The following content has been provided by the University of Erlangen-Nürnberg.

Okay, so I remind you...

...what we constructed this morning. So every time we had a surface sigma, which was oriented maybe with boundary...

...there was some...

...with some mark points on the boundary. Actually, each component of the boundary had to contain at least one mark point, the way we constructed those surfaces.

So there is sigma and also some subset V. Those are the mark points.

We considered some manifold which was constructed out of sigma and V, which is actually...

...now I can maybe write it in some mathematical terms. So those are homomorphisms from the fundamental group weight of sigma with respect to V...

...into G, which just means that for any path between these mark points we have an element of G...

...and this element doesn't change if you deform this path and we have a nice composition rule that if you take this path followed by that path...

...then the corresponding element is the product of these two. So that's what's written here.

And on this page we have an action of G power V, which just conjugates those homonyms, so to say.

So at each vertex we can act with separate copy of G. And we have also a map going from here.

I should really write something that goes to... that would be more appropriate.

So for every arc of the boundary we have the corresponding homonomy.

But we can identify those edges with vertices simply by saying that for every vertex we take the outgoing edge. So that's what we roughly did.

And then we notice that this is actually... this we call nu, and this is a quasi Hamiltonian G power V space...

...with some twisted permutation given by going around those boundary components.

Now why is this thing such a big deal? That's why we spent so much time deriving this result.

So why is it so interesting? The reason really is that if sigma does have boundary, then this moduli space M sigma V...

...I should write is of the form G power N for some M. For example, if our surface has just one boundary component on one marked point...

...and this of G Nus... I'm not sure whether it's safe to call G Nus G, if everything else was G, so say it's of G Nus P.

And in this case we just have this fundamental group which is free. So those generators are for each hole. We can go either like that or we can go inside and...

...so then we get just G to the power 2 P. So it's a nice smooth manifold, whereas if the boundary is empty...

...and then actually this description is not quite correct, then I should write something like home Y1 sigma into G modulo G.

So that's what we get using those triangulations. Then this modulum by G produces some space which might be relatively ugly.

So say if the group is compact, then it is not so bad, we're getting some more bifold, but still it might be some relatively difficult space compared to this one.

And now we can see if we again take surface of the same G Nus P, the two paths from here to there for the corresponding moduli spaces...

...this is simply the quasi-Hamiltonian reduction. So this is quasi-Hamiltonian G space, there is only one mark point, one boundary component.

So here is the action of G and here is the moment map. And to close off this hole, what we do is simply impose that the holonomy here is trivial...

...and then we divide by this action of G, which is exactly this quasi-Hamiltonian reduction that we saw, at G0 equal to 1.

So there is a nice, simple type of construction how to pass from a nice non-singular space to this very interesting but possibly singular space.

And this thing cannot be done using simplistic geometry. So this stuff, M sigma V is not...

...what do I mean by that? Imagine suppose for example that if G is compact and one connected, that's the most interesting case anyway.

So in this case, we have that...our manifold is just this, G2P, it is a compact manifold, but if you look at what is this, H2 of G2P is the same just...

...you just need to look at what is the second cohomology of G and this vanishes. That's impossible for a compact, simple manifold.

Because if you have a...maybe it's a nice thing to remind that if you have a...and it's compact and simplistic, and what happens? If it's of dimension 2n...

...you can take the simple form 2 to the power n, so you do this n times, and this thing is a volume form.

What is the...so the volume form, and its integral over M of this thing is well defined, right, and it's...since its volume form is definitely non-zero...

...which means that this thing has non-trivial cohomology clause, that certainly means...this implies that the cohomology clause of this is non-zero, which implies that the cohomology clause of omega is non-zero.

So one can say that this construction is somehow inherently quasi-Hamiltonian, there is no way how to repair this quasi-symplectic form to a symplectic form.

So let me now tell you...maybe one more thing, that there is also...it's also kind of very economical construction, because now we can start with simply two errors with one hole, I think like that.

And by fusing, I told you, you might remember we defined some kind of fusion product, so if you take...I'll remind you what it was...if you take two things of that kind and fuse them...what was fusion?

Fusion was that we also take a triangle, and now we glue one of the edges of the triangle to this thing, and the other edge we glue to this boundary component.

In the end what happens? That those two boundaries will be somehow partially glued, so if you glue this to that and this to that...so let you imagine that we're going to have just one more point.

So this fusion operation, geometrically, if you do that, it means that it fuses those two holes.

And by repeating this operation, so you can take several copies of those, so those two are each with just one hole, and you can produce moduli spaces for arbitrary genus for one hole, say.

So in a way, everything is already concentrated for the moduli space of the surface. So it's a relatively nice construction in the end for the symplatic form on the moduli space of closed surface.

So in the end we need to perform reduction and get the moduli space here. But now I should tell you something about the original, purely symplatic construction of the symplatic form on the moduli space of flat connections.

So let me...this is perhaps a bit too fast, but...now I'll try to give you a bit more details.