A potential theory for the Wess--Zumino--Witten equation in the space of K\"ahler potentials

We develop a potential theory for the Wess--Zumino--Witten (WZW) equation in the space of K\"ahler potentials which is parallel to the potential theory for the Hermitian--Yang--Mills equation. A concept called $\omega$-harmonicity on graphs is introduced which characterizes the WZW equation. We also show that, with respect to a Banach--Mazur type distance function, the distance between two solutions of the WZW equation is subharmonic. The harmonic map into the space of K\"ahler potentials, as a special case of the WZW equation, is also investigated. In particular, we show the solvability of the Dirichlet problem for the harmonic map, and the approximation/quantization by its finite dimensional counterparts.