Holographic R\'enyi entropy for two-dimensional N=(1,1) superconformal field theory

In this paper we investigate the holographic R\'enyi entropy in $\mathcal N=1$ supergravity (SUGRA) in AdS$_3$ spacetime, which is dual to the two-dimensional $\mathcal N=(1,1)$ superconformal field theory (SCFT). We consider both cases of two short intervals on a line with zero temperature and one interval on a circle with low temperature. In SUGRA side we consider contributions of both graviton and gravitino, and in SCFT side we consider contributions of both stress tensor $T$, $\bar T$ and their superpartners $G$, $\bar G$. We find matches between SUGRA and SCFT results.


Introduction
The investigation of entanglement entropy has been gaining more and more attention in the last decade. We give the definitions of entanglement entropy [1,2]. For a system with normalized density matrix ρ with trρ = 1, one can divide the system into a subsystem A and its complement B. The entanglement is defined as with the reduced density matrix being ρ A = tr B ρ. It encodes the quantum entanglement between A and B. To calculate the entanglement entropy, one can use the replica trick [3]. One firstly calculates the Rényi entropy S (n) and then takes n → 1 limit to get the entanglement entropy. For two subsystems A and B that are not necessarily complements of each other, one can define the mutual information A,B . The Rényi entropies in a two-dimensional conformal field theory (CFT) are much easier to calculate than their higher dimensional cousins. For the case of one interval on an infinite straight line with zero temperature, the Rényi entropy is exact and universal [4] S n = c(n + 1) 6n log l ǫ , (1.5) with c being central charge of the CFT, l being the length of the interval, and ǫ being the UV cutoff.
However, when the interval is short perturbative calculation is available [8][9][10]. Similarly, Rényi entropy of one interval on a circle with zero temperature is universal [4] S n = c(n + 1) 6n log R πǫ sin πl R , (1.6) with l and R being lengthes of the interval and the circle, respectively. When low temperature is turned on, there would be thermal corrections to the Rényi entropy that depend on field contents of the CFT [11][12][13][14][15].
For a subsystem A in the boundary of the AdS spacetime, the holographic entanglement entropy is proportional to area of the minimal surface Σ A in the bulk that is homogenous to A with G being the Newton constant. The area law of the Ryu-Takayanagi (RT) formula for the holographic entanglement entropy has been proved in terms of the generalized gravitational entropy [24].
The RT formula is proportional to the inverse of Newton constant and so is only the classical result, and there are also subleading quantum corrections [8,25,26].
It was proposed long time ago that quantum gravity in AdS 3 spacetime is dual to a two-dimensional CFT with central charge [27] c = 3ℓ 2G , (1.8) where ℓ is the AdS radius. Expansion of small Newton constant in gravity side corresponds to expansion of large central charge in CFT side. The RT formula in AdS 3 /CFT 2 correspondence was analyzed carefully in both the CFT and gravity sides. In CFT side the tree level Rényi entropy at large central charge is related to Virasoro vacuum block, which only depends on operators in conformal family of identity operator [28]. The corresponding classical gravitational configuration was constructed in [29].
Furthermore, 1-loop corrections to holographic Rényi entropy under this gravitational background were calculated in [25], and the field contents of gravity theory are relevant.
Various conditions have been investigated for the holographic entropy of two short intervals on a line with zero temperature [8,10,25,[30][31][32][33][34]. One could get the Rényi entropy in expansion of the small cross ration x. There were calculations in both the gravity and CFT sides, and perfect matches were found. In case of the correspondence between pure Einstein gravity and a large central charge CFT, there are contributions of graviton in gravity side and contributions of stress tensor T ,T in CFT side [8,10,25,30,34]. In case of higher spin gravity/CFT with W symmetry correspondence, with higher spin chemical potential being turned off, there are contributions of graviton and higher spin fields in gravity side and contributions of stress tensor T ,T and W ,W operators in CFT side [30,31,33,34]. In case of critical massive gravity/logarithmic CFT correspondence, there are contributions of graviton and logarithmic modes in gravity side and contributions of stress tensor T ,T and their logarithmic partners in CFT side [32]. A scalar field in the bulk corresponds to a scalar operator in the CFT, and the 1-loop holographic entanglement entropy of the case was considered in [33]. There is a similar story for the holographic Rényi entropy of one interval on a circle with low temperature [13][14][15]25].
In this paper we extend the previous results to supersymmetric AdS 3 /CFT 2 correspondence. In gravity side we consider N = 1 supergravity (SUGRA) in AdS 3 spacetime, where there is massless graviton as well massless gravitino. The N = 1 SUGRA is dual to a two-dimensional N = (1, 1) superconformal field theory (SCFT), where we have to consider the stress tensor T ,T and their superpartners G,Ḡ. We calculate the Rényi entropy of both cases of two short intervals on a line with zero temperature and one interval on a circle with low temperature in both SUGRA and SCFT sides. For the case of two intervals, we get the Rényi mutual information to order x 5 with x being the cross ratio. For the case of one interval on a circle, we get the Rényi entropy to order e −5πβ/R with β being the inverse temperature and R being the length of the circle. There are perfect matches between SUGRA and SCFT results for both cases. with R being the circumference of the cylinder, and in R sector vacuum one has H R = 0. In large central charge limit, R sector vacuum is highly excited compared to NS vacuum, and so contributions of R sector to Rényi entropy are highly repressed. In AdS 3 /CFT 2 correspondence, NS sector SCFT corresponds to quantum gravity in global AdS 3 spacetime, while on the other hand R sector SCFT corresponds to quantum gravity in background of zero mass BTZ black hole [35,36]. Thus the gravitational configuration in [25,29] only corresponds to the NS sector SCFT. So in this paper we only consider contributions of NS sector to Rényi entropy.
The rest of the paper is arranged as follows. In Section 2 we give some basic properties of the two-dimensional N = (1, 1) SCFT. In Section 3 we calculate the Rényi mutual information for the case of two intervals on a line with zero temperature in both the SUGRA and SCFT sides. In Section 4 we consider the case of one interval on a circle with low temperature. We conclude with discussion in Section 5. We collect some useful summation formulas in Appendix A.
In this section we give some basic properties of the two-dimensional N = (1, 1) SCFT on a complex plane that are useful in this paper. One can see details in, for example, the textbooks [37][38][39].
In the two-dimensional N = (1, 1) SCFT, one has the stress tensor T (z) andT (z) and their superpartners G(z) andḠ(z). Operator G(z) is a holomorphic primary operator with conformal weights h = 3/2,h = 0, andḠ(z) is an antiholomorphic primary operator with conformal weights h = 0,h = 3/2. Since the holomorphic and antiholomorphic parts are independent and similar, we will only discuss the holomorphic part below.
For holomorphic quasiprimary operators φ i we have two-point function with C denoting the complex plane, h i being conformal weight of φ i , and α i being the normalization factor. Note that the operators φ i are orthogonalized but not normalized. Of course we have α 1 = 1 for the identity operator 1. For a holomorphic quasiprimary operator φ with conformal weight h and normalization factor α φ , we have the mode expansion with r being integers or half-integers when h is an integer or a half-integer. Note that we only consider NS sector for fermionic operators in this paper. When φ is hermitian, we have φ † r = φ −r . We also have with |0 being the vacuum state. We have the correspondence between operators and states We have the bra states We have the normalization factors For examples, when m = 0, 1, 2 we have Table 1: The linearly independent holomorphic operators in N = (1, 1) SCFT.
with the limit z → ∞. Note that the products of bra and ket states can be written as correlation functions. For example we have This strategy has been used in [14,15,34] to calculate correlation functions.
For quasiprimary operator T and primary operator G, we adopt the usual normalization factors with c being the central charge of the SCFT. As stated in the introduction, we only consider NS sector of the SCFT, and so we expand G by half-integer modes. We use L m with m ∈ Z and G r with r ∈ Z + 1/2 to denote the modes of the T (z) and G(z), and then we have the N = 1 super Virasoro Note that every local operator in a two-dimensional unitary CFT can be written as linear combinations of quasiprimary operators and their derivatives. 1 We count the number of linearly independent holomorphic operators in the N = (1, 1) SCFT to level 5 as from which we get the number of holomorphic quasiprimary operators to level 5 as These operators can be written as quasiprimary operators or derivatives thereof, and they are listed in Table 1.
1 For the N = (1, 1) SCFT, we can also introduce a complex Grassmann variable θ and work in a superspace with coordinate (z, θ). The quasiprimary operators can be combined as super quasiprimary operators in superspace. Each holomorphic super quasiprimary operator is composed of two holomorphic quasiprimary operators Φ(z, θ) = φ(z)+θψ(z), and they are related by |ψ ∼ G −1/2 |φ . This may be useful in the search of quasiprimary operators in higer levels. Also for the SCFT n that will be introduced in Subsection 3.2, we may expand the twistor operators in global superconformal blocks instead of the ordinary global conformal blocks. This may be more convenient in the expansion to higher orders. We thank the anonymous referee for suggestion about this.
Here there are definitions of quasiprimary operators with the brackets denoting normal ordering. Note that C is not only a quasiprimary operator, but also a primary one. The normalization factors for these quasiprimary operators are and the normalization factors for the derivatives of the quasiprimary operators can be got easily from (2.7). In (2.14) we have added a factor i in the definition of D, and this makes that with α D being positive in large c limit. For the same reason we do not have factor i in the definition of C.
Also we need how these operators transform under a general conformal transformation z → f (z).
We have the Schwarz derivative and define the shorthand These quasiprimary operators transform as from which transformations of their derivatives can be got easily.

Two intervals on a line with zero temperature
In this section we investigate the Rényi entropy of two short intervals on a line with zero temperature.
In this case the CFT is located on a complex plane. We consider the case when the cross ratio x is small, and so we can get the first few orders of the Rényi entropy in both the gravity side and the CFT side. In the gravity side it is the N = 1 SUGRA, and there are contributions from both the graviton and the gravitino. In the CFT side, it is the N = (1, 1) SCFT, and there are contributions from both stress tensor T ,T and operators G,Ḡ.

Holographic Rényi entropy
The classical part of the holographic Rényi entropy is proportional to the central charge. It is related to the classical configuration of the gravity. The gravitino vanishes in classical SUGRA solution, and so we conclude that the gravitational configuration for pure Einstein gravity in [25,29] still applies to the N = 1 SUGRA case. We have the classical part of the holographic Rényi mutual information [25] with c being the central charge of the dual SCFT.

Rényi entropy in SCFT side
We use the replica trick in the SCFT side, and get an SCFT on an n-sheeted complex plane, which is a genus (n − 1)(N − 1) Riemann surface R n,N in the case of N intervals. Equivalently, this configuration can be viewed as n copies of the SCFT on a complex plane, with twist operators σ,σ being inserted at the boundaries of the intervals [4]. We denote the n copies of the SCFT as SCFT n . The twist operators are primary operators with conformal weights We choose the two intervals A = [0, y] ∪ [1, 1 + y] with y ≪ 1, and so the cross ratio x = y 2 ≪ 1.
In SCFT n we have the operator product expansion (OPE) [8][9][10] σ(z,z)σ(0, 0) = c n z 2hσz2hσ with c n being the normalization factor of the twist operators, summation K being over all the independent quasiprimary operators of SCFT n , and (3.10) Also, the OPE coefficient d K can be calculated as [9] with l being the length of the single interval [0, l] that results in the Riemann surface R n,1 in replica trick. To calculate the expectation value of Φ K on R n,1 , we use the conformal transformation [4,9] z → f (z) = z − l z that maps R n,1 with coordinate z to a complex plane with coordinate f . level quasiprimary operator degeneracy number · · · · · · · · · · · · Table 2: Holomorphic quasiprimary operators in SCFT n to level 5. Here j, j 1 , j 2 , j 3 are integers and take values from 0 to n − 1.
With the OPE (3.9), the partition function (3.8) becomes [10,30,31] tr A ρ n A = c 2 n x − c(n 2 −1) with summation K being over all the independent quasiprimary operators of SCFT n , and F being the hypergeometric function. When every quasiprimary operator we consider can be written as a product of holomorphic and antiholomorphic parts and there is one-to-one correspondence between operators in holomorphic and antiholomorphic sectors, the partition function can be further simplified as (3.14) with summation K being over all the independent holomorphic quasiprimary operators. In this case the Rényi mutual information is (1 − x) trx L 0 n + x = 1 + nx 3/2 + nx 2 + n(n − 1) 2 x 3 + n 2 x 7/2 + n(n + 1)x 4 (3.16) + n(n + 1)(n + 2) 6 x 9/2 + n(n − 1)(n + 2) 2 with trx L 0 being defined as (2.12). These holomorphic quasiprimary operators are listed in Table 2, where we have the definitions The factors i's in F j 1 j 2 and H j 1 j 2 are chosen to make α F > 0 and α H > 0. We would have α E > 0 and α I > 0 if G is bosonic. But in fact G is fermionic, and so we have α E < 0 and α I < 0 in our definitions.
Then we calculate the normalization factors α K and OPE coefficients d K . It is easy to see that The useful normalization factors are The useful OPE coefficients are (3.20) with the definitions s j 1 j 2 ≡ sin π(j 1 −j 2 ) n and c j 1 j 2 ≡ cos π(j 1 −j 2 ) with the tree part being The result is in accordance with the SUGRA/SCFT correspondence.

One interval on a circle with low temperature
In this section we investigate the Rényi entropy of one interval on a circle with low temperature. In this case the CFT is located on a torus. We calculate in both the SUGRA and SCFT sides, using the methods in [13][14][15]25].

Holographic Rényi entropy
We set that the length of the circle is R and the interval is A = [−l/2, l/2]. The temperature is T , and the inverse temperature is β = 1/T . In low temperature we have β ≫ R, and the holographic Rényi entropy can be expanded by exp(−2πβ/R) [14,15,25]. The procedure is that one firstly calculates the Rényi entropy at large temperature that is expanded by exp(−2πR/β) and then makes the modular transformation R → iβ, β → iR to get the Rényi entropy at low temperature.
At zero temperature, the holographic Rényi entropy for one interval with length l on a circle with length R is [20,21,25] S n = c(n + 1) 6n log R πǫ sin πl R , (4.1) with ǫ being the UV cutoff. This is the same as the CFT result (1.6) in [4]. At low temperature, there would be thermal correction to the Rényi entropy. Similar to the case of two intervals on a line, the classical Holographic Rényi entropy in SUGRA is the same as that in pure Einstein gravity. One can find the classical part of the correction to the holographic Rényi entropy in [14,15,25] δS cl For the 1-loop part, (3.3) and (3.5) still apply, but now the Schottky group is parameterized differently.
The 1-loop correction to Rényi entropy from graviton can be found in [14,15,25]   We also get the 1-loop correction to the Rényi entropy from gravitino We take the n → 1 limit, and get the 1-loop correction to the entanglement entropy

Rényi entropy in SCFT side
We use the method in [13][14][15] and calculate the contributions of stress tensor T ,T and operators G,Ḡ to Rényi entropy in the SCFT side. As in the case of two intervals, we only analyze the holomorphic operators carefully, and we multiply the Rényi entropy by a factor 2 to account for the contributions from the antiholomorphic sector.
When the temperature is low β ≫ R, we has the SCFT that is located on a cylinder with a thermally corrected density matrix. The hamiltonian from the holomorphic sector is and without affecting the final result we shift it to be We have the unnormalized density matrix [13][14][15] with summation φ being over all the independent non-identity holomorphic quasiprimary operators of the SCFT. To the order of e −5πβ/R we only need to consider three states |G , |T , and |∂G .
the Rényi entropy Thus the thermal correction to Rényi entropy is with definitions of I, II and III being Originally we have the SCFT on a cylinder with coordinate w = x − it and of circumference R.
We denote the cylinder also by R. In replica trick we get an SCFT on an n-sheeted cylinder, which we denote by R n . Note that here we take the viewpoint that there is one copy of the SCFT and n copies of the cylinder. Firstly we make the transformation z = e 2πiw/R , (4.13) and this changes the n-sheeted cylinder R n with coordinate w to an n-sheeted complex plane C n with coordinate z. Then we make the transformation [13][14][15] 14) and this changes the n-sheeted complex plane C n to a complex plane C with coordinate f . To calculate (4.12), we adopt the strategy in [14,15]. Firstly they equal to correlation functions on C n , and then one uses (2.6), (4.14) and transforms them to correlation functions on C. Explicitly, we have

Conclusion and discussion
In this paper we investigated the holographic Rényi entropy for the two-dimensional N = (1, 1) SCFT, which is dual to N = 1 SUGRA in AdS 3 spacetime. We considered both cases of two short intervals on a line with zero temperature and one interval on a circle with low temperature. For the first case, we got the Rényi mutual information to order x 5 with x being the cross ratio. For the second case, we got the thermal correction of Rényi entropy to order e −5πβ/R with β being the inverse temperature and R being the length of the circle. We found perfect matches between SUGRA and SCFT results.
It would be nice to extend the results to higher orders, in terms of x for the two intervals case and in terms of e −πβ/R for the one interval case. In the SUGRA side, the so-called p-consecutively decreasing words and p-letter words of Schottky group with p ≥ 2 would be needed. In SCFT side one needs m-point correlation functions with m ≥ 4. It is also interesting to consider the holographic Rényi entropy of large interval at high temperature in the SUGRA/SCFT correspondence, as what was done for Einstein gravity in [45,46].
In this paper we have only considered the NS sector of the SCFT. It is an interesting question whether one can calculate Rényi entropy of the SCFT in R sector vacuum and compare it with the holographic result in some suitable gravitational background.
In the appendix we give some summation formulas that are used in our calculation. We define The above formulas are useful because they appear in the following summations 0≤j 1 <j 2 ≤n−1 1 s 2m with s j 1 j 2 ≡ sin π(j 1 −j 2 ) n . There are also two other useful summation formulas