Weyl connection

{{short description|Generalization of the Levi-Civita 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 {{harv|Weyl|1918}} in an attempt to unify general relativity and electromagnetism. His approach, although it did not lead to a successful theory,{{harvnb|Bergmann|1942|loc=Chapter XVI: Weyl's gauge-invariant geometry}} lead to further developments of the theory in conformal geometry, including a detailed study by Élie Cartan {{harv|Cartan|1943}}. They were also discussed in {{harvtxt|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

$\nabla h = (2\,d\gamma+\alpha_g)\otimes h$

so the one-form transforms by

$\alpha_{e^{2\gamma}g} = 2\,d\gamma+\alpha_g.$

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.{{harvnb|Higa|1993}}

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.

{{harvtxt|Hall|1992}} 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 {{harvtxt|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, {{harvtxt|LeBrun|Mason|2009}} 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.{{harvtxt|Ciambelli|Leigh|2020}}{{harvtxt|Jia|Karydas|2021}}

Weyl connections have been generalized to the setting of parabolic geometries, of which conformal geometry is a special case, in {{harvtxt|Čap|Slovák|2003}}.

Citations

References

{{citation|authorlink=Peter Bergmann|first=Peter|last=Bergmann|title=Introduction to the theory of relativity|publisher=Prentice-Hall|year=1942}}.
{{citation|first1=Andreas|last1=Čap|first2=Jan|last2=Slovák|title=Weyl structures for parabolic geometries|arxiv=math/0001166|year=2003|journal=Mathematica Scandinavica|volume=93|issue=1|pages=53–90|jstor=24492421|doi=10.7146/math.scand.a-14413|doi-access=free}}.
{{citation|first=Élie|last=Cartan|authorlink=Élie Cartan|title=Sur une classe d'espaces de Weyl|journal=Annales scientifiques de l'École Normale Supérieure|volume=60|issue=3|pages=1–16|year=1943|doi=10.24033/asens.901 |doi-access=free}}.
{{citation|title=Weyl connections and their role in holography|first1=Luca|last1=Ciambelli|first2=Robert|last2=Leigh|authorlink2=Robert Leigh (physicist)|journal=Physical Review D|year=2020|volume=101|issue=8 |page=086020 |arxiv=1905.04339|doi=10.1103/PhysRevD.101.086020|s2cid=152282710 }}
{{citation|first=A|last=Dikarev|title=On holonomy of Weyl connections in Lorentzian signature|journal=Differential Geometry and Its Applications|year=2021|volume=76|number=101759|doi=10.1016/j.difgeo.2021.101759|arxiv=2005.08166|s2cid=218673884 }}.
{{citation|authorlink=Luther P. Eisenhart|first=Luther|last=Eisenhart|title=Non-Riemannian geometry|publisher=AMS|year=1927}}.
{{citation|first=Gerald|last=Folland|authorlink=Gerald Folland|year=1970|title=Weyl manifolds|journal=Journal of Differential Geometry|volume=4|issue=2|pages=145–153|doi=10.4310/jdg/1214429379|doi-access=free}}.
{{citation|first=G.|last=Hall|title=Weyl manifolds and connections|journal=Journal of Mathematical Physics|volume=33|page=2633|year=1992|issue=7 |doi=10.1063/1.529582}}.
{{citation|first=Tatsuo|last=Higa|year=1993|title=Weyl manifolds and Einstein–Weyl manifolds|journal=Commentarii Mathematici Universitatis Sancti Pauli|volume=42|issue=2|pages=143–160}}.
{{citation|first1=W|last1=Jia|first2=M|last2=Karydas|title=Obstruction tensors in Weyl geometry and holographic Weyl anomaly|journal=Physical Review D|year=2021|volume=104|number=126031|page=126031 |doi=10.1103/PhysRevD.104.126031 |s2cid=238215186 |url=https://journals.aps.org/prd/pdf/10.1103/PhysRevD.104.126031|doi-access=free|arxiv=2109.14014}}
{{citation|authorlink1=Claude LeBrun|first1=Claude|last1=LeBrun|first2=Lionel J.|last2=Mason|title=The Einstein–Weyl equations, scattering maps, and holomorphic disks|journal=Mathematical Research Letters|volume=16|year=2009|issue=2 |pages=291–301|doi=10.4310/MRL.2009.v16.n2.a7|doi-access=free|arxiv=0806.3761}}.
{{citation|authorlink=Hermann Weyl|first=Hermann|last=Weyl|title=Reine Infinitesimalgeometrie|journal=Mathematische Zeitschrift|volume=2|year=1918|issue=3–4 |pages=384–411|doi=10.1007/BF01199420|s2cid=186232500 |url=https://zenodo.org/record/1447411 }}.

External links

{{citation|url=https://encyclopediaofmath.org/wiki/Weyl_connection|title=Weyl connection|publisher=Encyclopedia of Mathematics}}

Category:Conformal geometry

Category:Connection (mathematics)

Weyl connection

Citations

References

Further reading

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