  We characterise the chain rule symmetry for the geometric stochastic heat
equations in the full subcritical regime for Gaussian and non-Gaussian noises.
We show that the renormalised counter-terms that give a solution invariant
under changes of coordinates are generated by iterations of covariant
derivatives. The result was known only for space-time white noises, with a very
specific proof that so far could not be extended to the general case. The key
idea of the present paper is to change the perspective on several levels and to
use ideas coming from operad theory and homological algebra. Concretely, we
introduce the operad of Christoffel trees that captures the counter-terms of
the renormalised equation; our main new insight is to describe the space of
invariant terms homologically, using a suitable perturbation of the
differential of the operadic twisting of that operad. As a consequence, we
obtain the correct renormalisation for the quasi-linear KPZ equation in the
subcritical regime completing the programme started by Hairer and Gerencser.
Previously, the main algebraic tool used in the study of singular SPDEs were
Hopf algebras of decorated trees; our work shows that operad theory and
homological algebra add new powerful tools with immediate applications to open
problems that were out of reach by other methods.
