Recurrence relations for the $$ {\mathcal{W}}_3 $$ conformal blocks and $$ \mathcal{N}=2 $$ SYM partition functions

Abstract

Abstract Recursion relations for the sphere 4-point and torus 1-point $$ {\mathcal{W}}_3 $$ W 3 conformal blocks, generalizing Alexei Zamolodchikov’s famous relation for the Virasoro conformal blocks are proposed. One of these relations is valid for any 4-point conformal block with two arbitrary and two special primaries with charge parameters proportional to the highest weight of the fundamental irrep of SU(3). The other relation is designed for the torus conformal block with a special (in above mentioned sense) primary field insertion. AGT relation maps the sphere conformal block and the torus block to the instanton partition functions of the $$ \mathcal{N}=2 $$ N = 2 SU(3) SYM theory with 6 fundamental or an adjoint hypermul-tiplets respectively. AGT duality played a central role in establishing these recurrence relations, whose gauge theory counterparts are novel relations for the SU(3) partition functions with N f = 6 fundamental or an adjoint hypermultiplets. By decoupling some (or all) hypermultiplets, recurrence relations for the asymptotically free theories with 0 ≤ N f < 6 are found.