VEVs of Wilson loops in Chern-Simons theory with compact gauge groups give us colored Jones, HOMFLY and Kauffman polynomials. I have not seen the computation for Wilson loops in Chern-Simons theory with non-compact gauge groups. I think that what keep us from computing them is due to infinite dimensional representations of non-compact gauge groups.

Are there any references which describe this problem explicitly? What are the issues in computing Wilson loops in non-compact gauge groups? Is there any proposal to calculate them? Especially I am interested in the simplest cases of $SL(2,R)$ and $SL(2,C)$.

  • $\begingroup$ Hi @Qmechanic, are you doing the retagging of TP questions as from now? And should I then stop with this since you can do this much better than I can? $\endgroup$ – Dilaton Jan 7 '13 at 22:10
  • $\begingroup$ Hi @Dilaton: No, please continue your excellent much-needed retagging work. I just noticed that the CS tag was not used in some of the most popular CS questions. $\endgroup$ – Qmechanic Jan 7 '13 at 22:22
  • $\begingroup$ Ok @Qmechanic :-) $\endgroup$ – Dilaton Jan 7 '13 at 22:37

This is more of a comment.

The first obvious problem is that the partition function can sometimes be infinite. For example, $Z(T^3)$ is the dimension of the Hilbert space attached to $T^2$, which is infinite-dimensional for a non-compact group $G$.

The second problem is the choice of the representation. Skein relations in Chern-Simons arise due to finite-dimensionality of the Hilbert space attached to $S^2$ with 4 marked points (2 positively-oriented and 2 negatively-oriented). If the Hilbert space is, say, $n$-dimensional, the partition functions evaluated on $n+1$ different crossings should be linearly dependent, this is the skein relation. An easy computation shows that $n$ is the number of irreducible representations occurring in $V^{\otimes 2}$, where $V$ is the representation attached to the knot. So, on the one hand you want $V$ to be finite-dimensional (in which case there is a skein relation). On the other hand, finite-dimensional representations give the same answers as the compact form, at least on the perturbative level (this claim appears in Gukov's and Witten's papers), so people are not so interested in them.


How about using geometric quantization, as in here or here or here.


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