Highest-weight vectors and three-point functions in GKO coset decomposition

TL;DR

This paper revisits the GKO coset construction, deriving formulas for highest weight vectors, their norms, and matrix elements, linking these to conformal blocks, AGT correspondence, and Painlevé tau-functions.

Contribution

It provides explicit formulas for highest weight vectors and matrix elements in GKO cosets, connecting conformal blocks to Nekrasov partition functions and Painlevé equations.

Findings

Formulas for highest weight vectors and their norms

Relations on conformal blocks derived

Connections to Nekrasov functions and Painlevé tau-functions

Abstract

We revisit the classical Goddard-Kent-Olive coset construction. We find the formulas for the highest weight vectors in coset decomposition and calculate their norms. We also derive formulas for matrix elements of natural vertex operators between these vectors. This leads to relations on conformal blocks. Due to the AGT correspondence, these relations are equivalent to blowup relations on Nekrasov partition functions with the presence of the surface defect. These relations can be used to prove Kyiv formulas for the Painlev\'e tau-functions (following Nekrasov's method).