An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity

Curry, Sean; Gover, A. Rod

doi:10.48550/arxiv.1412.7559

Cited by 12 publications

(25 citation statements)

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“…A second motivation for our study is that this general setting allows us to study the extrinsic conformal geometry of the boundary geometry. Mathematically, our results are part of a general program to understand conformal hypersurface geometry [GW15] (see [CG15] for an overview), and to develop the calculus for integrated conformal hypersurface invariants begun in [GGHW15]. Indeed, we wish to initiate a new approach to geometric invariant theory based on holographic renormalization.…”

Section: Introductionmentioning

confidence: 99%

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Gover

Waldron

2017

Commun. Math. Phys.

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We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.Comment: 31 pages, LaTeX, 5 figures, anomaly formula generalized to any bulk geometry, improved discussion of hypersurfaces with boundar

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Section: Introductionmentioning

confidence: 99%

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Gover

Waldron

2017

Commun. Math. Phys.

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“…The above flat model for conformal geometry was generalized in [10] by Fefferman and Graham who replaced M d,2 by a d + 2 dimensional manifold equipped with a metric which admits a hypersurface orthogonal homothety. Note that there is a natural interpretation of this picture as curved Cartan geometry; moreover the Cartan approach naturally leads to a Weyl covariant differential calculus, known as tractor calculus, originally constructed in [11] (see also [12,13] for a physics oriented review) and then generalized to all parabolic geometries in [14]. It can be understood as the equivalent of the superfield formalism for conformal invariance.…”

Section: Introductionmentioning

confidence: 99%

The symplectic origin of conformal and Minkowski superspaces

Fioresi

Latini

2016

Journal of Mathematical Physics

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Supermanifolds provide a very natural ground to understand and handle supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and $10$ dimensions is also deeply related to the normed division algebras. In this paper we want to show the link between the conformal group and certain types of symplectic transformations over division algebras. Inspired by this observation we then propose a new\,realization of the real form of the 4 dimensional conformal and Minkowski superspaces we obtain, respectively, as a Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ and a big cell inside it. The beauty of this approach is that it naturally generalizes to the 6 dimensional case (and possibly also to the 10 dimensional one) thus providing an elegant and uniform characterization of the conformal superspaces.Comment: 15 pages, references added, minor change

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“…Let us call defining densities that obey Equation (3.6) unit. Then from Proposition 3.3 and reference [4] (see also [25,10]) we have the following result.…”

Section: Scanned With Camscannermentioning

confidence: 91%

“…where Φ indicates any tensor product of tractor and conformal density bundles. In a self-explanatory matrix notation (see [10])…”

Section: Scanned With Camscannermentioning

confidence: 99%

“…In fact this is easily established explicitly, as the conformal d-sphere arises as the ray projectivisation P + of the future null cone N + of the (Lorentzian signature) Minkowski metric g := diagonal(−1, 1, • • • , 1) on R d+2 (see Example 2.6 above). Then the tractor connection arises from parallel transport in R d+2 viewed as an affine space (see [7,10] for more detail). It follows that each (constant) vector Ĩ in R d+2 determines a corresponding parallel tractor I on the sphere and vice versa.…”

Section: Scanned With Camscannermentioning

confidence: 99%

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Singular Yamabe and Obata Problems

Gover¹,

Waldron²

2019

Preprint

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A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded separating hypersurface that, in dimension three, is necessarily a Willmore energy minimiser or, in higher dimensions, satisfies a conformally invariant analog of the Willmore equation. In any case the zero locus is critical for a conformal functional that generalises the total Q-curvature by including extrinsic data. These observations lead to some interesting global problems that include natural singular variants of a classical problem solved by Obata.

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An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity

Cited by 12 publications

References 34 publications

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The symplectic origin of conformal and Minkowski superspaces

Singular Yamabe and Obata Problems

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