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

R Aros,D.E Diaz,F Bugini

doi:10.1007/jhep04(2020)080

Abstract

The aim of this note is to unveil a striking equivalence between the one-loop divergences in 7D Einstein and 6D Conformal Gravities.The particular combination of 6D pointwise Weyl invariants of the 6D Conformal Gravity corresponds to that of Branson’s Q-curvature and can be written solely in terms of the Ricci tensor and its covariant derivatives. The quadratic metric fluctuations of this action, 6D Weyl graviton, are endowed with a sixth-order kinetic operator that happens to factorize on a 6D Einstein background into product of three shifted Lichnerowicz Laplacians. We exploit this feature to use standard heat kernel techniques and work out in one go the UV logarithmic divergences of the theory that contains in this case the four Weyl anomaly coefficients.In a seemingly unrelated computation, we determine the one-loop IR logarithmic divergences of 7D Einstein Gravity in a particular 7D Poincaré-Einstein background that is asymptotically hyperbolic and has the above 6D Einstein manifold at its conformal infinity or boundary.We show the full equivalence of both computations, as an outgrowth of the IR/UV connection in AdS/CFT correspondence, and in this way the time-honoured one-loop calculations in Einstein and higher-derivative gravities take an interesting new turn.

Highlights

In a seemingly unrelated computation, we determine the one-loop IR logarithmic divergences of 7D Einstein Gravity in a particular 7D Poincare-Einstein background that is asymptotically hyperbolic and has the above 6D Einstein manifold at its conformal infinity or boundary
The quadratic metric fluctuations of this action, 6D Weyl graviton, are endowed with a sixth-order kinetic operator that happens to factorize on a 6D Einstein background into product of three shifted Lichnerowicz Laplacians
The one-loop partition function for the corresponding 6D Weyl graviton can be obtained by integrating out the quadratic metric fluctuations after fixing the Feynman-de Donder gauge and taking into account the ghost contribution

Summary

Conformally flat case: hyperbolic space

The heat kernels for the transverse vector and for the transverse-traceless symmetric ranktwo tensor in hyperbolic space has long been known [44,45,46]. After evaluation of the proper-time integral in terms of gamma functions, we obtain the numerical factor corresponding to the one-loop effective Lagrangian on hyperbolic space that accompanies the volume, following the prescription of [47]. The volume anomaly is given by the Q-curvature so that we can directly read off the type-A Weyl anomaly coefficient, it coincides with the result obtained from the boundary computation on the round six-sphere. This holographic result in the conformally flat situation was first derived in [14] and, established for the whole family of bulk gauge fields of higher spins dual to boundary conformal higher spins

Non-conformally flat case

W 2 180

Conclusion

A Second metric variation of the Q-curvature at 6D Einstein background

CiklCjkl

J δBij