Entanglement entropy at large-N

JHEP01(2025)182 Published for SISSA by Springer Received: November 15, 2024 Accepted: December 25, 2024 Published: January 29, 2025 Entanglement entropy at large-N P. Talavera Department of Physics, Polytechnic University of Catalonia, Diagonal 647, Barcelona, 08028, E, Spain E-mail: [email protected] Abstract: I show that at early times the evaporation process for a stack of NS5-branes at high energy is suppressed in the large-N limit. At much later times, the new saddles in the gravitational action are no longer suppressed at large-N, and evaporation proceeds as usual. Keywords: 1/NExpansion, Black Holes in String Theory, AdS-CFT Correspondence, D-Branes ArXiv ePrint: 2411.09427 Open Access,©The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP01(2025)182 JHEP01(2025)182 Contents 1 Motivations 1 2 NS5-branes and LST backgrounds 2 2.1 Kruskal coordinates 3 3 Entanglement entropy 4 3.1 Entanglement entropy for a single interval 5 3.2 Entaglement entropy for two-disjoint intervals 7 4 Invariance 9 5 Some remarks 10 6 Some generalizations 10 1 Motivations In this short note, I would like to explore a novel behaviour that occurs in black holes associated with a stack of NS5-branes at the Hagedorn temperature. This behaviour is a direct consequence of a large-N limit 1 and is rooted in the peculiarity of the system. The setup is quite appealing, first because Little String Theory (LST) is a non-local field theory without gravity, and one hopes to use this knowledge as a precursor to understanding general settings of string theory. And secondly, the thermodynamic variables indicate that the system is at the Hagedorn temperature, a point where the partition function is poorly defined. The theoretical framework is defined on the world-volume of NS5-branes in the limit of vanishing string coupling, gα→ 0 . It was shown in [ 1 ] that LST is holographically dual to string theory on CGHS black hole. We shall rely heavily on this fact. A Semi-classical analysis [ 2 ], shows that the radiation associated with the black hole inside the LST is purely thermal and therefore indistinguishable from white noise [ 3 ]. This fact has deeper implications [ 2 ]: the probability of emitting a shell of energy ω1 + ω2 is equal to the probability of emitting two independent shells with the same total amount of energy, ω1 + ω2 . As a direct consequence the radiation always comes as a pure state, the Hilbert space can be factorized into two disjoint parts, H=Hin ⊕Hout ,(1.1) corresponding to states located at the inner and outer sides of the event horizon respectively. Note that in (1.1) is missing an interaction piece Hint which is mandatory if an outside observer located at the spatial infinity wants to extract information about the system [ 4 ]. 1From now on, N will denote the number of NS5 branes. – 1 – JHEP01(2025)182 The aim of this work is to clarify whether this pattern is due to the setup temperature or to some deeper cause. To do this, I look at the entanglement entropy in the spirit of [ 5 ]. Our results are unambiguous: the large-N limit suppresses semi-classical configurations, and as a consequence the system does not interact with the environment. Corrections, i.e. non-trivial saddles of the Euclidean path integral [ 6 , 7 ], are not suppressed in this limit and are fully responsible for the unitary evaporation of the black hole. In the spirit of [ 8 ] this new configurations will be complex solutions of the gravitational equations and correspond to saddles relevant to the unitary Page curve. For those familiar with QCD, this effect has a close parallel in the decay ρ→ππ at large-N c . Inserting the hadronic spectrum of large-N c QCD as a proxy for the real hadronic spectrum provides rather good approximation. In such a model the spectrum of the theory in the large-N c limit consists of an infinite number of narrow stable meson states [ 9 ]. 2 NS5-branes and LST backgrounds Holography relates LST to string theory in the near-horizon geometry of NS5-branes [ 10 ]. Our starting point is the supergravity solution for N coincident near-extremal NS5-branes in the string frame [ 11 ] ds2=− 1−r2 0 r2!dx2 1+ 6 X j=2 dx2 j+1 + N m2 sr2 dr2 1−r2 0/r2+r2dΩ2 3!,(2.1) e2ϕ=g2 s1 + N m2 sr2, where r0 is the location of the horizon, gs is the asymptotic string coupling constant and m2 s is essentially the string tension. The index i = 2 ,··· , 6corresponds to the flat directions along the five-brane, and d Ω 3 is the line element of the unit 3 sphere. The thermodynamics of the NS5-brane is ENS5 V5 =1 (2π)5(α′)3 N g2 s +r2 0 g2 sα′!,(2.2) βNS5 = 2πsN m2 ss1 + m2 sr2 0 N.(2.3) The first term, proportional to N g2 s , in (2.2) is the tension between the extremal NS5-branes. There are several limits that can be run on (2.1) . One of them is a high energy limit, for which we approach the near horizon in the decoupling limit r0→0, gs→0,r2 0 g2 sα′≡fixed .(2.4) The resulting theory is conjectured to be dual to a Little String Theory [ 10 ]. To take the limit, it is more convenient to change the variables to u:= r gsl2 s , u0:= r0 gsl2 s ,(2.5) – 2 – JHEP01(2025)182 after which (2.1) becomes ds2=− 1−u2 0 u2!dx2 1+ 6 X j=2 dx2 j+N m2 su2 du2 1−u2 0/u2+u2dΩ2 3!, e2ϕ=Nm2 s u2.(2.6) The change (2.4) is motivated as follows: in Type IIB string theory u corresponds to the mass of a string stretching between two NS5-branes, while in Type IIA ul−1 s is the string tension of an open D2-brane stretched between two D5-branes. It is the thermodynamics associated with this setup (2.6) [ 12 ] that will be more relevant to our discussion below ELST V5 =1 (2π)5(α′)3N g2 s +u2 0l2 s, βLST = 2πsN m2 s ,SLST V5 =1 (2π)4(α′)2√Nu2 0ls. (2.7) Note that the temperature is independent of the energy. This fact allows to tune independently both, energy and temperature, and makes the free energy to vanish. It is clear, from (2.7) , that if u2 0l2 s≫N/g2 s follows that S = βLSTELST which is the leading behaviour for the entropy of a gas of weakly interacting closed strings at the Hagedorn temperature, the latter identified with that in (2.7) [ 13 ]. Since we are interested in describing the time evolution across the horizon, which is necessary to study the two-sided correlation functions, we shall adopt Kruskal coordinates for the description of (2.1) and (2.6) . This is the task for the rest of this section. Before we start, we shall write (2.1) and (2.6) generically as ds2=−f1(r)dx2 1+ 6 X j=2 dx2 j+Ai(r) dr2 f1(r)+r2dΩ2 3!,(2.8) with f1(u)=1−u2 0 u2,(2.9) and with Ai ( r )chosen properly in each case. 2.1 Kruskal coordinates As is customary we shall write the line element (2.1) in terms of the U ( x1, r )and V ( x1, r ) Kruskal coordinates 2 U=−eci(Fi(x)−ax1), V =eci(Fi(x)+ax1),(2.10) where ci , 3 is a constant chosen on a case-by-case basis, see below, and F ( x )refers to the Tortoise coordinate in (2.8) Fi(x) = Zdx pAi(x) f1(x).(2.11) 2The parameter ais introduced to make the arguments in the functions dimensionless. 3The subindex idenotes the model (2.1) or (2.6). – 3 – JHEP01(2025)182 After using (2.10) one obtains for (2.8) [ 14 ] ds2=−e−2ciFi(ar)|f1(ar)| a2c2 i dU dV + 6 X j=2 dx2 j+Ai(r)r2dΩ2 3=−Ω2(r)dU dV +..., (2.12) where the arguments inside the functions emphasise that the relations (2.10) cannot be analytically inverted in most situations. Both solutions, (2.1) and (2.6) , have a S3 term which plays a trivial role and we are free to ignore these directions for the time being. 4 Note that at this stage the prefactor Ωin (2.12) vanishes or becomes infinite at the event horizon, except for a specific value of the constant ci , which is dictated precisely so that Ω does not vanish in the entire patch. The results for cLST and cNS5 are given in the equations below. For both models we can perform the integral (2.11) analytically, obtaining up to an arbitrary additive constant 5 LST FLST(y) = b 2log(y2−1) , cLST =1 b,Ω2(y) = b2r2 0 y2.(2.13) And NS5 FNS5(y)=qy2+b2+1 2p1+b2log √1+b2−py2+b2 √1+b2+py2+b2 , cNS5 =1 √b2+1 ,(2.14) Ω2(y)= r2 0 y2e−2qb2+y2 b2+1 (1+b2)2 1+sb2+y2 b2+1   2 , where we have defined b2:= N m2 sr2 0 .(2.15) Note that F ( y )in (2.13) and (2.14) diverge as we approach the event horizon. This is a consequence of the Kretschmann scalar of (2.1) or (2.6) having a singularity at r→ 0. Furthermore, Ω( y )is finite on the horizon or at large distances. As a consistency check on our setting we have verified that in both cases U V → 0, (2.10) , at the event horizon. In figure (1) we represent the conformal diagram, {U, V } coordinates (2.10) , for (2.8) . 3 Entanglement entropy We are going to follow the now well-known standard analysis in order to find the entanglement entropy. We take a two-sided black hole, as shown in figure (1) , which is initially in a pure 4This is equivalent to considering only s-wave emission. 5Henceforth ywill be a dimensionless variable y=: r/r0. – 4 – JHEP01(2025)182 Singularity Singularity UV t−+→&r−+→ t− ∞ → &r−+→ χ+ χ∞y∞ 2y+ 2 y+ 1 y∞ 1 I Horizon Figure 1. Conformal diagram for either (2.1) or (2.6) . Each point in the diagram represents a two-dimensional Euclidean space, in our case extending in the directions {x1, u} of (2.8) . Although the conformal factors (2.13), (2.14) are different, their causal structures are the same. Red curves show the singularity at r→ 0. Dashed lines represent the horizon, r→r0 . Finally, solid black lines represent the asymptotic. One salient point is that the r→ 0surfaces bend the conformal diagram, as does the AdS Schwarzschild black hole [ 21 ]. The region χ , with states identified with the Haking radiation, is divided into two parts, χ− and χ+ . The boundary surfaces of χ± are y± 2 respectively. We show the configuration containing an island, I, that extends outside the horizon. Its boundaries are located at y± 1. The goal is to compute the entanglement entropy of the union I∪χ+. state. Over time, it will become increasingly entangled with the thermal bath. 6 Thus its entropy is initially given by that of the matter entropy in the bulk while the black hole is there to provide the fixed curved background. At the Page time, the O ( G−1 N )entanglement between the left and right black holes is replaced by the O ( G−1 N )entanglement between the individual black holes and the bath [ 16 ]. So there are at least two competing surfaces and the generalised entropy is given by [ 5 ] S(χ) = minIextIA(∂I) 4GN +Smatter(χ∪I).(3.1) The first term of (3.1) corresponds to the area of the edge of the island contribution while the second is the von Neumann entropy computed with the quantum field theory formalism in the absence of gravity. 7 This term depends on the relative location of the radiation with respect to the black hole horizon. 3.1 Entanglement entropy for a single interval In a 2 d CFT there is a universal expression for the entanglement entropy of an interval of length L , where the edges are very far apart [ 17 ]. This expression, in turn, coincides with the matter entropy per unit area of the metric (2.12) if we ignore the role of the S3 [ 18 ], 8 Sb matter =c 3log L ϵ≈c 6log "1 ϵ2 z2 y+y− Ω(y+)Ω(y−)#(3.2) 6 It is evident that the Page curve we shall obtain is that of a non-gravitating theory which satisfies the split property of local QFT [15] and not that of the black hole. 7See the final remark at the end of subsection 3.2.2. 8See figure (1) for notation. – 5 – JHEP01(2025)182 where ϵ is a uv regulator, c is the central charge and z2 ab = Ω2(a)Ω2(b) [U(b)−U(a)] [V(b)−V(a)] (3.3) is the geodesic distance between two boundary points in Kruskal coordinates with y+ = ( ty, y ) and y− = ( −ty + iπ 2, y ). 9 Although (3.2) has been found in 2 d , there are actually several numerical checks on its consistency, under some restrictions, in higher dimensional spacetimes [ 19 ]. The expression (3.2) contains both ultraviolet and infrared divergences. The former are taken into account with the standard renormalization technique and the overall outcome is the renormalization of the Newton’s constant GN . The latter divergences arise when one of the y± approaches the horizon. Due to the general change of coordinates (2.10) , (3.2) can be written as Sb matter ≈c 3log eciFi(y)cosh ci ty r0+c 3log [2 Ω(y)] .(3.4) The details of the setup, the form of F ( x )and Ω( x ), are not important at this point. With (3.4) at face, it is not surprising that, with ci as a constant, all the gravitational models will behave linearly at large time, once we take into account the proper subtractions. In this regime, and provided that (3.2) is valid only for spacetime points far from the horizon, the remaining terms in (3.4) which do not directly involve time explicitly, are subleading. We pause to motivate the subsequent steps in our study before proceeding with the standard analysis, see for example [ 20 ]. Note that the relevant quantity within the argument of the time-dependent part of (3.4) is cit . In most of the cases studied in the literature, the constant ci is a harmless numerical factor 10 but due to the presence of the function Ai ( r )in (2.12) it now plays a crucial role, since it introduces the free parameters that define the theory. Thus, in the sequel, and in contrast to previous studies, it is not an early/late epoch expansion what characterises the study of the black hole emission but the interplay of the factors in the cit relation. More specifically, we are interested in the behaviour of the entanglement entropy with the dependence of ci on the number of branes ci ( N ). 3.1.1 The NS5 and LST entanglement entropies To make the point clearer, let us have a closer look at the possible values of ci ( N )for the (2.1) and (2.6) models. First of all, because they are a supergravity solution, the effective string coupling must be bounded at its maximum value, the horizon. This leads to the constraint N≪m2 sr2 0.(3.5) Second, the scalar curvature should be small if the classical geometry is to hold R ∼ 1 N, N ≫1.(3.6) Combining (3.5) and (3.6) we get 1≪N≪m2 sr2 0.(3.7) 9 By this convention, the signs of U and V in the left wedge of the Kruskal diagram picks and extra minus sign. 10We have checked this claim in several models either with single or multi-horizons. – 6 – JHEP01(2025)182 With (3.7) at hand, we can conclude that although the two models are formally identical, see (2.8) , the behaviour of the associated ci functions, (2.13) and (2.14) , is completely different at large-N cLST ≫1, cNS5 →1.(3.8) Bearing in mind that approximation the late time entanglement entropies (3.4) are LST S(χ) = Sb matter ≈c 3 1 √N ty ls ,(3.9) NS5 S(χ) = Sb matter ≈c 3 ty r0 .(3.10) As expected, the entanglement entropy grows linearly with time in both models. A few words are in order. Roughly speaking, entropy, (2.7) , gives the amount by which a system, let us call it A , interacts with itself. Whereas the entanglement entropy, (3.9) , captures the interactions between two of its subsystems, A1,A2∈ A . What (3.9) shows is that if you have a parametrically large number of NS5-branes, O ( N ) ≫ O (1), but with (3.7) still valid, the system becomes non-interacting, the Hilbert space factorises and therefore we cannot recover information from outside [ 2 ]. This is equivalent to saying that the system becomes non-interacting with its boundary where the radiation is collected. In addition, in this limit the system can be considered as consisting of non-interacting strings [ 22 , 23 ], where the string tension is related to the usual string tension as [ 24 ]. ˆτ=τ N.(3.11) Thus the Hagedorn string tension is quantized to a fractional unit of the ordinary sting tension and vanishes at large-N. Just for completeness, we roughly estimate the so-called Page time: as time goes on, (3.9) has some conflict with the von Neumann entropy finiteness because the entropy radiation becomes larger than that of the black hole, which has only a finite number of degrees of freedom. This happens at ty⪆Nm2 sr2 0,(3.12) which proves to be extraordinarily long for the (2.6) model. 3.2 Entaglement entropy for two-disjoint intervals Now it’s time to turn to the island contribution and see if the previous pattern has been washed out or still remains. We now look for non-trivial configurations for the bath at late times. We consider the left ↔ right symmetric island AB in figure 1. Then the entanglement entropy is given by S(χ) = A(∂I) 4GN +Sa matter .(3.13) These islands can be located either inside or outside the horizon depending on the specific gravitational model. This is equivalent to look for the entanglement entropy of two disjoint – 7 – JHEP01(2025)182 intervals inside R2 [ 25 ] between the points ( y+ 1, y− 1 )and ( y+ 2, y− 2 )in vacuum state. The contribution of the matter can be factorised as follows Sa matter =c 3log "zy+ 1y− 1zy+ 2y− 2zy+ 1y+ 2zy− 1y− 2 ϵ4zy+ 1y− 2zy− 1y+ 2#.(3.14) From a physical perspective (3.14) gives the entanglement entropy for the matter fields located between the radiation and the island regions. Using (2.10) we again obtain the same functional dependence on time for both models, assuming ci ( N )as numerical constant, Sa matter ≈c 3log4eci(N)(F(y1)−F(y2))coshci(N)ty1 r0coshci(N)ty2 r0 (3.15) +c 3log  cosh[ci(N)(F(y1)−F(y2))]−coshci(N) r0(ty1−ty2) cosh[ci(N)(F(y1)−F(y2))]+coshci(N) r0(ty1+ty2) +c 3log[Ω(y1)Ω(y2)]. The fact that (3.15) holds independently of any gravitational model, indicates that the radiation-bath coupling is somehow universal in all of them. As we have already seen, (3.10) , the NS5-brane model follows the standard behaviour with respect to the N dependence, and so from now on we shall concentrate mainly on the LST model. The next step is to consider that the island is formed near the horizon, 11 y1≈ 1 + ϵ , and that y2 is near the boundary, y2≫ 1. Then (3.15) becomes 12 Sa matter =c 6log32ϵy2 2cosh2cLST(N)ty1 r0cosh2cLST(N)ty2 r0 +c 3log  1−2√2ϵ y2coshcLST(N) r0(ty1−ty2) 1+2√2ϵ y2coshcLST(N) r0(ty1+ty2) +c 3log[Ω(y1)Ω(y2)].(3.16) We shall now comment on the limiting behaviour, (3.8) , of (3.16) . 3.2.1 O(cLST(N)) ≲O(t) This first limit is the standard one. We shall consider a large, but finite, time evolution and also a cLST ( N )at most of the same order as time with the constraint 1≫2√2ϵ y2 cosh cLST(N) r0 (ty1−ty2).(3.17) With this approximation in mind and a little of algebra, we get Sa matter =2 3clogy2−2 3c√2ϵ y2 coshcLST(N) r0 (ty1−ty2)+1 3clog[Ω(1+ϵ)Ω(y2)],(3.18) which has an extreme for the values ty1=ty2, ϵ =1 2y2 2 .(3.19) 11The ϵparameter, ϵ≪1, can be either positive or negative. 12The expression (3.16) coincides with the results of [26] once we set the function Ω = 1 and cLST = 1. – 8 –