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Chern-Simons theory is of topological nature. If we put it on a compact 3-manifold this should yield a topological invariant. As any TQFT it should not have any time dependence, in other words, the notion of time is not defined.

My question is if this is correct or not. If we put the theory on compact $M_3$ then in principle we should be able to deform it such that it looks (locally)a tube with initial and final regions $M_3$ and in the middle stretched like $\mathbb{R} \times M_2$. Another choice would be $S^1 \times M_2$. Is there any notion of time periodic or not?

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    $\begingroup$ If you put the theory on a compact $M_3$ I would be very disturbed if it can be deformed to $\mathbb{R}\times M_2$ since that's not compact. I'm not sure what exactly you're trying to ask here. $\endgroup$ – ACuriousMind Mar 28 '17 at 21:29
  • $\begingroup$ Yes, because I made a small typo. Let me correct the question $\endgroup$ – Marion Mar 28 '17 at 21:32
  • $\begingroup$ Not all manifolds are tori, you get slightly more manifolds by allowing some twisting before regluing the ends of the cylinder, these are called mapping tori. Still, only very small part of compact 3D manifolds are mapping tori, most are hyperbolic, see geometrization You can certainly consider CS theory on a torus, or even a cylinder, Dunne explicitly treats it as a theory on 2+1 spacetime. $\endgroup$ – Conifold Mar 29 '17 at 0:08
  • $\begingroup$ But cylinder or torus is 2d. We need to put CS on a compact 3 manifold. What is then the notion of time if all the dimensions are compact? Physically time has to be non-compact. $\endgroup$ – Marion Mar 29 '17 at 10:51
  • $\begingroup$ And note that Dunne always considers non-compact manifolds of the form 2+1. I am interested in the case where we have a compact manifold without boundary, so closed. What is the notion of time there? $\endgroup$ – Marion Mar 29 '17 at 11:07

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