In the gauge theoretic approach, the geometric Langlands correspondence is understood as a mirror symmetry.

Let Y(X;G) be the moduli space of flat G-bundles over an oriented 2-manifold X, where G is a complex reductive group. It is a symplectic manifold and it has a complex structure coming from the complex structure of G. Using the complex structure one can define a B-model and using the real symplectic structure one can define an A-model.

Now, let G^L be the Langlands dual group of G. It turns out that there is a mirror symmetry between the B-model of Y(X;G^L) and the A-model of Y(X;G). To establish this mirror symmetry one needs the fact that these spaces have another interpretation as moduli spaces of Higgs bundles.

In this talk we plan to introduce the audience to the moduli space of parabolic (ramified) Higgs bundles by studying an explicit example, of parabolic SL(2,C)-Higgs bundles over an elliptic curve, given by Frenkel and Witten in http://arxiv.org/abs/0710.5939v3

The Langlands dual group of SL(2,C) is PGL(2,C) we plan to continue the example in a second talk where we will get to the SYZ mirror symmetric picture.