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I am trying to understand how one derives the dilaton monopole interaction. In "Black holes and membranes in higher-dimensional theories with dilaton fields", Gibbons and Maeda mentioned that one could obtain the dilaton monopole interaction as such:

where the action is given by

However, I do not understand their reasoning for introducing ##\Psi## to define ##\Sigma## in order to derive Eq. (4.8). Could someone explain it?

where the action is given by

However, I do not understand their reasoning for introducing ##\Psi## to define ##\Sigma## in order to derive Eq. (4.8). Could someone explain it?