Weyl connection

In differential geometry, a Weyl connection (also called a Weyl structure) is a generalization of the Levi-Civita connection that makes sense on a conformal manifold. They were introduced by Hermann Weyl (Weyl 1918) in an attempt to unify general relativity and electromagnetism. His approach, although it did not lead to a successful theory,^[1] lead to further developments of the theory in conformal geometry, including a detailed study by Élie Cartan (Cartan 1943). They were also discussed in Eisenhart (1927).

Specifically, let ${\displaystyle M}$ be a smooth manifold, and ${\displaystyle [g]}$ a conformal class of (non-degenerate) metric tensors on ${\displaystyle M}$, where ${\displaystyle h,g\in [g]}$ iff ${\displaystyle h=e^{2\gamma }g}$ for some smooth function ${\displaystyle \gamma }$ (see Weyl transformation). A Weyl connection is a torsion free affine connection on ${\displaystyle M}$ such that, for any ${\displaystyle g\in [g]}$, $${\displaystyle \nabla g=\alpha _{g}\otimes g}$$ where ${\displaystyle \alpha _{g}}$ is a one-form depending on ${\displaystyle g}$.

If ${\displaystyle \nabla }$ is a Weyl connection and ${\displaystyle h=e^{2\gamma }g}$, then ${\textstyle \nabla h=(2\,d\gamma +\alpha _{g})\otimes h}$ so the one-form transforms by ${\textstyle \alpha _{e^{2\gamma }g}=2\,d\gamma +\alpha _{g}.}$ Thus the notion of a Weyl connection is conformally invariant, and the change in one-form is mediated by a de Rham cocycle.

An example of a Weyl connection is the Levi-Civita connection for any metric in the conformal class ${\displaystyle [g]}$, with ${\displaystyle \alpha _{g}=0}$. This is not the most general case, however, as any such Weyl connection has the property that the one-form ${\displaystyle \alpha _{h}}$ is closed for all ${\displaystyle h}$ belonging to the conformal class. In general, the Ricci curvature of a Weyl connection is not symmetric. Its skew part is the dimension times the two-form ${\displaystyle d\alpha _{g}}$, which is independent of ${\displaystyle g}$ in the conformal class, because the difference between two ${\displaystyle \alpha _{g}}$ is a de Rham cocycle. Thus, by the Poincaré lemma, the Ricci curvature is symmetric if and only if the Weyl connection is locally the Levi-Civita connection of some element of the conformal class.^[2]

Weyl's original hope was that the form ${\displaystyle \alpha _{g}}$ could represent the vector potential of electromagnetism (a gauge dependent quantity), and ${\displaystyle d\alpha _{g}}$ the field strength (a gauge invariant quantity). This synthesis is unsuccessful in part because the gauge group is wrong: electromagnetism is associated with a ${\displaystyle U(1)}$ gauge field, not an ${\displaystyle \mathbb {R} }$ gauge field.

Hall (1993) harvtxt error: no target: CITEREFHall1993 (help) showed that an affine connection is a Weyl connection if and only if its holonomy group is a subgroup of the conformal group. The possible holonomy algebras in Lorentzian signature were analyzed in Dikarev (2021).

A Weyl manifold is a manifold admitting a global Weyl connection. The global analysis of Weyl manifolds is actively being studied. For example, Mason & LeBrun (2008) harvtxt error: no target: CITEREFMasonLeBrun2008 (help) considered complete Weyl manifolds such that the Einstein vacuum equations hold, an Einstein–Weyl geometry, obtaining a complete characterization in three dimensions.

Weyl connections also have current applications in string theory and holography.^[3][4]

Weyl connections have been generalized to the setting of parabolic geometries, of which conformal geometry is a special case, in Čap & Slovák (2003).