Holographic entanglement entropy in open-closed string duality

0.63 MB·

by  Vasilis Niarchos

DCPT-2017/01
Holographic entanglement entropy in open-closed string duality
Vasilis Niarchos
Department of Mathematical Sciences and Center for Particle Theory
Durham University, Durham, DH1 3LE, UK
We study minimal co-dimension-2 surfaces in the asymptotically flat background of extremal 3-
brane solutions in ten-dimensional type IIB supergravity. A conjectured open-closed string duality
combined with the Ryu-Takayanagi prescription implies that the area of the surfaces we consider
could be interpreted as the entanglement entropy of a dual (3+1)-dimensional large-N, strongly-
coupled open string field theory on D3-branes. As the size of the surface is varied we observe a
transitionfromavolumelawtoanarealawinagreementwithexpectationsfromnon-localityinan
openstringfieldtheory. Someofthespecificsofthistransitionbearaqualitativeresemblancewith
the behaviour of holographic entanglement entropy in non-commutative super-Yang-Mills theory.
7
1 INTRODUCTION mass motion, singleton) dynamics of the branes could
0 be identified in gravity within the blackfold approach
2 [20,21],andsystematicallyrecastintermsoftheabelian
A holographic correspondence between theories of
n Dirac-Born-Infeld (DBI) effective action. The latter is a
gravity in asymptotically flat backgrounds and open
a characteristicfeatureofopenstringtheory. Forotherre-
string theories, generalizing the AdS/CFT correspon-
J centdiscussionsoftheemergenceoftheDBIactionfrom
dence beyond the near-horizon limit [1], has been con-
1 (super)gravity we refer the reader to [16, 22, 23].
1 jectured by several authors in the past, see e.g. [2–16].
In this note we are looking for a quantity that has the
For D3-branes in ten-dimensional type IIB supergravity
powertoprobemoreefficientlythefullnon-abeliansector
] the conjecture proposes a holographic relation between
h of the dual U(N) string field theory. For concreteness,
closedstringtheoryin3-branesolutionsandthelarge-N,
t
- we will focus on the canonical example of D3-branes in
strongly-coupled U(N) string field theory on D3-branes.
p
flat space. The quantity we would like to consider is en-
e Morerecentlyin[15]thefollowingpointswereputfor-
tanglement entropy (EE). This is an observable that can
h
ward/emphasized: (i) General open-closed string dual-
[ probeinteractionsanddegreesoffreedomacrossdifferent
ities may arise for string field theories on D-brane se-
scales. Ithasbeenproposedtohaveasimpleholographic
1 tups(withsuitableco-dimension)ingenericclosedstring
manifestation in gravity as the area of a corresponding
v
backgrounds. The asymptotic closed string background
3 minimal co-dimension-2 surface [24, 25]. Although this
plays the role of an arbitrary external source for open
1 proposal is better understood in the case of AdS space
1 string fields. (ii) The absence of a near-horizon decou- [26–29],herewewillassumethattheprescriptionismore
3 pling limit in these dualities may be justifiable as a con-
generally valid and applies also to asymptotically flat
0 sequence of the open string completeness conjecture by
spacetimes. Our main goal is to examine the proper-
.
1 A.Sen[17,18]. Non-trivialchecksofthisconjecturewere tiesofholographicentanglemententropy(HEE)inasym-
0 presented in open string tachyon condensation and non-
potically flat 3-brane solutions where open-closed string
7 criticalstringtheory. (iii)Inthismoregeneralversionof
duality could operate according to the above-mentioned
1
the holographic principle it is proposed that holography
: conjectures.
v works as a tomographic principle, where different open
Aclearfeatureofopenstringtheoryweshouldbelook-
i
X string field theories capture/reconstruct holographically ingforisnon-locality. ThisismanifestinEEasavolume
different subsectors of closed string theory/gravity.
r law, rather than an area law, which is characteristic of
a
Finding evidence in favor of this proposal, and set- local quantum field theory [30–32]. The volume law in
ting up a solid holographic dictionary, is a highly non- HEEhasbeennotedbeforeinarelatedcontextinapro-
trivial task, largely hindered at the present stage by posal for flat space holography in [33]. However, unlike
the lack of powerful computational tools in interacting [33]inthisnotewearenotconsideringHEEsolelyinflat
open string field theories. This is unlike the AdS/CFT space; we are considering it in a solution that interpo-
correspondence where many non-trivial checks can be lates between a near-horizon AdS space and the asymp-
found by matching explicit (super)gravity computations toticflatspace,wherewehaveamorespecificconjecture
tocorrespondingcomputationsinstrongly-coupledlarge- for the anticipated dual non-gravitational theory. Be-
N quantum field theories. With such technical limita- sides the volume law, which is mainly due to flat space
tionsitisusefultoidentifystructuresinclassicalgravity (in analogy to [33]), our case also exhibits a transition
thatreproducefamiliarfeaturesfromopenstringtheory. to the more standard area law as the size of the entan-
In [15] (see also [19]) we focused on the long-wavelength gling region is varied. We argue that the main features
dynamics of asymptotically flat p-brane solutions, and of this result are consistent with expectations from an
argued that the sector related to the abelian (center-of- interacting open string field theory.
2
The volume law has also been noted in the (3+1)- furtherinthelastsection. Inthenextsectionweexplain
dimensional holographic duals of the dipole theory how (3) is derived from gravity. For economy and clar-
and non-commutative super-Yang-Mills (NCSYM) the- ity of the presentation we will focus exclusively on the
ory [30, 34, 35]. We will observe that the asymptotically derivation of the HEE on the straight belt geometry (1).
flat 3-brane HEE exhibits similarities with the HEE of
the NCSYM theory.
HEE FROM CLASSICAL GRAVITY
MAIN RESULT We focus on N parallel overlapping D3-branes in flat
space in ten-dimensional type IIB string theory. In the
We consider the HEE, S , of a prospective dual (3+ large-N ’t Hooft limit, where λ = g N is kept fixed and
AS s
1)-dimensionalopenstringfieldtheoryonastraightbelt large (g being the string coupling), this setup is most
s
conveniently described by the extremal 3-brane super-
A =(cid:26)xi(cid:12)(cid:12)x1 ∈(cid:20)−(cid:96), (cid:96)(cid:21) , x2,x3 ∈(cid:20)−L,L(cid:21)(cid:27) . (1) gravity solution
S (cid:12) 2 2 2 2
ds2 =H−1/2η dxµdxν +H1/2(dr2+r2dΩ2) , (5)
µν 5
Lisaninfraredcutoffthatistakeneventuallytoinfinity. eΦ =g , C =g−1(H−1−1)dx0∧dx1∧dx2∧dx3 .
s 4 s
(cid:96) isthe tunablewidthof the belt. Wewill alsointroduce
ThedilatonΦisconstantandthe4-formC yieldsaself-
anultravioletlengthcutoffεthatiskeptfixedthroughout 4
dual 5-form flux. The solution is expressed here in the
thecomputation. Wearemainlyinterestedintheregime
string frame. The function H is
ε(cid:28)(4πλ)14(cid:96)s , (2) H(r)=1+ rH4 , r4 =4πλα(cid:48)2 . (6)
r4 H
wheretheasymptoticradialcutoffr inthegravitational
ε
bulk is outside the near-horizon region (rH (cid:28)√rε = rεH2 r →In∞whiasttfhoellaowsysmwpetowtiicllfleaxtprsepsascaellrelegniognt.h scales of the
—see eq. (6) for the definition of r ). (cid:96) = α(cid:48) is the
H s problem in terms of the length scale r that is intrinsic
string length. When the cutoff is well inside the near- H
to the solution (5). In particular, for the quantities in
horizon throat we recover the familiar divergent piece
(1) we set r = ur , and (cid:96) = lr , L = Lr , ε = (cid:15)r ,
of the N = 4 SYM EE, S (cid:39) N2L2, which is less H H H H
AS 2π ε2 where u,l,L and (cid:15) are now dimensionless quantities.
interesting for our purposes.
FollowingtheRyu-Takayanagiprescription[24,36,37]
Themainresultofthisnoteisthefollowingprediction
we are looking for a constant time, co-dimension-2 min-
for the HEE of the large-N string field theory on D3- imal surface A¯ in the background (5) that asymptotes
S
branes in the regime (2)
(atlargeradialdistanceu)totheboundaryoftheregion
√ (1). The volume of this minimal surface,
SALS2((cid:96)) =(cid:40) N2N√2π2λλααε(cid:48)(cid:48)52(cid:96)++...... ,, (cid:96)(cid:96)(cid:29)(cid:28)(cid:96)(cid:96)c . (3) Vol(A¯S)=(cid:90) d8σe−2Φ(cid:113)G(in8)d , (7)
3 ε6 c
√ can be used to express the HEE as
(cid:96) ∼ λα(cid:48) is a critical length where a transition be-
c ε Vol(A¯ )
tween different branches of extremal co-dimension-2 sur- S = S . (8)
AS 32π6(α(cid:48))4
faces occurs. It is estimated numerically in eq. (13) and
with an analytic extrapolation in eq. (18). The dots in- Ineq.(7)G(8) isthedeterminantoftheinducedmetric
dicate less divergent terms. ontheco-dimeinndsion-2surfaceA¯ . Wrappingthissurface
S
Eq. (3) exhibits a transition between a short distance
aroundthe5-sphereofthebackground(5)andthedirec-
regimewithacharacteristicvolumedependencetoalong
tions x2,x3, and assuming a single dependence on the
distanceregimewiththemorestandardareadependence. coordinate x ≡ x1, we can express it as a solution u(x)
We find evidence that this is a robust feature of the sys- rH
that extremizes the volume (7), written more explicitly
temindependent ofthechoiceofthe geometryof theen-
as
tangling region A . A similar computation of the HEE
for the cylinder S Vol(A¯ )= π3rH8 L2 (cid:90) dxu5H1/2(cid:113)1+H(u˙)2 , (9)
S g2
(cid:26) (cid:12) (cid:20) L L(cid:21)(cid:27) s
AS = xi(cid:12)(cid:12)(x1)2+(x2)2 =(cid:96)2 , x3 ∈ −2, 2 (4) whereH =1+u−4 andu˙ = ∂∂ux. Theextremizationequa-
tions of (9) can be recast as the conservation equation
verifies the same type of transition (see eq. (22)) consis- (cid:115)
tent with an interpretation based on open-closed string u5 1+u−4 =u5(cid:113)1+u−4 ≡c , (10)
duality. The physical content of eq. (3) will be discussed 1+(1+u−4)u˙2 ∗ ∗
3
u(cid:42) u(cid:42)
50 10
40 8
30 6
20 4
10 2
l l
5 10 15 20 25 2 1 2 3 4 5 6 2
FIG. 1. A plot of the turning point u of the solutions of eq. (10) as a function of the dimensionless belt half-width l. Both
∗ 2
plots represent the solution for the same cutoff value (cid:15)=0.02 (blue thick line). The right plot zooms around the point where
the middle and bottom branches meet. The red curve is an analytic prediction based on the approximation (14). The brown
curve is an analytic prediction based on the approximation (19).
where u is the turning point of the surface in the bulk. is the only existing branch. The existence of a similar
∗
c is essentially the value of the conserved Hamiltonian pattern of three co-existing extremal surfaces was also
of the system (9). We solve the non-linear first-order observed in the NCSYM theory in [35].
differential equation (10) with an explicit UV cutoff (cid:15) Numerically, we observe that the critical width is
(cid:18) (cid:19)
l 1 0.4 0.4r2 √ (cid:96)
u x=± = . (11) l ∼ ⇔ (cid:96) ∼ H =0.8 πλ s (cid:96) . (13)
2 (cid:15) c (cid:15) c ε ε s
As was noted in previous work [35], it is important to The explicit presence of the UV cutoff scale ε in (cid:96) was
c
keep the cutoff fixed thoughout the computation, other- also noted in the case of the NCSYM theory [30, 34, 35].
wiseimportantbranchesofthesolutioncanbemissedas An analytic rough estimate of (cid:96) will be discussed soon.
c
(cid:15)→0. We proceed to analyse each of these branches ana-
After a few trivial algebraic manipulations, the HEE lytically with approximations performed in two opposite
(8) evaluated on the on-shell profile of the function u(x) regimes: u ∼ 1 (cid:29)1 and u (cid:28) 1.
∗ (cid:15) ∗ (cid:15)
can be written as
Approximations in the flat space regime: u (cid:29)1.
(cid:115) ∗
S = L2N2 (cid:90) 1(cid:15) duu10 (1+u−4)3 .(12) In this regime the function H(u) is 1 to first approxima-
AS πc u10c−2(1+u−4)−1 tion and c ∼ u5. Then, we can easily find an analytic
u∗ ∗
solution for u(x) in the form
The equations (10), (11) are easily solved numerically.
√
Tl hiseptluortnteindginpoFinitg.u1∗ faosr athfeunccuttiooffn voafluthee1(cid:15)be=lt5w0i,dtohr x=− ΓπΓ(cid:0)−(cid:0)251(cid:1)(cid:1)u∗ − 4uu5∗4 2F1(cid:18)25,12,75,uu1∗100(cid:19) . (14)
(cid:15)=0.02 (other small values of (cid:15) were checked to exhibit 10
the same behavior). We observe an intermediate range
The relation between l and 1 follows immediately from
of l, where there are three separate branches of extermal (cid:15)
this equation by implementing the boundary condition
surfacesolutions,letuscallthemupper,middleandbot-
(11). We notice that the resulting curve (red curve in
tom branches for decreasing values of u . In the upper
∗ both plots in Fig. 1) agrees very well with the numeri-
branch the extremal surface is mainly embedded in the
cal solution in both the upper and middle branches and
asymptoticflatspaceregion. Inthelowerbranchtheex-
starts deviating only when u ∼1 and l∼1.
tremal surface is well embedded inside the AdS throat. ∗
Under the same assumptions, for the HEE we obtain
We will analyse each of these branches analytically with
the expression
perturbative methods in a moment.
√
In Fig. 2 we plot the value of the HEE for each of L2N2u6(cid:20)3 πΓ(cid:0)−3(cid:1)
these branches. We observe that the upper branch is SAS = π 30∗ Γ(cid:0)− 1 (cid:1)5 (15)
the dominant (or single) branch for all values of l in the 10
interval[0,l ). Forl>l thebottom(AdS)branchtakes (cid:18) 1 (cid:19)6 (cid:18) 3 1 2 (cid:19)(cid:21)
c c +5 F − , , ,((cid:15)u )10 .
overanddominates. Atverylargevaluesofl thisbranch (cid:15)u 2 1 5 2 5 ∗
∗
4
HEE
HEE
l
l
5 10 15 20 2 2 4 6 8 10 12 14 2
FIG. 2. A plot of the profile of the HEE as a function of the belt half-width l. The same cutoff value (cid:15)=0.02 is used in this
2
figure as in Fig. 1. The right plot zooms into the region of the transition at l=l of the left plot. The dotted line that passes
c
throughtheoriginintheleftplotdepictsthevolumelaw (16),whichislinearinl. Thedashedhorizontallinedepictsthearea
law (17). The blue points refer to the numerical data for the upper and middles branches, and the red points to the data for
the bottom branch.
It is not hard to show by a series expansion of this equa- InformationfromtheAdSregime: u (cid:28)1. Assume
∗
tion that when u ∼ 1 to leading order for a moment that we take the radial cutoff r inside the
∗ (cid:15) ε
near-horizon region, i.e. we take r (cid:28)r . In this regime
ε H
N2L2l we recover the minimal surface as a deformation of the
S (cid:39) , (16)
AS 2π (cid:15)5 more familiar AdS ×S5 results. The function H(u) is
5
to first approximation u−4 and c ∼ u3. The analytic
which is proportional to the volume of the belt AS, and ∗
solution for the function u(x) in this case implies at the
representstheleadingcontributiontotheHEEwhen(cid:96)(cid:28)
cutoff
(cid:96) . This result provides the upper line term on the r.h.s.
c
ofeq.(3)). Thelinearexpressioninl,(16),isdepictedin (cid:96) 1 (cid:34)√πΓ(cid:0)2(cid:1) (cid:18)1 2 5 (cid:19)(cid:35)
theleftplotofFig.2bythedottedlinethatgoesthrough = 3 −((cid:15)u )6 F , , ,((cid:15)u )6 .
the origin. 2 u∗ Γ(cid:0)16(cid:1) ∗ 2 1 2 3 3 ∗
Inadifferentlimit,whereu (cid:28) 1,wecandeducefrom (19)
∗ (cid:15)
the series expansion of the expression (15) the leading One can see from the numerical solutions of the minimal
HEE contribution surface in the opposite regime r (cid:28) r (i.e. the regime
H ε
(2) of interest in this note) that the position x(u) of the
N2L2
S (cid:39) . (17) surfacemovesveryslowlyasudecreasesuntilthesurface
AS 6π (cid:15)6
reachesthenear-horizonthroat. Asaresult,theequation
Strictly speaking this result is valid only under the addi- (19)isnotonlyagoodapproximationintheAdSregime,
tionalassumptionu (cid:29)1alongthesubdominantmiddle r (cid:28)r ,butalsointheregime(2). Thisresultisclearly
∗ ε H
branch. It is clear, however, from the numerical data visibleinthenumericalresultsofFig.1,wherethebrown
depicted in Figs. 1, 2 that the solution (14) works im- curve depicting (19) is in excellent agreement with the
pressively well even for order 1 values of u . It is also bottom branch from the region u∼1, l∼1 and towards
∗
clearfromthedataofFig.2thatoncethemiddlebranch higher values of l.
asymptotes to the bottom branch, it reaches the value
(17) which remains constant as l is increased. This con-
stantisthesameastheleadingdivergenceofthebottom INTERPRETATION OF RESULTS
branch (see red curve in the plots of Fig. 2).
This observation allows us to obtain a rough estimate Theabove-mentionedconjecturesofopen-closedstring
of lc by equating the HEEs in eqs. (16) and (17) duality imply, if correct, that the HEE (3) is holograph-
ically related to the EE of a dual large-N open string
N2L2l N2L2 1
c ∼ ⇔ l ∼ . (18) field theory. We cannot verify (3) directly in a strongly
2π (cid:15)5 6π (cid:15)6 c 3(cid:15)
interacting open string field theory, but it is still natural
Comparing with (13) we observe that the factors ∼ 0.4 to ask if a transition like (3) could be envisaged in an
(from the numerics) and 1 ∼ 0.3 (from the analytic ap- open string theory. Here we would like to argue that the
3
proximation), compare well with each other. answer to this question is naturally a positive one.
5
Let us start with the estimate of the transition scale It would be interesting to reproduce the above expec-
√ √
(cid:96) ∼ λα(cid:48). The λ factor could clearly be an effect of tations with an explicit computation in weakly coupled
c ε
the strong ’t Hooft coupling limit. In a weakly coupled open string theory, perhaps along similar lines to the
string theory one would anticipate a transition scale of computation performed in [39]. A related computation
the form (cid:96) ∼ α(cid:48). Could this be consistent? From previ- worth exploring has to do with the corrections of the en-
c ε
ous discussions of EE in non-local theories (see e.g. [30]) tanglemententropyofN =4SYMtheoryinthepresence
we know that (cid:96) is naturally associated with the scale ofirrelevantdeformationsinducedbyopenstringtheory.
c
of non-locality. In the case of an open string theory the Inthebulkthisinvolvesacomputationinaregimewhere
characteristic size of a rotating string with energy E is the radial cutoff r is comparable to r . In this paper
ε H
(cid:96) ∼ Eα(cid:48) [38]. Consequently, in the presence of a UV we focused on the regime r (cid:29)r .
E ε H
cutoff ε−1 the maximum value of (cid:96)E, which should be Finally, note that in the computations of this paper
associated with the non-locality scale, is (cid:96)ε−1 = (cid:96)c ∼ αε(cid:48) the volume law came essentially from the effects of the
exactlyasanticipatedabove. Therelevanceofthisscale, asymptotic flat space —the scaling of eq. (16) was de-
instead of (cid:96)s, is a sign of UV/IR mixing as pointed out rived exclusively in flat space. We would like to empha-
in the context of NCSYM theories in [30]. size two points related to this feature that are concep-
We can now ask about the more precise transition im- tually close to the discussion of holography as a tomo-
plied by (3). First, we notice that eq. (3) can be recast graphic principle in [15]. First, we note that although
(up to an overall numerical factor for each line on the flat space itself does not have a preferred radial direc-
r.h.s.) as tion, the 3-brane solution naturally defines one via the
splitting ds2 = η dxµdxν +dr2 +r2dΩ2 that leads to
(cid:40)N2L2ε(cid:96)4c(cid:96) +... , (cid:96)(cid:28)(cid:96)c the minimal co-dµimνension-2 surface we c5onsidered. For
S ((cid:96))∼ . (20)
AS a p-brane solution of different worldvolume dimension a
N2Lε24(cid:96)2c +... , (cid:96)(cid:29)(cid:96)c differenttransversespacewouldbechosenleadingtoan-
otherminimalsurface. Second, noticethatwewouldnot
Moreover, in this language the condition (2) becomes
have obtained sensible results had we restricted only to
simply ε (cid:28) (cid:96) . In a weakly coupled open string theory
c the flat space part of our computation. In flat space the
with a UV cutoff ε−1 and U(N) Chan-Paton indices one
minimal surface would not exhibit the bottom branch of
expectsthefollowingbehaviouroftheEE.When(cid:96)(cid:28)(cid:96) ,
c Fig. 1 and above some width (cid:96) no minimal surface
all the degrees of freedom in the region A can interact max
S would exist. In the 3-brane setup this potential issue is
(and entangle) with the outside degrees of freedom. In a
remedied by the non-trivial geometry in the bulk that
unit cell of volume ε3 there are roughly N2(cid:96)c degrees of
ε deviates from flat space.
freedom, so one expects the EE to scale as
(cid:96) 1 (cid:96) (cid:96) I am grateful to Francesco Aprile, Aristos Donos, Si-
S ∼(cid:96)L2·N2 c · =N2 c (21)
AS ε ε3 ε4 mon Ross and Tadashi Takayanagi for useful comments.
reproducingthevolumelawinthefirstlineofther.h.s.of
eq. (20). Similarly, when (cid:96)(cid:29)(cid:96) only degrees of freedom
c
inside a strip of size (cid:96) around the boundary ∂A can
c S
at most entangle with the outside, which leads to the
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