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I know that usually, the way to compare the Hilbert spaces arising from $SU(2)$ Chern-Simons theory with different Kähler polarizations is via the Hitchin connection. However, it should be possible, I would think, to try to use a BKS pairing between them. Are any results known on whether this pairing is unitary? (My guess would be that the answer is no, based upon the results of Kirwin et al for toric varieties, but then the character varieties that are quantized in Chern-Simons theory aren't really toric varieties).
Likewise for real polarizations, I know Jeffrey and Weitsman did some work on BKS pairings but I don't recall seeing any theorems in their papers about whether the pairing was unitary for real polarizations in general. Has there been further work done on the pairings for real polarizations?

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is it known that BKS and Hitchin-KZ are different ? In the context of CS Hitchin=KZ are so natural that I would guess any natural construction should coincide with it... By the way BKS is Blattner-Kostant-Sternberg, am I right ? – Alexander Chervov Apr 25 2012 at 6:31
Yes, sorry, BKS is Blattner-Kostant-Sternberg. They two constructions cannot be exactly equal. Hitchin depends on a path through Teichmuller space (although it is projectively path-independent). BKS doesn't, so they are at most projectively equivalent. – Blake Apr 25 2012 at 19:31

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