A potential theory for the Wess-Zumino-Witten equation in the space of Kähler potentials

We develop a potential theory for the Wess–Zumino–Witten (WZW) equation in the space of Kähler potentials which is parallel to the potential theory for the Hermitian–Yang–Mills equation. A concept called ω-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 Kähler 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.