Irregular SLE(4) martingales and isomonodromic deformations

TL;DR

This paper derives Loewner evolution for isomonodromic deformations with irregular singularities and constructs SLE(4) martingale observables linked to conformal field theory.

Contribution

It introduces a novel connection between isomonodromic deformations with irregular singularities and SLE(4) martingale observables, including explicit Loewner evolution.

Findings

Derived Loewner evolution for isomonodromic parameters with irregular singularities.

Constructed SLE(4) martingale observables involving Schwarzian derivatives.

Characterized observables via confluent BPZ equations in CFT.

Abstract

We consider non-Fuchsian monodromy preserving deformations on a Riemann sphere. The associated isomonodromic deformation parameters on this surface comprise the positions of the singularities, together with the Birkhoff (spectral) invariants owing to the presence of irregular singularities. Our first main result is the derivation of the Loewner evolution of these isomonodromic deformation parameters. Using this result, we construct martingale observables for Schramm-Loewner evolution (SLE(4)) processes in the presence of double poles. Geometrically, the expressions contain the pre-Schwarzian and Schwarzian of the Loewner evolution, arising from conformal covariance of the observable. Furthermore, we characterize these SLE(4) observables uniquely in terms of confluent BPZ equations of a CFT with central charge c=1.