Holographic Interpolation between $a$ and $F$

We define the stress-energy tensor by T μ ​ ν ≡ 2 g ​ δ ​ I δ ​ g μ ​ ν T_{\mu\nu}\equiv\frac{2}{\sqrt{g}}\frac{\delta I}{\delta g^{\mu\nu}} for an action I I . The Euler density is normalized to be ∫ S d d d ​ x ​ g ​ E d = 2 \intop\nolimits_{S^{d}}d^{d}x\sqrt{g}E_{d}=2

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There is no sign factor ( − 1 ) d / 2 (-1)^{d/2} in the right hand side because our convention of the a a -anomaly ( 1

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S. Ryu and T. Takayanagi, Phys.Rev.Lett. 96

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S. Ryu and T. Takayanagi, JHEP 0608

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The null energy condition T μ ​ ν ​ ξ μ ​ ξ ν ≥ 0 T_{\mu\nu}\xi^{\mu}\xi^{\nu}\geq 0 is crucial in the proof of the holographic c c -theorem [ 16 , 17 , 18 ] where the d d -dimensional null vector ξ \xi has only two non-zero components ξ z \xi^{z} and ξ t \xi^{t} . Thus defining a formal null vector ξ = ( ξ z , ξ t , 0 , ⋯ , 0 ) \xi=(\xi^{z},\xi^{t},0,\cdots,0) in continuous d d dimensions, the proof can be carried over for d ≥ 1 d\geq 1

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