Are dualities in QFT just change of variables? Although a lot is usually said about dualities in QFT, this question doesn't seem to have any straightforward answer in most of the references. When we talk about dualities between QFTs what do we mean? The easiest ones, the ones with proven duality, is always a change of variable. Now I understand that to find this change (or relating fields of the two sides) may not be trivial. Hence we may need other ways(correlators etc). But is it always that?
There is also the question of phase space. For instance in (2+1)D Gravity-Chern-Simons duality, the action reduces from one to the other but the phase spaces are different. Gravity does not allow degenerate metric but CS theory does allow for zero gauge field solutions. So when can we say dualities are exact?
Any references with clear and concise explanation would be enough.
 A: Describing the the physical observables of a quantum theory in terms of deformations of a partition function $Z$ one can ask about a Lagrangian. If a Lagrangian exist
$$
Z=\int [d\phi^{\alpha}]e^{-S[\phi^{\alpha}]}\,\qquad S[\phi]=\int d^{D}x L(\phi,\partial\phi).
$$
we compute correlation functions of $\phi^{\alpha}$ in (space-)time by deforming the Lagrangian
$$
Z[j_{\alpha}]=\int [d\phi^{\alpha}]e^{-S[\alpha]+\int d^D x j_{\alpha}\phi^{\alpha}}
$$
and then taking functional derivatives of $j_{\alpha}$ at $j_{\alpha}=0$. This does not cover the whole space of deformations since there is a counter-example: one can deform the partition function by restricting the field configurations to be singular at some locus in (space-)time. This types of physical observables are known as deffects.
Sometimes we do not known if a Lagrangian exist, as in the case of $d=6$ $\mathcal{N}=(2,0)$ superconformal field theory, however a  partition function still exist, and the map between deformations and computable quantities are more hard to establish.
Dualities comes from different ways of writing the partition function. In particular there are more than one Lagrangian description for a given partition function
$$
Z=\int [d\phi^{\alpha}]e^{S[\phi^{\alpha}]} = \int [d\varphi^{\alpha}]e^{-\tilde S[\varphi^{\alpha}]}
$$
and in order to establish a dual map one should provide a dictionary that translate deformations of one side into deformations of the other side of the equality above.
An example: in $d=4$ Maxwell theory the Lagrangian description of the partition function is a path integral over a one-form $A$ modulo gauge transformations $\delta A=d\phi$. Local functionals of A that are gauge invariant are functions of the Faraday two form $F=dA$, the field strenght of $A$. The quantum mechanical equations of motion (Dyson-Schwinger) (up to factors of $i$ and signs) is
$$
d*F=\hbar \frac{\delta}{\delta A}
$$
Note that $dF=0$ is an identity since $F=dA$, so one cannot deform $dF=0$ by coupling the Lagrangian with sources. In order to deform this identity one should introduce magnetic monopoles, which are deffects.
There is a dual Lagrangian description of Maxwell theory. The partition function of this dual description is a path integral over the one-form $\tilde A$ where now $*F= d\tilde A$ and so $d*F = 0$ is automatically solved. The quantum mechanical equations of montion of this dual description is, up to signs and factos of $i$
$$
dF=\hbar \frac{\delta}{\delta A}\,.
$$
Simple deformations of the path integral over $A$ will translate into complicate deformation of the path integral over $\tilde A$ and vice-versa. Is more easy to couple this theory with a an electric charge using the path integral over $A$, while coupling this theory with magnetic charges might be simpler with the path integral over $\tilde A$. Section 3 of this paper explain well this duality.
The example above ilustrate that dualities are a changing of variables, but non-trivial ones, which are usually non-local in the space of fields, and involve performing integrals to get from one side to the other.
The space-time is also a variable that can change under duality. An example is the AdS/CFT duality: the partition function of a quantum gravitational theory in $AdS_{d+1}$ is equal to a partition function of a CFT that lives at the $d$ dimensional boundary of $AdS_{d+1}$. The map between deformations is
$$
\langle e^{j_{\alpha}\phi^{\alpha}}\rangle_{CFT} = Z_{AdS}[j_{\alpha}]
$$
where $j_{\alpha}$ determines the boundary conditions of fields in $AdS_{d+1}$. There are other types of deformations that we can do in both sides, and keep track of the dictionary of these deformations is a at some extent a conjectural task.
Finally, note that this duality can be used to describe CFTs in situtations where a Lagrangian formulation is not yet avaliable. For example, $d=6$ $\mathcal{N}=(2,0)$ superconformal field theory, see this.
Even the statistic of the "fundamental fields", the fields in which the path integral is performed, may change under duality, like bosonization in $d=2$, see also this.
