Correlation functions in conformal Toda field theory II

TL;DR

This work advances exact analytic control of correlation functions in $\mathfrak{sl}(3)$ Toda field theory with $\mathW{W_3}$ symmetry by showing that the three-point function with a semi-degenerate field and the four-point function with one degenerate and one semi-degenerate field admit finite-dimensional Coulomb-integral representations. The authors derive explicit integral representations for the semi-degenerate three-point function $C_m(\alpha_1,\alpha_2|\varkappa)$ of dimension $4m$, and a corresponding four-point function of dimension $4(m+1)$, using a network of Liouville-reduction relations, kernel identities, and Weyl/reflection symmetries. Central to the approach are detailed integral identities (transformations) that relate residues from screening charges to those from semi-degenerate fields, and the systematic reduction of high-dimensional Coulomb integrals to standard forms. These results illuminate the analytic structure of Toda correlators, connect Toda and Liouville theories, and provide practical representations for both analytic and numerical investigations of $W_N$-symmetric CFTs beyond the Liouville (N=2) case.

Abstract

This is the second part of the paper 0709.3806v2. Here we show that three-point correlation function with one semi-degenerate field in Toda field theory as well as four-point correlation function with one completely degenerate and one semi-degenerate field can be represented by the finite dimensional integrals.

Figures (1)

• Figure 1: This picture represents integral property \ref{['Nogi']}. In the l.h.s. we have a stack which consists of $m$ variables $t$ and stack of $n$ variables $s$ with Coulomb interaction between them associated with Cartan matrix of the Lie algebra $\mathfrak{sl}(3)$. In the r.h.s. we have three stacks of variables: $m-k$ variables $t$, $n$ variables $s$ and $k$ variables $w$ with Coulomb interaction associated with the Lie algebra $\mathfrak{sl}(4)$.