Deriving the Hilbert spaces for Chern-Simons TQFTs with complex gauge group - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:27:54Z http://mathoverflow.net/feeds/question/79666 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79666/deriving-the-hilbert-spaces-for-chern-simons-tqfts-with-complex-gauge-group Deriving the Hilbert spaces for Chern-Simons TQFTs with complex gauge group Blake 2011-10-31T23:58:37Z 2011-11-01T23:40:43Z <p>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)\$.<br> 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\$?</p> http://mathoverflow.net/questions/79666/deriving-the-hilbert-spaces-for-chern-simons-tqfts-with-complex-gauge-group/79703#79703 Answer by Satoshi Nawata for Deriving the Hilbert spaces for Chern-Simons TQFTs with complex gauge group Satoshi Nawata 2011-11-01T11:51:57Z 2011-11-01T11:51:57Z <p>The quantization procedure is proposed by Gukov by using A-polynomials</p> <p><a href="http://arxiv.org/abs/hep-th/0306165" rel="nofollow">http://arxiv.org/abs/hep-th/0306165</a></p> <p>This quantization is shown to be true for \$SL(2,\mathbb{C})\$ character variety of hyperbolic knots in \$S^3\$.</p> <p><a href="http://arxiv.org/pdf/math/0604057v2" rel="nofollow">http://arxiv.org/pdf/math/0604057v2</a></p> http://mathoverflow.net/questions/79666/deriving-the-hilbert-spaces-for-chern-simons-tqfts-with-complex-gauge-group/79781#79781 Answer by algori for Deriving the Hilbert spaces for Chern-Simons TQFTs with complex gauge group algori 2011-11-01T23:40:43Z 2011-11-01T23:40:43Z <p>This was done in a paper by E. Witten: <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.cmp/1104202513" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.cmp/1104202513</a></p> <p>The paper was written before the arxiv came to be, so unfortunately it is not on the arxiv.</p> <p>Another paper by Witten may be relevant here: <a href="http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.2933v4.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.2933v4.pdf</a> but I haven't studied it in detail.</p>