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

If $\nabla$ is a Weyl connection and $h=e^{2\gamma }g$ , 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 $[g]$ , with $\alpha _{g}=0$ . This is not the most general case, however, as any such Weyl connection has the property that the one-form $\alpha _{h}$ is closed for all $h$ 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 $d\alpha _{g}$ , which is independent of $g$ in the conformal class, because the difference between two $\alpha _{g}$ 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 $\alpha _{g}$ could represent the vector potential of electromagnetism (a gauge dependent quantity), and $d\alpha _{g}$ the field strength (a gauge invariant quantity). This synthesis is unsuccessful in part because the gauge group is wrong: electromagnetism is associated with a $U(1)$ gauge field, not an $\mathbb {R}$ 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).

Citations

^ Bergmann 1975, Chapter XVI: Weyl's gauge-invariant geometry harvnb error: no target: CITEREFBergmann1975 (help)
^ Higa 1993
^ Ciambelli & Leigh (2020)
^ Jia & Karydas (2021)

References

Bergmann, Peter (1942), Introduction to the theory of relativity, Prentice-Hall.
Čap, Andreas; Slovák, Jan (2003), "Weyl structures for parabolic geometries", Mathematica Scandinavica, 93 (1): 53–90, arXiv:math/0001166, doi:10.7146/math.scand.a-14413, JSTOR 24492421.
Cartan, Élie (1943), "Sur une classe d'espaces de Weyl", Annales scientifiques de l'École Normale Supérieure, 60 (3): 1–16, doi:10.24033/asens.901.
Ciambelli, Luca; Leigh, Robert (2020), "Weyl connections and their role in holography", Physical Review D, 101 (8): 086020, arXiv:1905.04339, doi:10.1103/PhysRevD.101.086020, S2CID 152282710
Dikarev, A (2021), "On holonomy of Weyl connections in Lorentzian signature", Differential Geometry and Its Applications, 76 (101759), arXiv:2005.08166, doi:10.1016/j.difgeo.2021.101759, S2CID 218673884.
Eisenhart, Luther (1927), Non-Riemannian geometry, AMS.
Folland, Gerald (1970), "Weyl manifolds", Journal of Differential Geometry, 4 (2): 145–153, doi:10.4310/jdg/1214429379.
Hall, G. (1992), "Weyl manifolds and connections", Journal of Mathematical Physics, 33 (7): 2633, doi:10.1063/1.529582.
Higa, Tatsuo (1993), "Weyl manifolds and Einstein–Weyl manifolds", Commentarii Mathematici Universitatis Sancti Pauli, 42 (2): 143–160.
Jia, W; Karydas, M (2021), "Obstruction tensors in Weyl geometry and holographic Weyl anomaly", Physical Review D, 104 (126031): 126031, arXiv:2109.14014, doi:10.1103/PhysRevD.104.126031, S2CID 238215186
LeBrun, Claude; Mason, Lionel J. (2009), "The Einstein–Weyl equations, scattering maps, and holomorphic disks", Mathematical Research Letters, 16 (2): 291–301, arXiv:0806.3761, doi:10.4310/MRL.2009.v16.n2.a7.
Weyl, Hermann (1918), "Reine Infinitesimalgeometrie", Mathematische Zeitschrift, 2 (3–4): 384–411, doi:10.1007/BF01199420, S2CID 186232500.