Holographic Rényi entropy for two-dimensional N=(1,1) superconformal field theory
aa r X i v : . [ h e p - t h ] D ec Holographic R´enyi entropy for two-dimensional N = ( , ) superconformal field theory Jia-ju Zhang ∗ Theoretical Physics Division, Institute of High Energy Physics, Chinese Academy of Sciences,19B Yuquan Rd, Beijing 100049, P. R. ChinaTheoretical Physics Center for Science Facilities, Chinese Academy of Sciences,19B Yuquan Rd, Beijing 100049, P. R. China
Abstract
In this paper we investigate the holographic R´enyi entropy in N = 1 supergravity (SUGRA)in AdS spacetime, which is dual to the two-dimensional N = (1 ,
1) superconformal field theory(SCFT). We consider both cases of two short intervals on a line with zero temperature and oneinterval on a circle with low temperature. In SUGRA side we consider contributions of both gravitonand gravitino, and in SCFT side we consider contributions of both stress tensor T , ¯ T and theirsuperpartners G , ¯ G . We find matches between SUGRA and SCFT results. ∗ [email protected] ontents N =(1,1) SCFT 53 Two intervals on a line with zero temperature 7 The investigation of entanglement entropy has been gaining more and more attention in the lastdecade. We give the definitions of entanglement entropy [1, 2]. For a system with normalized densitymatrix ρ with tr ρ = 1, one can divide the system into a subsystem A and its complement B . Theentanglement is defined as S A = − tr A ρ A log ρ A , (1.1)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 calculatesthe R´enyi entropy S ( n ) A = − n − A ρ nA , (1.2)and then takes n → A and B that arenot necessarily complements of each other, one can define the mutual information I A,B = S A + S A − S A ∪ B , (1.3)and the R´enyi mutual entropy I ( n ) A,B = S ( n ) A + S ( n ) B − S ( n ) A ∪ B . (1.4)When there is no ambiguity, we write for short S = S A , S n = S ( n ) A , I = I A,B , and I n = I ( n ) A,B .The R´enyi entropies in a two-dimensional conformal field theory (CFT) are much easier to calculatethan their higher dimensional cousins. For the case of one interval on an infinite straight line withzero temperature, the R´enyi entropy is exact and universal [4] S n = c ( n + 1)6 n log lǫ , (1.5)2ith c being central charge of the CFT, l being the length of the interval, and ǫ being the UV cutoff.For cases of multiple intervals, there are no universal results and details of the CFT are needed [5–9].However, when the interval is short perturbative calculation is available [8–10]. Similarly, R´enyientropy of one interval on a circle with zero temperature is universal [4] S n = c ( n + 1)6 n log (cid:16) Rπǫ sin πlR (cid:17) , (1.6)with l and R being lengthes of the interval and the circle, respectively. When low temperature isturned on, there would be thermal corrections to the R´enyi entropy that depend on field contents ofthe CFT [11–15].As an application of the AdS/CFT correspondence [16–19], one can calculate the entanglement en-tropy in a CFT using the holographic entanglement entropy [20–23] in anti-de Sitter (AdS) spacetime.For a subsystem A in the boundary of the AdS spacetime, the holographic entanglement entropy isproportional to area of the minimal surface Σ A in the bulk that is homogenous to AS A = Area(Σ A )4 G , (1.7)with G being the Newton constant. The area law of the Ryu-Takayanagi (RT) formula for the holo-graphic 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 spacetime is dual to a two-dimensionalCFT with central charge [27] c = 3 ℓ G , (1.8)where ℓ is the AdS radius. Expansion of small Newton constant in gravity side corresponds to expan-sion of large central charge in CFT side. The RT formula in AdS /CFT correspondence was analyzedcarefully in both the CFT and gravity sides. In CFT side the tree level R´enyi entropy at large centralcharge is related to Virasoro vacuum block, which only depends on operators in conformal family ofidentity operator [28]. The corresponding classical gravitational configuration was constructed in [29].Furthermore, 1-loop corrections to holographic R´enyi entropy under this gravitational backgroundwere 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 ona line with zero temperature [8, 10, 25, 30–34]. One could get the R´enyi entropy in expansion of thesmall cross ration x . There were calculations in both the gravity and CFT sides, and perfect matcheswere found. In case of the correspondence between pure Einstein gravity and a large central chargeCFT, there are contributions of graviton in gravity side and contributions of stress tensor T , ¯ T in CFTside [8,10,25,30,34]. In case of higher spin gravity/CFT with W symmetry correspondence, with higherspin chemical potential being turned off, there are contributions of graviton and higher spin fields ingravity side and contributions of stress tensor T , ¯ T and W , ¯ W operators in CFT side [30,31,33,34]. Incase of critical massive gravity/logarithmic CFT correspondence, there are contributions of graviton3nd logarithmic modes in gravity side and contributions of stress tensor T , ¯ T and their logarithmicpartners in CFT side [32]. A scalar field in the bulk corresponds to a scalar operator in the CFT, andthe 1-loop holographic entanglement entropy of the case was considered in [33]. There is a similarstory for the holographic R´enyi entropy of one interval on a circle with low temperature [13–15, 25].In this paper we extend the previous results to supersymmetric AdS /CFT correspondence. Ingravity side we consider N = 1 supergravity (SUGRA) in AdS spacetime, where there is masslessgraviton as well massless gravitino. The N = 1 SUGRA is dual to a two-dimensional N = (1 , T , ¯ T and theirsuperpartners G , ¯ G . We calculate the R´enyi entropy of both cases of two short intervals on a linewith zero temperature and one interval on a circle with low temperature in both SUGRA and SCFTsides. For the case of two intervals, we get the R´enyi mutual information to order x with x beingthe cross ratio. For the case of one interval on a circle, we get the R´enyi entropy to order e − πβ/R with β being the inverse temperature and R being the length of the circle. There are perfect matchesbetween SUGRA and SCFT results for both cases.When dealing with a two-dimensional SCFT, one should be careful with the boundary conditionsof fermionic operators. For an SCFT on a cylinder, one can consider antiperiodic boundary conditionof fermionic operators, and it is called Neveu-Schwarz (NS) sector of the SCFT. Or one can considerperiodic boundary condition, and it is called Ramond (R) sector. One can use a conformal transfor-mation and map an SCFT on a cylinder to an SCFT on a complex plane. For an SCFT on a complexplane, in NS sector fermionic operators are periodic when circling around the origin, while in R sectorfermionic operators are antiperiodic. Fermionic operators are expanded by half-integer modes in NSsector, and by integer modes in R sector. In NS sector vacuum of an SCFT on a complex plane onehas the conformal weights h NS = ¯ h NS = 0, and in R sector vacuum one has h R = ¯ h R = c with c being the central charge. In NS vacuum of an SCFT on a cylinder one has the energy H NS = − πc R with R being the circumference of the cylinder, and in R sector vacuum one has H R = 0. In largecentral charge limit, R sector vacuum is highly excited compared to NS vacuum, and so contributionsof R sector to R´enyi entropy are highly repressed. In AdS /CFT correspondence, NS sector SCFTcorresponds to quantum gravity in global AdS spacetime, while on the other hand R sector SCFTcorresponds to quantum gravity in background of zero mass BTZ black hole [35, 36]. Thus the grav-itational configuration in [25, 29] only corresponds to the NS sector SCFT. So in this paper we onlyconsider contributions of NS sector to R´enyi entropy.The rest of the paper is arranged as follows. In Section 2 we give some basic properties of thetwo-dimensional N = (1 ,
1) SCFT. In Section 3 we calculate the R´enyi mutual information for the caseof two intervals on a line with zero temperature in both the SUGRA and SCFT sides. In Section 4we consider the case of one interval on a circle with low temperature. We conclude with discussion inSection 5. We collect some useful summation formulas in Appendix A.4
Basics of two-dimensional N =(1,1) SCFT In this section we give some basic properties of the two-dimensional N = (1 ,
1) SCFT on a complexplane that are useful in this paper. One can see details in, for example, the textbooks [37–39].In the two-dimensional N = (1 ,
1) SCFT, one has the stress tensor T ( z ) and ¯ T (¯ z ) and theirsuperpartners G ( z ) and ¯ G (¯ z ). Operator G ( z ) is a holomorphic primary operator with conformalweights h = 3 /
2, ¯ h = 0, and ¯ G (¯ z ) is an antiholomorphic primary operator with conformal weights h = 0, ¯ h = 3 /
2. Since the holomorphic and antiholomorphic parts are independent and similar, wewill only discuss the holomorphic part below.For holomorphic quasiprimary operators φ i we have two-point function h φ i ( z ) φ j ( w ) i C = α i δ ij ( z − w ) h i , (2.1)with C denoting the complex plane, h i being conformal weight of φ i , and α i being the normalizationfactor. Note that the operators φ i are orthogonalized but not normalized. Of course we have α = 1for the identity operator 1. For a holomorphic quasiprimary operator φ with conformal weight h andnormalization factor α φ , we have the mode expansion φ ( z ) = X r φ r z r + h , (2.2)with r being integers or half-integers when h is an integer or a half-integer. Note that we only considerNS sector for fermionic operators in this paper. When φ is hermitian, we have φ † r = φ − r . We alsohave φ r | i = 0 , r > − h, (2.3)with | i being the vacuum state. We have the correspondence between operators and states ∂ m φ ↔ | ∂ m φ i ≡ ∂ m φ (0) | i = m ! φ − h − m | i , m = 0 , , , · · · . (2.4)We have the bra states h ∂ m φ | = | ∂ m φ i † = m ! h | φ h + m = h | ∂ m φ ( ∞ ) , (2.5)with the definitions ∂ m φ ( ∞ ) ≡ lim z →∞ ( − z ∂ z ) m [ z h φ ( z )] . (2.6)We have the normalization factors α ∂ m φ ≡ h ∂ m φ | ∂ m φ i = m !(2 h + m − h − α φ . (2.7)For examples, when m = 0 , , φ ( ∞ ) = z h φ ( z ) , ∂φ ( ∞ ) = − z h +2 ∂φ ( z ) − hz h +1 φ ( z ) , (2.8) ∂ φ ( ∞ ) = z h +4 ∂ φ ( z ) + 2(2 h + 1) z h +3 ∂φ ( z ) + 2 h (2 h + 1) z h +2 φ ( z ) , · · · operator 1 G T ∂G ∂T B , ∂ G A , C , ∂ T D , ∂ B , ∂ G ∂ A , ∂ C , ∂ T · · · 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 correlationfunctions. For example we have h ∂ m φ i | ∂ n φ j i = h ∂ m φ i ( ∞ ) ∂ n φ j (0) i C = α ∂ m φ i δ ij δ mn . (2.9)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 α T = c , α G = 2 c , (2.10)with c being the central charge of the SCFT. As stated in the introduction, we only consider NSsector 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 / T ( z ) and G ( z ), and then we have the N = 1 super Virasoroalgebra [ L m , L n ] = ( m − n ) L m + n + c m ( m − δ m + n , [ L m , G r ] = (cid:16) m − r (cid:17) G m + r , (2.11) { G r , G s } = 2 L r + s + c (cid:16) r − (cid:17) δ r + s . Note that every local operator in a two-dimensional unitary CFT can be written as linear combina-tions of quasiprimary operators and their derivatives. We count the number of linearly independentholomorphic operators in the N = (1 ,
1) SCFT to level 5 astr x L = ∞ Y m =0 x m +3 / − x m +2 = 1 + x / + x + x / + x + 2 x / + 3 x + 3 x / + 3 x + O ( x / ) , (2.12)from which we get the number of holomorphic quasiprimary operators to level 5 as(1 − x )tr x L + x = 1 + x / + x + x / + 2 x + x / + O ( x / ) . (2.13)These operators can be written as quasiprimary operators or derivatives thereof, and they are listedin Table 1. For the N = (1 ,
1) SCFT, we can also introduce a complex Grassmann variable θ and work in a superspace withcoordinate ( z, θ ). The quasiprimary operators can be combined as super quasiprimary operators in superspace. Eachholomorphic super quasiprimary operator is composed of two holomorphic quasiprimary operators Φ( z, θ ) = φ ( z )+ θψ ( z ),and they are related by | ψ i ∼ G − / | φ i . This may be useful in the search of quasiprimary operators in higer levels. Alsofor the SCFT n that will be introduced in Subsection 3.2, we may expand the twistor operators in global superconformalblocks 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. A = ( T T ) − ∂ T, B = ( T G ) − ∂ G, C = ( G∂G ) + 12(5 c + 22) [34( T T ) − (7 c + 41) ∂ T ] , (2.14) D = ( T i ∂G ) −
34 (i ∂T G ) −
15 i ∂ G, with the brackets denoting normal ordering. Note that C is not only a quasiprimary operator, butalso a primary one. The normalization factors for these quasiprimary operators are α A = c (5 c + 22)10 , α B = c (4 c + 21)12 , (2.15) α C = c (4 c + 21)(10 c − c + 22) , α D = 7 c (10 c − , 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 hD ( z ) D ( w ) i C = α D ( z − w ) , (2.16)with α D being positive in large c limit. For the same reason we do not have factor i in the definitionof C .Also we need how these operators transform under a general conformal transformation z → f ( z ).We have the Schwarz derivative s ( z ) ≡ f ′′′ ( z ) f ′ ( z ) − (cid:18) f ′′ ( z ) f ′ ( z ) (cid:19) , (2.17)and define the shorthand f = f ( z ) , f ′ = f ′ ( z ) , f ′′ = f ′′ ( z ) , s = s ( z ) . (2.18)These quasiprimary operators transform as T ( z ) = f ′ T ( f ) + c s, G ( z ) = f ′ / G ( f ) , A ( z ) = f ′ A ( f ) + 5 c + 2230 s (cid:16) f ′ T ( f ) + c s (cid:17) , (2.19) B ( z ) = f ′ / B ( f ) + 4 c + 2148 f ′ / sG ( f ) , C ( z ) = f ′ C ( f ) , D ( z ) = f ′ / D ( f ) + i(10 c − f ′ / (cid:0) f ′′ s − f ′ s ′ ) G ( f ) + 4 f ′ sG ′ ( f ) (cid:1) , from which transformations of their derivatives can be got easily. In this section we investigate the R´enyi 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 is7mall, and so we can get the first few orders of the R´enyi entropy in both the gravity side and theCFT side. In the gravity side it is the N = 1 SUGRA, and there are contributions from both thegraviton and the gravitino. In the CFT side, it is the N = (1 ,
1) SCFT, and there are contributionsfrom both stress tensor T , ¯ T and operators G , ¯ G . The classical part of the holographic R´enyi entropy is proportional to the central charge. It is relatedto the classical configuration of the gravity. The gravitino vanishes in classical SUGRA solution, andso we conclude that the gravitational configuration for pure Einstein gravity in [25, 29] still applies tothe N = 1 SUGRA case. We have the classical part of the holographic R´enyi mutual information [25] I (cl) n = c ( n − n + 1) x n + c ( n − n + 1) x n + c ( n − n + 1) (11 n + 1)(119 n − x n + c ( n − n + 1) (589 n − n − x n + O ( x ) , (3.1)with c being the central charge of the dual SCFT.The 1-loop part of the holographic R´enyi entropy depends on the field contents of the gravitytheory, and one considers the fluctuation of the fields around the classical background. The procedurewas given in [25], and it is related to the 1-loop partition function in [40,41]. The 1-loop R´enyi entropyis S n = − n − (cid:16) log Z n − n log Z (cid:17) , (3.2)with Z n being the 1-loop partition function around a genus n − by a Schottky group Γ, the1-loop partition function for the spin-2 massless graviton is [40–42] Z = Y γ ∈P ∞ Y m =0 | − q m +2 γ | , (3.3)with P being a set of representatives of the primitive conjugacy classes of Γ. Here q γ is defined inthe way that the eigenvalues of γ is q ± / γ with | q γ | <
1. For the case of two short intervals on a linewith zero temperature, q γ can be written as expansion of the cross ration x , and so the 1-loop R´enyientropy can be expanded by x too. To order x the 1-loop R´enyi mutual information is [25] I n, (2) = ( n + 1)( n + 11)(3 n + 10 n + 227) x n (3.4)+ ( n + 1)(109 n + 1495 n + 11307 n + 81905 n − x n + O ( x ) . In the N = 1 SUGRA in AdS background, there is also the superpartner of the graviton, themassless spin-3/2 gravitino. The 1-loop partition function (3.3) should be multiplied by [43, 44] Z / = Y γ ∈P ∞ Y m =0 | q m +3 / γ | . (3.5)8hen we get the additional 1-loop R´enyi mutual information from the gravitino I n, (3 / = ( n + 1)(2 n + 23 n + 191) x n + ( n + 1)(33 n + 358 n + 2857 n − x n (3.6)+ ( n + 1)(32422 n + 336385 n + 2606961 n − n − x n + O ( x ) . We use the replica trick in the SCFT side, and get an SCFT on an n -sheeted complex plane, which is agenus ( n − N −
1) Riemann surface R n,N in the case of N intervals. Equivalently, this configurationcan be viewed as n copies of the SCFT on a complex plane, with twist operators σ , ˜ σ being insertedat the boundaries of the intervals [4]. We denote the n copies of the SCFT as SCFT n . The twistoperators are primary operators with conformal weights h σ = ¯ h σ = h ˜ σ = ¯ h ˜ σ = c ( n − n . (3.7)We choose the two intervals A = [0 , y ] ∪ [1 , y ] with y ≪
1, and so the cross ratio x = y ≪ R n, is equivalent of the four-point function of SCFT n on a complex plane [4] tr A ρ nA = h σ (1 + y, y )˜ σ (1 , σ ( y, y )˜ σ (0 , i C . (3.8)We use the OPE of the twist operators to do short interval expansion [8–10, 30–33]. We denote theorthogonalized quasiprimary operators in SCFT n by Φ K ( z, ¯ z ), and a general Φ K has normalizationfactor α K and conformal weights h K , ¯ h K .In SCFT n we have the operator product expansion (OPE) [8–10] σ ( z, ¯ z )˜ σ (0 ,
0) = c n z h σ ¯ z h σ X K d K X m,r ≥ a mK m ! ¯ a rK r ! z h K + m ¯ z ¯ h K + r ∂ m ¯ ∂ r Φ K (0 , , (3.9)with c n being the normalization factor of the twist operators, summation K being over all the inde-pendent quasiprimary operators of SCFT n , and a mK ≡ C mh K + m − C m h K + m − , ¯ a rK ≡ C r ¯ h K + r − C r h K + r − . (3.10)Also, the OPE coefficient d K can be calculated as [9] d K = 1 α K l h K +¯ h K lim z →∞ z h K ¯ z h K h Φ K ( z, ¯ z ) i R n, , (3.11)with l being the length of the single interval [0 , l ] that results in the Riemann surface R n, in replicatrick. To calculate the expectation value of Φ K on R n, , we use the conformal transformation [4, 9] z → f ( z ) = (cid:16) z − lz (cid:17) /n , (3.12)that maps R n, with coordinate z to a complex plane with coordinate f .9evel quasiprimary operator degeneracy number0 1 1 13/2 G j n n T j n n G j G j with j < j n ( n − n ( n − B j n n T j G j with j = j n ( n − A j n C j n n ( n + 1) T j T j with j < j n ( n − E j j with j < j n ( n − D j n G j G j G j with j < j < j n ( n − n − n ( n +1)( n +2)6 F j j with j = j n ( n − G j B j with j = j n ( n − T j G j G j with j = j , j = j , j < j n ( n − n − n ( n − n +2)2 H j j with j < j n ( n − I j j with j < j n ( n − · · · · · · · · · · · · Table 2: Holomorphic quasiprimary operators in SCFT n to level 5. Here j , j , j , j are integers andtake values from 0 to n − A ρ nA = c n x − c ( n − n X K α K d K x h K +¯ h K F ( h K , h K ; 2 h K ; x ) F (¯ h K , ¯ h K ; 2¯ h K ; x ) , (3.13)with summation K being over all the independent quasiprimary operators of SCFT n , and F being thehypergeometric function. When every quasiprimary operator we consider can be written as a productof holomorphic and antiholomorphic parts and there is one-to-one correspondence between operatorsin holomorphic and antiholomorphic sectors, the partition function can be further simplified astr A ρ nA = c n x − c ( n − n (cid:16) X K α K d K x h K F ( h K , h K ; 2 h K ; x ) (cid:17) , (3.14)with summation K being over all the independent holomorphic quasiprimary operators. In this casethe R´enyi mutual information is I n = 2 n − (cid:16) X K α K d K x h K F ( h K , h K ; 2 h K ; x ) (cid:17) . (3.15)10n SCFT n , we count the number of independent holomorphic quasiprimary operators as(1 − x ) (cid:0) tr x L (cid:1) n + x = 1 + nx / + nx + n ( n − x + n x / + n ( n + 1) x (3.16)+ n ( n + 1)( n + 2)6 x / + n ( n − n + 2)2 x + O ( x / ) , with tr x L being defined as (2.12). These holomorphic quasiprimary operators are listed in Table 2,where we have the definitions E j j = G j i ∂G j − i ∂G j G j , F j j = T j i ∂G j −
34 i ∂T j G j , (3.17) H j j = T j i ∂T j − i ∂T j T j , I j j = ∂G j ∂G j −
38 ( G j ∂ G j + ∂ G j G j ) . The factors i’s in F j j and H j j are chosen to make α F > α H >
0. We would have α E > α I > if G is bosonic. But in fact G is fermionic, and so we have α E < α I < α K and OPE coefficients d K . It is easy to see that d G = d B = d C = d D = d T G = d GGG = d F = 0 . (3.18)The useful normalization factors are α = 1 , α T = c , α GG = − c , α A = c (5 c + 22)10 ,α T T = c , α E = − c , α G B = − c (4 c + 21)18 , (3.19) α T GG = − c , α H = 2 c , α I = − c . The useful OPE coefficients are d = 1 , d T = n − n , d j j GG = − n c s j j ,d A = ( n − n , d j j T T = 18 n c s j j + ( n − n ,d j j E = − n c c j j s j j , d j j G B = − i( n − n c s j j , (3.20) d j j j T GG = i64 n c (cid:18) s j j s j j s j j − ( n − cs j j (cid:19) ,d j j H = 116 n c c j j s j j , d j j I = − i448 n c (cid:18) s j j − n + 7) s j j (cid:19) , with the definitions s j j ≡ sin π ( j − j ) n and c j j ≡ cos π ( j − j ) n .Finally, using the formula (3.15), normalization factors (3.19), OPE coefficients (3.18) and (3.20),as well as the summation formulas in Appendix A, we obtain the R´enyi mutual information I n = I tree n + I n + I n + · · · , (3.21)11ith the tree part being I tree n = c ( n − n + 1) x n + c ( n − n + 1) x n + c ( n − n + 1) (11 n + 1)(119 n − x n + c ( n − n + 1) (589 n − n − x n + O ( x ) , (3.22)the 1-loop part being I n = ( n + 1)(2 n + 23 n + 191) x n + ( n + 1)(201 n + 2191 n + 17479 n + 289) x n + ( n + 1)(11098 n + 116115 n + 899139 n + 40985 n − x n + O ( x ) , (3.23)and the 2-loop part being I n = ( n + 1)( n − n + 19)( n + 19 n + 628) x cn + O ( x ) . (3.24)The result in the SCFT side can be compared to the SUGRA one. The tree part of the R´enyimutual information (3.22) equals the classical part of the holographic R´enyi mutual information (3.1) I tree n = I cl n . (3.25)The 1-loop part of the R´enyi mutual information (3.23) equals the summation of the 1-loop holographicR´enyi mutual information from the graviton (3.4) and gravitino (3.6) I n = I n, (2) + I n, (3 / . (3.26)The result is in accordance with the SUGRA/SCFT correspondence. In this section we investigate the R´enyi entropy of one interval on a circle with low temperature. Inthis case the CFT is located on a torus. We calculate in both the SUGRA and SCFT sides, using themethods in [13–15, 25].
We set that the length of the circle is R and the interval is A = [ − l/ , l/ T , andthe inverse temperature is β = 1 /T . In low temperature we have β ≫ R , and the holographic R´enyientropy can be expanded by exp( − πβ/R ) [14, 15, 25]. The procedure is that one firstly calculates theR´enyi entropy at large temperature that is expanded by exp( − πR/β ) and then makes the modulartransformation R → i β , β → i R to get the R´enyi entropy at low temperature.At zero temperature, the holographic R´enyi entropy for one interval with length l on a circle withlength R is [20, 21, 25] S n = c ( n + 1)6 n log (cid:16) Rπǫ sin πlR (cid:17) , (4.1)12ith ǫ being the UV cutoff. This is the same as the CFT result (1.6) in [4]. At low temperature, therewould be thermal correction to the R´enyi entropy. Similar to the case of two intervals on a line, theclassical Holographic R´enyi entropy in SUGRA is the same as that in pure Einstein gravity. One canfind the classical part of the correction to the holographic R´enyi entropy in [14, 15, 25] δS cl n = − (cid:18) c ( n − n + 1) n sin πlR (cid:19) e − πβ/R + O (e − πβ/R ) . (4.2)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 R´enyi entropy from graviton can be found in [14, 15, 25] δS n, (2) = − n − " n sin πlR sin πlnR − n ! e − πβ/R + O (e − πβ/R ) . (4.3)We also get the 1-loop correction to the R´enyi entropy from gravitino δS n, (3 / = − n − (cid:26)(cid:18) n sin πlR sin πlnR − n (cid:19) e − πβ/R + (cid:20) n sin πlR sin πlnR (cid:18) n cos πlR sin πlnR (4.4) − n sin 2 πlR sin 2 πlnR + sin πlR (cid:16) πlnR + 5 (cid:17)(cid:19) − n (cid:21) e − πβ/R + O (e − πβ/R ) (cid:27) . We take the n → δS = 8 (cid:16) − πlR cot πlR (cid:17) e − πβ/R + O (e − πβ/R ) , (4.5) δS / = 6 (cid:16) − πlR cot πlR (cid:17) e − πβ/R + 10 (cid:16) − πlR cot πlR (cid:17) e − πβ/R + O (e − πβ/R ) . We use the method in [13–15] and calculate the contributions of stress tensor T , ¯ T and operators G , ¯ G to R´enyi entropy in the SCFT side. As in the case of two intervals, we only analyze the holomorphicoperators carefully, and we multiply the R´enyi entropy by a factor 2 to account for the contributionsfrom the antiholomorphic sector.When the temperature is low β ≫ R , we has the SCFT that is located on a cylinder with athermally corrected density matrix. The hamiltonian from the holomorphic sector is H = 2 πR (cid:16) L − c (cid:17) , (4.6)and without affecting the final result we shift it to be H = 2 πR L . (4.7)We have the unnormalized density matrix [13–15] ρ = | ih | + X φ ∞ X m =0 e − π ( m + h φ ) β/R α ∂ m φ | ∂ m φ ih ∂ m φ | , (4.8)with summation φ being over all the independent non-identity holomorphic quasiprimary operators ofthe SCFT. To the order of e − πβ/R we only need to consider three states | G i , | T i , and | ∂G i .13e trace the degree of freedom of B , and get the reduced density matrix ρ A = tr B ρ . Then we getthe R´enyi entropy S n = − n − A ρ nA (tr A ρ A ) n , (4.9)with the additional factor 2 accounting for contributions from the antiholomorphic sector. Note thatwe have tr A ρ A = tr ρ , as well as [4] log tr A (tr B | ih | ) n = − c ( n − n log (cid:16) Rπǫ sin πlR (cid:17) . (4.10)Thus the thermal correction to R´enyi entropy is δS n = − nn − (cid:2) ( I − − πβ/R + ( II − − πβ/R + ( III − − πβ/R + O (e − πβ/R ) (cid:3) , (4.11)with definitions of I , II and III being I = tr A (cid:2) tr B | G ih G | (tr B | ih | ) n − (cid:3) α G tr A (cid:0) tr B | ih | (cid:1) n ,II = tr A (cid:2) tr B | T ih T | (tr B | ih | ) n − (cid:3) α T tr A (cid:0) tr B | ih | (cid:1) n , (4.12) III = tr A (cid:2) tr B | ∂G ih ∂G | (tr B | ih | ) n − (cid:3) α ∂G tr A (cid:0) tr B | ih | (cid:1) n . Originally we have the SCFT on a cylinder with coordinate w = x − i t and of circumference R .We denote the cylinder also by R . In replica trick we get an SCFT on an n -sheeted cylinder, whichwe 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 π i w/R , (4.13)and this changes the n -sheeted cylinder R n with coordinate w to an n -sheeted complex plane C n withcoordinate z . Then we make the transformation [13–15] f ( z ) = (cid:18) z − e i πl/R z − e − i πl/R (cid:19) /n , (4.14)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 thenone uses (2.6), (4.14) and transforms them to correlation functions on C . Explicitly, we have I = h G ( ∞ ) G (0) i C n α G = 1 n sin πlR sin πlnR ,II = h T ( ∞ ) T (0) i C n α T = c ( n − n sin πlR + 1 n sin πlR sin πlnR , (4.15) III = h ∂G ( ∞ ) ∂G (0) i C n α ∂G = 12 n sin πlR sin πlnR (cid:18) n cos πlR sin πlnR − n sin 2 πlR sin 2 πlnR + sin πlR (cid:16) πlnR + 5 (cid:17)(cid:19) . Note that here we have only incorporated contributions from holomorphic sector, and to get the full result we needto multiply a factor 2 on the right-hand side of the equation. δS tree n = − (cid:18) c ( n − n + 1) n sin πlR (cid:19) e − πβ/R + O (e − πβ/R ) , (4.16)which equals to the classical part of the correction to holographic R´enyi entropy δS cl n (4.2). The 1-looppart of correction to R´enyi entropy is δS n = − n − (cid:26)(cid:18) n sin πlR sin πlnR − n (cid:19) e − πβ/R + n sin πlR sin πlnR − n ! e − πβ/R + (cid:20) n sin πlR sin πlnR (cid:18) n cos πlR sin πlnR − n sin 2 πlR sin 2 πlnR (4.17)+ sin πlR (cid:16) πlnR + 5 (cid:17)(cid:19) − n (cid:21) e − πβ/R + O (e − πβ/R ) (cid:27) , and this equals the summation of contributions of graviton and gravitino to the 1-loop holographicR´enyi entropy δS n, (2) (4.3) and δS n, (3 / (4.4). In this paper we investigated the holographic R´enyi entropy for the two-dimensional N = (1 ,
1) SCFT,which is dual to N = 1 SUGRA in AdS spacetime. We considered both cases of two short intervalson a line with zero temperature and one interval on a circle with low temperature. For the first case,we got the R´enyi mutual information to order x with x being the cross ratio. For the second case,we got the thermal correction of R´enyi entropy to order e − πβ/R with β being the inverse temperatureand 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 caseand in terms of e − πβ/R for the one interval case. In the SUGRA side, the so-called p -consecutivelydecreasing words and p -letter words of Schottky group with p ≥ m -point correlation functions with m ≥
4. It is also interesting to consider the holographicR´enyi entropy of large interval at high temperature in the SUGRA/SCFT correspondence, as whatwas done for Einstein gravity in [45, 46].In this paper we have only considered the NS sector of the SCFT. It is an interesting questionwhether one can calculate R´enyi entropy of the SCFT in R sector vacuum and compare it with theholographic result in some suitable gravitational background.
Acknowledgments
The author would like to thank Bin Chen and Jun-Bao Wu for careful reading of the manuscript andvaluable suggestions. Special thanks Matthew Headrick for his Mathematica code
Virasoro.nb thatcould be downloaded at http://people.brandeis.edu/~headrick/Mathematica/index.html . Thework was in part supported by NSFC Grants No. 11222549 and No. 11575202.15
Some useful summation formulas
In the appendix we give some summation formulas that are used in our calculation. We define f m = n − X j =1 (cid:16) sin πjn (cid:17) m , (A.1)and explicitly we need f = n − , f = ( n − (cid:0) n + 11 (cid:1) ,f = ( n − (cid:0) n + 23 n + 191 (cid:1) , (A.2) f = ( n − (cid:0) n + 11 (cid:1) (cid:0) n + 10 n + 227 (cid:1) ,f = ( n − (cid:0) n + 35 n + 321 n + 2125 n + 14797 (cid:1) . The above formulas are useful because they appear in the following summations X ≤ j Quantum computation and quantum information . CambridgeUniversity Press, Cambridge, UK, 2010.[2] D. Petz, Quantum information theory and quantum statistics . Springer, Berlin, German, 2008.[3] C. G. Callan Jr. and F. Wilczek, “On geometric entropy,” Phys.Lett. B333 (1994) 55–61, arXiv:hep-th/9401072 [hep-th] .[4] P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J.Stat.Mech. (2004) P06002, arXiv:hep-th/0405152 [hep-th] .[5] M. Caraglio and F. Gliozzi, “Entanglement Entropy and Twist Fields,” JHEP (2008) 076, arXiv:0808.4094 [hep-th] .[6] S. Furukawa, V. Pasquier, and J. Shiraishi, “Mutual information and boson radius in a c = 1critical system in one dimension,” Phys.Rev.Lett. (2009) 170602, arXiv:0809.5113 [cond-mat] . 167] P. Calabrese, J. Cardy, and E. Tonni, “Entanglement entropy of two disjoint intervals inconformal field theory,” J.Stat.Mech. (2009) P11001, arXiv:0905.2069 [hep-th] .[8] M. Headrick, “Entanglement R´enyi entropies in holographic theories,” Phys.Rev. D82 (2010) 126010, arXiv:1006.0047 [hep-th] .[9] P. Calabrese, J. Cardy, and E. Tonni, “Entanglement entropy of two disjoint intervals inconformal field theory II,” J.Stat.Mech. (2011) P01021, arXiv:1011.5482 [hep-th] .[10] B. Chen and J.-j. Zhang, “On short interval expansion of R´enyi entropy,” JHEP (2013) 164, arXiv:1309.5453 [hep-th] .[11] C. P. Herzog and M. Spillane, “Tracing Through Scalar Entanglement,” Phys.Rev. D87 no. 2, (2013) 025012, arXiv:1209.6368 [hep-th] .[12] C. P. Herzog and T. Nishioka, “Entanglement Entropy of a Massive Fermion on a Torus,” JHEP (2013) 077, arXiv:1301.0336 [hep-th] .[13] J. Cardy and C. P. Herzog, “Universal Thermal Corrections to Single Interval EntanglementEntropy for Two Dimensional Conformal Field Theories,” Phys.Rev.Lett. no. 17, (2014) 171603, arXiv:1403.0578 [hep-th] .[14] B. Chen and J.-q. Wu, “Single interval R´enyi entropy at low temperature,” JHEP (2014) 032, arXiv:1405.6254 [hep-th] .[15] B. Chen, J.-q. Wu, and Z.-c. Zheng, “Holographic R´enyi entropy of single interval on Torus:With W symmetry,” Phys.Rev. D92 (2015) 066002, arXiv:1507.00183 [hep-th] .[16] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv.Theor.Math.Phys. (1998) 231–252, arXiv:hep-th/9711200 [hep-th] .[17] S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from noncriticalstring theory,” Phys.Lett. B428 (1998) 105–114, arXiv:hep-th/9802109 [hep-th] .[18] E. Witten, “Anti-de Sitter space and holography,” Adv.Theor.Math.Phys. (1998) 253–291, arXiv:hep-th/9802150 [hep-th] .[19] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, “Large N field theories,string theory and gravity,” Phys.Rept. (2000) 183–386, arXiv:hep-th/9905111 [hep-th] .[20] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys.Rev.Lett. (2006) 181602, arXiv:hep-th/0603001 [hep-th] .[21] S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP (2006) 045, arXiv:hep-th/0605073 [hep-th] .1722] T. Nishioka, S. Ryu, and T. Takayanagi, “Holographic Entanglement Entropy: An Overview,” J.Phys. A42 (2009) 504008, arXiv:0905.0932 [hep-th] .[23] T. Takayanagi, “Entanglement Entropy from a Holographic Viewpoint,” Class.Quant.Grav. (2012) 153001, arXiv:1204.2450 [gr-qc] .[24] A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy,” JHEP (2013) 090, arXiv:1304.4926 [hep-th] .[25] T. Barrella, X. Dong, S. A. Hartnoll, and V. L. Martin, “Holographic entanglement beyondclassical gravity,” JHEP (2013) 109, arXiv:1306.4682 [hep-th] .[26] T. Faulkner, A. Lewkowycz, and J. Maldacena, “Quantum corrections to holographicentanglement entropy,” JHEP (2013) 074, arXiv:1307.2892 .[27] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of AsymptoticSymmetries: An Example from Three-Dimensional Gravity,” Commun.Math.Phys. (1986) 207–226.[28] T. Hartman, “Entanglement Entropy at Large Central Charge,” arXiv:1303.6955 [hep-th] .[29] T. Faulkner, “The Entanglement R´enyi Entropies of Disjoint Intervals in AdS/CFT,” arXiv:1303.7221 [hep-th] .[30] B. Chen, J. Long, and J.-j. Zhang, “Holographic R´enyi entropy for CFT with W symmetry,” JHEP (2014) 041, arXiv:1312.5510 [hep-th] .[31] E. Perlmutter, “Comments on R´enyi entropy in AdS /CFT ,” JHEP (2014) 052, arXiv:1312.5740 [hep-th] .[32] B. Chen, F.-y. Song, and J.-j. Zhang, “Holographic R´enyi entropy in AdS /LCFT correspondence,” JHEP (2014) 137, arXiv:1401.0261 [hep-th] .[33] M. Beccaria and G. Macorini, “On the next-to-leading holographic entanglement entropy in AdS /CF T ,” JHEP (2014) 045, arXiv:1402.0659 [hep-th] .[34] M. Headrick, A. Maloney, E. Perlmutter, and I. G. Zadeh, “R´enyi entropies, the analyticbootstrap, and 3D quantum gravity at higher genus,” JHEP (2015) 059, arXiv:1503.07111 [hep-th] .[35] O. Coussaert and M. Henneaux, “Supersymmetry of the (2+1) black holes,” Phys.Rev.Lett. (1994) 183–186, arXiv:hep-th/9310194 [hep-th] .[36] J. M. Maldacena and A. Strominger, “AdS(3) black holes and a stringy exclusion principle,” JHEP (1998) 005, arXiv:hep-th/9804085 [hep-th] .1837] P. Di Francesco, P. Mathieu, and D. Senechal, Conformal field theory . Springer, New York,USA, 1997.[38] J. Polchinski, String theory: Vol. 2, Superstring theory and beyond . Cambridge University Press,Cambridge, UK, 1998.[39] R. Blumenhagen and E. Plauschinn, “Introduction to conformal field theory,” Lect.Notes Phys. (2009) 1–256.[40] X. Yin, “Partition Functions of Three-Dimensional Pure Gravity,” Commun.Num.Theor.Phys. (2008) 285–324, arXiv:0710.2129 [hep-th] .[41] S. Giombi, A. Maloney, and X. Yin, “One-loop Partition Functions of 3D Gravity,” JHEP (2008) 007, arXiv:0804.1773 [hep-th] .[42] B. Chen and J.-q. Wu, “One loop partition function in AdS /CFT ,” arXiv:1509.02062 [hep-th] .[43] J. R. David, M. R. Gaberdiel, and R. Gopakumar, “The Heat Kernel on AdS(3) and itsApplications,” JHEP (2010) 125, arXiv:0911.5085 [hep-th] .[44] H.-b. Zhang and X. Zhang, “One loop partition function from normal modes for N = 1supergravity in AdS ,” Class.Quant.Grav. (2012) 145013, arXiv:1205.3681 [hep-th] .[45] B. Chen and J.-q. Wu, “Large Interval Limit of R´enyi Entropy At High Temperature,” arXiv:1412.0763 [hep-th] .[46] B. Chen and J.-q. Wu, “Holographic calculation for large interval R´enyi entropy at hightemperature,” Phys. Rev. D92 (2015) 106001, arXiv:1506.03206 [hep-th]arXiv:1506.03206 [hep-th]