  Solar flares and coronal mass ejections are associated with rapid changes in
field connectivity and powered by the partial dissipation of electrical
currents in the solar atmosphere. A critical unanswered question is whether the
currents involved are induced by the motion of pre-existing atmospheric
magnetic flux subject to surface plasma flows, or whether these currents are
associated with the emergence of flux from within the solar convective zone. We
address this problem by applying state-of-the-art nonlinear force-free field
(NLFFF) modeling to the highest resolution and quality vector-magnetographic
data observed by the recently launched Hinode satellite on NOAA Active Region
10930 around the time of a powerful X3.4 flare. We compute 14 NLFFF models with
4 different codes and a variety of boundary conditions. We find that the model
fields differ markedly in geometry, energy content, and force-freeness. We
discuss the relative merits of these models in a general critique of present
abilities to model the coronal magnetic field based on surface vector field
measurements. For our application in particular, we find a fair agreement of
the best-fit model field with the observed coronal configuration, and argue (1)
that strong electrical currents emerge together with magnetic flux preceding
the flare, (2) that these currents are carried in an ensemble of thin strands,
(3) that the global pattern of these currents and of field lines are compatible
with a large-scale twisted flux rope topology, and (4) that the ~10^32 erg
change in energy associated with the coronal electrical currents suffices to
power the flare and its associated coronal mass ejection.
