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

## TL;DR

This paper develops recursion relations for ${
m W}_3$ conformal blocks on sphere and torus, connecting them to ${
m N}=2$ $SU(3)$ SYM partition functions via AGT duality, and extends these relations to asymptotically free theories.

## Contribution

It introduces new recursion relations for ${
m W}_3$ conformal blocks and links them to ${
m N}=2$ $SU(3)$ SYM partition functions, generalizing Zamolodchikov's relation.

## Key findings

- Derived recursion relations for sphere ${
m W}_3$ conformal blocks.
- Established relations for torus ${
m W}_3$ conformal blocks.
- Extended relations to asymptotically free theories with fewer flavors.

## Abstract

Recursion relations for the sphere $4$-point and torus $1$-point ${\cal W}_3$ 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 ${\cal N}=2$ $SU(3)$ SYM theory with 6 fundamental or an adjoint hypermultiplets 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 $0\le N_f<6$ are found.