for these objects in this
answer. Their gauge property suggests that we must treat them as background gauge fields (they are not integrated on in the path integral). Thus, different trivializations are equivalent to different backgrounds. Quantum mechanics tells us that they correspond to inequivalent quantizations, and we know that these inequivalent quantizations are bound to show up as in the case of the Aharonov-Bohm effect.


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