/sci/ - Science & Math

>Actually allowing /sci/ to dictate you on what to study

Alright you fucks. I have a physical model [math]\mathcal{L}[/math] on a manifold with boundary [math]M[/math] and a symmetry [math]G[/math] of [math]\mathcal{L}[/math]. I know the generators of the Lie algebra [math]\mathfrak{g}[/math] of [math]G[/math] and that the manifold [math]M[/math] requires three patches [math]U, U_\pm[/math]. I have written down the Chern-Simons term on the interior and the boundary of the model
[eqn]
\int_{\partial U}\operatorname{tr}\left(AdA+\frac{2}{3}A^3\right) + \sum_{\sigma = \pm}\int_{\partial U_{\sigma}}\operatorname{tr}\sigma\left(A_\sigma dA_\sigma +\frac{2}{3}A_\sigma^3\right),
[/eqn]
the goal is to use this to characterize the fractional quantum Hall effect. However, under gauge transformations the Chern-Simons terms give
[eqn]
\int_{\partial U}\operatorname{tr}\left(g^{-1}dg\right)^3 + \sum_{\sigma = \pm}\int_{\partial U_\sigma}\operatorname{tr}\left(g_\sigma^{-1}dg_\sigma\right)^3
[/eqn]
where [math]g, g_\pm[/math] are the coordinate transition maps on the three patches. Now [math]M[/math] is 3-dimensional so these gauge transformations are characterized by [math]\pi_3(G)[/math], which I'm inclined to believe is [math]\mathbb{Z}[/math], which doesn't contain any fractions.
How do I get fractional topological quantum numbers from Chern-Simon terms?