What is a "statistical" gauge field? In the Fractional Quantum Hall Effect (FQHE), one introduces a Chern-Simons (CS) gauge field and it is called statistical. Why? Another main question is below (*), but maybe I should state some things I think first.
I understand the CS theory gives an electromagnetic response which reproduces the expected Hall conductivity, and I (think) the gauge field itself was first realized as a singular gauge transformation on the wave function. I understand this transformation leads to a composite Fermion picture as well.
(*)I see Landau-Ginzburg (LG) approaches which just introduce these CS gauge fields. They have a role of "eating" up the e&m Goldstone modes, which is important to the phenomenology of FQHE. Is there anything in particular which calls for the introduction of coupling your theory to a statistical gauge field? Is this just a mapping from one hard problem to a more workable one, or is there more here I am missing?
 A: The statistical gauge field is an emergent field that is a better suited variable to describe the low-energy degrees of freedom in the fractional quantum hall state. It is called 'statistical' because it is a dynamical degree of freedom, i.e., it must be integrated over to get the partition function for the system. There are various ways in which one gets this dynamical gauge field in FQH phenomena.

*

*To describe gapped $\nu = 1/m$ Laughlin states ($m$ even for boson states, odd for fermonic states), one can write the particle current in terms of a new gauge field $a_\mu$
$$
j^\mu = \frac{1}{2\pi}\epsilon^{\mu\nu\lambda}\partial_\nu a_\lambda .
$$
Note that, ignoring topological obstructions (assume we are in infinite flat spacetime), any current field satisfying the continuity equation $\partial_\mu j^\mu = 0$ can be represented in the above form. Moreover, there is a gauge structure/redundancy as $a_\mu$ and $a_\mu + \partial_\mu \chi$ denote the same current configuration, and therefore the same physical state. 
Next, we guess an effective action in terms of the new d.o.f., which leaves us with the action
$$
S[a,A] = \int d^3x \left( \frac{m}{4\pi}\epsilon^{\mu\nu\lambda}a_\mu\partial_\nu a_\lambda + \frac{1}{2\pi}\epsilon^{\mu\nu\lambda} A_\mu\partial_\nu a_\lambda\right).
$$
As you said, this action reproduces the correct Hall quantization. $a_\mu$ is an emergent dynamical/statistical d.o.f. that describes the microscopic particles, and must be integrated over to get the partition function. In contrast, $A_\mu$ is a background gauge field that describes the external electromagnetic field the system experiences. David Tong's notes (section 5.2) give a good introduction to this idea. 
I want to stress that the above is not a derivation from microscopic principles, instead it is more reminiscent of Landau-Ginzburg theory where we write down an effective action by looking at what the experiment is telling us (fractional Hall conductance). But, like L-G theory, it makes further predictions from this starting point (topological ground state degeneracy, for example).


*The composite fermion (CF) picture: The idea of using a dynamical gauge field to describe flux attachment to electrons was pioneered by Halperin, Lee, and Read. The idea here is to define a new composite fermion field
$$
\psi_{CF}^\dagger(\boldsymbol{r}) = \psi_{e}^\dagger(\boldsymbol{r}) \exp\left(-i n_\phi \int d^2 \boldsymbol{r}' \text{arg}(\boldsymbol{r}-\boldsymbol{r}') \rho(\boldsymbol{r}') \right), \qquad \rho(\boldsymbol{r}) = \psi_e^\dagger(\boldsymbol{r})\psi_e(\boldsymbol{r})= \psi_{CF}^\dagger(\boldsymbol{r})\psi_{CF}(\boldsymbol{r}),
$$
where $n_\phi$ is an even integer characterizing the number of fluxes attached to each electron. The exponent picks up a phase $2 \pi n_\phi$ when one CF winds around another. One way to implement this is by introducing a new gauge field $a_\mu \equiv (0,\boldsymbol{a})$ that couples to the CF field (with charge $n_\phi$) such that it satisfies $\nabla \times \boldsymbol{a}(\boldsymbol{r}) = 2 \pi \rho(\boldsymbol{r}) \hat{z}$. The required phase factor naturally arises as an Aharanov-Bohm phase once the CF is coupled to this gauge field. 
Right now, the gauge field $\boldsymbol{a}$ is not dynamical but is constrained in a very unnatural way. One way to make this better is by letting $\boldsymbol{a}$ be dynamical but constraining it using a Lagrange multiplier (which we will call $a_0$ for reasons that will become apparent). So, we add a term in the action that looks like
$$
S_{\text{constraint}} = n_\phi a_0\left(\frac{1}{2\pi}\epsilon^{ij}\partial_i a_j - \psi^\dagger_{CF} \psi_{CF}\right),
$$
where $a_0$ and $\boldsymbol{a} \equiv (a_1,a_2)$ are now dynamical. One can see that these terms can be interpreted as (1) a $U(1)_{n_\phi}$ Chern-Simons term for the dynamical gauge field $a_\mu \equiv (a_0,\boldsymbol{a})$, and (2) the usual coupling term between the potential and the charge density. Therefore the effective theory in our problem is that of a composite fermion interacting with a dynamical/statistical gauge field with a Chern-Simons term.


*The Parton construction: without going into too much detail, there is a different approach to the composite fermion picture using partons. The idea is to write down the electron field as that of a composite fermion and a boson (describing the flux attachment) $\psi_e^\dagger(\boldsymbol{r}) = \psi_{CF}^\dagger(\boldsymbol{r}) b^\dagger(\boldsymbol{r})$, with the constraint that the boson current matches the composite fermion current (in close analogy to the previous discussion). This constraint is enforced by a Lagrange multiplier which we call the parton gauge field
$$
S_{\text{constraint}} = a_\mu\left(j^\mu_{\text{boson}} - j^\mu_{CF}\right).
$$
You can read more about this approach in this talk by John McGreevy and this paper by Alicea, Motrunich, Refael, and Fisher.
In conclusion, various successful approaches to FQHE physics end up with an emergent dynamical gauge field with a Chern-Simons term simply because that turns out to be the correct degrees of freedom to describe the system with. This gauge field is called statistical as they are dynamical (my preferred term)/fluctuating. 
A slightly unrelated note, we can formally integrate out all gapped degrees of freedom to get an effective theory for the electromagnetic response of the system $S[A_\mu]$, and that will include a Chern-Simons term for the electromagnetic field. This theory will have gapped photons, which can be understood similar to the Higgs mechanism where the gapless photon modes are 'eaten up' by the emergent gauge fields in the system.
