Information flows in strongly coupled ABJM theory

JHEP01(2019)232
Published o SISSA by Sp inge
Recei ed:Decembe 16, 2018
Accep ed:Janua y 23, 2019
Published:Janua y 31, 2019
In o ma ion lows in s ongly coupled ABJM heo y
Vijay Balasub amanian,a,b Niko Jokela,c,d A u P¨onnic,d and Al onso V. Ramalloe,
aDa id Ri enhouse Labo a o y, Uni e si y o Pennsyl ania,
PA 19104 Philadelphia, U.S.A.
bTheo e ische Na uu kunde, V ije Uni e si ei B ussel and In e na ional Sol ay Ins i u es,
Pleinlaan 2, B-1050 B ussels, Belgium
cDepa men o Physics,
P.O. Box 64, FIN-00014 Uni e si y o Helsinki, Finland
dHelsinki Ins i u e o Physics,
P.O. Box 64, FIN-00014 Uni e si y o Helsinki, Finland
eDepa amen o de F´ısica de Pa ´ıculas, Uni e sidade de San iago de Compos ela,
E-15782 San iago de Compos ela, Spain
Ins i u o Galego de F´ısica de Al as Ene x´ıas, Uni e sidade de San iago de Compos ela,
E-15782 San iago de Compos ela, Spain
E-mail: [email p o ec ed],[email p o ec ed],
[email p o ec ed],[email p o ec ed]
Abs ac : We use holog aphic me hods o cha ac e ize he RG low o quan um in o ma-
ion in a Che n-Simons heo y coupled o massi e e mions. Fi s , we use en anglemen
en opy and mu ual in o ma ion be ween s ips o de i e he dimension o he RG-d i ing
ope a o and a mono onic c- unc ion. We hen display a scaling egime whe e, unlike
in a CFT, he mu ual in o ma ion be ween s ips changes non-mono onically wi h s ip
wid h, anishing in bo h IR and UV bu ising o a maximum a in e media e scales. The
associa ed in o ma ion ansi ions also con ibu e o non-mono onici y in he condi ional
mu ual in o ma ion which cha ac e izes he independence o neighbo ing s ips a e con-
di ioning on a hi d. Finally, we cons uc a measu e o ex ensi i y which es s o wha
ex en in o ma ion ha egion A sha es wi h egions B and C is addi i e. In gene al,
mu ual in o ma ion is supe -ex ensi e in holog aphic heo ies, and we migh expec supe -
ex ensi i y o be maximized in CFTs since hey a e scale- ee. Su p isingly, ou massi e
heo y is mo e supe -ex ensi e han a CFT in a ange o scales nea he UV limi , al hough
i is less supe -ex ensi e han a CFT a all lowe scales. Ou analysis equi es he ull en-
dimensional dual g a i y backg ound, and he ex emal su aces compu ing en anglemen
en opy explo e all o hese dimensions.
Keywo ds: AdS-CFT Co espondence, D-b anes
A Xi eP in : 1811.09500
Open Access,c
The Au ho s.
A icle unded by SCOAP3.h ps://doi.o g/10.1007/JHEP01(2019)232
JHEP01(2019)232
Con en s
1 In oduc ion 1
2 Backg ound solu ion 2
2.1 Holog aphic Callan-Symanzik equa ion 6
3 En anglemen en opy o s ips 9
3.1 N N: he ew la o limi 10
3.1.1 Tu ning poin ou side he ca i y (x∗≥1) 10
3.1.2 Tu ning poin inside he ca i y (x∗<1) 12
3.2 Dimension o he RG d i ing ope a o 13
4 The low o mu ual in o ma ion 17
4.1 Mu ual in o ma ion ansi ions 17
4.2 c- unc ions: low o he numbe o deg ees o eedom 20
4.3 Flow o ex ensi i y 24
4.4 Flow o condi ional mu ual in o ma ion 27
5 Discussion 30
A De ails o he backg ound 31
A.1 Expansion in la o 32
B UV expansion o he en anglemen en opy 33
1 In oduc ion
In e ac ions in a quan um ield heo y (QFT) cause quan um in o ma ion o be non-locally
dis ibu ed ac oss space h ough he phenomenon o en anglemen in he wa e unc ion. I
is o g ea in e es o unde s and how his s uc u e o sha ed in o ma ion changes wi h
ene gy scale, o equi alen ly, du ing eno maliza ion g oup (RG) low. Essen ially he only
ool we ha e o s udy his ques ion in s ongly coupled ield heo ies is holog aphy. E en
in hese se ings he e is a key challenge: he e a e e y ew exac solu ions in holog aphic
g a i y ha co espond o non- i ial eno maliza ion g oup lows in QFT.
Many “phenomenological” holog aphic ea men s o RG low a e cons uc ed so ha
ad-dimensional ield heo y is desc ibed by a (d+ 1)-dimensional heo y o g a i y in
which he ex a dimension ep esen s QFT scale. G a i a ional solu ions can be easie o
ob ain in his simpli ied se ing. Howe e , in cases whe e an ac ual gauge-g a i y duali y
has been cons uc ed in s ing heo y, he e a e always he en dimensions o s ing heo y
in he g a i a ional desc ip ion, wi h in e nal symme ies o he QFT ep esen ed by he
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JHEP01(2019)232
s uc u e o he ex a dimensions beyond he adial di ec ion which ep esen s QFT scale.
In such se ings which a e ully g ounded in s ing heo y, RG lows o QFT a e desc ibed
as comple e en-dimensional g a i a ional backg ounds whe e he “in e nal” and “AdS”
dimensions in e ac , mix and some imes exchange oles. Finding exac g a i a ional solu-
ions in his comple e se ing is di icul . Ou goal in his pape is o examine he RG low
o quan um in o ma ion in a s ongly coupled QFT unning be ween non- i ial in e ac ing
ixed poin s, in a scena io whe e we ha e he powe ul le e o a comple e en-dimensional
classical g a i a ional dual.
To his end, we ocus on he ABJM heo y [1] which is he (2+1)-dimensional analogue
o he 4d maximally supe symme ic Yang-Mills heo y. This 3d Che n-Simons gauge
heo y has a symme y g oup U(N)k×U(N)−kwi h le els (k, −k) coupled o ma e ields
in he bi undamen al ep esen a ion o he gauge g oup. When Nand ka e la ge, he
model has a dual holog aphic desc ip ion as an AdS4×CP3geome y wi h luxes in Type
IIA supe g a i y. The ABJM heo y can be gene alized by adding undamen al qua ks
ans o ming in he (N, 1) and (1, N) ep esen a ions o he gauge g oup [2,3]. In he
g a i y dual, hese qua ks a e inco po a ed by adding D6-b anes w apping an RP3inside
he in e nal CP3and ex ended along AdS4.
When he numbe o la o s N is su icien ly small, he ield heo y can be ea ed
in a quenched app oxima ion whe e he loops o undamen als a e supp essed. In he
same limi , he D6-b anes in he g a i y dual can be ea ed as p obes in he AdS4×CP3
backg ound. When N is la ge so ha he ield heo y is no quenched, he D6-b anes
in he g a i y desc ip ion will back eac on he geome y. Cons uc ing he back eac ed
supe g a i y solu ion is a di icul ask in gene al. Howe e , in he Veneziano limi [4] in
which bo h he numbe o colo s Nand la o s N a e aken o be la ge wi h N /N ixed,
we can employ a sys ema ic pe u ba i e app oxima ion [5]. Using his echnique one can
ind he comple e g a i y duals. In ou con ex , hese gene alized ABJM geome ies we e
ound in [6–8] o massless la o s and in [9] o massi e la o s, wi h a ecen ex ension [10]
o a non-commu a i e geome y. This is no wha some call a massi e de o ma ion o he
ABJM heo y [11,12]; en anglemen lows in ha con ex a e s udied in [13,14]. The
ex emal su aces ha holog aphically compu e en anglemen in ou se ing will explo e
all 10 dimensions o he dual g a i y heo y (see p e ious examples in, e.g., [15,16]).
2 Backg ound solu ion
The maximally supe symme ic h ee-dimensional U(N)k×U(N)−kChe n-Simons heo y
coupled o massi e bi undamen al qua ks should be con o mal in he UV, a scales well
abo e he qua k masses, and in he IR, a scales well below he qua k masses. Thus, we
expec a dual desc ip ion in Type IIA supe g a i y ha ends o AdS4 imes a compac
ac o bo h nea in ini y and in he deep in e io . The ele an solu ion, in e pola ing
smoo hly be ween wo AdS4space imes, was ound in [9]. The s ing ame me ic in his
solu ion is:
ds2
10 =h(x)−1/2dx2
1,2+h(x)1/2e2g(x)dx2
x2+q(x)ds2
S4+ds2
S2
,(2.1)
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JHEP01(2019)232
whe e ds2
1,2in he Minkowski me ic in 2 + 1 dimensions, ds2
S4is he s anda d me ic o a
uni ou -sphe e, and S2
is a ibe ed wo-sphe e (see below). We a e jus p esen ing he
me ic he e and no he luxes, since he la e do no play a ole in ou analyses. The
adial coo dina e xin (2.1) is ela ed o a canonical coo dina e in which he me ic is
asymp o ically ds2=− 2d 2+d 2/ 2+. . . as ollows
d =egdx
x.(2.2)
Th ough his ela ion, he IR ( = 0) and he UV ( =∞) co espond o x= 0 and x=∞,
espec i ely. As he geome y is no globally AdS, he dila on is also no a cons an (see
below), bu in he IR and UV he geome y ends owa ds AdS4in Poinca ´e coo dina es
imes a compac ac o . Thus x= 0 is he Poinca ´e ho izon.
The unc ion e2g(x), he local adius o he in e nal S2, quan i ies he ela i e squashing
be ween he in e nal and ex e nal mani olds. The adius o S4is gi en by he unc ion
e2 := q e2g. The unc ion q(x) measu es he ela i e squashing be ween he S2 ibe and
he S4base on he in e nal mani old. A con enien ep esen a ion o he me ic o S4is
ds2
S4=4
(1 + ξ2)2"dξ2+ξ2
3
X
i=1
(ωi)2#,(2.3)
whe e ωia e he SU(2) le -in a ian one- o ms which sa is y dωi=1
2ijkωj∧ωkand
0≤ξ < ∞. The me ic o he ibe ed wo-sphe e S2can be w i en in e ms o one- o ms
E1, E2:
ds2
S2
=E12+E22(2.4)
E1=dθ +ξ2
1 + ξ2sin ϕω1−cos ϕω2(2.5)
E2= sin θdϕ −ξ2
1 + ξ2ω3+ξ2
1 + ξ2cos θcos ϕω1+ sin ϕω2,(2.6)
whe e 0 ≤θ≤π, 0≤ϕ < 2π.
The massi e qua ks in he ield heo y equi e he inclusion o D6-b anes ha span
he AdS4pa o he geome y and w ap a pa o he in e nal mani old. The D6-b ane
cha ge densi y is measu ed by iola ion o he Bianchi iden i y o he RR wo- o m F2, and
his u ns ou o be p opo ional o η−1 [9] whe e
η(x) = 1 + ˆ1−1
x2Θ(x−1) ,(2.7)
in e ms o he Hea iside unc ion Θ(z). Thus, ηdic a es he dis ibu ion o smea ed la o
D6-b anes ha ac as sou ces o he SUGRA equa ions o mo ion. No ice ha η= 1 o
x < 1, meaning ha he D6-b anes do no span he whole adial coo dina e bu end abo e
he Poinca ´e ho izon. Thus, in he ollowing, we call he egion x≤1 he “ca i y”. The
pa ame e ˆencodes back eac ion o he massi e qua ks and ela es o he numbe N o
D6-b anes as:
ˆ=3N
4k=3
4
N
Nλ , (2.8)
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JHEP01(2019)232
whe e he ’ Hoo coupling is ela ed o he Che n-Simons le el kand he numbe o colo s
Nas λ=N/k.
The me ic ansa z (2.1) is gua an eed o p ese e N= 1 supe symme y in h ee
dimensions [6,9], i he ollowing mas e equa ion is sa is ied
W00 + 4η0+W0+ 4ηW0+ 10η
3W−W0+ 4η+ 6
x(W0+ 4η)= 0 (2.9)
o Wde ined as
W(x) = 4
kh1/4e2 −g−φ.(2.10)
Then, up o eedom in in eg a ion cons an s, he unc ions ,g,h, and he dila on φ
can be cons uc ed om W(see appendix A). Mo eo e , he squashing unc ion qcan be
w i en in e ms o he mas e unc ion Wand i s de i a i e as:
q=3W
x(W0+ 4η).(2.11)
The qua k mass mqencoded in his g a i a ional backg ound can be compu ed by
e alua ing he Nambu-Go o ac ion o a undamen al s ing s e ched along he holog aphic
di ec ion be ween x= 0 and x= 1 a ixed alues o he o he spacelike coo dina es. This
gi es mq= q/2πα0, whe e α0is he Regge slope (which we will se o 1), wi h qgi en by:
q=Z1
0
dxeg
x.(2.12)
Following (2.2), qis jus he canonical coo dina e o he ip o he la o D6-b anes.
The IR limi . In he egion x<1 ( he “ca i y” egion) he e a e no D6-b anes (see (2.7)).
Thus, he solu ion in his egion should sol e he ma e - ee equa ions wi h N →0, ˆ→0,
and η→1. In his limi , W= 2xgi es one solu ion o he mas e equa ion. This choice
ep oduces he o iginal ma e - ee ABJM solu ion a e ans o ming o he canonical
coo dina e [9]. In his case he squashing unc ion qin (2.11) becomes equal o one and
he in e nal me ic in (2.1) is he s anda d Fubini-S udy me ic o CP3gi ing a 10d me ic
ds2
10,=0 =L2
ABJM ds2
AdS4+ 4 L2
ABJM ds2
CP3,(2.13)
whe e ds2
AdS4and ds2
CP3a e espec i ely he AdS4and CP3me ics. The o me , in Poinca ´e
coo dina es, is gi en by:
ds2
AdS4= 2dx2
1,2+d 2
2.(2.14)
In (2.13)LABJM is he adius o he AdS4pa o he me ic and in s ing uni s is
L4
ABJM = 2π2N
k,(2.15)
whe e Nand ka e wo in ege s which co espond, in he gauge heo y dual, o he ank
o he gauge g oups and he Che n-Simons le el, espec i ely.
Fo ou pu pose o sol ing he ull equa ions wi h ma e , he pu e ABJM solu ion does
no su ice in he x < 1 egion. Ra he we need a solu ion ha will ma ch app op ia ely
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JHEP01(2019)232
on o he solu ion in he x > 1 egion whe e he D6-b anes back eac on he geome y.
Fo una ely, in he limi ˆ→0, he e is a one-pa ame e amily o analy ic solu ions
o (2.9):
W unning =4(1 + 4γx)x
1 + √1+4γx ,(2.16)
whe e γis a cons an . These “ unning solu ions” educe o ABJM in he deep IR
(W unning(x→0) →2x). Bu as xinc eases, he unning solu ion lows away om he
ABJM ixed poin and e en ually asymp o es a x→ ∞ o a me ic desc ibing a esol ed
Ricci la cone. Wi h a ini e numbe o qua k la o s and hence ˆ > 0, he unning solu ion
will apply in he “ca i y” x < 1 which is no pene a ed by he D6-b anes and hence is
locally desc ibed by an N = 0 solu ion. We will ma ch he unning solu ion in he x < 1
egion on o he app op ia e me ic in he x > 1 egion.
The UV limi . To gain con ol o e he x > 1 egion, we i s conside he asymp o ic
UV limi x→ ∞. In his limi , he D6-b ane dis ibu ion ends o a cons an since
η→1 + ˆ≡η0. Then one solu ion o he mas e equa ion (2.9) is
W(x→ ∞)≈q0(η0+q0)
2−q0
x , (2.17)
whe e he squashing unc ion qis also a cons an q0=3+3η0−p9η2
0−2η0+ 9/2.
The esul ing en-dimensional asymp o ic geome y is o he ype AdS4×M6, whe e M6
is a squashed e sion o CP3wi h cons an squashing ac o s. Indeed, he me ic akes
he o m
ds2
10 ≈L2
0ds2
AdS4+L2
0
b2hq0ds2
S4+ds2
S2
i,(2.18)
whe e L0is cons an and gi en by
L4
0= 128π2N
k
(2 −q0)q3
0
(η0+q0) (q0+ 1)5,(2.19)
and bis ano he cons an , which can be w i en in e ms o he asymp o ic UV squashing
q0as
b=2q0
q0+ 1 .(2.20)
The pa ame e bwill play an impo an ole in wha ollows. I s in e p e a ion is clea
om (2.18): i ep esen s he ela i e squashing o he CP3pa o he asymp o ic me ic
wi h espec o he AdS4pa . In he un la o ed case ˆ= 0 we ha e q0=b= 1. In gene al
q0and bg ow wi h ˆand each hei maximal alues q0= 5/3 and b= 5/4 when ˆ→ ∞.
In ac , (2.17) sol es he mas e equa ion o all xi ηis cons an e e ywhe e, and
co esponds o he limi o massless qua ks (mq∝ q→0) ha back eac in he AdS
desc ip ion. This case is discussed in dep h in [6–8,17], and gi es he desi ed asymp o ics
because a ene gies a abo e he mass scale o he qua ks hey should ac as i hey a e
massless. Holog aphically, his means ha he ull solu ion should asymp o e o (2.17).
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JHEP01(2019)232
The ull solu ion. The solu ion o he mas e equa ion ha we a e in e es ed in has
wo pieces. Inside he ca i y (x≤1) he mas e unc ion Wis gi en by (2.16). A he
bounda y o he ca i y x= 1 we need o glue his solu ion con inuously o ano he one
which asymp o es o (2.17). This can be ob ained nume ically by he s anda d shoo ing
echnique s a ing a x= 1, wi h bounda y condi ions W(1) and W0(1) gi en by he
unning solu ion (2.16). Shoo ing is done by a ying he cons an γin he unning solu ion
o x < 1 un il he desi ed asymp o ic solu ion o W(x→ ∞) is ob ained by in eg a ing
ou wa d wi h bounda y condi ions ha impose con inuous Wand W0a x= 1. A gene ic
alue o γwill p o ide a solu ion o which W∼x3/2as x→ ∞, i.e., no asymp o ically
AdS4bu a he a G2cone. Fo a gi en ˆ he e exis s a unique γ=γ(ˆ) such ha
W∼x, as x→ ∞, i.e., such ha he solu ion asymp o es o (2.17). Such a piecewise
me ic hen desc ibes an in e pola ion be ween wo AdS4space imes wi h di e en adii.
These solu ions we e cons uc ed nume ically in [9] and as a powe se ies in ˆin [18]. In
addi ion o he nume ical solu ions, we will also use his powe se ies, which is e iewed in
appendix A.
2.1 Holog aphic Callan-Symanzik equa ion
The solu ion gi en abo e is a holog aphic desc ip ion o he eno maliza ion g oup low o
a s ongly in e ac ing heo y o ma e and gauge ields. The AdS4geome ies a x→0
and x→ ∞ indica e ha he low uns be ween di e en con o mal ixed poin s. We can
cha ac e ize he low in e ms o a Callan-Symanzik equa ion o he qua k mass, ex ending
he discussion in [17].
The na u al mass scale in a op-down holog aphic se up such as ou s in ol ing la o
and colo b anes is he dis ance be ween hese wo se s o b anes. This dis ance is ela ed
o he mass o undamen al qua ks in he ield heo y ha a e dual o he open s ings
ending on he la o b ane [19]. In ou model, he la o b anes a e D6-b anes embedded
supe symme ically in he backg ound. The qua k mass is ob ained by e alua ing he
Nambu-Go o ac ion o a s ing ex ended om he o igin and ending on he b anes [20].
To e alua e he qua k mass, conside a p obe D6-b ane embedded along he ou AdS4
di ec ions and w apping a h ee-dimensional cycle inside he compac in e nal mani old.
The p ecise o m o his cycle o a supe symme ic embedding can be de e mined by
using kappa symme y. I was shown in [6] ha his equi es ha he pullbacks o he
wo ld olume o he SU(2) one- o ms ω1and ω2o (2.3) mus anish. Ex ending he b ane
along he coo dina e ϕo he S2 ibe , he induced me ic on he b ane wo ld olume akes
he o m:
ds2
7=1
ph( )dx2
1,2+ph( )(1 + e2g( )θ0( )2)d 2
+e2g( )ph( )q( )dα2+qsin2αdβ2+ sin2θ(dψ + cos αdβ)2,(2.21)
whe e, o con enience, we a e using he canonical AdS4coo dina e in oduced in (2.2).
The angula coo dina es α,β, and ψa e ela ed o ξ,ϕ, and o he one used o pa ame e ize
he pullback o ω3(see [6] o de ails). The kappa symme y condi ion hen educes o he
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JHEP01(2019)232
ollowing i s -o de di e en ial equa ion o θ=θ( ):
eg( )dθ
d = co θ . (2.22)
This equa ion can be in eg a ed exac ly a e a change o a iable o a new adial coo di-
na e u:
d
eg( )=du
u→u= exp Z
e−g( 0)d .(2.23)
Then, he embedding unc ion θis simply
cos θ=R0
u,(2.24)
whe e R0is an in eg a ion cons an . Thus, o ou SUSY embedding ucos θis cons an .
In o de o easily desc ibe he co esponding se en-dimensional su ace, i is con enien o
in oduce wo new coo dina es (R, ρ) as
R=ucos θ;ρ=usin θ . (2.25)
In hese coo dina es he ( , θ) sec o o he en-dimensional me ic akes he o m
ph( )d 2+e2g(u)dθ2=ph(u)e2g(u)
u2dR2+dρ2,(2.26)
whe e u2=R2+ρ2. Fo supe symme ic embeddings R=R0= cons an , which leads o
he induced wo ld olume me ic
ds2
7=1
√hdx2
1,2+√h e2g
pR2
0+ρ2dρ2
+e2g√hqdα2+qsin2αdβ2+ sin2θdψ2+ cos αdβ2,(2.27)
whe e all unc ions depend on ρ h ough he combina ion pR2
0+ρ2. The e o e, ρplays
he ole o he holog aphic coo dina e on he wo ld olume. Ac ually, in he UV egion
ρ≈u→ ∞ we ha e ha eg≈ /b and (2.23) can be in eg a ed as ∼ρ1
b. Since is he
canonical AdS4coo dina e we should iden i y he ene gy scale wi h ρ1
b.
Conside now a undamen al s ing loca ed a he poin ρ=ρ∗. The s ing ex ends
om R= 0 o i s bounda y alue R=R0, whe e i in e sec s wi h he D6-b ane. The
unning mass o he co esponding dual ( alence) qua k is equal o he Nambu-Go o ac ion
o his s ing pe uni ime. The induced me ic on he s ing wo ldshee ex ended in ( , R)
a ρ=ρ∗is gi en by
ds2
2=−d 2
√h+√h e2gdR2
R2+ρ2
∗
.(2.28)
The unning qua k mass is hen
mq=1
2π α03/2ZR0
0p−de g2dR =1
2πα03/2ZR0
0
eg(u)dR
pR2+ρ2
∗
.(2.29)
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JHEP01(2019)232
We wan o ind an e olu ion equa ion o mqwi h espec o he ene gy scale Λ. As
a gued abo e, we should iden i y he ene gy scale wi h ρ1/b
∗. The e o e, we de ine:
Λ := ρ1/b
∗.(2.30)
We can wo k ou he e olu ion equa ion o mq, which a e a s aigh o wa d calcula ion
boils down o:
2πα03/2∂mq
∂log Λ =bρ∗
∂mq
∂ρ∗
=bZR0
0pR2+ρ2
∗
R∂RegdR −bR0eg√R2
0+Λ2b
pR2
0+ Λ2b.(2.31)
The UV-limi o he abo e equa ions can be wo ked ou easily. Indeed, in his limi he
unc ion g o ρ=ρ∗is gi en by
eg≈(R2+ρ2
∗)1
2b
b,(2.32)
and we in e om (2.29) ha he unning mass is gi en by
mq≈1
2π α03/2bZR0
0
(R2+ρ2
∗)1
2b−1
2≈Λ1−bR0
2π α03/2b,(2.33)
whe e, in he las s ep, we ook he limi in which Λ is la ge. Mo eo e , plugging (2.32)
in o (2.31) we a i e a
∂mq
∂log Λ =mq−1
2πα03/2
R0
(R2
0+ Λ2b)b−1
2b
.(2.34)
Fu he mo e, in he deep UV limi in which Λ is la ge, we can neglec he R2
0in he
denomina o and using (2.33) we conclude ha he second e m in (2.34) is jus −b mq.
We inally ob ain he e olu ion equa ion o he qua k mass mq:
∂mq
∂log Λ =−γmmq,(2.35)
whe e we in oduced he anomalous dimension o he qua k mass
γm=b−1.(2.36)
The esul (2.35) is p ecisely he Callan-Symanzik equa ion o he e ec i e mass which
we we e looking o . We see in his way ha he g a i a ional backg ound desc ibes a
eno maliza ion g oup low o he ull heo y. No ice ha γm= 0 o he un la o ed case
(and hus he e is no unning o mqwhen he e a e no dynamical la o s) and γm→1/4
as ˆ→ ∞.
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JHEP01(2019)232
UV-beha io o he en anglemen en opy, p o iding a c oss-check on he o mulas de i ed
in he las sec ion.1
Conside he holog aphic dual o any h ee-dimensional ield heo y nea a UV ixed
poin . The ou -dimensional pa o he bulk me ic mus asymp o e o AdS4and can be
w i en nea he bounda y as
ds2
4=L2
UV
z2dz2
(z)−d 2+dx2+dy2,(3.31)
whe e (z) = 1 + O(zα) and he bounda y lies a z= 0. Depa u es om AdS4, and hus
he e ec s o RG low, a e encoded in he subleading beha io o (z). We now assume ha
he RG low is d i en by a ele an scala pe u ba ion Owi h a dimension 1/2<∆<3.
This is dual holog aphically o a scala ield φwi h mass m2= ∆(∆ −3). The bulk ac ion
will hen be
S4=1
16πG(4)
NZd4x√−g4R+6
L2
UV −1
2∂µφ∂µφ−1
2m2φ2+. . ..(3.32)
When φ(z, ~x) = 0 he co esponding backg ound solu ion educes o AdS4. The equa ion
o mo ion o he scala gi es he nea bounda y beha io
φ(z, ~x) = z3−∆(φ0(~x) + O(z2)) (3.33)
and co esponds o a pe u ba ion o he UV CFT by
S3=S(UV)
3+Zd3x φ0(~x)O(~x).(3.34)
Le us ocus on a uni o m pe u ba ion φ0(~x) = cons . = φ0. In his case he Eins ein
equa ions gi e he back eac ion o he scala ield on he me ic
(z) = 1 + 3−∆
4φ2
0z2(3−∆) .(3.35)
Gi en he me ic unc ions nea he UV his can be used o ind he ope a o dimension ∆.
The e ec o RG low is also encoded in he UV beha io o he en anglemen en opy.
Conside he en anglemen en opy o a s ip in a heo y wi h a g a i y dual (3.31). Fo
na ow s ips he holog aphic en anglemen en opy in he Eins ein ame is
S=1
4G(4)
NZ√g2,(3.36)
whe e
ds2
2=L2
UV
z2z02
(z)+ 1(dx1)2+ (dx2)2(3.37)
1Ou ea men pa ly ollows [22].
– 15 –

JHEP01(2019)232
is he induced me ic on a bulk su ace ancho ed o he s ip and he p ime deno es de i a-
ion wi h espec o x1. Explici ly, his gi es
S=LyL2
UV
2G(4)
NZl/2
0
dx
z2sz02
(z)+ 1 ,(3.38)
whe e Lyand la e he leng h and he wid h o he s ip, espec i ely. To ind he minimal
su ace we e alua e he Eule -Lag ange equa ions o z(x), z0=√ (z)√z4
∗−z4
z2, which, when
inse ed in (3.36), yields
S=LyL2
UV
2G(4)
NZz∗
0
z2
∗dz
z2p(z4
∗−z4) (z)(3.39)
=LyL2
UV
2G(4)
NZz∗
0 z2
∗
z2pz4
∗−z4−z2
∗z4−2∆
2pz4
∗−z4
3−∆
4φ2
0+. . .!dz . (3.40)
In addi ion o he usual di e gence o a eas o su aces ancho ed o asymp o ically AdS
bounda ies, he e is a di e gence coming om he scala ield-dependen e m when 5/2≤
∆<3. Sub ac ing hese di e gences,
Sdi = lim
→0
LyL2
UV
2G(4)
N1
+5−2∆
2(5 −∆)
3−∆
4φ2
0,(3.41)
he egula ed en anglemen en opy S eg =S−Sdi eads
S eg =LyL2
UV
2G(4)
N −√2π3/2
Γ1
42
1
z∗−√πΓ5
4−∆
2
8Γ 7
4−∆
2
3−∆
4φ2
0z5−2∆
∗+. . .!.(3.42)
To wo k ou he u ning poin as a unc ion o s ip wid h z∗(l), we e alua e
l= 2 Zz∗
0
z2dz
p (z)pz4
∗−z4= 2 Zz∗
0 z2
pz4
∗−z4−z2+2(3−∆)
pz4
∗−z4+. . .!dz (3.43)
=2√2π3/2
Γ1
42z∗−√πΓ9
4−∆
2
4Γ 11
4−∆
2
3−∆
4φ2
0z7−2∆
∗+. . . . (3.44)
This ela ion can be e e ed o gi e
z∗=Γ1
42
2√2π3/2l+2−14+3∆π−23/2+3∆Γ1
416−4∆ Γ9
4−∆
2
Γ11
4−∆
2
3−∆
4φ2
0l7−2∆ +. . . . (3.45)
Finally, he en anglemen en opy is
S eg =LyL2
UV
2G(4)
N −4π3
Γ1
4
1
l−2−32/2+3∆π−7+3∆Γ1
410−4∆ Γ5
4−∆
2
Γ11
4−∆
2
3−∆
4φ2
0l5−2∆ +. . .
!.
(3.46)
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JHEP01(2019)232
In ou massi e ABJM se ing we can iden i y
(z) = L2
0z2
ph(z)= 1 −κ2b
2h2−4g2
2b−1( qz)2b+. . . (3.47)
leading o
∆=3−b= 2 −γm,(3.48)
o he dimension o he leading ope a o d i ing he RG low. He e we used he exp ession
o he mass anomalous dimension (2.36) in he las s ep. We can likewise use (3.46)
o de i e he RG-d i ing ope a o dimension om he UV-beha io o he en anglemen
en opy. I u ns ou ha he leading co ec ion o he en anglemen en opy beha es as
δS eg
ABJM ∼l−1+2b(de ails in appendix B) which co ec ly implies (3.48), ∆ = 3 −b. This
con i ms he alidi y o ou analy ical exp essions o he en anglemen en opy.
4 The low o mu ual in o ma ion
The mu ual in o ma ion be ween wo en angling egions Aand Bis de ined as
I(A, B) = S(A) + S(B)−S(A∪B),(4.1)
and cha ac e izes he in o ma ion ha is sha ed be ween hese domains. The low o mu ual
in o ma ion wi h he size o he en angling egions cha ac e izes how in o ma ion is sha ed
ac oss space a di e en scales. We will cha ac e ize his low o he mu ual in o ma ion
be ween s ips in ou ways: (a) in e ms o a “phase ansi ion” de ined by he sepa a ion
dis ance be ween s ips o ixed wid h a which he mu ual in o ma ion anishes, (b) in
e ms o a c- unc ion ob ained om he mu ual in o ma ion ha coun s he deg ees o
eedom ha a e ac i e a he scale o he s ips, (c) in e ms o a measu e o “ex ensi i y”
ha cha ac e izes he addi i i y o he mu ual in o ma ion be ween a egion Aand wo
o he egions Band C, and (d) in e ms o mu ual in o ma ion be ween egions Aand B
condi ioned on knowledge o a hi d egion C.
4.1 Mu ual in o ma ion ansi ions
In heo ies wi h a holog aphic dual he e is ypically a scale-dependen phase ansi ion
in he mu ual in o ma ion: Iis ini e when wo egions o ixed sizes a e su icien ly close,
bu anishes when he sepa a ion g ows o a ce ain c i ical size. Holog aphically, his
ansi ion occu s when he minimal bulk su ace ha app oaches A∪Bon he bounda y
agmen s in o he union o minimal su aces o Aand B(see igu e 3). In he ew la o
limi we will ind an analy ic exp ession o he c i ical sepa a ion a which he phase
ansi ion occu s in he mu ual in o ma ion be ween s ips. The c i ical sepa a ion will be
a unc ion o he scale lde ined by he s ips hemsel es.
To his end, le us conside a sys em o wo pa allel s ips o wid h lsepa a ed by a
dis ance s. The ansi ion be ween he wo con igu a ions in igu e 3happens when hei
en opies a e equal:
2S eg(l) = S eg(2l+s) + S eg(s).(4.2)
– 17 –
JHEP01(2019)232
l ls
A B
bd y
bulk
(A)
l ls
A B
bd y
bulk
(B)
Figu e 3. Bulk ex emal su aces (blue) ha can con ibu e o he holog aphic mu ual in o ma ion
be ween bounda y s ips Aand Bo leng h l. (A) When he sepa a ion so he s ips is la ge
he minimal a ea su ace bounded by A∪Bis he union o he minimal su aces o Aand B
sepa a ely. Thus he mu ual in o ma ion (4.1) anishes because S(A)+S(B) = S(A∪B). (B) When
he sepa a ion so he s ips is small, he minimal a ea su ace bounded by A∪Bis he union
he pic u ed su aces. Thus he mu ual in o ma ion (4.1) is non- anishing because S(A) + S(B)>
S(A∪B). I he heo y is con o mal (ei he because he e a e no qua ks, o when hey a e massless)
he ex shows ha his mu ual in o ma ion phase ansi ion occu s when s/l|ˆ=0 = (√5−1)/2 =
1/ϕ [23,24] whe e ϕis he golden a io (√5 + 1)/2.
Subs i u ing he ˆexpansion S eg =S eg,0+ ˆS eg,ˆin (4.2) we ind
2(S eg,0(l) + ˆS eg,ˆ(l))
=S eg,0(2l+s) + ˆS eg,ˆ(2l+l/ϕ) + S eg,0(s) + ˆS eg,ˆ(l/ϕ),(4.3)
whe e we ha e in oduced he golden a io ϕ= (√5 + 1)/2:
1
ϕ=√5−1
2.(4.4)
Sol ing o s(l) gi es
s
l=1
ϕ+ ˆ3 (ϕ+ 2) Γ 1
44
40√2π3
λ2l
N2Ly2S eg,ˆ(l)−S eg,ˆ(2l+l/ϕ)−S eg,ˆ(l/ϕ)+O(2),(4.5)
whe e he en opies o indi idual con igu a ions a e gi en ei he by (3.21) o by (3.30)
depending on whe he hei leng hs a e smalle o la ge han lc i gi en in (3.22). Close
o he CFT ixed poin s in he UV and IR we ind ha
UV : s
l≈1
ϕ+ ˆΓ1
48
384(2ϕ−1)π7 ql
√λ2
,2l+l/ϕ < lc i (4.6)
IR : s
l≈1
ϕ+ ˆ32√2(5/2−ϕ)π6
15Γ 1
44 ql
√λ−3
, ql
√λ→ ∞ .(4.7)
In he UV limi he bulk minimal su aces emain in he egion x > x∗= 1, while in he
IR limi all he su aces pene a e o x<x∗.
To leading o de in ˆ,s(l) coincides wi h he esul o a con o mal heo y dual o
AdS4:s(l)|ˆ=0/l = 1/ϕ = (√5−1)/2≈0.618 [23,24].2I also ends o his alue in he
2In e es ingly, o he symme ic case o mequally sepa a ed s ips o equal wid h l, as also s udied
in [23,24], he c i ical sepa a ion l/s =(2
m)+q(2
m)2+4
2is he so-called “me allic mean”.
– 18 –
JHEP01(2019)232
0 5 10 15 20 25 30
l q
λ
0.618
0.620
0.622
0.624
0.626
0.628
c i ical s
l
Figu e 4. C i ical alues o s/l o ˆ= 0.1, a which he mu ual in o ma ion be ween s ips o
wid h lsepa a ed by a spacing sd ops o ze o. The dashed ho izon al o ange line deno es he
c i ical alue in a CFT. The nume ical esul s ( ed do s) closely ma ch he analy ical esul s (blue
cu e; eq. (4.5)). Highe o de in ˆco ec ions o (4.5) should p oduce an e en close ma ch. The
e ical ed dashed line is lq q/√λwhe e lqis he wid h o he s ip whose bulk minimal su ace
jus ouches he ca i y.
con o mal limi s well abo e and below he mass scale o he qua ks (l→0 and l→ ∞). A
in e media e scales l, he mu ual in o ma ion ansi ion occu s a a sepa a ion sbe ween
s ips ha is g ea e han in a con o mal ield heo y ( igu e 4), and is maximized a a scale
whe e he bulk minimal su ace almos eaches he “ca i y” associa ed o he qua k mass
scale. In e es ingly, and su p isingly, his sugges s ha in o ma ion is mo e non-locally
sha ed away om he con o mal ixed poin s.
The non-mono onic beha io in igu e 4also implies ha in a ange o ixed alues o
s/l (be ween he peak and he ho izon al dashed line), he mu ual in o ma ion be ween
s ips will anish bo h o la ge land o small l. In bo h hese cases he massi e ABJM
heo y is in a phase whe e he en opy o he union o s ips is gi en holog aphically by
he disconnec ed su aces in igu e 3A. Fo hese s/l he e is also an in e media e ange o
l(de e mined by he in e sec ion o a ho izon al line o ixed s/l wi h he c i ical cu e in
igu e 4) in which he mu ual in o ma ion is non-ze o, o equi alen ly, in which he en opy
o he union o s ips is gi en by he connec ed su aces in igu e 3B. This has a ema kable
implica ion abou he o ganiza ion o quan um in o ma ion in his heo y. Appa en ly,
wo s ips a a ixed s/l ha sha e mu ual in o ma ion can be mu ually disen angled by
inc easing he s ip wid h a ixed s/l. This so o pu i ica ion by expansion does no
happen in a CFT whe e s ips wi h a gi en s/l will always ei he sha e some mu ual
in o ma ion o none a all, independen ly o he alue o l.
As he numbe o qua ks o , equi alen ly, ˆ= (3/4)(N /N)λ, inc eases, he peak in
igu e 4becomes highe ( igu e 5) and na ows bu also acqui es a longe ail owa ds he
in a ed (la ge l). The ail means ha he de ia ion om he con o mal heo y pe sis s
– 19 –
JHEP01(2019)232
ϵ
=9
ϵ
=1
ϵ
=0.1
5 10 15 20 25 30
l q
λ
0.64
0.66
0.68
0.70
c i ical s
l
0.5 1 5 10 50
ϵ

0.64
0.66
0.68
0.70
peak heigh
(A) (B)
Figu e 5. (A) Nume ical analysis o he c i ical alue o s/l (s= s ip sepa a ion, l= s ip wid h)
a which he mu ual in o ma ion ansi ion occu s o a ying numbe s o la o s: (3/4)(N /N)λ=
ˆ= 0.1,1,9. As N inc eases, he peak de ia ion om he con o mal alue o s/l g ows, and occu s
u he in he UV (smalle s ip wid hs l). The e ical dashed lines a e lq q/√λwhe e lqis he
wid h o he s ip whose bulk minimal su ace jus ouches he ca i y o each ˆ. (B) Peak de ia ion
o he c i ical s/l as a unc ion o he numbe o la o s. The nume ical e idence con i ms ha in
he absence o la o s (ˆ→0) we e u n o he con o mal esul s/l = 1/ϕ whe e ϕis he golden
a io, and sugges s ha in he limi o many la o s (ˆ→ ∞) he de ia ion om he con o mal
alue sa u a es.
deepe in o he in a ed. When he ˆ > 1, he ansi ion occu s a an s/l a om he
con o mal alue e en a scales well in o he in a ed. O cou se in he deep in a ed limi
he heo y app oaches a con o mal ixed poin (an AdS4geome y in he bulk desc ip ion)
and so he ansi ion s/l e en ually e u ns o he con o mal alue. Meanwhile he scale
la which he e is he g ea es de ia ion om he con o mal esul (i.e., he peak o he
bump in igu e 4) mo es u he in o he UV as N inc eases ( igu e 6). A he same ime
his scale app oaches mo e closely he s ip wid h lq o which he co esponding minimal
su ace jus ouches he ca i y. I would be in e es ing o s udy he ˆ→ ∞ limi o see
whe he he peak heigh app oaches a ini e alue as sugges ed by he nume ical analysis
in igu e 5and whe he he peak loca ion coincides p ecisely wi h lq.
4.2 c- unc ions: low o he numbe o deg ees o eedom
In quan um ield heo y one seeks o de ine a c- unc ion ha coun s he numbe o deg ees
o eedom a ailable as a unc ion o he scale o measu emen l. Liu and Mezei [25,26]
p oposed ha a c- unc ion can be de ined in e ms o he de i a i e o he en anglemen
en opy wi h espec o scale.3In ou con ex , whe e we a e compu ing he en anglemen
3A spa ial in eg al o a simila quan i y, called he di e en ial en opy, was shown o ep oduce he a eas
o closed su aces in AdS space [27,28], see also [29].
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JHEP01(2019)232
lpeak
lq
0.1 0.5 1 5 10 50
ϵ

1
2
3
4
5
l q
λ
0.1 0.5 1 5 10 50
ϵ

0.050
0.075
0.100
0.125
0.150
lq-lpeak
lq
(A) (B)
Figu e 6. Nume ical analysis o he scale lpeak a which he c i ical alue o s/l maximally de ia es
om he con o mal alue. (A) Compa ison o lpeak q/√λwi h lq q/√λas a unc ion o he numbe
o la o s (ˆ= (3/4)(N /N)λ). He e lqis he s ip wid h a which he co esponding bulk minimal
su ace jus ouches he ca i y; i.e., his is he scale se by he qua k masses. The absolu e di e ence
be ween hese scales sh inks in he many la o limi . (B) The ela i e di e ence be ween hese
scales also sh inks in he many la o limi , bu may be app oaching a ini e alue.
o s ips, we can w i e his as
F(l) = l2
Ly
∂S eg
∂l .(4.8)
whe e lis he s ip wid h and Lyis he s ip leng h which can be aken o in ini y.4F(l)
is UV ini e and in e pola es in a eno malizable QFT be ween ixed, scale-independen
alues a he UV and IR ixed poin s. We can compu e F(l) analy ically using ou explici
exp essions o en anglemen en opy as a unc ion o s ip wid h. Plo ing he esul s
( igu e 7) indeed shows ha F(l) in ou massi e ABJM heo y app oaches a cons an
alue in he UV (small l q/√λ) and dec eases mono onically o ano he cons an in he IR
(la ge l q/√λ), consis en ly wi h he in e p e a ion ha F(l) coun s he deg ees o eedom
a a scale l. Simila s a emen also holds o disk egions in he same geome y [9].
In ou model Fis ex emized a he ixed poin s o he low, i.e., he i s de i a i e
∂F/∂l will anish in he la ge and small llimi s. Since Fis bo h con inuous and mono onic,
he i s non- anishing de i a i es should be nega i e in he UV (l q/√λ→0) and posi i e
4In mo e de ail, ollowing [26], he en anglemen en opy o he s ip will ha e he gene al beha io
S∼Ly(di e gen + ini e). The di e gence comes om sho dis ance modes s addling he edges o he
s ip and so will be p opo ional o Ly, bu will no depend on he s ip wid h l. Thus we can ex ac
he ini e pa by compu ing l∂S/∂l. Howe e , his quan i y will di e ge wi h he leng h o he s ip so
we should di ide by Ly o ge a ini e en anglemen pe uni leng h as (l/Ly)∂S/∂l. We can make his
quan i y dimensionless by mul iplying by l, o de ine F= (l2/Ly)∂S/∂l. A he con o mal poin s he e is
no scale, so Fshould be a cons an , independen ly o he s ip wid h l, bu away om con o mali y Fwill
be a dimensionless combina ion o he scales in he heo y.
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JHEP01(2019)232
0 5 10 15 20 25 30
l q
λ
0.340
0.345
0.350
0.355
0.360
0.365
0.370
λ ℱ
N2
Figu e 7.Fmono onically in e pola es be ween cons an alues a he UV (l q/√λ→0) and
IR (l q/√λ→ ∞) con o mal ixed poin s. The e ical dashed line is a lq q/√λwhe e lqis he
wid h o he s ip whose bulk minimal su ace jus ouches he ca i y. The poin s a e om he
ull nume ical solu ion and he con inuous cu e is om ou analy ic esul s in he ew la o limi
(small ˆ). He e ˆ= 0.1.
in he IR (l q/√λ→ ∞). Thus, we expec he second de i a i e F00(l) o change sign a
some in e media e scale. In he ew la o limi (small ˆ= (3/4)(N /N)λ) we can show
explici ly ha
∂2F(l)
∂l2=




−ˆΓ(1
4)4
72√2π4
N2 2
q
λ3/2, l →l−
c i
ˆΓ(1
4)2
12π7/2
N2 2
q
λ3/2l
lc i −1−1/2, l →l+
c i .
(4.9)
No ice ha since we a e wo king a i s o de in ˆ,lc i in (4.9) can be in e p e ed as lq, i.e.,
as he wid h o he s ip whose bulk minimal su ace jus ouches he ca i y in he g a i y
solu ion. In o he wo ds, F00(l) changes sign a a scale co esponding o he mass o he
undamen al qua ks. No e also ha he second de i a i e is discon inuous ac oss he ca i y.
This is expec ed, as he Eins ein equa ion o ou backg ound solu ion has a discon inui y
in he sou ce a he loca ion o he ca i y whe e he ene gy-momen um enso is ab up ly
u ned on. One can e ine he smea ing p ocedu e [30] by using embedded D6-b anes wi h
a dis ibu ion o ip posi ions ha a e no del a- unc ion peaked a x∗. This would lead
o a con inuous opological ansi ion in he geome y a he scale qwi h smoo h second
de i a i es o he backg ound me ic and hus o F(l).
Casini e al. [31] p oposed an al e na i e me hod o compu ing a c- unc ion o h ee-
dimensional QFTs om he mu ual in o ma ion I(A+, A−) be ween he in e io o a smalle
ci cle (A−) and he ex e io o a la ge ci cle (A+) wi h an annulus o wid h δbe ween
hem. The ad an age o wo king wi h mu ual in o ma ion is ha his quan i y is UV- ini e,
and is hus well-de ined wi hou any egula iza ion unlike en anglemen en opy. Since we
a e wo king wi h s ips, no ci cula egions, we conside he mu ual in o ma ion be ween
– 22 –
JHEP01(2019)232
δ δl
AB BC C
bd y
bulk
Figu e 8. S ip con igu a ion we conside in he de ini ion o he c- unc ion. We ha e a s ip o
wid h lbo de ed by wo hin s ips o wid h δ. We conside he mu ual in o ma ion be ween he
cen e s ip ( ed) and he ex e io (o ange).
a s ip o wid h l( egion A) and he emainde o he space ( egion B) ou side a bounda y
egion o wid h δon ei he side o he s ip ( egion C) — see igu e 8. We will ake δ→0
in he end. We hen de ine ou candida e c- unc ion o be
c(l) = lim
δ→0
l2
Ly
∂I(l, δ)
∂l ,(4.10)
whe e he mu ual in o ma ion is gi en by I(A, B) = S(A) + S(B)−S(A∪B), whe e S(X)
is he en anglemen en opy o egion X. In ac , we can show ha
c(l)=2F(l).(4.11)
To his end, conside a QFT pa i ioned in o h ee egions — A, cha ac e ized by leng h
scale l; C, bo de ing Acha ac e ized by a size δl; and B, which is he es o he space.
We now w i e ou i s candida e c- unc ion (4.8) in e ms o he di e en ial ope a o
Ll= (l2/Ly)∂/∂l as
F(l) = LlS(A).(4.12)
Ou second candida e c- unc ion is
c(l) = lim
δ→0LlI(A, B).(4.13)
The equali y c= 2F hen ollows o pu e s a es:
c(l) = lim
δ→0LlS(A) + lim
δ→0LlS(B)−lim
δ→0LlS(A∪B) (4.14)
=LlS(A) + lim
δ→0LlS(B)
| {z }
=LlS(A)
−lim
δ→0LlS(C)
| {z }
=0
= 2F(l).(4.15)
He e we used he ac ha Cis he complemen o A∪Band so S(C) = S(A∪B) o
pu e s a es. As δ anishes, so does he “bulk” con ibu ion o S(C), so ha S(C) becomes
equal o i s UV-di e gen piece. The la e is p opo ional o he leng h o he bounda y
be ween Aand Band does no depend on l. Thus LlS(C) anishes in he δ→0 limi .
– 23 –
JHEP01(2019)232
Likewise, LlS(B)→LlS(A) as δ→0, since Bis he complemen o Ain his limi , and
S(A) = S(¯
A) o pu e s a es. No e ha , as discussed ea lie , he di e en ial ope a o Ll
emo es he di e gences in he s ip en anglemen en opy so ha hese a gumen s a e
well-de ined. We s a ed his p oo o s ips, bu a simila a gumen holds o compac A.
4.3 Flow o ex ensi i y
Conside h ee en angling egions A,B, and Cand hei ipa i e in o ma ion
I3(A, B, C) = I(A, B) + I(A, C)−I(A, B ∪C).(4.16)
Despi e appea ances, his quan i y is symme ic be ween A,B, and Cas can be e i ied
by exp essing he mu ual in o ma ions in e ms o en anglemen en opies. I3is a measu e
o ex ensi i y o mu ual in o ma ion in he sense ha I3= 0 implies ha he in o ma ion
ha Asha es wi h B∪Cis he sum o he in o ma ion sha ed wi h Band Csepa a ely. In
a gene al quan um ield heo y I3can ake ei he sign depending on he choice o s a e and
en angling egions [32], bu i was shown in [33] ha in holog aphic heo ies I3≤0. This
means ha he in o ma ion ha egion Asha es wi h Band Cis in gene al supe -ex ensi e
in a holog aphic heo y — he e is in o ma ion in in insically 3-pa y en anglemen ha
canno be unco e ed jus om he 2-pa y en anglemen . To in es iga e how he ex ensi i y
o mu ual in o ma ion a ies wi h scale we de ine he a io
e=I(A, B ∪C)
I(A, B) + I(A, C),(4.17)
so ha e= 1 in an ex ensi e heo y, e > 1 in a supe -ex ensi e heo y, and e < 1 in a
sub-ex ensi e heo y. The ex ensi i y eis only well de ined when he denomina o is non-
anishing, i.e., i he egion Asha es a leas some mu ual in o ma ion wi h a leas one o
Bo Csepa a ely.5As we ha e seen, in some en angling egion con igu a ions he mu ual
in o ma ion can be iden ically ze o. I ha happens o I(A, B) and I(A, C) while I(A, B ∪
C) emains posi i e hen edi e ges, ep esen ing maximal supe -ex ensi i y, and his can
ce ainly happen in quan um heo ies.6Below we will i s de e mine con igu a ions whe e e
is ini e o a CFT, and hen ask how he massi e ABJM heo y beha es in hese si ua ions.
Conside h ee s ips A,B, and Co equal wid h l ha a e sepa a ed by dis ance s
wi h Ain he middle. We can s udy how he ex ensi i y o en anglemen changes wi h scale
by a ying lwhile keeping s/l ixed ( igu e 9). In he absence o massi e qua ks he heo y
is con o mal, and so he ex ensi i y eis cons an unde his scaling a ia ion. Applying
he s anda d o mulas o he en anglemen en opy o in e als in a (2 + 1)-dimensional
CFT (see, e.g., [21] o he i s e m on he igh hand side o eq. (B.13)) one inds a simple
esul :
e(s/l) = 1
2·3(s/l)3+ 8(s/l)2+ (s/l)−6
2(s/l)3+ 5(s/l)2+ (s/l)−3, o a CFT .(4.18)
5No e ha he mu ual in o ma ion is a non-nega i e quan i y so ha he wo e ms in he denomina o
canno cancel each o he .
6See an in e es ing example in he con ex o mul i-bounda y wo mholes in [34].
– 24 –
JHEP01(2019)232
o cyclic inequali ies
n
X
i=1
S(Ai|Ai+1 . . . Ai+k)≥S(A1. . . An),(5.1)
whe e nis he numbe o egions and n= 2k+ 1. We ha e nume ically checked his amily
o inequali ies in massi e ABJM heo y o pa allel s ip con igu a ions up o 9 sepa a e
egions.
The esul s in his pape depended on ha ing a ully en-dimensional g a i a ional
dual o a s ongly coupled QFT. In ac , he ex emal su aces compu ing en anglemen
en opy explo ed he ull 10 dimensions. Many ecen applica ions o holog aphy o he
s udy o quan um in o ma ion ha e ea ed he g a i a ional dual as e ec i ely ha ing one
ex a dimension compa ed o he ield heo y. Bu a ully ealized duali y in s ing heo y
equi es 10 dimensions, and, as we ha e seen he e, he de ails o he ull geome y mani es
hemsel es in he s uc u e o he RG low. Indeed, as discussed in [15] he “in e nal”
and AdS dimensions can in e ac signi ican ly, and e en some imes comple ely exchange
oles [16]. I would be use ul o de elop mo e such examples. Finally, i would be in e es ing
o pu he Che n-Simons ield heo y s udied in his pape on a 3-sphe e and es he ecen
p oposal o he c- unc ion [38].
Acknowledgmen s
We would like o hank Ma DeC oss, A jun Ka , Esko Keski-Vakku i, Onka Pa ika ,
and G. S´a osi o help ul discussions. A. V. R. is unded by he Spanish g an s FPA2014-
52218-P and FPA2017-84436-P by Xun a de Galicia (GRC2013-024), by FEDER and by he
Ma ia de Maez u Uni o Excellence MDM-2016-0692. V. B. was suppo ed in pa by he
Simons Founda ion (# 385592, V. B.) h ough he I F om Qubi Simons Collabo a ion,
and he US Depa men o Ene gy g an FG02-05ER-41367. V. B. also acknowledges
he hospi ali y o he Aspen Cen e o Physics which is suppo ed by Na ional Science
Founda ion g an PHY-1607611. This wo k was ini ia ed du ing he “Holog aphic me hods
o s ongly coupled sys ems” wo kshop o he Galileo Galilei Ins i u e in Flo ence.
A De ails o he backg ound
He e we will gi e he de ailed exp essions o he di e en unc ions appea ing in ou
backg ound ( o a de i a ion see [9]). Inside he ca i y, i.e., o x≤1, hese unc ions a e
analy ic and depend on he cons an γcha ac e izing he unning o he mas e unc ion
W(x) in he sou celess egion. The unc ions ,g, and he dila on φa e gi en by:
e
q
=1 + √1+4γ
√2
x
√1+4γx(1 + √1+4γx)1/2
eg
q
=1 + √1+4γ
2
x
√1+4γx
keφ=1 + p1+4γx
1+4γx( 4
qh)1/4,(A.1)
– 31 –

JHEP01(2019)232
whe e qis he coo dina e o he ip o he b ane (see (2.12)). Fo x≥1 he mas e
equa ion (2.9) o W(x) mus be in eg a ed nume ically (al hough an analy ic solu ion in
powe s o ˆcan be ob ained, see below). In e ms o Wand η, he di e en unc ions a e:
e
q
=(1 + √1+4γ)2
2√1+4γ1/3 3x
W0+ 4ηW1/6exp 2
3Zx
1
η(ξ)dξ
W(ξ)
eg
q
=(1 + √1+4γ)2
2√1+4γ1/3x
W1/3exp 2
3Zx
1
η(ξ)dξ
W(ξ)
keφ=(1 + √1+4γ)2
2√1+4γ1/312x( 4
qh)1/4
W1/3(W0+ 4η)exp 2
3Zx
1
η(ξ)dξ
W(ξ).(A.2)
The cons an γappea ing in (A.1) and (A.2) is ob ained by he shoo ing echnique de-
sc ibed in he main ex . The dila on in hese wo equa ions has been w i en in e ms o
he wa p ac o h. Fo x≤1 he unc ion h(x) can be ound analy ically:
k
N 4
qh=π2
2(p1+4γ−1)41 + 1
4γx"α+ 24 log √4γx
√1+4γx + 1
+1
2+ 6γx +1 + (1 −6γx)√1+4γx
4γx √1+4γx + 1
γ2x2#,(A.3)
whe eas o x≥1 i can be w i en in e ms o W(x) as:
k
N 4
qh= 4π2 4
qe−gW0+ 4η"Z∞
x
ξe−3g(ξ)dξ
W(ξ)2#.(A.4)
The in eg a ion cons an α ha appea s in he wa ping his ixed by con inui y a x= 1:
limx→1−h(x) = limx→1+h(x). Howe e , he exp ession o αis leng hy, so we do no w i e
i he e explici ly.
A.1 Expansion in la o
The mas e equa ion, and he co esponding unc ions o he ansa z, can be sol ed analy -
ically in a powe se ies expansion in he la o de o ma ion pa ame e ˆ[18]. I we w i e:
W(x) = X
n=0
Wn(x)ˆn= 2x+W1(x)ˆ+W2(x)ˆ2+W3(x)ˆ3+O(ˆ4),(A.5)
hen we ge o he i s h ee unc ions in (A.5) in he egion x≤1:
W1(x) = 12
5x2
W2(x) = 8
875 (171 −70 x)x2
W3(x) = 16
197071875 235688 + 135135 35 x−76) x.(A.6)
– 32 –
JHEP01(2019)232
Mo eo e , o x≥1 we ha e:
W1(x) = 7x2−2
2x−1
10x3
W2(x) = 13125x8+ 3500x6−3962x4−3360x4log x+ 300x2−35
14000x7
W3(x) = −23
208000x11 +89
184800x9−302
39375x7+8293
98000x5−2120717
8400000x3+37
96x
−log(x)10703x4+ 20160x4log(x)−3600x2+ 840
70000x7−15x
32 .(A.7)
The cons an γob ained om he shoo ing me hod can also be expanded in powe s o ˆ,
wi h he esul :
γ=2 ˆ
5+228 ˆ2
875 +18855104 ˆ3
591215625 +O(ˆ4).(A.8)
The emaining unc ions o he ansa z can be ob ained by plugging hese expansions
in o (A.2) and (A.4). The exp essions ound in his way a e leng hy and will no be
ep oduced he e.
B UV expansion o he en anglemen en opy
In his sec ion we wo k ou he UV expansion o he s ip en anglemen en opy. The
holog aphic en anglemen en opy on a s ip can be compu ed om equa ion (3.5). The
UV expansion is gi en by
S(l) = S∞(l) + δS(l),(B.1)
whe e S∞(l) is he en anglemen en opy compu ed a he UV ixed poin and δS(l) is he
i s pe u ba ion a ound he UV. Explici ly, hese a e gi en by
S∞(l) = −4π2FUV(S3)
Γ1
44
1
l(B.2)
δS =V6Ly
2G(10)
NZ∞
x∗∂L
∂H UV
δH +∂L
∂H∗UV
δH∗+∂L
∂GUV
δGdx . (B.3)
The UV expansions o H(x) and G(x) a e
H(x) = H0x4/b 1 + H2
x2+O(x−4)(B.4)
G(x) = G0x−2−2/b 1 + G2
x2+O(x−4),(B.5)
whe e he coe icien s a e
H0=L8
0κ4 4
qq4
0e−4φ0
b12 , G0=L4
0
b2 2
qκ2(B.6)
H2= 2(h2+ 4 2+ 2g2−2φ2), G2=h2+ 2g2.(B.7)
– 33 –
JHEP01(2019)232
Now he UV pe u ba ion δS is gi en by
δS =V6Ly
2G(N)
NpG0H0Z∞
x∗
(G2+H2)x4/b + (H2x2−(G2+ 2H2)x2
∗)x4/b
∗
2x∗(x4/b −x4/b
∗)3/2x−3+3/bdx
=V6Ly
2G(N)
N
√G0H0x−2+1/b
∗
2Z∞
1
(G2+H2)z4/b +H2(z2−2) −G2
(z4/b −1)3/2x−3+3/bdz
=V6Ly
2G10
b√G0H0√π
16 x−2+1/b
∗ (2G2+ (1 + 2b)H2)Γ −1
4+b
2
Γ1
4+b
2−H2Γ−1
42
4√2π!.(B.8)
We mus e-exp ess he abo e esul in e ms o he s ip wid h l. The s ip wid h nea
he UV is
l= 2pH∗Z∞
x∗pG(x)
pH(x)−H∗
dx =2√G0
x1/b
∗
b√2π3/2
Γ1
44+δl , (B.9)
whe e he i s e m on he las exp ession is he UV wid h and δl is he i s co ec ion.
The co ec ion is
δl =pG0x−2+2/b
∗Z∞
x∗
x4/b(H2x2+ (G2−H2)x2
∗)−G2x2+4/b
∗
(x4/b −x4/b
∗)3/2x−3−1/bdx
=pG0x−2−1/b
∗Z∞
1
z4/b(G2+H2(z2−1)) −G2
(z4/b −1)3/2z−3−1/bdz
=b√G0√π
8x−2−1/b
∗ (2G2+ (1 + 2b)H2)Γ 3
4+b
2
Γ5
4+b
2−H2Γ−1
42
4√2π!.(B.10)
A his poin we know l(x∗). We s ill need o e e his ela ion o use in equa ion (B.8).
x∗as a unc ion o lis
x1/b
∗=2√2b√G0π3/2
Γ1
42
1
l+δx1/b
∗(B.11)
δx1/b
∗=b1−2bG1/2−b
0Γ1
44b
23+3bπ−1/2+3bl−1+2b (2G2+ (1 + 2b)H2)Γ 3
4+b
2
Γ5
2+b
2−H2Γ−1
42
4√2π!.
(B.12)
Using hese esul s we conclude ha he egula ed en anglemen en opy on a s ip includ-
ing he i s UV co ec ion is
S(l)
Ly
=4π2FUV(S3)
Γ1
44−1
l+β(b)l−1+2b+. . .,(B.13)
whe e
β(b) = (b−1)(−5+3b)Γ 1
42+4bΓ−1
4+b
2
27/2+4bb(5 −4b)bπ1+5b((2 −b)b/κ)2bΓ5
4+b
2.(B.14)
One applica ion o his UV expansion is o he de e mina ion o he egion in he
{ˆ, s/l}plane whe e he heo y shows g ea e supe -ex ensi i y han in a CFT. As desc ibed
– 34 –
JHEP01(2019)232
in sec ion 4.3, we need o compu e he de i a i e o he ex ensi i y pa ame e e(4.17) in
he UV limi o anishing s ip wid hs l, and in a scaling limi whe e s/l is held ixed.
Because bo h sand la e small in his limi , he holog aphic minimal su aces compu ing
en anglemen en opy a e all localized in he deep UV egion. The compu a ion o ∂le(l)|l=0
hen in ol es de i a i es o (4.17), in e ms o he exp ession o en anglemen en opy gi en
in (B.13). This exp ession is alid o all ˆ, i.e., we do no need o ake he ew la o limi .
Open Access. This a icle is dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License (CC-BY 4.0), which pe mi s any use, dis ibu ion and ep oduc ion in
any medium, p o ided he o iginal au ho (s) and sou ce a e c edi ed.
Re e ences
[1] O. Aha ony, O. Be gman, D.L. Ja e is and J. Maldacena, N= 6 supe con o mal
Che n-Simons-ma e heo ies, M2-b anes and hei g a i y duals,JHEP 10 (2008) 091
[a Xi :0806.1218] [INSPIRE].
[2] S. Hohenegge and I. Ki sch, A No e on he holog aphy o Che n-Simons ma e heo ies wi h
la ou ,JHEP 04 (2009) 129 [a Xi :0903.1730] [INSPIRE].
[3] D. Gaio o and D.L. Ja e is, No es on adding D6 b anes w apping Rp3in AdS4×CP3,
JHEP 11 (2012) 015 [a Xi :0903.2175] [INSPIRE].
[4] G. Veneziano, Some Aspec s o a Uni ied App oach o Gauge, Dual and G ibo Theo ies,
Nucl. Phys. B 117 (1976) 519 [INSPIRE].
[5] C. N´u˜nez, A. Pa edes and A.V. Ramallo, Unquenched Fla o in he Gauge/G a i y
Co espondence,Ad . High Ene gy Phys. 2010 (2010) 196714 [a Xi :1002.1088] [INSPIRE].
[6] E. Conde and A.V. Ramallo, On he g a i y dual o Che n-Simons-ma e heo ies wi h
unquenched la o ,JHEP 07 (2011) 099 [a Xi :1105.6045] [INSPIRE].
[7] N. Jokela, J. Mas, A.V. Ramallo and D. Zoakos, The modynamics o he b ane in
Che n-Simons ma e heo ies wi h la o ,JHEP 02 (2013) 144 [a Xi :1211.0630]
[INSPIRE].
[8] Y. Bea, N. Jokela, M. Lippe , A.V. Ramallo and D. Zoakos, Flux and Hall s a es in ABJM
wi h dynamical la o s,JHEP 03 (2015) 009 [a Xi :1411.3335] [INSPIRE].
[9] Y. Bea, E. Conde, N. Jokela and A.V. Ramallo, Unquenched massi e la o s and lows in
Che n-Simons ma e heo ies,JHEP 12 (2013) 033 [a Xi :1309.4453] [INSPIRE].
[10] Y. Bea, N. Jokela, A. P¨onni and A.V. Ramallo, Noncommu a i e massi e unquenched
ABJM,In . J. Mod. Phys. A 33 (2018) 1850078 [a Xi :1712.03285] [INSPIRE].
[11] K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Pa k, N= 5,6Supe con o mal Che n-Simons
Theo ies and M2-b anes on O bi olds,JHEP 09 (2008) 002 [a Xi :0806.4977] [INSPIRE].
[12] J. Gomis, D. Rod iguez-Gomez, M. Van Raamsdonk and H. Ve linde, A Massi e S udy o
M2-b ane P oposals,JHEP 09 (2008) 113 [a Xi :0807.1074] [INSPIRE].
[13] K.K. Kim, O.-K. Kwon, C. Pa k and H. Shin, Reno malized En anglemen En opy Flow in
Mass-de o med ABJM Theo y,Phys. Re . D 90 (2014) 046006 [a Xi :1404.1044] [INSPIRE].
– 35 –
JHEP01(2019)232
[14] K.K. Kim, O.-K. Kwon, C. Pa k and H. Shin, Holog aphic en anglemen en opy o
mass-de o med Aha ony-Be gman-Ja e is-Maldacena heo y,Phys. Re . D 90 (2014) 126003
[a Xi :1407.6511] [INSPIRE].
[15] P.A.R. Jones and M. Taylo , En anglemen en opy in op-down models,JHEP 08 (2016)
158 [a Xi :1602.04825] [INSPIRE].
[16] V. Balasub amanian, A. Law ence, A. Rolph and S. Ross, En anglemen shadows in LLM
geome ies,JHEP 11 (2017) 159 [a Xi :1704.03448] [INSPIRE].
[17] N. Jokela, A.V. Ramallo and D. Zoakos, Magne ic ca alysis in la o ed ABJM,JHEP 02
(2014) 021 [a Xi :1311.6265] [INSPIRE].
[18] Y. Bea, Holog aphic duali y and applica ions, Ph.D. Thesis, San iago de Compos ela
Uni e si y, San iago de Compos ela Spain (2016) [a Xi :1612.00247] [INSPIRE].
[19] A. Ka ch and E. Ka z, Adding la o o AdS/CFT,JHEP 06 (2002) 043 [hep- h/0205236]
[INSPIRE].
[20] J. E dmenge , N. E ans, I. Ki sch and E. Th el all, Mesons in Gauge/G a i y Duals — A
Re iew,Eu . Phys. J. A 35 (2008) 81 [a Xi :0711.4467] [INSPIRE].
[21] S. Ryu and T. Takayanagi, Holog aphic de i a ion o en anglemen en opy om AdS/CFT,
Phys. Re . Le . 96 (2006) 181602 [hep- h/0603001] [INSPIRE].
[22] I.R. Klebano , T. Nishioka, S.S. Pu u and B.R. Sa di, On Shape Dependence and RG Flow o
En anglemen En opy,JHEP 07 (2012) 001 [a Xi :1204.4160] [INSPIRE].
[23] M. Alishahiha, M.R. Mohammadi Moza a and M.R. Tanhayi, On he Time E olu ion o
Holog aphic n-pa i e In o ma ion,JHEP 09 (2015) 165 [a Xi :1406.7677] [INSPIRE].
[24] O. Ben-Ami, D. Ca mi and J. Sonnenschein, Holog aphic En anglemen En opy o Mul iple
S ips,JHEP 11 (2014) 144 [a Xi :1409.6305] [INSPIRE].
[25] H. Liu and M. Mezei, A Re inemen o en anglemen en opy and he numbe o deg ees o
eedom,JHEP 04 (2013) 162 [a Xi :1202.2070] [INSPIRE].
[26] H. Liu and M. Mezei, P obing eno maliza ion g oup lows using en anglemen en opy,
JHEP 01 (2014) 098 [a Xi :1309.6935] [INSPIRE].
[27] V. Balasub amanian, B.D. Chowdhu y, B. Czech, J. de Boe and M.P. Helle , Bulk cu es
om bounda y da a in holog aphy,Phys. Re . D 89 (2014) 086004 [a Xi :1310.4204]
[INSPIRE].
[28] M. Head ick, R.C. Mye s and J. Wien, Holog aphic Holes and Di e en ial En opy,JHEP
10 (2014) 149 [a Xi :1408.4770] [INSPIRE].
[29] P.A.R. Jones and M. Taylo , En anglemen en opy and di e en ial en opy o massi e
la o s,JHEP 08 (2015) 014 [a Xi :1505.07697] [INSPIRE].
[30] E. Conde, J. Gailla d and A.V. Ramallo, On he holog aphic dual o N= 1 SQCD wi h
massi e la o s,JHEP 10 (2011) 023 [E a um ibid. 1308 (2013) 082] [a Xi :1107.3803]
[INSPIRE].
[31] H. Casini, M. Hue a, R.C. Mye s and A. Yale, Mu ual in o ma ion and he F- heo em,
JHEP 10 (2015) 003 [a Xi :1506.06195] [INSPIRE].
[32] H. Casini and M. Hue a, Rema ks on he en anglemen en opy o disconnec ed egions,
JHEP 03 (2009) 048 [a Xi :0812.1773] [INSPIRE].
– 36 –

JHEP01(2019)232
[33] P. Hayden, M. Head ick and A. Maloney, Holog aphic Mu ual In o ma ion is Monogamous,
Phys. Re . D 87 (2013) 046003 [a Xi :1107.2940] [INSPIRE].
[34] V. Balasub amanian, P. Hayden, A. Maloney, D. Ma ol and S.F. Ross, Mul ibounda y
Wo mholes and Holog aphic En anglemen ,Class. Quan . G a . 31 (2014) 185015
[a Xi :1406.2663] [INSPIRE].
[35] M. Head ick and T. Takayanagi, A Holog aphic p oo o he s ong subaddi i i y o
en anglemen en opy,Phys. Re . D 76 (2007) 106013 [a Xi :0704.3719] [INSPIRE].
[36] N. Bao, S. Nezami, H. Oogu i, B. S oica, J. Sully and M. Wal e , The Holog aphic En opy
Cone,JHEP 09 (2015) 130 [a Xi :1505.07839] [INSPIRE].
[37] M. F eedman and M. Head ick, Bi h eads and holog aphic en anglemen ,Commun. Ma h.
Phys. 352 (2017) 407 [a Xi :1604.00354] [INSPIRE].
[38] J.K. Ghosh, E. Ki i sis, F. Ni i and L.T. Wi kowski, Holog aphic RG lows on cu ed
mani olds and he F- heo em,a Xi :1810.12318 [INSPIRE].
– 37 –