A compilation of dualities between 4D N=2 supersymmetric gauge theories and 2D conformal field theories

In this thesis AGT-W conjecture is reviewed. AGT conjecture is a phenomenon which is shown by Luis F. Alday, Davide Gaiotto and Yuji Tachikawa in their paper published in 2009. It relates the partition functions of both N=2 supersymmetric U(2), SU(2) gauge theories and Liouville field theory which is a type of conformal field theory. One month later, Niclas Wyllard extended the conjecture and proved the connection between N=2 supersymmetric SU(N) gauge theories and A_{N-1} Toda field theories. In addition to AGT-W conjecture, gauge/Liouville Triality, spectral duality and 2D/4D duality through Nekrasov-Shatashvili limit are summarized. N=2 supersymmetric gauge theories are also studied in type IIA, IIB and topological string theory background.