Conformal field theory on the Riemann sphere and its boundary version for SLE

TL;DR

This paper extends conformal field theory to boundary domains using the Schottky double, linking it to SLE martingale-observables and exploring boundary conditions.

Contribution

It introduces a boundary version of conformal field theory on the Riemann sphere and connects it to SLE processes with boundary conditions.

Findings

Correlation functions form martingale-observables for SLE with force points and spins.

Boundary conformal field theory relates to backward SLE via Neumann boundary conditions.

The approach uses background charge modifications of the Gaussian free field.

Abstract

From conformal field theory on the Riemann sphere, we implement its boundary version in a simply-connected domain using the Schottky double construction. We consider the statistical fields generated by background charge modification of the Gaussian free field with Dirichlet boundary condition under the OPE multiplications. We prove that the correlation functions of such fields with symmetric background charges form a collection of martingale-observables for (forward) chordal/radial SLE with force points and spins. We also present the connection between conformal field theory with Neumann boundary condition and the theory of backward SLE.