Rigid Fuchsian systems in 2-dimensional conformal field theories

TL;DR

This paper applies Katz theory of Fuchsian rigid systems to solve differential equations in 2D conformal field theories, specifically in W3 Toda CFT, enabling explicit solutions and insights into fusion rules and correlation functions.

Contribution

It introduces a novel application of Katz theory to derive explicit Fuchsian differential equations and solutions in W3 Toda CFT, advancing the mathematical understanding of conformal blocks.

Findings

Derived explicit differential equations for four-point conformal blocks.

Constructed fusion matrices and shift relations for structure constants.

Provided integral solutions and monodromy representations for the equations.

Abstract

We investigate Fuchsian equations arising in the context of 2-dimensional conformal field theory (CFT) and we apply the Katz theory of Fucshian rigid systems to solve some of these equations. We show that the Katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions. We focus on the case of W3 Toda CFT: we argue that the differential equations arising for four-point conformal blocks with one n-th level semi-degenerate field and a fully-degenerate one in the fundamental sl3 representation are associated to Fuchsian rigid systems. We show how to apply Katz theory to determine the explicit form of the differential equations, the integral expression of solutions and the monodromy group representation. The theory of twisted homology is also used in the analysis of the integral expression. This approach allows to construct the corresponding fusion matrices and to perform the whole bootstrap program: new explicit factorization of W3 correlation functions as well shift relations between structure constants are also provided.