First of all it's necessary to correct an error: 2 spacetime dimensions means 1 spacelike and 1 time-like dimension; 3 spacetime dimensions means, usually, 2 spacelike and 1 time-like dimension (though one should be careful with AdS.).  

For the former one needs QED2 for the latter QED3. They're different theories. 

In 2+1 dimensions the Chern-Simons term can be obtained as the counterterm that arises when computing the vacuum polarization, for the abelian case. 

It might be useful to read http://inspirehep.net/record/214349

Hey, thanks for the quick answer. Sorry, I just forgot the +1. This was simply a typo. We deal with 2+1 spacetime dimensions. Well, I want to make sure, that I want to have a lagrangian of the full theory. Obviously, integrating out the fermions give me a chern simons term in the effective theory. However, thank you for the link! I think this will help a lot!

Let us assume that the fermions of the theory have a nonzero mass. If I calculate the photon polarisation tensor  I also obtain the chern simons term. However,  the renormalization of fields and couplings would not generale a counterterm which cancels this term. Thus I need to implement it per hand. Is this allowed?

Last but not least, in pseudo QED one changes the F_mu_nu F^mu^nu term to get the right electron-electron interaction (1/r). Should I take also this ansatz for the unrenormalized lagrangian?

In 2+1 dimensions, the fermion mass term is pseudoscalar, so if it's present-the fermions are massive-then, in fact, the photon is massive, too-this was noted by Deser, Jackiw and Templeton and is reviewed by Coleman and Hill. 

There's no question of adding it by hand-it's there because the symmetries of the theory require it. It can be canceled however, by fermion flavors of the appropriate properties. 

There's no such notion as ``pseudo QED'', this is just a term that doesn't mean anything. In any dimension QED describes the appropriate interaction of fermions with an Abelian gauge field that has the corresponding static limit, which is the corresponding Coulomb term. It's not 1/r in 2+1 dimensions, of course, but that's expected. 

I unterstand. So to describe massive fermions, which live in 2+1 D, due to a broken party symmetry (fermion mass term) I have to add a chern simons term even in the bare lagrangian, right? Do you have a paper, describing how one could cancel the term due to fermion flavours?  Is then the right lagrangian describing the (integer) quantum Hall effect? 

People like E.C. Marino and Sergio Alves claim that confining fermions to the plane does not mean that the gauge fields are restricted to the plane in experiments, thus pseudo QED.  Do you agree? They say to obtain the experimental right coulomb interaction, the dynamic gauge term must change. Do you unterstand the difference here to the high energy ansatz?

Of course-that's the point: if you don't add it, you'll find it as a counterterm, just like in any field theory: you include in the Lagrangian all terms consistent with symmetries and power counting. 

One can cancel it by having flavors of opposite sign of the mass, since the coefficient of the Chern-Simons term is proportional to the sign of the mass. 

 Start by reading Coleman and Hill.  Essentially all these issues and more are discussed there. 

It's wrong to talk about confinement here: the theory *is* defined  in 2+1 dimensions, so the magnetic field is a pseudoscalar, not a pseudovector. And the matter fields, too, are defined in 2+1 dimensions. The theory isn't, necessarily,  a reduction from 3+1. 

The ``right'' Coulomb interaction is that that's defined by the theory, when it's a consistent quantum field theory-which can be shown in many different ways.  So there isn't any ``high energy Ansatz'' involved here. 

There's no meaning to the word ``restriction'', because what matters is that the 2+1-dimensional theory is well-defined. How it can be obtained from a 3+1-dimensional theory is another issue-there are many ways. But it doesn't make sense, if one is interested in the theory in 2+1 dimensions. What matters is, whether it's a consistent quantum field theory. 

Thanks for this precise answer! That means that in our case the fermions are always described by anyons, due to flux charge coupling (if I include a minimal coupling to the hamiltonian)? However, why people then introduce the concept of pseudo QED gets not very clearify for me. Did you unterstand their argumentation  in detail?  

The spinor field in 2+1 isn't, necessarily, an anyon-that's an additional phase to the Wilson loop factor and it's independent of the gauge field, of course. The spin-statistics theorem simply allows for any phase, under exchange, when constructing many-particle states-it doesn't imply that the phase will be affected, inevitably. Fermions and bosons can be defined unambiguously. 

I don't understand what ``pseudo QED'' means. The mathematical description is what's relevant. What is true, once more, is that, in 2+1 dimensions, fields, that transform as spinors under the Lorentz group, and carry charges  in certain representations of the gauge group, don't remain massless; therefore the corresponding gauge field doesn't remain  massless, either-and vice versa. One can choose which term to start with and find the other one. Therefore, in the U(1) case,  the Coulomb interaction is screened. 

Well, I think now I got the right understanding of the problem. Please correct me if I am wrong.

The IQH as well as the FQH effect deal with electrons, living in 2+1 D. Due to current conservation and symmetries of the system (broken parity and time reversal), their gauge field has a mass, lives in 2+1 D, and comes with the CS term. This is its most important term in the long-range limit (small momenta, low energy). The CS coefficient charaterizes then the statistics of the particles. However the minimal coupling to the external magnetic field (lives in 3+1 D) is characterized by a minimal coupling of the CS current and the vector potential in 3+1D  (this is treated as a pertubation). Integrating out the internal gauge fields gives then the effective action, just depending on the external vector potential, which gives raise to the QH response.

The coefficient of the CS isn't inevitably related to the statistics of gauge field or spinor excitations. Nor are the statements about the 3+1 dimensional theory obviously correct. And there aren't any internal vs. external vector potentials either. 

The 2+1 dimensional theory can be studied on its own merits-it doesn't need any relation to a 3+1 dimensional theory. 

If one does wish to investigate such a relation, then the 2+1 manifold is a boundary of some  3+1 dimensional manifold and one needs to integrate out more than just one component of the gauge field-and it is by no means obvious that one will get a sensible theory at all. One way to attempt something that could be reasonable is to introduce defects, e.g. https://arxiv.org/abs/0710.1714 , https://arxiv.org/abs/1001.1885 and https://arxiv.org/abs/1010.5281