I found in physics that Chern-Simons theory is closely related with three dimensional gravity.

From this paper Three Dimensional Gravity Revisited, the author talks about the Chern-Simons for

$$\mathrm{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)$$

with the gauge field $A\in\Omega^{1}(M,\mathfrak{g})$. However, in the gravity case, the author is talking about gauge groups $SL(2,\mathbb{R})$, $SO(2,2)$, or $SO(2,1)$.

Usually, in mathematics textbooks, the Chern-Simons theory is defined as the secondary class and the structure groups are usually $U(1)$, $SU(N)$ and $SL(2,\mathbb{C})$, so that it makes sense to talk about Chern classes of complex vector bundles.

Topologically, the group $SL(2,\mathbb{R})$ is $S^{1}\times\mathbb{H}^{2}$, where $\mathbb{H}^{2}$ is the two dimensional Poincare disc.

1. How does this group act on a complex vector space?

2. How to defined the second Chern-class with this structure group?

In the beginning, I thought that the $SL(2,\mathbb{R})$-Chern-Simons theory is defined via the first Pontryagin class. However, on page 13, the author claims that the quantization of the first Chern class of $SL(2,\mathbb{R})$ is equal to that of $U(1)$-line bundle because $SL(2,\mathbb{R})$ is contractible to $U(1)$.

3. Why doesn't such a "shrinking" of structure group affect the quantization of the first Chern-class?

  • $\begingroup$ I guess $SL(2,\mathbb{R})$ Chern-Simons theory should be equivalent to $SL(2,\mathbb{C})$ Chern-Simons theory with structure group restricted to $SL(2,\mathbb{R})$. So given a representation of $SL(2,\mathbb{R})$, we can complexify that representation, and there must be the associated complex vector bundle. Then we can talk about second Chern class, etc. $\endgroup$
    – Henry
    Aug 2 '18 at 0:13
  • 1
    $\begingroup$ Moreover, $SL(2,\mathbb{R})$ being contractible to $U(1)$ means that any vector bundle with structure group $SL(2,\mathbb{R})$ can be smoothly deformed to a vector bundle with structure group $U(1)$. Because first Chern classes take discrete values, statements on page 13 about $U(1)$ bundles should carry over immediately to the case of $SL(2,\mathbb{R})$ bundles. $\endgroup$
    – Henry
    Aug 2 '18 at 0:29
  • $\begingroup$ Hi @ Sunghyuk Park Would you give me some references talking about details of the deformation? $\endgroup$ Aug 2 '18 at 9:15
  • $\begingroup$ A relevant paper - projecteuclid.org/download/pdf_1/euclid.cmp/1104249532 $\endgroup$
    – Mtheorist
    Apr 4 '19 at 15:50
  • $\begingroup$ @Mtheorist Thank you for the link. I had read that paper. $\endgroup$ Apr 4 '19 at 17:32

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