The Correlator Toolbox, Metrics and Moduli

TL;DR

This paper develops a framework for generalized SLE processes using boundary conformal field theories, emphasizing the role of moduli and geometric interpretations involving polygons and gravitational backgrounds.

Contribution

It introduces a set of operators for boundary CFTs to construct generalized SLE($, ho$), highlighting the importance of moduli and providing a geometric derivation involving polygons and gravitational coupling.

Findings

Generalized SLE($, ho$) can be derived from boundary CFT operators.

Moduli are essential for a consistent kinematic description of these processes.

A geometric interpretation relates parameters _j to polygon exterior angles.

Abstract

We discuss the possible set of operators from various boundary conformal field theories to build meaningful correlators that lead via a Loewner type procedure to generalisations of SLE($\kappa,\rho$). We also highlight the necessity of moduli for a consistent kinematic description of these more general stochastic processes. As an illustration we give a geometric derivation of $\text{SLE}(\kappa,\rho)$ in terms of conformally invariant random growing compact subsets of polygons. The parameters $\rho_j$ are related to the exterior angles of the polygons. We also show that $\text{SLE}(\kappa,\rho)$ can be generated by a Brownian motion in a gravitational background, where the metric and the Brownian motion are coupled. The metric is obtained as the pull-back of the Euclidean metric of a fluctuating polygon.