So, I'm new to gauge theories and symplectic reduction and was trying to analyze the Chern Simons theory in three dimensions.
I have a few questions regarding the steps towards reduction.
First off, is it necessary for the bundle to be trivial?
Second, how does one explicitly calulate the extremal values of the functional? I know the solution is the curvature but don't even know how to formally work out the variational derivatives.
Third, when analyzing the Hamiltonian for a manifold decomposed into a 2-D surface and a time interval, how does the Legendre transform work exactly?
Fourth, Atiyah-Bott says that the space of connections of the surface has a symplectic structure and somehow the curvature represents its momentum map. Any insight into this?
Finally, the solutions to the system are represented as a moduli space of flat connections. How does this fit in with the previous steps?
Thank you very much in advance! Any help is welcome.
So, a very nice set of exercises at the end of Chapter 5 of Mark J.D. Hamilton's "Mathematical Gauge Theory" directly answers the questions posed here.
Indeed, the whole book is brilliant and has cleared up a lot more doubts than the ones raised in the questions posted here. I found it to be all-in-all an excellent primer on Gauge Theories.