to analyze, we will study the coset from the point of view of Kähler geometry. Because of the sign flips compared to the Minkowskian version, the target space will no longer be Kähler. It is therefore not possible to define complex coordinates together with a Kähler potential that determines the metric. As we show below, it is still possible to define coordinates and a potential from which the metric can be computed. To see this, we first define the fields $$a = \sigma + {\tfrac{1}{2}}\chi \varphi\ ,\quad C_\pm = {\tfrac{1}{2}}(\varphi \pm
  \chi)\ ,$$ in terms of which the sigma model part of the Euclidean Lagrangian reads $${\mathcal{L}}_\mathrm{UH}^E = {\tfrac{1}{2}}|{\mathrm{d}}\phi|^2 - 2 {\mathrm{e}}^{-\phi} *\! {\mathrm{d}}C_+ {\wedge}{\mathrm{d}}C_- - {\tfrac{1}{2}}{\mathrm{e}}^{-2\phi}\, |{\mathrm{d}}a + C_+ {\overset{\leftrightarrow}{{\mathrm{d}}}} C_-|^2\ .$$ If we further pass to coordinates $u^1_\pm,u^2_\pm\in\fieldR$ via the relations $$S_\pm = {\mathrm{e}}^{\phi} \mp a - C_+ C_- = {\frac{\raisebox{-2pt}{$1 \mp u^1_\pm$}}{1 \pm
  u^1_\pm}}\ ,\quad C_\pm = {\frac{\raisebox{-2pt}{$u^2_\pm$}}{1 \pm u^1_\pm}}\ ,$$ then the Lagrangian can be written as $${\mathcal{L}}^E_\mathrm{UH} = 2 g_{ij} *\! {\mathrm{d}}u^i_+ {\wedge}{\mathrm{d}}u^j_-$$ with a metric $$\label{K+-}
  g_{ij} = - {\frac{\raisebox{-2pt}{${\partial}^2$}}{{\partial}u^i_+ {\partial}u^j_-}}\, \ln\! \big( 1 + u^1_+
  u^1_- + u^2_+ u^2_- \big)\ .$$ The $u^i_\pm$ are inhomogeneous coordinates of the coset space , transforming under 