Some steps towards noncommutative mirror symmetry

We start by reviewing some small aspects of the recent work of Kapustin and Witten [KW] on S-duality and the Geometric Langlands Program, especially how this leads to the question of an extension of mirror symmetry to noncommutative spaces. Next, we consider the application of one of the steps of [KW] to the gauge theory on the D5-brane worldvolume in Type IIB string theory. We will see that in this case the inclusion of noncommutative deformations is no longer just an option but is necessary in the generic case. We will show that in a special case one can factorize the fields in such a way that one can reduce everything to questions on mirror symmetry on the noncommutative 2-torus. Following the - by now classical - approach of Dijkgraaf to mirror symmetry on elliptic curves, we will see how q-deformations of special functions arise. We will pose the qustion how this is related to field theories on the noncommutative 2-torus.