# Crossing invariant correlation functions at $c=1$ from isomonodromic $\tau$ functions

## TL;DR

This paper constructs crossing invariant correlation functions at central charge c=1 in conformal field theories using isomonodromic tau functions and flat connections, providing rigorous foundations for several known theories.

## Contribution

It introduces a rigorous method to derive crossing invariant functions in c=1 CFTs from moduli space distributions, connecting isomonodromic tau functions to physical correlation functions.

## Key findings

- Constructed crossing invariant functions for c=1 CFTs.
- Derived correlation functions for Ashkin-Teller and Runkel-Watts theories.
- Provided a rigorous foundation for analytic Liouville theory.

## Abstract

We present an approach that gives rigorous construction of a class of crossing invariant functions in $c=1$ CFTs from the weakly invariant distributions on the moduli space $\mathcal M_{0,4}^{SL(2,\mathbb{C})}$ of $SL(2,\mathbb{C})$ flat connections on the sphere with four punctures. By using this approach we show how to obtain correlation functions in the Ashkin-Teller and the Runkel-Watts theory. Among the possible crossing-invariant theories, we obtain also the analytic Liouville theory, whose consistence was assumed only on the basis of numerical tests.