This in reference to this fascinating lecture by Nicolai Reshetikhin-

http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf

- Given what is said on page 13 in section 4.1 its not clear to me why the ``partition function" should be a vector in the same space of states which is being assigned to the boundary of the space-time manifold.

I am confused to see the claim that $Z(M) \in H(\partial M)$.I would have thought that the partition function is a linear function on the space of states which takes in a boundary configuration and gives back a number.

I would like to know what is meant by saying (on the same page) that "..(this vector space of states) may depend on the extra structure at the boundary (it can be a vector bundle over the moduli space of such structures).."

How does the above relate to the claim on page 47 that for Chern-Simons theory the "..space of states assigned to the boundary is the space of holomorphic sections of the geometric quantization line bundle over the moduli space of flat connections in a trivial principal G-bundle over the boundary (provided we made a choice of complex structure)..."

I would like to know what the above means and I would be happy to get back some further references about this...especially what is a "geometric quantization bundle"?

- Is the above somehow purely an effect of quantization? Thinking classically intuitively i would have thought that the space of states assigned to the boundary is the space of gauge equivalence classes of flat connections on the manifold which can be extended to a flat connection on the whole space-time. Is the above wrong?

I can't see where the structures of a vector bundle and may be even its sections over the above moduli space seem to be getting involved.

- Is there a general argument to see that the space of states attached to the boundary is always a symplectic manifold and that the subspace of that will ever be picked up by solving the Euler-lagrangian equations in the interior will be some Lagrangian submanifold of it?

In these lectures some comment seems to be made about how the gauge invariance may complicate the above scenario.I would like to know more about that.

- At the start of the lecture the author seems to demand that the space-times belong to a category of manifolds where the morphism is that of cobordism. I would like to know what was the intuition behind making this choice. Why not some other simpler morphism?

Does working in this category of cobordisms somehow help in justifying the functoriality demand on the partition function with respect to gluing of manifolds?