Reconstructing the Bulk Dual of ABJM from Holographic Entanglement Entropy
RReconstructing the Bulk Dual of ABJM fromHolographic Entanglement Entropy
Ashton Lowenstein ∗ and Avik Chakraborty † Department of Physics and AstronomyUniversity of Southern CaliforniaLos Angeles, CA 90089, USA
Abstract
Recent work has shown that entanglement and the structure of spacetime are inti-mately related. One way to investigate this is to begin with an entanglement entropy ina conformal field theory (CFT) and use the AdS/CFT correspondence to calculate thebulk metric. We perform this calculation for ABJM, a particular 3-dimensional super-symmetric CFT (SCFT), in its ground state. In particular we are able to reconstructthe pure
AdS metric from the holographic entanglement entropy of the boundaryABJM theory in its ground state. Moreover, we are able to predict the correct AdSradius purely from entanglement. We also address the general philosophy of relatingentanglement and spacetime through the Holographic Principle, as well as some of thephilosophy behind our calculations. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] J a n ontents The discovery of the AdS/CFT correspondence [1–4] has proven to be one of the mostfruitful and interesting events in modern theoretical physics. The correspondence providesa realization of the Holographic Principle [5, 6], which states that the degrees of freedom ofa ‘bulk’ theory can be described by degrees of freedom on its boundary. Another attractivefeature of the correspondence is its connection between strongly coupled field theory andweakly coupled gravity.In its most basic form, the correspondence relates quantities computable in a theory ofgravity, specifically a spacetime which is asymptotically AdS, to ones calculated in a ‘dual’quantum field theory through what’s called a ‘holographic dictionary.’ For example, thebehavior of scalar fields in the bulk gravity theory can be determined by studying correlationfunctions of operators in the boundary field theory [3]. The metric of the bulk gravity theoryis also determined in part by the stress tensor of the dual conformal field theory (CFT) [7].In [8, 9] Ryu and Takayanagi proposed a relationship between entanglement entropy in thedual field theory and the area of an extremal surface in the bulk gravity theory.In particular, suppose the field theory is defined on a manifold M which has the topology R × Σ, where Σ is a spacelike hypersurface. Let ρ be a global pure state in the field theoryand A ⊂ Σ a constant-time subregion of M . The boundary ∂A is called the entanglingsurface. The reduced density matrix ρ A is defined as the partial trace of ρ over degrees offreedom in the complement A C . The entanglement entropy of the state ρ A is then definedas S A ≡ S [ ρ A ] = − tr A ( ρ A log ρ A ) . (1)The dual spacetime geometry will be an asymptotically AdS manifold whose conformalboundary is M . Let Γ be the constant-time surface which is anchored along ∂A on theboundary with the extremal volume A [Γ] (calculated with the bulk metric). Then the Ryu-Takayanagi (RT) formula states that S A = A [Γ]4 G N . (2)1n practice it is easiest to use this formula to compute the entanglement entropy for a CFTin some global state (usually the vacuum or a thermal state) reduced to a region A . Thatis, one gains information about the boundary field theory by performing a calculation usinggeometric information in the bulk. Recent studies of entanglement entropy in the contextof AdS/CFT indicate that entanglement is intimately related to the structure of spacetimeand gravity theories [10–18]. From this perspective, one might ask how much can be learnedabout the bulk geometry entanglement in the CFT. This question has been explored inseveral ways, including the ER = EPR proposal, entanglement wedge reconstruction withand without using the Petz map, reconstructing bulk geodesics, and reconstructing the bulkmetric.To philosophically motivate the calculations later in the paper, we can begin by sum-marizing an argument in [11]. As a thought experiment, picture a field theory defined on asphere S , and split this sphere into two regions A, B . The extremal surface anchored alongthe boundary of A is a sheet that splits the bulk ball into two distinct regions ˜ A, ˜ B . Nowsuppose we can calculate the entanglement entropy of some state ρ reduced to A , and sup-pose that the entanglement entropy is tuneable with some parameter. That is, if we increasethe parameter the entanglement goes up, and vice versa. Imagine we tune the entanglemententropy down to 0. By the RT formula, the area of of the bulk extremal surface will alsoshrink to 0, meaning the regions ˜ A, ˜ B become totally disjoint. The effect of having no en-tanglement entropy between the two regions in the CFT is to have two separate spacetimes.In this sense, the bulk spacetime is held together by entanglement entropy. A cartoon ofthis process in one less dimension is presented below in fig. (1). AB ˜ A ABRT RT AB RTB A ˜ B ˜ B ˜ B ˜ B ˜ A ˜ A ˜ A Figure 1: A cartoon demonstrating how tuning the entanglement entropy to zero producestwo disjoint spacetimes. The RT surface is shown in red.The rest of the paper is organized as follows. In section 2 we present a qualitativeargument demonstrating the relationship between entanglement and spacetime. In section 3we review the basics of ABJM [19] and present results for the entanglement entropy of severalentangling surfaces. We argue that when the entangling surface is a finite length rectangularstrip the entanglement entropy should match the RT result. In section 4 we introducethe framework for computing the bulk metric from entanglement entropy and perform the2alculation for ABJM in Minkowski space. We successfully reconstruct the pure
AdS metricwith the correct AdS radius from the entanglement entropy of the ABJM vacuum reducedto a strip in Minkowski space. Examples of this methodology for 2-dimensional CFTs invarious states and N = 4 SYM in the vacuum were explored in [20]. The ABJM theory [19] is a three dimensional supersymmetric Chern-Simons-matter the-ory with gauge group U ( N ) k × U ( N ) − k where ( k, − k ) are the Chern-Simons levels of theaction. The theory is superconformal, generally with N = 6 supersymmetry, and is weaklycoupled in the limit k (cid:29) N . The supersymmetry is enhanced to N = 8 when N = 2, as wellas when k = 1 , N branes is probing a C / Z k singularity,the field theory description of the branes is an N = 3 predecessor of ABJM, namely asupersymmetric Chern-Simons-matter theory with massive dynamical gauge fields. One canintegrate out the gauge fields in the low energy limit to recover the N = 6 ABJM theory. Inthe limit of large N this brane construction can be thought of M-theory on AdS / Z k . Fromthis perspective, the Chern-Simons level k is related to the circle one would dimensionallyreduce M-theory on to get type IIA string theory. Thus this pairing of AdS and ABJMgives a realization of the AdS/CFT correspondence.There has been a great deal of success calculating the partition function for ABJM on(squashed and branched) 3-spheres using localization techniques (for a review, see [21]). Thepartition function of ABJM on S can then be used to calculate entanglement entropy forABJM in a different context. Using conformal transformations, it can be shown that [22]the entanglement entropy of the vacuum reduced to a disk is given by S disk = log Z S , (3)where Z S is the S partition function. To leading order in N the universal part of the resultis [23] log Z S = √ π k / N / . (4)Including the divergent area term and choosing the UV regulator appropriately, the entan-glement entropy becomes [8] S disk = √ π k / N / Å (cid:96)a − ã , (5)where (cid:96) is the radius of the disk and a is the UV regulator. This is expected for a 3-dimensional CFT (see e.g. [24] and references therein).The construction of Casini et al. [22] which make the entanglement entropy calculationtractable on the field theory side rely heavily on the spherical symmetry of the entangling3urface. However, using a spherical entangling surface renders the bulk metric reconstructioncalculation very difficult. It will be much easier to use a rectangular strip as our entanglingsurface. In fact, it is possible to calculate the entanglement entropy of the vacuum reducedto a rectangular strip using holography [8]. If the strip has width (cid:96) and length L (understoodto be very large), the result is S strip = √ k / N / Å La − π L Γ(1 / (cid:96) ã . (6) The general philosophy for using entanglement entropy to calculate the metric of thebulk spacetime is as follows. Suppose we start with just the idea of a holographic dictionary.In particular, putting the boundary CFT in its vacuum state should be dual to a relatively‘calm’ bulk geometry. This is exemplified by Fefferman-Graham coordinates [25, 26], in thesense that the geometry is affected by the expectation value of the stress tensor of thefield theory. Further, we want the conformal boundary of our bulk spacetime to matchthe spacetime the CFT is defined on. Along with this, we are free to use diffeomorphisminvariance of the bulk gravity theory to choose coordinates which are convenient. The finalpart of the holographic dictionary that we need is the RT formula (2). Using these principleswe write down an ansatz for the bulk metric.For the methodology of the calculation we follow [16, 17]. Consider a CFT defined on d -dimensional Minkowski space. Suppose we know the entanglement entropy of this theory’svacuum reduced to a rectangular strip. Schematically, we expect that since the bulk geometryis not too perverse ( i.e. calm), the RT surface should share some of the symmetry of theentangling surface. As such, we choose the metric ansatz ds = R z (cid:32) − h ( z ) dt + f ( z ) dz + d − (cid:88) i =1 dx i (cid:33) . (7)Arrange the entangling region to be A : (cid:110) x i | − (cid:96)/ < x < (cid:96)/ , < x , x , . . . , x d − < L (cid:111) . (8)An example is shown in fig. 2.The symmetry between x , . . . , x d − informs us to assume the RT surface can be parametrizedin the bulk by z = z ( x ). The induced metric on the RT surface is ds = R z (cid:32)(cid:16) z (cid:48) f + 1 (cid:17) dx + d − (cid:88) i =2 dx i (cid:33) , (9)where z (cid:48) ≡ dz/dx . Call the RT surface γ . The volume of γ is A [ γ ] = R d − L d − (cid:90) (cid:96)/ (cid:96)/ dx (cid:112) z (cid:48) ( x ) f ( z ( x )) z ( x ) d − , (10)4 Lyz
BB A x z = z ∗ Figure 2: A sketch of a the entangling region in a 3d CFT and its corresponding RT surfacein the bulk.where we have integrated out the extra coordinates. This volume is a functional of theembedding z ( x ) and the metric function f ( z ).According to the RT prescription, we must extremize (10) with respect to f, z . We canuse the calculus of variations to solve the problem. Call the integrand the Lagrangian L ( z, z (cid:48) ; x ) = (cid:112) z (cid:48) f ( z )) z d − , (11)where z, z (cid:48) are the generalized coordinates and x is the ‘time’. Since L does not dependexplicitly on x , its Legendre transform (the Hamiltonian) is conserved H = z (cid:48) ∂ L ∂z (cid:48) − L , d H dx = 0 . (12)The Hamiltonian is given by H = − z d − (cid:112) z (cid:48) f ) . (13)Because the Hamiltonian is conserved, we can set it equal to something that does not dependon x . In particular, we choose to relate it to the turning point of the RT surface, called z ∗ : H = − z d − ∗ . (14)The next steps, which will be displayed specifically for the case of interest below, are torewrite the volume functional using z ∗ , relate the field theory entanglement entropy to thearea functional using the RT formula, and then solve the resulting integral equation for themetric function f ( z ). Note that this procedure does not give us the time component of themetric h ( z ). 5 .1 The bulk dual of ABJM in its ground state Write the metric ansatz ds = R z (cid:16) − h ( z ) dt + f ( z ) dz + dx + dy (cid:17) . (15)Assume the RT surface is given by z = z ( x ) with x ∈ ( − (cid:96)/ , (cid:96)/
2) and y ∈ (0 , L ). Since theRT surface γ is 2-dimensional, we will start referring to A as the area. The area functionalis A = 2 LR (cid:90) (cid:96)/ dx (cid:112) z (cid:48) f ( z )) z . (16)In the calculus of variations problem the Lagrangian is L ( z, z (cid:48) ; x ) = (cid:112) z (cid:48) f ( z )) z , (17)and the conserved Hamiltonian is H = − z (cid:112) z (cid:48) f ) . (18)Let z ∗ be the turning point of the surface, so that dzdx (cid:12)(cid:12)(cid:12) z = z ∗ = 0. Then z (cid:48) = (cid:112) z ∗ − z z f ( z ) , (19)which implies (cid:96) = 2 (cid:90) z ∗ a dz z f ( z ) (cid:112) z ∗ − z , (20)where a is a UV cutoff. In terms of z ∗ the Lagrangian is L = z ∗ /z , and our area functionalbecomes A = 2 LR (cid:90) z ∗ a dz z ∗ f ( z ) z (cid:112) z ∗ − z , (21)Next, we view the entanglement entropy in eqn. (6) and the area functional in eqn. (21)as functions of the parameter (cid:96) . Their (cid:96) derivatives are dS strip d(cid:96) = √ k / N / π L Γ(1 / (cid:96) , (22) d A d(cid:96) = dAdz ∗ dz ∗ d(cid:96) . (23)Because A does not explicitly depend on (cid:96) we must use the chain rule. Write A = 2 LR (cid:90) z ∗ a F ( z ∗ , z ) dz & (cid:96) = 2 (cid:90) z ∗ a z z ∗ F ( z ∗ , z ) dz, (24)6here F ( z ∗ , z ) = z ∗ f ( z ) z (cid:112) z ∗ − z . (25)By the Leibniz rule for differentiating an integral d A dz ∗ = 2 LR ñ lim z → z ∗ F ( z ∗ , z ) − (cid:90) z ∗ a z ∗ z f ( z )( z ∗ − z ) / dz ô d(cid:96)dz ∗ = 2 z ∗ ñ lim z → z ∗ F ( z ∗ , z ) − (cid:90) z ∗ a z ∗ z f ( z )( z ∗ − z ) / dz ô . Hence d A d(cid:96) = LR z ∗ . (26)Upon differentiating the RT formula with respect to (cid:96) and solving for (cid:96) in terms of z ∗ ,the entanglement entropy can be rewritten as S strip ( z ∗ ) = √ k / N / La − LRπ / k / N / / » G (4) N Γ(1 / z ∗ . (27)The RT formula then gives us the integral equation S strip ( z ∗ ) = A ( z ∗ ) / G (4) N for the function f ( z ). The details of solving the integral equation are presented in Appendix A. The resultis lim a/z → f ( z ) = G (4) N k / N / √ R . (28)We are only interested in the regime a/z (cid:28) z = 0.After rescaling z , the metric is ds = ˜ R ˜ z (cid:16) − h ( z ) dt + dz + dx + dy (cid:17) , (29)where the AdS radius is ˜ R = Ç G (4) N k / N / √ å / . (30)As a sanity check, notice that from [9] we find G (4) N = 48 π (cid:96) p (2 R AdS ) , R AdS = (cid:96) p (32 π k / N ) . (31)Solving these two gives us R AdS = G (4) N k / N / √ , (32)which is exactly what we have found! 7 .2 Fixing the last metric component The time-independent RT prescription is only enough to determine one of the unknownfunctions in the metric ansatz. With our assumptions about the holographic dictionary atthe beginning of section 4 and some mild assumptions about the entanglement entropy inABJM, it is safe to assume (see for example [27]) that the bulk metric satisfies Einstein’sequations.Using Λ = − / ˜ R , the three unique Einstein’s equations are12 hz + 2 h (cid:48)(cid:48) − ( h (cid:48) ) h − h (cid:48) z = 12 z (33) z (cid:16) h (cid:0) h (cid:48) − h (cid:48)(cid:48) )+ z ( h (cid:48) ) (cid:17) h − z = − z (34) zh (cid:48) h − z = − z . (35)The final one implies h = const. while the first two are consistent only if h = 1. Thus wehave recovered the metric of AdS ds = ˜ R z ( − dt + dz + dx + dy ) (36)with ˜ R given above. In this paper we have demonstrated again the relevance of entanglement entropy tothe structure of spacetime. In particular, we have used general ideas from holography toreconstruct the metric of the bulk spacetime dual to ABJM in its ground state in flat space.This result enlarges the list of AdS/CFT pairs that have been investigated through thisspecific methodology.It is worth noting an apparent tension between our philosophy and the actual sourceof the entanglement entropy in eqn. (6). This result was derived in the original paper byRyu and Takayanagi [8]. That is to say, they predicted the entanglement entropies for atheory dual to
AdS × S by doing a calculation in the bulk (they did not know that theywere computing the entanglement for ABJM with k = 1 because the RT paper predated theoriginal ABJM paper [19]). There appears to be a circularity in using a bulk calculation toreconstruct the bulk metric. However, there is a good reason to overlook that. It was shownby direct computation on the field theory side [23, 24] that the RT prediction for a sphericalentangling surface can be trusted. Moreover, the authors of [20] used a result from RT [8]to reconstruct the bulk metric dual to N = 4 SYM in flat space.It would be interesting to test the methodology of this paper with a time dependententanglement entropy in the CFT. We suspect that the covariant generalization of the RT8rescription [28] would be necessary. In the static examples considered here and in [20]the time component of the metric is determined by demanding that Einsten’s equations besatisfied, i.e. not using the entanglement entropy. It would further probe the connectionbetween entanglement and spacetime by trying to fully reconstruct a dynamic bulk metricusing a dynamic entanglement entropy. Acknowledgements
This work is supported by the DOE grant DE-SC0011687. We would like to thank FelipeRosso, Clifford Johnson, and Kryzystof Pilch for helpful discussions.
Appendix A
We now begin to solve the integral equation for the function f . We write F ( z ∗ ) = 4 G (4) N S EE ( z ∗ )2 LR z ∗ , p ( z ) = f ( z ) z , g ( z ) = z (37)so that F ( z ∗ ) = (cid:90) z ∗ a dz p ( z ) (cid:112) g ( z ∗ ) − g ( z ) . (38)This integral equation has the solution p ( z ) = 1 π ddz (cid:90) za dz ∗ F ( z ∗ ) g (cid:48) ( z ∗ ) (cid:112) g ( z ) − g ( z ∗ ) , (39)where g (cid:48) ( z ) = dgdz >
0. So, we find f ( z ) = 8 G (4) N z πLR ddz (cid:90) za dz ∗ z ∗ S EE ( z ∗ ) (cid:112) z − z ∗ . (40)Lets us define: C = √ N / La , (41) C = LRπ / N / / » G (4) N Γ(1 / z ∗ . (42)Thus f ( z ) = 8 G (4) N z πLR ddz (cid:90) za dz ∗ C z ∗ − C (cid:112) z − z ∗ ≡ G (4) N z πLR dMdz . (43)9ne finds M ( z ) = C ï π −
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