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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.

MrHolmes
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    Moduli space :) – Ted Shifrin Apr 11 '22 at 03:02
  • Thanks. Fixed it. – MrHolmes Apr 11 '22 at 10:55
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    Moment map interpretations of gauge equations are very niece because then the moduli space becomes (formally, b/c they are usually infinite-dimensional) a symplectic manifold. This comes with a plethora of tools and information regarding the moduli space itself! – topolosaurus Apr 11 '22 at 10:59
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    P.S.: I would very much like to learn about Chern-Simons theory, do you have any reference? – topolosaurus Apr 11 '22 at 10:59
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    I was reading section 2 Mnev's lecture notes on Batalin Vilkovsky https://arxiv.org/abs/1707.08096 . He leaves out some key calculations about the extremal values of the funcyional and the corresponding Hamiltonian. Those are the parts I'd like to fill in. – MrHolmes Apr 11 '22 at 11:02

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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.

MrHolmes
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