  There is an interesting dichotomy between a space-time metric considered as
external field in a flat background and the same considered as an intrinsic
part of the geometry of space-time. We shall describe and compare two other
external fields which can be absorbed into an appropriate redefinition of the
geometry, this time a noncommutative one. We shall also recall some previous
incidences of the same phenomena involving bosonic field theories. It is known
that some such theories on the commutative geometry of space-time can be
re-expressed as abelian-gauge theory in an appropriate noncommutative geometry.
The noncommutative structure can be considered as containing extra modes all of
whose dynamics are given by the one abelian action.
