The subject of dualities in physics has become a very popular topic these days. The examples that immediately come to mind are: AdS/CFT correspondence, AGT duality, mirror symmetry, non-perturbative dualities in string and M-theory. The latter are better known as T and U-dualities and are briefly reviewed in this post.
The most transparent way of understanding T-duality is dimensional reduction by compactification. It is useful to start with the Kaluza-Klein reduction of the ordinary field theory which leads to an infinite tower of massive states. Consider a theory of a massless scalar field on a $D+1$ dimensional space with coordinates $x^{\mu}$ which parametrize $D$ directions and one coordinate $z$ which is compact. Equations of motion for this theory are given by the Klein-Gordon equation
\begin{equation}
\partial_M\partial^M \phi =0.
\end{equation}
\partial_M\partial^M \phi =0.
\end{equation}
In the most general form the field $\phi(x^\mu,z)$ can be written as a Fourier transform with respect to the compact coordinate $z$
\begin{equation}
\phi(x^a,z)=\sum_{n}\chi_n(x^a)e^{2\pi i\frac{nz}{R}}.
\end{equation}
\phi(x^a,z)=\sum_{n}\chi_n(x^a)e^{2\pi i\frac{nz}{R}}.
\end{equation}
Substituting this back to the initial equation we see that the modes $\chi_n(x)$ satisfy the massive Klein-Gordon equation
\begin{equation}
(\partial_\mu\partial^\mu -m_n^2)\chi_n(x)=0
\end{equation}
(\partial_\mu\partial^\mu -m_n^2)\chi_n(x)=0
\end{equation}
with masses $m_n=2\pi n/R$. The only massless mode left is the one with $n=0$ while the others form the Kaluza-Klein tower of states with masses proportional to $1/R$. There are no any signs of T-duality symmetry so far.
The main feature of string theory is that it describes motion of one-dimensional objects rather than point-like particles. Point particles and string-like objects probe geometry of a background in very different ways. For example, point particles can feel all singularities of the space-time while closed strings can't distinguish between small and large circles. This allows to consider singular manifolds like orbifolds as appropriate backgrounds for string theory.
The reason for such behaviour of a closed string is T-duality that exchanges translational and winding modes and thus replaces radius of a compact direction $R$ with its inverse $\alpha'/R$. For an appropriate gauge choice the equations of motion for a string read
\begin{equation}\partial_a\partial^aX^{M}(\tau,\sigma)=0,\end{equation}
where $\sigma^a=\{\tau,\sigma\}$ are the coordinates on the worldsheet. Boundary condition on a closed string on the background with one compact direction of the radius $R$ differ from the ordinary ones and read
\begin{equation}\begin{split}&X^{\mu}(\tau,\sigma+2\pi)= X^{\mu}(\tau,\sigma),\\&X^{D}(\tau,\sigma+2\pi)= X^{D}(\tau,\sigma)+2\pi m R.\end{split}\end{equation}
Here the parametrisation of the string worldsheet was chosen in such a way that $\sigma$ runs from 1 to $2\pi$. The integer $m$ denotes the winding number of the string or in other words it shows how many times the string has wrapped around the compact direction. The ordinary field theory doesn't have an analogue of winding modes.
The requirement that the string wavefunction is uniquely defined leads to the quantization of momentum. The string wavefunction is given by the vertex operator acting on the ground state
\begin{equation}|\zeta_{\mu_1\ldots\mu_{l}},p^{\mu}\rangle = \int d\sigma \Pi(\zeta,X^{\mu})e^{ip_{M}X^{M}}|\mbox{ground}\rangle,\end{equation}
where $\Pi(\zeta,X^\mu)$ is some irrelevant for the discussion function that depends on the polarization tensor $\zeta_{\mu_1\ldots\mu_n}$. Since $X^{D}$ is a cyclic coordinate that is defined up to $2\pi m R$ the corresponding momentum $p_{D}$ should be discrete and proportional to $m/R$.
Hence, we have two integer numbers in our theory, one comes from the quantisation of momentum and corresponds to the Kaluza-Klein mode, the other is related to the boundary condition and represents the winding mode. The latter doesn't have an analogue in the ordinary field theory since it is related to the extended nature of strings.
T-duality symmetry can be easily seen from the mass spectrum of a closed string
\begin{equation}M^2=\frac{n^2}{R^2}+\frac{m^2 R^2}{\alpha'^2}+\frac{2}{\alpha'}(N+\tilde{N}-2),\end{equation}
where $N$ and $\tilde{N}$ are the level operators and the constant $\alpha'$ is related to the string tension. One immediately concludes that the mass spectrum is invariant under the replacement of the radius $R$ by its inverse $\alpha'/R$ and the translational mode $n$ by the winding mode $m$. As it was mentioned above this leads to the equivalence of small and large circles from point of view of a closed string and provides a natural smallest distance $\sqrt{\alpha'}$ that can be probed by a string.
The other way to come up with the invariance of string theory under T-duality transformations is using the so-called Buscher rules. These exploit the same idea of compactification on tori introducing a gauge fields corresponding to the action of cycles of these tori. The last step is to do some change of variables and to see that the obtained action is the same up to change of variables. This approach is pretty straightforward and I do not want to go through all these transformations here. Instead, I refer to the thesis by Daniel Thompson that contains clear explanation of the stuff as well as the spectral approach briefly reviewed above.
What is more interesting and more relevant to the duality invariant formalism of Doubled Field Theory is the so-called Duffs' procedure. Alternatively to the previous approaches it is not concerned with compactifications but reveals the symmetry that mixes equations of motion and Bianchi identities. The idea is to introduce a dual Lagrangian of some dual fields $\tilde{X}_\mu$ whose equations of motion give Biacnhi identities for the fields $X^{\mu}$ and whose Bianchi identities give equations of motion for $X^\mu$. I'll live the details to the next post.
