Chiral Decomposition For Non-Abelian Bosons
  We study the non-abelian extension for the splitting of a scalar field into chiral components. Using this procedure we find a non ambiguous way of coupling a non abelian chiral scalar field to gravity. We start with a (non-chiral) WZW model covariantly coupled to a background metric and, after the splitting, arrive at two chiral Wess-Zumino-Witten (WZW) models coupled to gravity. 
Nelson R. F. Braga and Clovis Wotzasek