Correlation functions in conformal Toda field theory I
V. A. Fateev, A. V. Litvinov
TL;DR
Addresses the computation of correlation functions in $\mathfrak{sl}(n)$ conformal Toda field theory, including exact three-point functions and four-point functions with degenerate insertions. The authors derive (i) explicit Coulomb-integral expressions for structure constants in degenerate regimes, (ii) higher-order differential equations for four-point functions when degenerate fields are present, and (iii) generalizations to $\mathfrak{sl}(n)$ yielding $(n,n-1)$ hypergeometric-type equations and their monodromy structures. They show that semiclassical (heavy) and light/minisuperspace limits reproduce the quantum results and often reduce to finite-dimensional integrals, providing cross-checks of the proposed formulas. The work advances analytic control over non-rational CFTs with $\mathW{W}_n$ symmetry and connects to affine Toda and WZNW models via Coulomb representations.
Abstract
Two-dimensional sl(n) quantum Toda field theory on a sphere is considered. This theory provides an important example of conformal field theory with higher spin symmetry. We derive the three-point correlation functions of the exponential fields if one of the three fields has a special form. In this case it is possible to write down and solve explicitly the differential equation for the four-point correlation function if the fourth field is completely degenerate. We give also expressions for the three-point correlation functions in the cases, when they can be expressed in terms of known functions. The semiclassical and minisuperspace approaches in the conformal Toda field theory are studied and the results coming from these approaches are compared with the proposed analytical expression for the three-point correlation function. We show, that in the framework of semiclassical and minisuperspace approaches general three-point correlation function can be reduced to the finite-dimensional integral.