I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360).

There, it was mentioned, the gravitational Chern Simons action:

$$I(g)=\frac{1}{4\pi} \int_M Tr(\omega d \omega + \frac{2}{3} \omega^3)$$

depends on the choice of framing, i.e. trivialization of tangent bundle of M, in the way that $I(g)\rightarrow I(g)+2\pi s$, where s is how many "units" you twisted the framing.

My question is: how to imagine the twist of framing on this 3 manifold, and further compute the number 's' for given two tangent bundle?

A related, and might be more interesting question is about (2.25) of the same paper:

$$Z\rightarrow Z \exp(2\pi i s \frac{c}{24})$$

Which says, if you twist the frame by s unit, the total contribute to the partition function of Chern Simons theory of gauge group G at level k will be a phase proportion to $c/24$, where c is the central charge of the corresponding current algebra.

Again, how to under to understand this formula, e.g., is there a CFT derivation of this phase shift?


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