The dynamics of metric-affine gravity
  Metric-affine theories of gravity provide an interesting alternative to
General Relativity: in such an approach, the metric and the affine (not
necessarily symmetric) connection are independent quantities. Furthermore, the
action should include covariant derivatives of the matter fields, with the
covariant derivative naturally defined using the independent connection. As a
result, in metric-affine theories a direct coupling involving matter and
connection is also present. The role and the dynamics of the connection in such
theories is explored. We employ power counting in order to construct the action
and search for the minimal requirements it should satisfy for the connection to
be dynamical. We find that for the most general action containing lower order
invariants of the curvature and the torsion the independent connection does not
carry any dynamics. It actually reduces to the role of an auxiliary field and
can be completely eliminated algebraically in favour of the metric and the
matter field, introducing extra interactions with respect to general
relativity. However, we also show that including higher order terms in the
action radically changes this picture and excites new degrees of freedom in the
connection, making it (or parts of it) dynamical. Constructing actions that
constitute exceptions to this rule requires significant fine tuned and/or extra
a priori constraints on the connection. We also consider f(R) actions as a
particular example in order to show that they constitute a distinct class of
metric-affine theories with special properties, and as such they cannot be used
as representative toy theories to study the properties of metric-affine
gravity.
