  In this note we demonstrate that the algebra associated with coordinate
transformations might contain the origins of a scalar field that can behave as
an inflaton and/or a source for dark energy. We will call this particular
scalar field the diffeomorphism scalar field. In one dimension, the algebra of
coordinate transformations is the Virasoro algebra while the algebra of gauge
transformations is the Kac-Moody algebra. An interesting representation of
these algebras corresponds to certain field theories that have meaning in any
dimension. In particular the so called Kac-Moody sector corresponds to
Yang-Mills theories and the Virasoro sector corresponds to the diffeomorphism
field theory that contains the scalar field and a rank-two symmetric, traceless
tensor. We will focus on the contributions of the diffeomorphism scalar field
to cosmology. We show that this scalar field can, qualitatively, act as a
phantom dark energy, an inflaton, a dark matter source, and the cosmological
constant Lambda.
