Weyl doubling
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Springer
Received: July 14, 2020 Accepted: August 18, 2020 Published: September 21, 2020
Weyl doubling
Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, U.K.
E-mail: [email protected], [email protected], [email protected] Abstract: We study a host of spacetimes where the Weyl curvature may be expressed algebraically in terms of an Abelian field strength. These include Type D spacetimes in four and higher dimensions which obey a simple quadratic relation between the field strength and the Weyl tensor, following the Weyl spinor double copy relation. However, we diverge from the usual double copy paradigm by taking the gauge fields to be in the curved spacetime as opposed to an auxiliary flat space. We show how for Gibbons-Hawking spacetimes with more than two centres a generalisation of the Weyl doubling formula is needed by including a derivative-dependent expression which is linear in the Abelian field strength. We also find a type of twisted doubling formula in a case of a manifold with Spin(7) holonomy in eight dimensions. For Einstein Maxwell theories where there is an independent gauge field defined on spacetime, we investigate how the gauge fields determine the Weyl spacetime curvature via a doubling formula. We first show that this occurs for the Reissner-Nordstr¨om metric in any dimension, and that this generalises to the electrically-charged Born-Infeld solutions. Finally, we consider brane systems in supergravity, showing that a similar doubling formula applies. This Weyl formula is based on the field strength of the p-form potential that minimally couples to the brane and the brane world volume Killing vectors. Keywords: Classical Theories of Gravity, Gauge-gravity correspondence ArXiv ePrint: 2007.03264
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)127
JHEP09(2020)127
Rashid Alawadhi, David S. Berman and Bill Spence
Contents 1
2 Weyl doubling 2.1 Spacetime classification 2.2 Invariants 2.2.1 Taub-NUT 2.2.2 Plebanski-Demianski 2.2.3 Eguchi-Hanson 2.2.4 Singly rotating Myers-Perry
3 3 4 5 6 7 8
3 The Gibbons-Hawking metrics
9
4 An eight-dimensional example with Spin(7) holonomy
12
5 Reissner-Nordstr¨ om and Born-Infeld 5.1 Reissner-Nordstr¨ om 5.2 Born-Infeld
14 14 15
6 Brane solutions 6.1 String in five dimensions 6.2 M2 brane 6.3 D3 brane 6.4 M5 brane
17 17 18 19 19
7 Discussion
20
A Myers-Perry pentad
22
1
Introduction
The “double copy” and its inverse the “single copy” have received considerable attention in the past few years, as they provide an intriguing link between gauge theories and gravity. The double copy refers to moving from gauge theory to gravity while the single copy is the inverse map (there is also a “zeroth copy” to a scalar theory). This relationship, as a map between perturbative scattering amplitudes in gauge theory and gravity, was first studied in [1–3]. A tree-level proof has been given [3–11], where it has a strin
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