Transformations of Spherical Blocks

We further explore the correspondence between N=2 supersymmetric SU(2) gauge theory with four flavors on epsilon-deformed backgrounds and conformal field theory, with an emphasis on the epsilon-expansion of the partition function natural from a topological string theory point of view. Solving an appropriate null vector decoupling equation in the semi-classical limit allows us to express the instanton partition function as a series in quasi-modular forms of the group Gamma(2), with the expected symmetry Weyl group of SO(8) semi-direct S_3. In the presence of an elementary surface operator, this symmetry is enhanced to an action of the affine Weyl group of SO(8) semi-direct S_4 on the instanton partition function, as we demonstrate via the link between the null vector decoupling equation and the quantum Painlev\'e VI equation.