One-loop divergences in 7D Einstein and 6D conformal gravities

It has been recently shown that there is a particular combination of conformal invariants in six dimensions which accepts a generic Einstein space as a solution. The Lagrangian of this Conformal Gravity theory — originally found by Lu, Pang and Pope (LPP) — can be conveniently rewritten in terms of products and covariant derivatives of the Weyl tensor. This allows one to derive the corresponding Noether prepotential and Noether-Wald charges in a compact form. Based on this expression, we calculate the Noether-Wald charges of six-dimensional Critical Gravity at the bicritical point, which is defined by the difference of the actions for Einstein-AdS gravity and the LPP Conformal Gravity. When considering Einstein manifolds, we show the vanishing of the Noether prepotential of Critical Gravity explicitly, which implies the triviality of the Noether-Wald charges. This result shows the equivalence between Einstein-AdS gravity and Conformal Gravity within its Einstein sector not only at the level of the action but also at the level of the charges.

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“…The same combination of conformal invariants defines the type-B anomaly of 7D Einstein-AdS gravity[14].…”

mentioning

confidence: 89%

Noether-Wald charges in six-dimensional Critical Gravity

Anastasiou

Araya

Corral

et al. 2021

J. High Energ. Phys.

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“…It suffices to keep track on the dependence of the bulk curvature invariants on the Fefferman-Graham invariantΦ 7 if we are only interested in the c 3 coefficient of the holographic anomaly. Again, from the dictionary in table 2 of reference [15], the contribution toΦ 7 from every bulk curvature invariant is pinned down…”

Section: Jhep05(2021)241mentioning

confidence: 99%

“…The aim in the present work is to proceed with the analysis of the holographic formula for boundary conformal higher spin fields in six dimensions and to extend the dictionary already established in the conformally flat situation [13]. In particular, we explore Ricciflat but non-conformally flat boundaries as was done in 4D [14] for CHS and also in 6D for GJMS [12] and for the Weyl graviton [15]. In doing so, a crucial technical breakthrough is achieved by the computation of the heat coefficient b 6 for Lichnerowicz Laplacians on an Einstein manifold background.…”

Section: Introductionmentioning

confidence: 99%

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A calculation of the Weyl anomaly for 6D conformal higher spins

Aros

Bugini

Díaz

2021

J. High Energ. Phys.

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In this work we continue the study of the one-loop partition function for higher derivative conformal higher spin (CHS) fields in six dimensions and its holographic counterpart given by massless higher spin Fronsdal fields in seven dimensions.In going beyond the conformal class of the boundary round 6-sphere, we start by considering a Ricci-flat, but not conformally flat, boundary and the corresponding Poincaré-Einstein space-filling metric. Here we are able to match the UV logarithmic divergence of the boundary with the IR logarithmic divergence of the bulk, very much like in the known 4D/5D setting, under the assumptions of factorization of the higher derivative CHS kinetic operator and WKB-exactness of the heat kernel of the dual bulk field. A key technical ingredient in this construction is the determination of the fourth heat kernel coefficient b6 for Lichnerowicz Laplacians on both 6D and 7D Einstein manifolds. These results allow to obtain, in addition to the already known type-A Weyl anomaly, two of the three independent type-B anomaly coefficients in terms of the third, say c3 for instance.In order to gain access to c3, and thus determine the four central charges independently, we further consider a generic non Ricci-flat Einstein boundary. However, in this case we find a mismatch between boundary and bulk computations for spins higher than two. We close by discussing the nature of this discrepancy and perspectives for a possible amendment.

show abstract

“…The aim in the present work is to proceed with the analysis of the holographic formula for boundary conformal higher spin fields in six dimensions and to extend the dictionary already established in the conformally flat situation [13]. In particular, we explore Ricci-flat but nonconformally flat boundaries as was done in 4D [14] for CHS and also in 6D for GJMS [12] and for the Weyl graviton [15]. In doing so, a crucial technical breakthrough is achieved by the computation of the heat coefficient b 6 for Lichnerowicz Laplacians on an Einstein manifold background.…”

Section: Introductionmentioning

confidence: 99%

A calculation of the Weyl anomaly for 6D Conformal Higher Spins

Aros

Bugini

Díaz

2021

Preprint

Self Cite

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In this work we continue the study of the one-loop partition function for higher derivative conformal higher spin (CHS) fields in six dimensions and its holographic counterpart given by massless higher spin Fronsdal fields in seven dimensions.In going beyond the conformal class of the boundary round 6-sphere, we start by considering a Ricci-flat, but not conformally flat, boundary and the corresponding Poincaré-Einstein spacefilling metric. Here we are able to match the UV logarithmic divergence of the boundary with the IR logarithmic divergence of the bulk, very much like in the known 4D/5D setting, under the assumptions of factorization of the higher derivative CHS kinetic operator and WKB-exactness of the heat kernel of the dual bulk field. A key technical ingredient in this construction is the determination of the fourth heat kernel coefficient b6 for Lichnerowicz Laplacians on both 6D and 7D Einstein manifolds. These results allow to obtain, in addition to the already known type-A Weyl anomaly, two of the three independent type-B anomaly coefficients in terms of the third, say c3 for instance.In order to gain access to c3, and thus determine the four central charges independently, we further consider a generic non Ricci-flat Einstein boundary. However, in this case we find a mismatch between boundary and bulk computations for spins higher than two. We close by discussing the nature of this discrepancy and perspectives for a possible amendment.

show abstract

One-loop divergences in 7D Einstein and 6D conformal gravities

Cited by 7 publications

References 68 publications

Noether-Wald charges in six-dimensional Critical Gravity

Noether-Wald charges in six-dimensional Critical Gravity

A calculation of the Weyl anomaly for 6D conformal higher spins

A calculation of the Weyl anomaly for 6D Conformal Higher Spins

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