A Wess--Zumino--Witten type equation in the space of K\"ahler potentials in terms of Hermitian--Yang--Mills metrics

We prove that the solution of a Wess-Zumino-Witten type equation from a domain $D$ in ${\mathbb{C}}^m$ to the space of K"ahler potentials can be approximated uniformly by Hermitian-Yang-Mills metrics on certain vector bundles. The key is a new version of Berndtsson’s theorem on the positivity of direct image bundles.