Weyl cohomology and the effective action for conformal anomalies

Abstract

We present a general method of deriving the effective action for conformal anomalies in any even dimension, which satisfies the Wess-Zumino consistency condition by construction. The method relies on defining the coboundary operator of the local Weyl group, g ab →exp(2)g ab , and giving a cohomological interpretation to counterterms in the effective action in dimensional regularization with respect to this group. Nontrivial cocycles of the Weyl group arise from local functionals that are Weyl invariant in and only in the physical even integer dimension dϭ2k. In the physical dimension the nontrivial cocycles generate covariant nonlocal action functionals characterized by sensitivity to global Weyl rescalings. The nonlocal action so obtained is unique up to the addition of trivial cocycles and Weyl invariant terms, both of which are insensitive to global Weyl rescalings. These distinct behaviors under rigid dilations can be used to distinguish between infrared relevant and irrelevant operators in a generally covariant manner. Variation of the dϭ4 nonlocal effective action yields two new conserved geometric stress tensors with local traces equal to the square of the Weyl tensor and the Gauss-Bonnet-Euler density, respectively. The second of these conserved tensors becomes (3) H ab in conformally flat spaces, exposing the previously unsuspected origin of this tensor. The method may be extended to any even dimension by making use of the general construction of conformal invariants given by Fefferman and Graham. As a corollary, conformal field theory ͑CFT͒ behavior of correlators at the asymptotic infinity of either anti-de Sitter or de Sitter spacetimes follows, i.e., AdS dϩ1 -or deS dϩ1 -CFT d correspondence. The same construction naturally selects all infrared relevant terms ͑and only those terms͒ in the low energy effective action of gravity in any even integer dimension. The infrared relevant terms arising from the known anomalies in dϭ4 imply that the classical Einstein theory is modified at large distances.

The unitary conformal field theory behind 2D Asymptotic Safety

“…Considerations of nontrivial cocycles of the Weyl group show that the corresponding Wess-Zumino action in d = 4 is generated by the E-and the F -term [43], analogous to the generation of ∆I in section 3.3 due to the R-term. It may thus be expected that there would be a mechanism to take the 4D limit, similar to the one of section 3.3 but now for E and F instead of R, if the couplings a k and b k were of first order in ε.…”

“…As shown in appendix A, the change of I under a finite Weyl transformation of the metric in its argument equals precisely −8 ∆I which therefore has the interpretation of a WessZumino term, a 1-cocycle related to the Abelian group of Weyl transformations [43]: 6…”

The (2+1)-dimensional gravastars

“…Thus, we study the work of Mazur and Mottola with chargeless gravastar in (3+1)-dimensions [2,3] under the (2+1)-dimensional space-time. We show that the (2+1)-dimensional neutral gravastars do exist without any curvature singularity at the origin, which may be considered as an alternative to BTZ black holes as presented by Bañados, Teitelboim and Zanelli (BTZ) [4].…”

“…Very recently, we proposed a charged (3+1)-dimensional gravastar admitting conformal motion [1] in the frame work of Mazur and Mottola model of a chargeless gravastar [2,3]. The model implies that the space of a gravastar has three different regions defined by different equations of state (EOS) as: (i) interior: 0 < r ≤ r 1 , p = −ρ, (ii) shell: 0 < r 1 < r < r 2 , p = +ρ and (iii) exterior: r 2 < r, p = ρ = 0.…”

Inflation and the dS/CFT Correspondence

“…There is no direct construction of the boundary field theory, although some general characteristics can be indirectly deduced such as the relation between conformal weights and particle masses and the absence 1 This conjecture followed related observations in [5][6][7][8][9][10][11][12][13][14][15][16][17], and was inspired by an analogy to related early [18] and modern [19][20][21] results obtained in AdS (anti-de Sitter space) . See also [22,23].…”