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One method for finding the Hilbert spaces corresponding to surfaces in Chern-Simons TQFT is by geometrically quantizing the phase space, which is just the character variety of the surface. I know that this has been done by Anderson and others in the holomorphic polarization and by Jeffrey and Weitsman in the real polarization when the gauge group is $SU(2)$.
Is there a derivation of the state spaces for Chern-Simons TQFTs with gauge group $SL(n,\mathbb{C})$ that uses geometric quantization? Equivalently, is there a geometric quantization of the $SL(n,\mathbb{C})$ character varieties of surfaces? Or at least for $n=2$?

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2 Answers 2

This was done in a paper by E. Witten: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104202513

The paper was written before the arxiv came to be, so unfortunately it is not on the arxiv.

Another paper by Witten may be relevant here: http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.2933v4.pdf but I haven't studied it in detail.

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The quantization procedure is proposed by Gukov by using A-polynomials

http://arxiv.org/abs/hep-th/0306165

This quantization is shown to be true for $SL(2,\mathbb{C})$ character variety of hyperbolic knots in $S^3$.

http://arxiv.org/pdf/math/0604057v2

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