Topological Recursion in the Seiberg-Witten partition function via AGT correspondence

A thorough, self-contained review of Seiberg–Witten theory and instanton calculus for $\mathcal{N}=2$ $SU(N)$ gauge theory is presented. The necessity of algebro-geometric techniques—such as non-commutative resolutions and localization—for computing the instanton partition function more efficiently is discussed. Finally, a remarkable duality between 2D CFT(Liouville field theory) and 4D $\mathcal{N}=2$ SYM is used to bridge the lesser-known classical Zamolodchikov recursive relation [Zamolodchikov, 1987] and Nekrasov’s partition function. This provides an instance of Topological Recursion, which would otherwise require highly sophisticated mathematics to uncover.