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

Abstract

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.

Addendum: Weyl cocycles, (1986 Class. Quantum Grav. 3 635)

“…In particular in 6 dimensions, four 0-and four 1-cocycles were found. Subsequently other similar or related papers appeared, [3][4][5][6][7][8][9][10][11][12], with not always overlapping results. The preparation of the book [13] has prompted me to redo the calculations from scratch for the six-dimensional case by solving, in particular, the relative linear systems with Mathematica.…”

“…δ ω ∆ (6) = Ω (1) − 4 Ω (2) − 4 Ω (3) + 12 Ω (4) + 2 Ω (5) − 12 Ω (9) δ ω ∆ (7) = 3 Ω (1) − 12 Ω (2) − 12 Ω (3) + 24 Ω (4) + 3 Ω (5) − 24 Ω (9) δ ω ∆ (8) = 3 4 Ω (1) − 3 Ω (2) − 3 Ω (3) + 6 Ω (4) + 3 4 Ω (5) δ ω ∆ (9) = 2 Ω (1) + 10 Ω (6) + 10 Ω (7) δ ω ∆ (10) = −Ω (1) + 6 Ω (2) + 4 Ω (3) + 4Ω (4) + 3 Ω (6) + Ω (7) + 4 Ω (8) + 16 Ω (9) δ ω ∆ (11) = 4Ω (1) + 12 Ω (2) + 16 Ω (3) − 16Ω (4) + 4 Ω (5) + 2 Ω (6) + 4 Ω (8) + 32 Ω (9) δ ω ∆ (12) = Ω (1) + 5 Ω (6) + 10Ω (8) δ ω ∆ (13) = Ω (1) − 6 Ω (2) − 4 Ω (3) − 4Ω (4) − 3 Ω (6) + Ω (7) − 8 Ω (8) − 16 Ω (9) δ ω ∆ (14 8) − 32 Ω (9) δ ω ∆ (16) = 0…”

“…δ ω C (3) = 4 ∆ (9) − 20 ∆ (11) − 8 ∆ (12) − 20 ∆ (13) − 4 ∆ (16) δ ω C (4) = −12 ∆ (4) − 12 ∆ (5) + 3∆ (9) − 6 ∆ (10) − 12 ∆ (12) − 6 ∆ (13)…”

“…δ ω C (5) = −10 ∆ (4) − 10 ∆ (5) − 4 ∆ (6) + 2 ∆ (7) − 8 ∆ (8) − 1 2 ∆ (9) + 12 ∆ (10) + 2 ∆ (11) − 4 ∆ (12) + 20 ∆ (13) − 18 ∆ (14) + 3 4 ∆ (16) δ ω C (6) = 6 ∆ (6) − 3 ∆ (7) + 12∆ (8) + ∆ (9) − 4 ∆ (10) − 7 ∆ (11) − 2 ∆ (12) − 16 ∆ (13) − 12 ∆ (14) − 4 ∆ (15) δ ω C (7) = 12 ∆ (6) − 6 ∆ (7) + 24∆ (8) − 12 ∆ (11) − 24 ∆ (13) + 24 ∆ (14) − 6 ∆ (15)…”

Analytic meronic black holes, gravitating solitons and higher spins in the Einstein $SU(N)$-Yang-Mills theory

“…However, the problem to define consistent renormalizable interactions between higher spin fields on (asymptotically) flat space-times remained. A situation with negative cosmological constants (due to its role as an effective infrared cutoff) was disclosed in [97,98] (an in-depth analysis of the current situation can be found in [99,100,101,102,103,104,105] and references therein). It is worth mentioning that there are also no-go theorems also in AdS (see for instance [106] [107] [108]): the present proposal, to be described here below, can be very useful also in order to avoid these AdS no-go theorems.…”