Holographic Interpolation between $a$ and $F$

An interpolating function $\tilde F$ between the $a$-anomaly coefficient in even dimensions and the free energy on an odd-dimensional sphere has been proposed recently and is conjectured to monotonically decrease along any renormalization group flow in continuous dimension $d$. We examine $\tilde F$ in the large-$N$ CFT's in $d$ dimensions holographically described by the Einstein-Hilbert gravity in the AdS$_{d+1}$ space. We show that $\tilde F$ is a smooth function of $d$ and correctly interpolates the $a$ coefficients and the free energies. The monotonicity of $\tilde F$ along an RG flow follows from the analytic continuation of the holographic $c$-theorem to continuous $d$, which completes the proof of the conjecture.


I. INTRODUCTION
A measure of degrees of freedom in a quantum field theory (QFT) remains to be elucidated in arbitrary d dimensions. Physically, it decreases monotonically as the energy scale is lowered because of the decoupling of massive particles. Implementation of such a measure in any QFT in diverse dimensions is intriguing and desirable to characterize the behavior under a renormalization group (RG) flow.
For even d, the conformal anomaly in the stress-energy tensor [1] defines the unique a coefficient for the Euler density E d and several b i coefficients for the Weyl invariants I i labeled by an integer i. The a coefficients are believed to be monotonically decreasing along any RG flow, namely the value a UV at the ultra-violet (UV) fixed point is equal or greater than that a IR at the infra-red (IR) fixed point, a UV ≥ a IR . This statement was established in two dimensions by the Zamolodchikov's c-theorem [2] and in four dimensions by the a-theorem [3-5]. On the other hand, the F -theorem asserts that the free energy, log Z S d , defined by the conformal invariant partition function Z S d on S d of radius R, decreases under any RG flow in odd dimensions [6,7]. A proof for d = 3 was presented by [8] through the relation of the free energy to the entanglement entropy S across an entangling surface S d−2 of radius R in R 1,d−1 [9] that holds for odd d up to UV divergences. These two proposals look quite different at first sight, but share the fact that both the a coefficient and the free energy can be read off on S d ; the former arises from the integration of the trace of the stress-energy tensor (1) and the latter from the partition function. To interpolate between the a coefficient and the free energy, Giombi and Klebanov define a new function [10] which correctly reduces to the free energies for odd d.
They show as d approaches to even integers [11] (see also [12] as a related work)F Note that the partition function Z S d used in (3) is conformal invariant and UV divergent for even d. The relation (4) follows from the fact that the conformal invariant partition function in d = 2n + dimensions behaves as log Z S d = (−1) d 2 a 2 + O(1) for small . This is because one has to add a local counter term to the partition function to obtain the renormalized partition function log Z The functionF is also defined for non-integer d and therefore smoothly interpolates between the a coefficients in even dimensions and the free energies in odd dimensions. They conjecture thatF is positive and decreases along any RG flow in arbitrary d dimensions, based on several examples including a double-trace deformation of the large-N conformal field theory (CFT). We will call their proposal theF -theorem.
In this letter, we provide a further evidence to thẽ F -theorem from the holographic viewpoint. To this end, we take advantage of the relation (2) and calculate the holographic entanglement entropy [13, 14] across a sphere S d−2 in the Einstein-Hilbert gravity on the AdS d+1 space. We perform the dimensional regularization in the bulk and obtain the analytic result ofF that is a positive and smooth function of dimension d. We show arXiv:1410.5973v2 [hep-th] 31 Oct 2014 that the equality (4) holds for even d and furthermore prove theF -theorem that follows from the holographic c-theorem [15][16][17][18] assuming the dimensional continuation of the null energy condition.

II. HOLOGRAPHIC PROOF OF THẼ F -THEOREM
We will evaluateF with the relation (2) between the free energy on S d and the entanglement entropy across S d−2 . The latter can be holographically calculated by the Ryu-Takayanagi formula in the Einstein-Hilbert gravity [13, 14] where G (d+1) N is the Newton constant, and γ stands for the (d − 1)-dimensional minimal surface in the AdS d+1 space, whose boundary is the entangling surface S d−2 . Since the boundary of the AdS d+1 space is the flat space R 1,d−1 , we will use the Poincaré coordinates where L is the AdS radius. The entangling surface is located at t = 0 and r = R at the boundary z = 0. In these coordinates, the minimal surface γ in the bulk is a hemi-hypersphere satisfying r 2 + z 2 = R 2 [13,14]. This solution leads the entanglement entropy across S d−2 where we introduced a small cutoff at z = to regularize the UV divergence and Vol(S d−2 ) is the volume of a unit (d − 2)-dimensional round sphere. Expanding the integrand with respect to y and performing the integration, one obtains the UV divergent parts of the entanglement entropy. We, however, want to employ the dimensional regularization instead of putting the UV cutoff at z = for our purpose. So we take = 0 and carry out the integral in the range 1 < d < 2, that yields Then we analytically continue d to any real value. It is clear that there are poles at even d in the entanglement entropy (9) corresponding to the conformal anomalies. Finally, using the relations (2) and (3), and the formula Γ(z)Γ(1−z) = π/ sin(πz), we obtainF in the holographic theoriesF This is manifestly a positive and smooth function of dimension d without poles at even d.
Now let us extrapolate the holographic values ofF to even dimensions and see if the relation (4) holds. The a coefficients holographically computed in the Einstein-Hilbert gravity are known to be [17][18][19][20] Combining it with (10), we confirm the relation (4) be-tweenF and a. Moreover, imposing the null energy condition in the bulk, the holographic c-theorem states that the a coefficient given by (11) satisfies the monotonicity, a UV ≥ a IR , for positive integer d [15][16][17][18]. Assuming the analytic continuation of dimension d in the gravity, the holographic c-theorem holds for d ≥ 1 [21] which assures theF -theorem due to the relation (4).

ACKNOWLEDGMENTS
We are grateful to Y. Tachikawa and K. Yonekura for valuable discussions and to S. Giombi