Quantum Correction of the Wilson Line and Entanglement Entropy in the Pure AdS 3 Einstein Gravity Theory
aa r X i v : . [ h e p - t h ] M a y NORDITA 2019-102
Quantum Correction of theWilson Line and Entanglement Entropyin thePure AdS Einstein Gravity Theory
Xing Huang a,b , Chen-Te Ma c,d,e , and Hongfei Shu f,g a Institute of Modern Physics, Northwest University, Xi’an 710069, China. b Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China. c Guangdong Provincial Key Laboratory of Nuclear Science,Institute of Quantum Matter, South China Normal University, Guangzhou 510006,Guangdong, China. d School of Physics and Telecommunication Engineering,South China Normal University, Guangzhou 510006, Guangdong, China. e The Laboratory for Quantum Gravity and Strings,Department of Mathematics and Applied Mathematics,University of Cape Town, Private Bag, Rondebosch 7700, South Africa. f Nordita, KTH Royal Institute of Technology and Stockholm University,Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden. g Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan. e-mail address: [email protected] e-mail address: [email protected] e-mail address: [email protected] bstract We calculate the expectation value of the Wilson line in the pure AdS Einsteingravity theory and also the entanglement entropy in the boundary theory. Ourone-loop calculation of entanglement entropy shows a shift of the central charge26. Finally, we show that the Wilson line provides the equivalent description tothe boundary entanglement entropy. This equivalence leads to a concrete exampleof the building of “minimum surface=entanglement entropy”. Introduction
The holographic principle states that physical degrees of freedom in quantum gravitytheory is fully encoded by the boundary [1]. Because Einstein gravity theory [2] losesrenormalizability, a direct study in quantum gravity theory is not easy. The holo-graphic principle provides the boundary perspective to study quantum gravity theory.In this direction, the most well-studied is the Anti-de Sitter/ Conformal Field Theory(AdS/CFT) correspondence [3].The AdS/CFT correspondence was motivated and conjectured by the ultraviolet com-plete theory, string theory. Therefore, this conjecture is quite realizable. The AdS blackhole solution [4] also provides an application to condensed matter systems [5]. A newapplication of the AdS/CFT correspondence is the equivalence between entanglemententropy of CFT [6] and the codimension-two minimum surface at a given time slice inthe AdS background [7]. The replica trick [8] is the most general method for computingentanglement entropy, but it is still hard to obtain an exact solution. The holographicmethod [9] provides usefulness to studying entanglement entropy in strongly coupledCFT [10]. The proposal becomes more realizable by using the replica trick in the bulkgravity theory to reach the same conclusion [11]. Because the entanglement entropyneeds to be defined by the decomposition, the gravity and gauge theories should sufferfrom the non-gauge invariant cutting. Nevertheless, borrowing the von Neumann al-gebra gives an interpretation for doing a partial trace operation without breaking thegauge symmetry in each sub-region [12]. Hence this holographic study gives a con-cretely useful application to the AdS/CFT correspondence.What people are mostly interested in the holographic principle is pure Einstein gravitytheory. Nevertheless, people cannot probe a quantum region due to the well-known is-sue, renormalizability. Nowadays, the closest route is the AdS Einstein gravity theory[13], defined by a gauge theory [14], not by a metric formulation. The gauge formula-tion is the SL(2) Chern-Simons gravity theory [14]. This theory can be quantized andis renormalizable [15]. The gauge formulation is equivalent to the metric formulationonly up to the classical and perturbation level [16]. Two formulations are not equiva-lent exactly. This gauge formulation [17] is also interesting for the simple extension ofhigher spins [18] with a unified study from the SL( M ) groups [19].Since the three-dimensional Einstein gravity theory does not have the local gravitation1uctuation, a direct derivation of the boundary theory is possible. The original deriva-tion points to CFT , the Liouville theory [20]. This theory does not have a normalizablevacuum. It contradicts a known truth. This contradiction goes away after one foundtwo-dimensional Schwarzian theory [21]. In a quantum regime, this boundary theorybreaks conformal symmetry in the sense that modular invariance is absent.The Wilson line [22] provides the universal contribution [23] to the holographic entan-glement entropy at the classical level [24]. The central question that we would like toaddress in this letter is the following: What is the quantum deformation of the minimumsurface?
Since the minimum surface is only defined at the classical level, we would liketo study the quantum deformation using the computable gauge formulation.We precisely compute [25] entanglement entropy for one-interval using the n -sheet par-tition function [26]. Then we show that the bulk operator, Wilson line, is exactly dualto the boundary entanglement entropy. We provide the quantum deformation of theminimum surface [27] from the Wilson line with the dual of entanglement entropy [28]. The action of the SL(2) Chern-Simons gravity theory is given by [14] S G = k π Z d x ǫ trθ Tr (cid:18) A t F rθ − (cid:0) A r ∂ t A θ − A θ ∂ t A r (cid:1)(cid:19) − k π Z d x ǫ trθ Tr (cid:18) ¯ A t ¯ F rθ − (cid:0) ¯ A r ∂ t ¯ A θ − ¯ A θ ∂ t ¯ A r (cid:1)(cid:19) − k π Z dtdθ Tr( A θ ) − k π Z dtdθ Tr( ¯ A θ ) , (1)in which we assume that the boundary conditions of the gauge fields A and ¯ A are: A − ≡ A t − A θ = 0 and ¯ A + = A t + A θ = 0. The variable k is defined by l/ (4 G ), where1 /l ≡ − Λ. The cosmological constant is denoted by Λ, and the three-dimensionalgravitational constant is denoted by G . The gauge fields are defined by the vielbein2 µ and spin connection ω µ : A µ ≡ A aµ J a ≡ J a (cid:18) l e aµ + ω aµ (cid:19) , ¯ A ν ≡ ¯ A aν ¯ J a ≡ ¯ J a (cid:18) l e aν − ω aν (cid:19) , (2)in which the Lie algebra indices are labeled by a , and the indices are raised or loweredby η ≡ diag( − , , µ and ν . This bulk termsin this theory are equivalent to the Chern-Simons theory up to a boundary term. Themeasure in this gravitation theory is R D A D ¯ A .The SL(2) × SL(2) generators are given by the followings: J ≡ − ! , J ≡ ! ,J ≡ − ! , ¯ J ≡ − ! , ¯ J ≡ − − ! , ¯ J ≡ − ! . (3)The generators satisfy the algebra:[ J a , J b ] = ǫ abc J c , Tr (cid:0) J a J b (cid:1) = η ab / J a , ¯ J b ] = − ǫ abc ¯ J c , Tr (cid:0) ¯ J a ¯ J b (cid:1) = η ab / . (4)The AdS geometry is ds = − ( r + 1) dt + dr / ( r + 1) + r dθ , in which the rangesof coordinates are defined by that −∞ < t < ∞ , 0 < r < ∞ , and 0 < θ ≤ π . We alsochoose the unit Λ = −
1. The metric is defined by the vielbein g µν ≡ · Tr( e µ e ν ).We substitute the solution ( F rθ = 0 and ¯ F rθ = 0) into the action, and use the asymptoticboundary condition to get: g − ∂ θ g SL(2) | r →∞ = A θ | r →∞ , ¯ g − ∂ θ ¯ g SL(2) | r →∞ = ¯ A θ | r →∞ , (5)3hich are fixed by the metric of AdS boundary. Using the SL(2) transformations: g SL(2) = F ! λ λ ! ! , ¯ g SL(2) = − ¯ F ! λ
00 ¯ λ ! − ¯Ψ 1 ! , (6)we obtain the boundary conditions: λ ∂ θ F = 2 r , ∂ θ F/∂ θ F = − r Ψ, ¯ λ ∂ θ ¯ F = 2 r ,and ∂ θ ¯ F /∂ θ ¯ F = − r ¯Ψ, which eventually give the boundary theory, two-dimensionalSchwarzian theory [21], S G = k π Z dtdθ (cid:18)
32 ( ∂ − ∂ θ F )( ∂ θ F )( ∂ θ F ) − ∂ − ∂ θ F∂ θ F (cid:19) − k π Z dtdθ (cid:18)
32 ( ∂ + ∂ θ ¯ F )( ∂ θ ¯ F )( ∂ θ ¯ F ) − ∂ + ∂ θ ¯ F∂ θ ¯ F (cid:19) , (7)where x + ≡ t + θ, x − ≡ t − θ, (8) ∂ + = 12 ∂ t + 12 ∂ θ , ∂ − = 12 ∂ t − ∂ θ . (9)The measure is R dF d ¯ F (cid:0) / ( ∂ θ F ∂ θ ¯ F ) (cid:1) . We first write down the boundary theory on the sphere manifold [21] and then calculatethe n -sheet partition function to obtain the entanglement entropy for one-interval, whichshows a shift of the central charge. The bulk Euclidean AdS metric can be asymptotically written as ds a = r ds + dr /r ,where ds = dψ + sin ψdθ , 0 ≤ ψ < π , and 0 ≤ θ < π . The ψ is the Euclidean4ime defined by ψ ≡ it . The line element ds is for the unit sphere. The asymptoticbehaviors of the gauge fields for the Lorentzian AdS metric are: A r →∞ = dr r rE + − dr r ! , ¯ A r →∞ = − dr r − rE − dr r ! , (10)where E + ≡ E θ + E t and E − ≡ E θ − E t are the boundary zweibein. Then we canfind the below boundary condition by replacing r → rE ± θ : λ = p rE + θ /∂ θ F , Ψ = − ( ∂ θ F/∂ θ F ) / (4 rE + θ ), ¯ λ = p rE − θ /∂ θ ¯ F , and ¯Ψ = − ( ∂ θ ¯ F /∂ θ ¯ F ) / (4 rE − θ ). For the spheremanifold, we have E ψ = dψ and E θ = sin ψdθ . Because we did the Wick rotation, weuse the following coordinates: x + = − iψ + θ , x − = − iψ − θ , ψ = i ( x + + x − ) /
2, and θ = ( x + − x − ) /
2. The θ -component of the boundary zweibein is defined by the E ± θ .Therefore, we have E + θ = E − θ = sin ψ . The boundary gauge-field in the Lorentzianmanifold satisfies the conditions: E + θ A t − E + t A θ = 0 and E − θ ¯ A t − E − t ¯ A θ = 0. Therefore,the AdS gravitation action with the spherical asymptotic boundary is [21] S GS = k π Z d x ǫ trθ Tr (cid:18) A t F rθ − (cid:0) A r ∂ t A θ − A θ ∂ t A r (cid:1)(cid:19) − k π Z d x ǫ trθ Tr (cid:18) ¯ A t ¯ F rθ − (cid:0) ¯ A r ∂ t ¯ A θ − ¯ A θ ∂ t ¯ A r (cid:1)(cid:19) + k π Z dtdθ Tr (cid:18) E + t E + θ A θ (cid:19) − k π Z dtdθ Tr (cid:18) E − t E − θ ¯ A θ (cid:19) . (11)Then we use the conditions λ ∂ θ F = 2 E + θ r and ¯ λ ∂ θ ¯ F = 2 E − θ r to obtain the boundaryeffective action on the sphere manifold [21] S GS = kπ Z dtdθ (cid:18) ( ∂ θ λ )( D − λ ) λ − ( ∂ θ ¯ λ )( D + ¯ λ )¯ λ (cid:19) , (12)where D + ≡ ∂ t + 12 E − t E − θ ∂ θ , D − ≡ ∂ t + 12 E + t E + θ ∂ θ . (13)5rom the field redefinition: F ≡
F/E + θ and ¯ F ≡ ¯ F /E − θ , the gravitation action on thesphere manifold becomes [21]: S GS = k π Z dtdθ (cid:18) ( ∂ θ F )( D − ∂ θ F )( ∂ θ F ) − ( ∂ θ ¯ F )( D + ∂ θ ¯ F )( ∂ θ ¯ F ) (cid:19) = k π Z dtdθ (cid:20) ( ∂ θ φ )( D − ∂ θ φ )( ∂ θ φ ) − ( ∂ θ φ )( D − φ ) (cid:21) − k π Z dtdθ (cid:20) ( ∂ θ ¯ φ )( D + ∂ θ ¯ φ )( ∂ θ ¯ φ ) − ( ∂ θ ¯ φ )( D + ¯ φ ) (cid:21) , (14)in which we used F ≡ tan( φ/
2) and ¯
F ≡ tan( ¯ φ/ Now we want to compute the R´enyi entropy S n ≡ (ln Z n − n ln Z ) / (1 − n ) (15)from the replica trick [8] on the θ -direction, where Z n is the n -sheet partition function,and Z is same as the partition function. Because we only consider the computa-tion up to the one-loop correction, we obtain that the partition function is a prod-uct of the classical partition-function ( Z c ) and the one-loop partition-function ( Z q ) Z n = Z n,c · Z n,q . When we take the logarithmic on the n -sheet partition function, weobtain ln Z n = ln Z n,c + ln Z n,q . Hence we can treat the classical and one-loop partition-functions separately in the computation of the R´enyi entropy. Our motivation is tocompare the entanglement entropy to the expectation value of the Wilson line. There-fore, we will take the limit n → Z n on the sphere manifold, and then the result corre-sponds to the entanglement entropy for one-interval.We first perform the coordinate transformation to get ds s = sech ( y )( dy + dθ ),in which we used sech y = sin ψ . In the n -sheet manifold, the range of the θ is0 < θ ≤ πn . The periodicity of this theory for θ is 2 πn . When we do the computa-tion, we need to regularize the range of the y -direction. The range of the y -direction is − ln( L/ǫ ) < y ≤ ln( L/ǫ ). The periodicity of this theory for the y -direction is 4 ln( L/ǫ )6ecause we assume the Dirichlet boundary condition in the y -direction. The L is thelength of an interval, and ǫ is the cut-off on the ending point of the interval.Finally, we identify the sphere from the torus to determine the complex structure τ n on the sphere. The coordinates of torus z ≡ ( θ + iy ) /n satisfy the identification: z ∼ z + 2 π and z ∼ z + 2 πτ n . The boundary condition of the fields, φ and ¯ φ is given by φ ( y/n, θ/n +2 π ) = φ ( y/n, θ/n )+2 π , φ (cid:0) y/n +2 π · Im( τ n ) , θ/n +2 π · Re( τ n ) (cid:1) = φ ( y/n, θ/n ),¯ φ ( y/n, θ/n + 2 π ) = ¯ φ ( y/n, θ/n ) + 2 π , and ¯ φ (cid:0) y/n + 2 π · Im( τ n ) , θ/n + 2 π · Re( τ n ) (cid:1) =¯ φ ( y/n, θ/n ). Therefore, we can quickly find that the complex structure on the sphereis τ n = (cid:0) i/ ( nπ ) (cid:1) ln( L/ǫ ). The fields on the sphere can be expanded from the way: φ = θ/n + ǫ ( y, θ ) and ¯ φ = − θ/n + ¯ ǫ ( y, θ ), where ǫ ( y, θ ) ≡ X j,k ǫ j,k e i jn θ − kτ y , ǫ ∗ j,k ≡ ǫ − j, − k , ¯ ǫ ( y, θ ) ≡ X j,k ¯ ǫ j,k e i jn θ − kτ y , ¯ ǫ ∗ j,k ≡ ¯ ǫ − j, − k , (16)and τ = nτ n . Because this theory has the SL(2) redundancy, the variables have theconstraints: ǫ j,k = 0 and ¯ ǫ j,k = 0 when j = − , ,
1. To compute the partition functionon the sphere, we need to do the Wick rotation t = − iψ , and then the derivativebecomes: D + = 12 ∂ t + 12 E − ψ E − θ ∂ θ = − i y ) ∂ y −
12 cosh( y ) ∂ θ ,D − = 12 ∂ t + 12 E + ψ E + θ ∂ θ = − i y ) ∂ y + 12 cosh( y ) ∂ θ , (17)and the gravitation action becomes S GS = k π Z π · Im( τ ) − π · Im( τ ) dy Z πn dθ × sech( y ) (cid:20) ( ∂ θ φ )( D − ∂ θ φ )( ∂ θ φ ) − ( ∂ θ φ )( D − φ ) (cid:21) − k π Z π · Im( τ ) − π · Im( τ ) dy Z πn dθ × sech( y ) (cid:20) ( ∂ θ ¯ φ )( D + ∂ θ ¯ φ )( ∂ θ ¯ φ ) − ( ∂ θ ¯ φ )( D + ¯ φ ) (cid:21) . (18)7ubstituting the saddle-points into the action, we obtain S GS = − c n ln (cid:18) Lǫ (cid:19) , (19)where c = 6 k is the central charge of the CFT [23]. Therefore, we obtain ln Z n,c = (cid:0) c/ (6 n ) (cid:1) ln( L/ǫ ). The R´enyi entropy (15) from the saddle-points is given by: S n,c = c − n (cid:18) n − n (cid:19) ln Lǫ = c (1 + n )6 n ln Lǫ , (20)in which we used − n ln Z ,c = − n ( c/ · ln( L/ǫ ). When we take n →
1, we obtain S ,c = ( c/
3) ln(
L/ǫ ).Now we consider the perturbation ǫ ( y, θ ) to obtain the one-loop effect. Because the φ -part and ¯ φ -part are the same, we can only consider the field φ to obtain the one-loopcorrection in the R´enyi entropy. The expansion from the ǫ in the gravitation action forthe φ -part is k π Z π · Im( τ ) − π · Im( τ ) dy Z πn dθ × (cid:18) n (cid:0) ∂ θ ǫ ( y, θ ) (cid:1)(cid:0) ¯ ∂∂ θ ǫ ( y, θ ) (cid:1) − (cid:0) ∂ θ ǫ ( y, θ ) (cid:1)(cid:0) ¯ ∂ǫ ( y, θ ) (cid:1)(cid:19) = − i k π X j,k j ( j − (cid:18) k + jn τ (cid:19) | ǫ j,k | , (21)where ¯ ∂ ≡ ( − i∂ y + ∂ θ ) /
2. Therefore, we obtain ∂ τ ln Z n,q = − X j =0 , ± ∞ X k = −∞ jn k + jn τ . (22)Now we use the following useful equation ˜ ψ (1 − x ) − ˜ ψ ( x ) = π cot( πx ), in which thedigamma function is defined by˜ ψ ( a ) ≡ − ∞ X n =0 n + a . (23)Therefore, we obtain: ∞ X m = −∞ m − x = − ∞ X m =0 m + x + ∞ X m =0 m + 1 − x = ˜ ψ ( x ) − ˜ ψ (1 − x ) = − π · cot( πx ) . (24)8ence we get ∂ τ ln Z n,q = − π ∞ X j =2 (cid:18) jn (cid:19) · cot (cid:18) jπτn (cid:19) . (25)Then we do the re-summation for the above series: ∂ τ ln Z n,q = − π ∞ X j =2 (cid:18) jn (cid:19) · cot (cid:18) jπτn (cid:19) = − π ∞ X j =2 jn · (cid:20) cot (cid:18) jπτn (cid:19) + i (cid:21) + 2 πi ∞ X j =2 jn . (26)By using the regularization ∞ X j =1 j → − , (27)we obtain: ∂ τ ln Z n,q = − π ∞ X j =2 jn · (cid:20) cot (cid:18) jπτn (cid:19) + i (cid:21) + 2 πi ∞ X j =2 jn → − π ∞ X j =2 jn · (cid:20) cot (cid:18) jπτn (cid:19) + i (cid:21) − i π n . (28)After integrating out the τ , we obtain:ln Z n,q = − ∞ X j =2 (cid:20) ln sin (cid:18) πjτn (cid:19) + i jπτn (cid:21) − i πτ n + · · · , (29)where · · · is independent of the τ . The above series is convergent for the Im( τ ) > L/ǫ → ∞ , we obtain ln Z n,q = (cid:0) / (3 n ) (cid:1) ln( L/ǫ ). The R´enyientropy for the one-loop correction is: S n,q = (cid:0) ln Z n,q,CS − n ln Z ,q,CS (cid:1) / (1 − n ) = 13( n + 1)3 n ln (cid:18) Lǫ (cid:19) , (30)where Z n,q,CS is the n -sheet partition of the φ and ¯ φ . Therefore, we obtain the R´enyientropy S n = (cid:0) ( c + 26)( n + 1) / (6 n ) (cid:1) ln( L/ǫ ) and the entanglement entropy S EE = (cid:0) ( c + 26) / (cid:1) ln( L/ǫ ). We obtain a shift of the central charge by 26, instead of 13 [21],because φ and ¯ φ each contributes 13, and the sum gives the shift 26.9 Wilson Line
The entanglement entropy in the two-dimensional Schwarzian theory gives the confor-mal deviation from the quantum correction. Here we want to obtain a bulk descriptionof the entanglement entropy. Since the Wilson lines [24] W ( P, Q ) ≡ Tr (cid:20) P exp (cid:18) Z PQ ¯ A (cid:19) P exp (cid:18) Z PQ A (cid:19)(cid:21) , (31)can provide the entanglement entropy in the CFT , we begin from this operator tostudy. The P denotes the path-ordering, P and Q are the two-ending points of theWilson lines at a time slice. Here the trace operation acts on the representation.We extend the Wilson line to the following form [22] W R ( C )= Z DU DP Dλ × exp (cid:20) Z C ds (cid:18) Tr(
P U − D s U )+ λ ( s ) (cid:0) Tr( P ) − c (cid:1)(cid:19)(cid:21) , (32)where U is an SL(2) element, P is its conjugate momentum, √ c ≡ c (1 − n ) / D s U ≡ dU/ds + A s U + U ¯ A s and A s ≡ A µ · ( dx µ /ds ). The equations of motion are: i (cid:0) k/ (2 π ) (cid:1) F µ µ = − Z ds ( dx µ /ds ) ǫ µ µ µ δ (cid:0) x − x ( s ) (cid:1) U P U − ,i (cid:0) k/ (2 π ) (cid:1) ¯ F µ µ = Z ds ( dx µ /ds ) ǫ µ µ µ δ (cid:0) x − x ( s ) (cid:1) P. (33)A solution of the equations of motion is that [22]: A = g − ag + g − dg , g = exp( L z ) exp( ρL );¯ A = − ¯ g − a ¯ g − ¯ g − d ¯ g , ¯ g = exp( L − ¯ z ) exp( − ρL ), where the gauge field is given as a = p c / · (1 /k ) · ( dz/z − d ¯ z/ ¯ z ) L . The SL(2) algebra is defined by that: [ L j , L k ] =( j − k ) L j + k , j, k = 0 , ±
1; Tr( L ) = 1 /
2, Tr( L − L ) = −
1, and the traces of otherbilinears vanish. Here we choose z ≡ r exp( i Φ) and ¯ z ≡ r exp( − i Φ). Then the space-time interval is ds = dρ + exp(2 ρ )( dr + n r d Φ ) [22]. This solution corresponds to U ( s ) = 1, P ( s ) = √ c L with the curve z ( s ) = 0 and ρ ( s ) = s . Hence we find thatincluding the Wilson line directly gives the n -sheet geometry [22]. When this geometry10pproaches the boundary, it is the n -sheet cylinder ( dt + n d Φ ) up to a scale trans-formation by using r ≡ exp( t ).Let us then comment about the quantum correction of the Wilson line. The backre-action of the Wilson line leads to the n -sheet manifold, and only the Chern-Simonsterm survives when n →
1, because the right hand sides of (33) vanish in that case.Hence this implies that entanglement entropy can be calculated from the Chern-Simonsterm on the n -sheet manifold and the analytical continuation. The expectation valueof Wilson line h W R i is Z n /Z n + · · · , where · · · vanishes when n →
1, and Z n is the n -sheet partition function of the two-dimensional Schwarzian theory, which is equivalentto the Chern-Simons theory with the same boundary condition. In other words, theentanglement entropy is [22] S EE = lim n → − n ln h W R i , (34)where h W R i is the expectation value of the Wilson line. Substituting the classicalsolution of the two-dimensional Schwarzian theory into the Wilson line, it provides theentanglement entropy of CFT [24], which implies that the Wilson line can be seen asthe geodesic line at the on-shell level. Moreover, the equivalence between the Wilsonline and the entanglement entropy is exact. As explained earlier, the reason is thatthe Chern-Simons partition function Z n for computing h W R i reduces to its boundaryversion of a Schwarzian theory, the latter of which is essentially the same partitionintroduced in Sec 3 (due to the same boundary condition). Hence the Wilson linecan be seen as the appropriate operator to provide one equivalent description of theminimum surface even at the quantum level. The Wilson line was used in the AdS Einstein gravity theory for obtaining the entan-glement entropy of CFT [8] at the classical level [22, 24]. It will be promoted to anoperator at a quantum level. We first computed the entanglement entropy in the bound-ary theory, two-dimensional Schwarzian theory [21]. Then we used the Wilson line toobtain the bulk description for the boundary entanglement entropy. This shows thatthe Wilson line is a suitable operator providing an equivalent description of the mini-mum surface in the usual correspondence of “minimum surface=entanglement entropy”.11ne should observe that entanglement entropy is related to the expectation value ofthe Wilson line, not logarithm of the Wilson line. Because the logarithm of Wilsonis an area operator, the quantum contribution of entanglement entropy will give thenon-area term. Hence this should justify that area term is not enough for holographicentanglement entropy [7]. Our study is for the Chern-Simons formulation [14]. Henceour result possibly may not be applied to the metric formulation at a quantum level.The concept of metric is only defined at the classical level but no common sense atthe quantum level. We know that entanglement entropy in CFT is dual to the AdSminimum surface [7]. By observing the quantum correction of entanglement entropy,quantum deformation of the minimum surface is given by the fluctuation of a Wilsonline or a gauge field, not a metric field. Acknowledgments
We would like to thank Chuan-Tsung Chan, Bartlomiej Czech, Jan de Boer, KristanJensen, and Ryo Suzuki for their useful discussion.Xing Huang acknowledges the support of NWU Starting Grant No.0115/338050048 andthe Double First-class University Construction Project of Northwest University. Chen-Te Ma was supported by the Post-Doctoral International Exchange Program and ChinaPostdoctoral Science Foundation, Postdoctoral General Funding: Second Class (GrantNo. 2019M652926), and would like to thank Nan-Peng Ma for his encouragement.Hongfei Shu was supported by the JSPS Research Fellowship 17J07135 for Young Sci-entists, from Japan Society for the Promotion of Science (JSPS) and the grant “ExactResultsin Gauge and String Theories” from the Knut and Alice Wallenberg founda-tion. 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