Highest-weight vectors and three-point functions in GKO coset decomposition
Mikhail Bershtein, Boris Feigin, Aleksandr Trufanov
TL;DR
The paper advances the explicit understanding of the GKO coset construction by deriving closed formulas for the highest-weight vectors in coset decomposition, their norms, and the matrix elements of vertex operators, enabling concrete relations between Virasoro and $ rak{sl}_2$ conformal blocks. It then translates these coset-based relations into blowup equations for Nekrasov partition functions with surface defects via the AGT correspondence, and uses this framework to obtain Kyiv-type formulas for Painlevé III$_3$ tau-functions, including Whittaker-limit analyses. The approach combines Wakimoto free-field realizations, degenerate fields, and operator-state techniques to produce recurrence relations for conformal blocks and to express three-point functions in terms of triangle and Gamma-function data. Further, the work establishes integral representations and Selberg-type identities for these objects, linking 2d CFT structures to gauge-theory partition functions and isomonodromic tau-functions, with potential generalizations to broader corner algebras and isomonodromic systems.
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).