Integrable structure, W-symmetry and AGT relation

TL;DR

This paper explores the integrable structure of a specific conformal field theory with W-algebra and Heisenberg symmetry, revealing a simple spectrum and factorized matrix elements that match Nekrasov partition functions.

Contribution

It identifies a system of commuting Integrals of Motion with simple spectral properties and explicit factorized matrix elements, linking conformal field theory to gauge theory partition functions.

Findings

System of commuting Integrals of Motion with simple spectrum

Factorized matrix elements matching Nekrasov bifundamental contributions

Explicit construction of integrable structure in W-algebra-based CFT

Abstract

In these notes we consider integrable structure of the conformal field theory with the algebra of symmetries $\mathcal{A}=W_{n}\otimes H$, where $W_{n}$ is $W-$algebra and $H$ is Heisenberg algebra. We found the system of commuting Integrals of Motion with relatively simple properties. In particular, this system has very simple spectrum and the matrix elements of special primary operators between its eigenstates have nice factorized form coinciding exactly with the contribution of the bifundamental multiplet to the Nekrasov partition function for U(n) gauge theories.