Exact partition functions for deformed N = 2 $$ \mathcal{N}=2 $$ theories with N f = 4 $$ {\mathcal{N}}_f=4 $$ flavours

We consider the Ω-deformed N = 2 SU(2) gauge theory in four dimensions with Nf = 4 massive fundamental hypermultiplets. The low energy effective action depends on the deformation parameters ε1, ε2, the scalar field expectation value a, and the hypermultiplet masses m = (m1, m2, m3, m4). Motivated by recent findings in the N = 2∗ theory, we explore the theories that are characterized by special fixed ratios ε2/ε1 and m/ε1 and propose a simple condition on the structure of the multi-instanton contributions to the prepotential determining the effective action. This condition determines a finite set ΠN of special points such that the prepotential has N poles at fixed positions independent on the instanton number. In analogy with what happens in the N = 2∗ gauge theory, the full prepotential of the ΠN theories may be given in closed form as an explicit function of a and the modular parameter q appearing in special combinations of Eisenstein series and Jacobi theta functions with well defined modular properties. The resulting finite pole partition functions are related by AGT correspondence to special 4-point spherical conformal blocks of the Virasoro algebra. We examine in full details special cases where the closed expression of the block is known and confirms our Ansatz. We systematically study the special features of Zamolodchikov’s recursion for the ΠN conformal blocks. As a result, we provide a novel effective recursion relation that can be exactly solved and allows to prove the conjectured closed expressions analytically in the case of the Π1 and Π2 conformal blocks.