## Writing determinant as traces

I am reading the paper http://arxiv.org/abs/hep-th/9701037.
In equation (2), the author write the determinat as traces, but I don't know how to do this.
I know that ##det(e^A)=e^{tr(A)}##, where ##A## is a complex matrix, and ##det(A)=e^{tr(L)}##, where ##e^L=A## and ##L## is also a complex matrix.
The problem becomes how to find the logrithm of the matrix ##g_{mn}+i##$\tilde{F}_{mn}$.
Above is what I can figure out now.

Any help would be vary appreciated!
Regards.
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