CFT exercises for the needs of AGT
Andrei Mironov, Sergei Mironov, Alexei Morozov, Andrey Morozov
TL;DR
The paper confronts the need for explicit Shapovalov matrices and triple-vertex data to validate AGT relations between W_N conformal blocks and Nekrasov functions, focusing on the W^{(3)} case at N=3 and the two lowest descendant levels. It develops a free-field framework to derive and test both Virasoro and W^{(3)} barΓ and Γ vertices, and shows how conformal blocks can be decomposed into these building blocks via the Shapovalov form, including recursion relations and special-state restrictions. The main contribution is a concrete collection of universal triple-vertex expressions and their free-field checks, enabling practical AGT applications and guiding higher-level extensions. The results offer a rigorous, computable route to implement AGT beyond Virasoro, with explicit formulas, checks, and guidance for extending to higher descendants and more general W-algebras.
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.