Consider $U(1)$ Chern-Simons theory on some three-manifold M: $$S = \frac{k}{4\pi}\int_M A \wedge dA.$$ The standard argument for why we require $k\in \mathbb{Z}$ comes from demanding invariance under large gauge transformations when $M$ is some manifold like $S^1\times S^2$, and a single unit of magnetic flux is placed through the $S^2$.

I'm curious why we still require that $k\in \mathbb{Z}$ when $M$ is some (physically relevant) manifold like $\mathbb{R}^3$, or even $T^2 \times \mathbb{R}$.

Furthermore, to derive the quantization of $k$ we had to assume that the bundle was non-trivial, i.e. that there was a net magnetic flux through $M$ (assuming now that $M$ is compact). In real-life manifestations of Chern-Simons theory (like the QHE), we will never have a net magnetic flux. One might say that we need to sum over all magnetic flux values in the path integral, but certainly we do not do this in e.g. the context of the QHE, since non-zero net fluxes through $M$ are unphysical if $A$ is the $A$ of electromagnetism.

An argument I could imagine making would be "the theory should be the same regardless of what manifold we put it on", and so if $k$ needs to be quantized for one choice of $M$ then it should be quantized for all $M$. This isn't super convincing to me though (what prevents me with working with a theory that is only defined on certain types of manifolds?), and plus, it seems like we don't require this of other theories — for example, we don't require that all theories (e.g. ones with fermions) make sense on non-orientable manifolds.

  • $\begingroup$ The EM field is not dynamical in the usual treatment of QHE. For FQHE there is an emergent dynamical gauge field (or several) which exists only on the surface and may be in any flux sector. $\endgroup$ – Ryan Thorngren May 20 '18 at 2:13
  • $\begingroup$ I'm fine with emergent gauge fields being in any flux sector, but the level is supposed to be quantized for the IQHE, where we just have an effective action for the EM field alone — in this case, being in a nontrivial flux sector is unphysical. $\endgroup$ – user3521569 May 20 '18 at 16:56
  • $\begingroup$ The quantization of the level comes from the proper definition of the AdA term, which is only locally a 3-form, and must be glued together by gauge transformations. arxiv.org/abs/hep-th/0505235 $\endgroup$ – Ryan Thorngren May 20 '18 at 18:29
  • $\begingroup$ Are you talking about how the "proper" definition is by doing $\int_{M_4} F\wedge F$? I agree that if you define things this way then the level is quantized. But this definition is not possible when the 3-manifold you're integrating $A\wedge dA$ over is not closed. For a physically relevant spacetime like $D^2 \times \mathbb{R}$, I can just use a single coordinate patch and $AdA$ seems like a fine 3-form to use globally. $\endgroup$ – user3521569 May 21 '18 at 0:22
  • $\begingroup$ You should choose gauges in local patches and write AdA in each patch and try to glue them. Even in contractible space the level will need to be quantized. $\endgroup$ – Ryan Thorngren May 21 '18 at 18:21

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