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

TL;DR

This paper develops a potential theory for the Wess--Zumino--Witten equation within the space of K"ahler potentials, introducing new concepts like $\,\omega$-harmonicity and analyzing solution properties.

Contribution

It introduces a novel potential theory for the WZW equation in K"ahler geometry, including $\,\omega$-harmonicity and subharmonic distance properties between solutions.

Findings

Distance between solutions is subharmonic.

Solvability of the Dirichlet problem for harmonic maps.

Finite dimensional approximation and quantization of solutions.

Abstract

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.