Let $(M,g)$ be a riemannian manifold with the Levi-Civita connection $\nabla$. I consider a PDE over the vector fields $X$ such that for any vector fields $Y$:

$$\nabla_Y X=\frac{1}{2} \frac{Y(g(X,X))}{g(X,X)}X$$

The gauge group ${\cal C}^{\infty}(M,{\bf R}^*)$ acts over the solutions of the equation $(f,X)\mapsto fX$.

Is such an equation the definition of a moduli space with good properties (finite dimensionality and compactness)? Has such a moduli space a meaning in physics?

For an hermitian fiber bundle $(E,h)$ and a section $s \in \Gamma (M,E)$, there is also an equation of moduli space:

$$\nabla^C_X s=\frac{1}{2} \frac{X(h(s,s))}{h(s,s)}s$$

with $\nabla^C$, the Chern connection.