# Highest-Weight Vectors and Three-Point Functions in GKO Coset Decomposition

## TL;DR

This paper explores mathematical structures in conformal field theory and connects them to partition functions in theoretical physics.

## Contribution

The paper derives new formulas for highest weight vectors and their norms in coset decomposition and relates them to blowup relations in Nekrasov partition functions.

## Key findings

- Formulas for highest weight vectors and their norms in coset decomposition are derived.
- Matrix elements of vertex operators are calculated and linked to conformal block relations.
- These relations are connected to blowup relations in Nekrasov partition functions via AGT correspondence.

## 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 Painlevé tau-functions (following Nekrasov’s method).