6.1 Logarithmic terms in 4-dimensional conformal field theory

In four dimensions the bulk conformal anomaly is a combination of two terms, the topological Euler term and the square of the Weyl tensor,
bulk a2 = AE (4∫) + BI (4), 1-- αβμν μν 2 E(4) = 64 E(R αβμνR − 4R μνR + R ), 1 ∫ αβμν μν 1 2 I(4) = − --- (RαβμνR − 2R μνR + --R ) . (274 ) 64 E 3
These are, respectively, the conformal anomalies of type A and B. In a theory with ns particles of spin s, one finds [76] (the contributions of fields of spin 3/2 and 2 can be obtained from Table 2 on p. 180 of the book of Birrell and Davies [22])
A = --1--(n0 + 11n + 62n1 + 0n + 0n2), 90π2 1∕2 3∕2 --1-- 233- 424- B = 30π2 (n0 + 6n1∕2 + 12n1 − 6 n3∕2 + 3 n2) . (275 )
The surface contribution to the conformal anomaly can be calculated directly by, for example, the heat kernel method, as in [101]. Although straightforward, the direct computation is technically involved. However, one has a short cut: there is a precise balance, observed in [196] and [112Jump To The Next Citation Point], between the bulk and surface anomalies; this balance is such that, to first order in (1 − α ), one can take α bulk α 2 a2(E ) = a2 (E ) + O (1 − α ) and use for the Riemann tensor of α E the representation as a sum of regular and singular (proportional to a delta-function concentrated on surface Σ) parts. The precise expressions are given in [112Jump To The Next Citation Point, 111Jump To The Next Citation Point]. However, this representation is obtained under the assumption that the surface Σ is a stationary point of an abelian isometry and thus has vanishing extrinsic curvature. Under this assumption, one finds that [112, 111Jump To The Next Citation Point] (see also [188Jump To The Next Citation Point])
a2(E α) = αab2ulk(E ) + (1 − α)aΣ2 + O (1 − α)2, aΣ = Aa Σ+ Ba Σ, 2 ∫A B aΣA = π- (Rijij − 2Rii + R ), 8 Σ∫ Σ π- 1- aB = − 8 Σ(Rijij − Rii + 3R ), (276 )
where α β μ ν Rijij = R αβμνni n jni nj, α β Rii = R αβni n i.

Each surface term in Eq. (276View Equation) is invariant under a sub-class of conformal transformations, g → e−2ωg, such that the normal derivatives of ω vanish on surface Σ. Moreover, the surface term due to the bulk Euler number is a topological invariant: using the Gauss–Codazzi equation

R = R Σ + 2Rii − Rijij − kiki + Trk2, (277 )
where RΣ is the intrinsic Ricci scalar of the surface and i kαβ is the extrinsic curvature, and in the assumption of vanishing extrinsic curvature the aΣA term, as shown in [111], is proportional to the Euler number of the 2D surface Σ,
∫ aΣ = π- R , (278 ) A 8 Σ Σ
where R Σ is the intrinsic curvature of Σ.

For completeness we note that this result can be generalized to an arbitrary codimension 2 surface in 4-dimensional spacetime. Then, the conformal transformation is generalized to any function ω with non-vanishing normal derivative at Σ. The terms with the normal derivatives of ω in the conformal transformation of a Σ 2 can be canceled by adding the quadratic combinations of extrinsic curvature, 2 Trk and kaka. The analysis presented by Solodukhin [204] (this analysis is based on an earlier consideration by Dowker [70]) results in the following expressions

aΣ2 = AaΣA + Ba ΣB , Σ π ∫ 2 π ∫ aA = -- (Rijij − 2Rii + R − Trk + kiki) = -- R Σ, 8 Σ∫ 8 Σ aΣB = − π- (Rijij − Rii + 1R − (Trk2 − 1kiki)). (279 ) 8 Σ 3 2
This is the most general form of the logarithmic term in the entanglement entropy in four spacetime dimensions.

Thus, as follows from Eq. (274View Equation), the logarithmic term in the entanglement entropy of a black hole in four dimensions is

(d=4) π-∫ π-∫ 1- s0 = A 8 Σ R Σ − B 8 Σ(Rijij − Rii + 3R ). (280 )
For conformal fields of various spin the values of A and B are presented in Eq. (275View Equation).

Consider some particular examples.

Extreme static geometry.
For an extreme geometry, which has the structure of the product H2 × S2 and characterized by two dimensionful parameters l (radius of H2) and l1 (radius of S2), the logarithmic term in the entropy

2 2 sext= Aπ2 − B-π--(1 − l1-) (281 ) 0 3 l2
is determined by both the anomalies of type A and B. In the case of the extreme Reissner–Nordström black hole one has l = l1 and the logarithmic term (281View Equation) is determined only by the anomaly of type A. For a conformal scalar field one has that 2 A = B ∕3 = 1∕90 π and this equation reduces to Eq. (258View Equation). As we already discussed, the geometry H2 × S2 for l = l1 is conformal to flat 4-dimensional space. Thus, the Weyl tensor vanishes in this case as does its projection to the subspace orthogonal to horizon S 2. That is why the type B anomaly does not contribute in this case to the logarithmic term.

The Schwarzschild black hole.
In this case, the background is Ricci flat and the logarithmic term is determined by the difference of A and B,

Sch 2 s0 = (A − B )π . (282 )
The same is true for any Ricci flat metric. For a conformal scalar field, Eq. (282View Equation) reduces to Eq. (88View Equation). For a scalar field the relation of the logarithmic term in the entropy and the conformal anomaly was discussed by Fursaev [103Jump To The Next Citation Point]. The logarithmic term vanishes if A = B. In this case the Riemann tensor does not appear in the conformal anomaly (274View Equation) so that the anomaly vanishes if the metric is Ricci flat. In particular, the relation A = B can be found from the 𝒩 = 4 super-conformal gauge theory, dual to supergravity on AdS5, according to the AdS/CFT correspondence of Maldacena [167Jump To The Next Citation Point]. The conformal anomaly in this theory was calculated in [131Jump To The Next Citation Point].

Non-extreme and extreme Kerr black hole.
For a Kerr black hole (characterized by mass m and rotation a) the logarithmic term does not depend on the parameter of the rotation and it takes the same form

sK0err= (A − B )π2 (283 )
as in the case of the Schwarschild metric. In the extreme limit a = m the logarithmic term takes the same value as (283View Equation).
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