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Take a spin Chern-Simons TQFT, such as $U(N)$ or $SO(N)$ with odd level. Such system depends on the spin structure of the underlying manifold.

But how exactly does the theory depend on the spin structure? Say we compute the partition function (or some correlator of Wilson loops) using two different spin structures. How do these two objects differ? I would expect a rather mild dependence, such as an overall $\pm1$ sign (or some other phase). Is this correct?

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  • $\begingroup$ In my humble opinion, a TQFT partition function sums over all even spin structures. A spin TQFT is a sub-theory that counts only a certain spin structure. In two dimensonal conformal field theory, for example, the Ising model (free Majorana fermion) sums over its spin structures. As far as I understand, the twist field $\sigma$ changes the boundary condition of the free fermion along the compactifed cycle. i.e. choosing a spin structure. So if you consider the spin Ising model, you must specify a certain spin structure, then you have only two fields $1$ and $\psi$. $\endgroup$ – Libertarian Monarchist Bot Nov 13 '18 at 1:33

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