We start by putting a disclaimer that Langlands program is a massive program and contains many abstract subtleties which we neither address here nor we think of them, at least when the focus is on theoretical physics. Langlands program is the study of the map from automorphic forms and representation theory to Galois theory. If we wish to talk about the Langlands correspondence for $GL_n/K$ where $K$ is a finite field extension of $Q_p$, then the correspondence is a canonical bijection between the $\infty$-dim irreducible $\mathbb{C}$ representation of $GL_n/K$ and $n$-dim complex representation of Galois group $Gal(\bar{K}/n)$. This was a local Langlands correspondence but such a duality is of non-interest to physicists. There exist many manifestations of the Langlands program in mathematics and physics. Instead, we are interested in geometric Langlands correspondence. It is surprising to note that mirror symmetry appears when we talk about geometric Langlands correspondence between gauge theories {Kapustin:2006pk}.
Monopoles were first subjected in a paper by Dirac in 1931 {dirac1931quantised}, in which there is a monopole sitting at the origin. Such a thing is without consequence and thus we find a quantization condition, also called {\it Dirac's quantization condition}. For subsequent developments in physics of it, refer to {preskill1984magnetic}. In this paper, we are concerned (mostly) with mathematical developments. We now fast forward to the paper by Goddard, Nuyts and Olive (GNO) in 1976 {Goddard:1976qe}.
GNO puts forward the idea that for a compact gauge group $G$, the electric charges take values in the weight lattice of the gauge group $G$ while magnetic charges take values in the weight lattice of gauge group $G^L$ where $G^L$ is the Langlands dual of the gauge group $G$. This is a very profound result in mathematical physics. The dual group could also be called GNO dual group. When we move ahead, we have the Monotonen-Olive conjecture {montonen1977magnetic,Osborn:1979tq} which states that the coupling constants of the group $G$ and $G^L$ are equivalent if
$$ \alpha \longleftrightarrow \frac{1}{\alpha'} $$
which might look familiar to people who have seen S-duality in string theory. Well, there are quite many similarities between string theory and such rich mathematics but we would not be able to cover all of them in this note. One interesting, however, is the statement that categories of A-branes defined over the moduli space of Higgs bundle over some Riemann surface $C$ for $G$ and $G^L$ are equivalent. A similar statement can be made for categories of B-branes defined over the moduli space of flat connections that are equivalent for $G_\mathcal{C}$ and $G^L_\mathcal{C}$ {Kapustin:2008pk,Kapustin:2006pk}.
In other words, the irreducible representations of the gauge group $G$ are known to be the Wilson loops, then by GNO duality, there exist irreducible representations of a dual group $G^L$ which are called `t hooft loops. Monotonen-Olive conjectured that group $G$ and $G^L$ are isomorphic if the Yang-Mills coupling constant are inverse to each other. For physicists, it same as S-duality of coupling constants and in the right context, indeed, the geometric Langlands correspondence is S-duality of $\mathcal{N}=4$ super Yang-Mills theory.
$\mathcal{N}=4$ super Yang-Mills theory is a perfect setting for this exposition. It means that we have four copies of the representation of supersymmetry (SUSY) algebra. The minimum degree of SUSY is $\mathcal{N}=1$. Note that $\mathcal{N}=4$ is the maximum number of SUSY we can have in six dimensions. The reason why we start with a six-dimensional manifold $\mathcal{M}_6$ is that it naturally leads to S-duality action when compactified {Kapustin:2006pk,Witten:2009at}. We can compactify $\mathcal{M}_6$ on a 2-torus (product space of $S^1 \times S^1$)
$$ \mathcal{M}_6 = \mathcal{M}_4 \times T^2. $$
Recall that $\mathcal{M}_6$ does not easily admit an action. But $\mathcal{M}_4$ does. However, $\mathcal{M}_6$ admits a 3-form $H$ (similar to 2-form $F$ in four-dimensional Maxwell's equation) where $H$ is a curvature term of some ($U(1)$) gerbe connection. This 3-form is related to the 2-form $F$ over $\mathcal{M}_4$ after compactification. On $T^2$, the conformal structure is provided by a complex parameter $\tau$ in the upper half of $\mathbb{c}^2$ (as the imaginary part of $\tau$ is always positive). One can now write the 3-form $H$ after the dimensional reduction as
$$ H = F \wedge dx + \star F \wedge dy $$
and because of the Bianchi identity $dH=0$, we have $dF=0$ and $d \star F =0$. One can similarly generalize it by relating a self-dual theory in $4k+2$ dimensions with $2k$ form curvature in $4k$ dimensions. In this case, a six-dimensional quantum field theory admits a self-dual curvature, which is also the reason for not having a definite action attached to it. See {Witten:2009at} for a discussion over this.
The parameter $\tau$ is given as
$$ \tau = \frac{\theta}{2\pi} + \frac{4 \pi i}{e^2} $$
and it has two symmetries in QFT, where one is classical
$$ T \colon \tau \rightarrow \tau +1 $$
and the other one is quantum symmetry
$$ S \colon \tau \rightarrow - \frac{1}{\tau} $$
and together these two generators $S$ and $T$ generates the modular group $SL(2,\mathbb{Z})$ which is a subgroup of $SL(2,\mathbb{R})$. This group is very special in string theory and unsurprisingly also in mathematics, especially in representation theory and Langlands program itself. The symmetry $S$ relates the coupling constants and thus we have a S-duality in the four-dimensional reduction.
Let us now start with a ten-dimensional manifold $\mathcal{M}_{10}$ and compactify six dimensions, then the action of the four-dimensional theory becomes
$$ S = \frac{1}{g} \int d^4x {\rm Tr} \frac{1}{2}\sum_{\mu, \nu=0}^3 F_{\mu \nu}F^{\mu \beta}\\ + \sum_{\mu=0}^3 \sum_{i=0}^6 D_\mu \phi_i D^\mu \phi_i + \frac{1}{2} \sum_{i,j=1}^{6}[\phi_i,\phi_j]^2 + \cdots $$
where $\cdots$ represents the fermionic part of the action. $g$ in the action is the Yang-Mills coupling constant. For brevity, the fermionic part can be ignored until the SUSY becomes important. In this action, we add a topological term
$$ S_\theta = -\frac{\theta}{8\pi^2} \int d^4x\ {\rm Tr}(F \wedge F) $$
and thus we have the complex parameter with us
$$ \tau = \frac{\theta}{2 \pi} + \frac{4\pi i}{e^2} $$
and the S-duality action with this.
There will be more posts for the Langlands program in the coming weeks.
