6.2 Logarithmic terms in 6-dimensional conformal field theory

In six dimensions, omitting the total derivative terms, the conformal anomaly is a combination of four different conformal invariants [17Jump To The Next Citation Point]
∫ abulk = (B I + B I + B I + AE ), (284 ) 3 M6 11 2 2 3 3 6
where I 1, I 2 and I 3 are cubic in the Weyl tensor
imnj kl kl mn ij I1 = CkijlC Cm n , I2 = Cij Ckl C mn , 2 i i 6- i jklm I3 = Ciklm (∇ δj + 4R j − 5R δj)C , (285 )
and E6 is the Euler density (60View Equation)
---1--- ν1ν2...ν6 μ1μ2 μ5μ6 E6 = 3072 π3𝜖μ1μ2...μ6𝜖 R ν1ν2...R ν5ν6 . (286 )
As was shown in [17Jump To The Next Citation Point] in a free conformal field theory with n0 scalars, n1∕2 Dirac fermions and nB 2-form fields, one has that10
8 ⋅ 3! ( 5 191 221 ) A = ----- − ---n0 − ---n1 ∕2 − ----nB , (287 ) 7! (72 72 4 ) B = ---1--- 28n + 896-n + 8008-n , (288 ) 1 (4π)37! 3 0 3 1∕2 3 B 1 ( 5 2378 ) B2 = ----3-- − -n0 + 32n1∕2 + -----nB , (289 ) (4π) 7! 3 3 ---1---( ) B3 = (4π)37! − 2n0 − 40n1 ∕2 − 180nB . (290 )
Applying the formulas (55View Equation) to I1, I2 and I3 and using the relation (62View Equation) for the Euler number, one finds for the logarithmic term in the entanglement entropy of 4-dimensional surface Σ in a 6-dimensional conformal field theory
sd0=6 = B1 sB1 + B2 sB2 + B3 sB3 + A sA , (291 )
where
sA = χ [Σ ] (292 )
is the Euler number of the surface Σ, and
jαβi ij jαβj ii 1- iαβμ i 1-- αβμν sB1 = 6 π(C C α β − C C α β − 4C C αβμ + 20C C αβμν), (293 ) 1 sB2 = 6 π(2CijαβC αiβj− Ci αβμCiαβμ + --CαβμνC αβμν), (294 ) 5 s = 8 π(∇2Cijij + 4Ri C αjij − R C αiβi − 6RCijij + Ci Ciαβμ − 3-CαβμνC ), (295 ) B3 α αβ 5 αβμ 5 αβμν
where tensors with Latin indices are obtained by contraction with components of normal vectors i n α, i = 1,2. Note that in Eq. (295View Equation) we used for brevity the notation 2 ijij i j i j 2 αβμν ∇ C ≡ nαnβn μnν∇ C. Eqs. (293View Equation), (294View Equation), and (295View Equation) agree with the results obtained in [134].

Let us consider some examples.

6-dimensional Schwarzschild black hole.
The 6-dimensional generalization of the Schwarzschild solution is [180]

r3 ds2 = g(r)d τ2 + g−1(r)dr2 + r2dω2S4 , g(r) = 1 − -+3-, (296 ) r
where dω2S4 is a metric of unit 4-sphere. The area of horizon is 2 A+ = 8π3-r4+. The Euler number of the horizon χ [S4] = 2. This metric is Ricci flat so that only the Riemann tensor contributes to the Weyl tensor. The logarithmic term in this case is
sSch = 16π3(− 51B1 + 156B2 − 192B3 ) + 2A . (297 ) 0
It is interesting to note that this term vanishes in the case of the interacting (2,0) conformal theory, which is dual to supergravity on AdS7. Indeed, in this case one has [131Jump To The Next Citation Point, 17Jump To The Next Citation Point]
b 8 ⋅ 3! Bi = ---i---, A = -----a, (4π)37! 7! 35- a = − 2 , b1 = 1680, b2 = 420, b3 = − 140 (298 )
so that (2,0) s0 = 0. This is as expected. The Riemann tensor does not appear in the conformal anomaly of the strongly interacting (2,0) theory so that the anomaly identically vanishes if the spacetime is Ricci flat. This property is not valid in the case of the free (2,0) tensor multiplet [17] so that the logarithmic term of the free multiplet is non-vanishing.

Conformally-flat extreme geometry.
In conformally-flat spacetime the Weyl tensor C αβμν = 0 so that terms (293View Equation), (294View Equation) and (295View Equation) identically vanish. The logarithmic term (291View Equation) then is determined by the anomaly of type A only. In particular, this is the case for the extreme geometry H2 × S4 with equal radii l = l1 of two components. One has

se0xt= 2A (299 )
for this extreme geometry. This geometry is conformal to flat spacetime and the logarithmic term (299View Equation) is the same as for the entanglement entropy of a round sphere in flat 6-dimensional spacetime. This generalizes the result discussed in Section 5.3.2 for the entropy of a round sphere due to a conformal scalar field. The result (299View Equation), as is shown in [39, 181], generalizes to a spherical entangling surface in a conformally-flat spacetime of any even dimension.
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