Schramm--Loewner-evolution-type growth processes corresponding to Wess--Zumino--Witten theories

A group theoretical formulation of Schramm--Loewner-evolution-type growth processes corresponding to Wess--Zumino--Witten theories is developed that makes it possible to construct stochastic differential equations associated with more general null vectors than the ones considered in the most fundamental example in [Alekseev et al., Lett. Math. Phys. 97, 243-261 (2011)]. Also given are examples of Schramm--Loewner-evolution-type growth processes associated with null vectors of conformal weight $4$ in the basic representations of $\widehat{\mathfrak{sl}}_{2}$ and $\widehat{\mathfrak{sl}}_{3}$.