Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it. Is there any known construction of a bundle with a connection such that it's curvature is $\alpha$?
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1$\begingroup$ I do not know an answer for the general linear case, but for compact groups like $SU(2)$ there is a nice answer, see for example "Some comments on CHern-Simons Gauge Theory" by Ramada, Singer and Weitsman. $\endgroup$– SebastianOct 7, 2014 at 11:40
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1$\begingroup$ What do you mean by the "natural" symplectic form? $M$ has a natural hyperkähler structure, hence a whole family of symplectic structures. arXiv:math-ph/0605026 contains some information on quantization of $M$. $\endgroup$– Xin NieOct 7, 2014 at 13:01
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$\begingroup$ @XinNie The OP is not fixing a complex structure on the surface $\endgroup$– Peter DalakovOct 7, 2014 at 13:55
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