which produces another metric in the same
conformal class. A theory or an expression invariant under this transformation is called
conformally invariant, or is said to possess Weyl invariance or Weyl symmetry. The Weyl symmetry is an important
symmetry in
conformal field theory. It is, for example, a symmetry of the
Polyakov action. When quantum mechanical effects break the conformal invariance of a theory, it is said to exhibit a
conformal anomaly or Weyl anomaly.
The ordinary
Levi-Civita connection and associated
spin connections are not invariant under Weyl transformations.
Weyl connections are a class of affine connections that is invariant, although no Weyl connection is individual invariant under Weyl transformations.
Conformal weight
A quantity has
conformal weight if, under the Weyl transformation, it transforms via
Thus conformally weighted quantities belong to certain
density bundles; see also
conformal dimension. Let be the
connection one-form associated to the Levi-Civita connection of . Introduce a connection that depends also on an initial one-form via
Then is covariant and has conformal weight .
Formulas
For the transformation
We can derive the following formulas
Note that the Weyl tensor is invariant under a Weyl rescaling.
References
Weyl, Hermann (1993) [1921]. Raum, Zeit, Materie [Space, Time, Matter]. Lectures on General Relativity (in German). Berlin: Springer.
ISBN3-540-56978-2.