CFT exercises for the needs of AGT

TL;DR

This paper provides explicit formulas for W-algebra conformal blocks and correlators for N=3, facilitating computational verification of the AGT relation between conformal field theory and gauge theory.

Contribution

It presents the simplest expressions and derivations of the Shapovalov matrix and triple correlators for W_3 algebra at low levels, enabling further studies of AGT correspondence.

Findings

Explicit formulas for W_3 conformal blocks at low levels

Derivations of Shapovalov matrix elements and correlators

Tools for computerizing AGT relation checks

Abstract

An explicit check of the AGT relation between the W_N-symmetry controlled conformal blocks and U(N) Nekrasov functions requires knowledge of the Shapovalov matrix and various triple correlators for W-algebra descendants. We collect simplest expressions of this type for N=3 and for the two lowest descendant levels, together with the detailed derivations, which can be now computerized and used in more general studies of conformal blocks and AGT relations at higher levels.