Multiple backward Schramm–Loewner evolution and coupling with Gaussian free field

It is known that a backward Schramm–Loewner evolution (SLE) coupled with free boundary Gaussian field (GFF) perturbation to give conformal welding of quantum surfaces. Motivated by generalization for surfaces multiple marked points, we propose notion SLE. To this aim, investigate the commutation relation between two Loewner chains, and consequently, find driving process each chain has have drift term given logarithmic derivative partition function, which determined system Belavin–Polyakov–Zamolodchikov-like equations so these chains are commutative. After observation, define SLE as tuple mutually commutative chains. immediately follows in obtained Girsanov transform We also discuss coupling GFF function uniquely they other.