Integrable structure, W-symmetry and AGT relation

TL;DR

This work develops an integrable-structure approach to the AGT correspondence for W_n symmetry by constructing a commuting set of Integrals of Motion in the algebra A = W_n ⊗ H and identifying an eigenbasis with a remarkably simple spectrum. The resulting matrix elements of carefully chosen primary fields factorize and reproduce the bifundamental Nekrasov function Z_bif, with the free-field realization guiding a bridge to Jack polynomials and Selberg integrals. The authors demonstrate that these factorized matrix elements underpin a combinatorial expansion of multipoint conformal blocks, and that the classical limit reveals a Calogero–Sutherland–type integrable system with a Burgers-type reduction under a two-field truncation. The results provide a robust, rank-n extension of the AGT framework on the sphere, and suggest avenues for generalizations to other CFTs and potential supersymmetric extensions of the correspondence.

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.

Figures (3)

• Figure 1: Contours of integration in the integral \ref{['ev']}.
• Figure 2: Embedding diagram for representation $P_{r_{1},r_{1}^{\prime};r_{2},r_{2}^{\prime}}^{\prime}$.
• Figure 3: Embedding diagram for $sl(4)$.