Conformal field theory for annulus SLE: partition functions and martingale-observables

TL;DR

This paper develops a conformal field theory framework in doubly connected domains to connect with various types of annulus SLE, deriving equations and identifying martingale-observables relevant for understanding these stochastic processes.

Contribution

It introduces a CFT approach for annulus SLE, deriving Ward and BPZ equations, and identifies martingale-observables and Coulomb gas solutions for partition functions.

Findings

Derived Ward's and BPZ equations for annulus SLE

Identified martingale-observables via OPE family correlations

Obtained Coulomb gas solutions for partition functions

Abstract

We implement a version of conformal field theory in a doubly connected domain to connect it to the theory of annulus SLE of various types, including the standard annulus SLE, the reversible annulus SLE, and the annulus SLE with several force points. This implementation considers the statistical fields generated under the OPE multiplication by the Gaussian free field and its central/background charge modifications with a weighted combination of Dirichlet and excursion-reflected boundary conditions. We derive the Eguchi-Ooguri version of Ward's equations and Belavin-Polyakov-Zamolodchikov equations for those statistical fields and use them to show that the correlations of fields in the OPE family under the insertion of the one-leg operators are martingale-observables for variants of annulus SLEs. We find Coulomb gas (Dotsenko-Fateev integral) solutions to the parabolic partial differential equations for partition functions of conformal field theory for the reversible annulus SLE.