Correlation functions for some conformal theories on Riemann surfaces

TL;DR

This paper explores the deep connections between 2D conformal field theories, hyperbolic geometry, and knot theory, focusing on how four-point correlation functions relate to topological invariants on Riemann surfaces.

Contribution

It introduces a novel geometric framework linking CFT correlation functions with topological invariants derived from random walks on Riemann surfaces with hyperbolic geometry.

Findings

Correlation functions are connected to topological invariants.

The framework applies to CFTs with monodromies in discrete subgroups of SL(2,R).

Provides a geometric method for calculating four-point functions.

Abstract

We discuss the geometrical connection between 2D conformal field theories, random walks on hyperbolic Riemann surfaces and knot theory. For the wide class of CFTs with monodromies being the discrete subgroups of SL(2,R), the determination of four-point correlation functions are related to construction of topological invariants for random walks on multipunctured Riemann surfaces