Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity
Received: August
Universality in the shape dependence of holographic
Chong-Sun Chu 0 1 2 3
Rong-Xin Miao 0 1 2
0 Open Access , c The Authors
1 101 Section 2 Kuang Fu Road, Hsinchu 30013 , Taiwan
2 Department of Physics, National Tsing-Hua University
3 Physics Division, National Center for Theoretical Sciences, National Tsing-Hua University
We consider higher derivative gravity and obtain universal relations for the shape coefficients (fa, fb, fc) of the shape dependent universal part of the R´enyi entropy for four dimensional CFTs in terms of the parameters (c, t2, t4) of two-point and three-point functions of stress tensors. As a consistency check, these shape coefficients fa and fc satisfy the differential relation as derived previously for the R´enyi entropy. Interestingly, these holographic relations also apply to weakly coupled conformal field theories such as theories of free fermions and vectors but are violated by theories of free scalars. The mismatch of fa for scalars has been observed in the literature and is due to certain delicate boundary contributions to the modular Hamiltonian. Interestingly, we find a combination of our holographic relations which are satisfied by all free CFTs including scalars. We conjecture that this combined relation is universal for general CFTs in four dimensional spacetime. Finally, we find there are similar universal laws for holographic R´enyi entropy in general dimensions. ArXiv ePrint: 1608.00328
AdS-CFT Correspondence; Classical Theories of Gravity; Conformal Field
1 Introduction
2 Holographic R´enyi entropy for higher derivative gravity 2.1 Gauss-Bonnet gravity 2.1.1
2.2 General higher curvature gravity
3 The story of free CFTs
4 Universality of HRE in general dimensions 4.1 CFTs in three dimensions
4.2 CFTs in higher dimensions
5 Conclusions A Equivalence between two stress tensors B Solutions in general higher curvature gravity C Universal laws in general dimensions
Introduction
One of the most mysterious features of quantum mechanics is the phenomena of
entanmeasured in terms of the entanglement entropy and the R´enyi entropy
For any integer n > 1, the R´enyi entropy Sn may be obtained from
1 − n
Sn =
Sn =
log Zn − n log Z1
1 − n
where Zn is the partition function of the field theory on a certain n-fold branched cover
manlabeled by an integer n, from which entanglement entropy SEE can be obtained as a limit
SEE = nli→m1 Sn
if Sn is continued to real n.
The study of entanglement entropy and the nature of quantum nonlocality has brought
new insights into our understandings of gravity. It is found that entanglement plays an
Generally, for a spatial region A in a d-dimensional spacetime, the R´enyi entropy for A
is UV divergent. If one organizes in terms of the short distance cutoff ǫ, one finds it contain
a universal term in the sense that it is independent on the UV regularization scheme one
choose. In odd spacetime dimensions, the universal term is ǫ independent. In even
spaceuniversal term of the R´enyi entropy has the following geometric expansion [14, 15],
Snuniv = log ǫ
Here the conformal invariants are
2 √
2 √
part of extrinsic curvature and the contraction of the Weyl tensor projected to directions
described by the coefficients fa, fb, fc, which depend on n and the details of CFTs in
general. The coefficient fa can be obtained by studying the thermal free energy of CFTs
on a hyperboloid [6]. The coefficients fc and fb are determined by the stress tensor
oneit is found in [16] that fc is completely determined by fa:
n − 1
fc(n) =
fa(1) − fa(n) − (n − 1)fa′ (n) .
It was conjectured in [17] that
fb(n) = fc(n)
holds for general 4d CFTs. This conjecture has passed numerical test for free scalar and
free fermion [17]. According to [12], it seems that the relation (1.8) holds only for free
CFTs. Evidence includes an analytic proof for free scalar. However, it is found to be
violated by strongly coupled CFTs with Einstein gravity duals [9].
In this paper, we apply the holographic approach developed in [9, 10, 13] to study the
general higher derivative gravity duals. For 4d CFTs, expanding the coefficients (fa, fb, fc)
(c, t2, t4) of two point and three point functions of stress tensors [18, 19]:
fa(n) = a − 2
(n − 1) + c
81 t4 (n − 1)2 + O(n − 1)3
fb(n) = c − c
fc(n) = c − c
45 t4 (n − 1) + O(n − 1)2
27 t4 (n − 1) + O(n − 1)2.
It should be mentioned that the expansion (1.9) of fa has been obtained in [20] by using
twoproof of it. We note that (1.9) and (1.11) satisfy the relation (1.7). This can be regarded as
and the eqs. (1.10), (1.11) reduce to the results obtained in [9] in this case. To the best of
our knowledge, the universal dependence of fb on the coefficients t2, t4 as obtained in the
relation (1.10) is new. This is one of the main results of this paper.
We remark that our holographic relations eqs. (1.9)–(1.11) are also satisfied by free
fermions and vectors.1 However, mismatch appears for free scalars. Actually, the
discrepa (...truncated)