New inequality for Wilson loops from AdS/CFT

0.22 MB·English

by  Tomoyoshi Hirata

KUNS-2120
New inequality for Wilson loops from AdS/CFT
Tomoyoshi Hirata
E-mail address: [email protected]
8
0 Department of Physics, Kyoto University, Kyoto 606-8502,Japan
0
2

n
a Abstract
J

1 The strong subadditivity is the most important inequality which entanglement entropy satisfies. Based on
2
the AdS/CFT conjecture,entanglemententropyinCFT is equalto the areaofthe minimal surfaceinAdS

space. It is known that a Wilson loop can also be holographically computed from the minimal surface in
]
h AdS space. In this paper, we argue that Wilson loops also satisfy a similar inequality, and find several
t
- evidences of it.
p
e
h
[ 1 Introduction

2
v Based on the AdS/CFT correspondenceat the large t’Hooft coupling λ, the expectation value of a Wilson
3 loop C in D = 4, N = 4 Super Yang Mills theory is related to the area A of the minimal surface whose
6 boundary is the loop C [1]:
8 W(C) =exp( √λA). (λ 1) (1)
2 h i − ≫
1. Recently another object was found to have a connection with minimal surfaces in AdS space. That is
0 entanglement entropy. Based on the AdS/CFT correspondence, when λ 1, the entanglement entropy of
8 region A in CFT is calculated by replacing the horizon area in the Beke≫nstein-Hawking formula with the
0
area of the minimal surface in AdS space whose boundary is the same as that of the region A [2] [3],
:
v
i Area(γ )
X S = C (λ 1), (2)
h Ai 4G(d+2) ≫
r N
a
where G(d+2) is the Newton constant in d+2 dimensional AdS space.
N
Entanglement entropy always follows a characteristic relation known as the strong subadditivity [4]
S +S S +S . (3)
A B A∪B A∩B
≥
As Wilson loops and entanglement entropy are related in (1) and (2) through minimal surfaces, Wilson
loops should also obey the strong subadditivity at large λ,
W W W W . (4)
∂A ∂B ∂(A∪B) ∂(A∩B)
h ih i ≤ h ih i
Indeed in this paper we point out that the strong subadditivity of Wilson loops is satisfied if we assume
minimal surface condition (1). We also expect Wilson loops to obey the strong subadditivity in arbitrary
coupling regions and find severalevidences of it.
This inequality of Wilson loops includes many physical properties, for example, the convexity of quark
potentials, and the convexity of cusp renormalizationfunction.
1
In this paper, we describe a profound feature of the strong subadditivity of Wilson loops and study
whether or not the strong subadditivity of Wilson loops is satisfied in any coupling region. To do this, we
firstly checked the strong subadditivity in the strong coupling region assuming minimal surface conjecture
(1). Secondly, usign Bachas inequality we found that the strong subadditivity is satisfied with symmetric
Wilson loops with arbitrary coupling constants in any dimensional space. Thirdly, we found that the
inequality is satisfied with small-deformed Wilson loops in small coupling regions in any dimension.
These evidences cause us to conjecture that the strong subadditivity for Wilson loops is satisfied in
arbitraryWilson loops, an arbitrarycoupling constant, and an arbitrarydimension. In addition, they give
us a criterion of AdS/CFT conjecture (1).
2 The strong subadditivity in entanglement entropy and Wilson
loops
2.1 Entanglement entropy and its character
Consider a quantum mechanical system with many degrees of freedom like a field theory. If the system is
put at zerotemperature, then the totalquantum systemis describedby the groundstate Ψ . When there
| i
is no degeneracy of the ground state, the density matrix is that of the pure state
ρ = Ψ Ψ. (5)
tot
| ih |
ThevonNeumannentropyofthetotalsystemisclearlyzero: S = trρ logρ =0. Nextwedivide
tot tot tot
thetotalsystemintotwosubsystems,AandA¯. Inthefieldtheorycase,−wecandothisbydividingphysical
spaceintotworegionsanddefiningAasthefieldinoneregionandA¯asthefieldinthe otherregionNotice
that physically we do not do anything to the system and the cutting procedure is an imaginary process.
cAocrcroersdpionngdlyin,gthteo ttohtoasleHofilsbuebrtsyssptaemcescAananbde Aw¯.ritten as a direct product of two spaces Htot = HA ⊗HA¯
Now we define the reduced density matrix ρA by tracing out the Hilbert space HA¯
ρA =trA¯ρtot. (6)
The observer who is only accessible to the subsystem A feels as if the total system were described by the
reduced density matrix ρ . Because if O is an operator which acts non-trivially only on A, then its
A A
expectation value is
O =trO ρ =tr O ρ (7)
A A tot A A A
· ·
where the trace tr is taken only over the Hilbert space H .
A A
Then we define entanglement entropy of the subsystem A as the von Neumann entropy of the reduced
density matrix ρ
A
S = tr ρ logρ . (8)
A A A A
−
This entropy measures the amount of information lost by tracing out the subsystem A¯. One can define
entanglement entropy by choosing another total density matrix than (5). However, this choice is sufficient
for the purposes of this paper.
Entanglement entropy satisfies many inequalities: the most important one being the the strong subad-
ditivity
S +S S +S . (9)
A B A∪B A∩B
≥
This inequality is also satisfied by any general density matrix. The strong subadditivity is the strongest
inequality of the von-Neumann entropy. Indeed, it has mathematically been shown that the strong sub-
additivity in conjunction with severalother more obvious conditions (such as the invariance under unitary
transformations and the continuity with respect to the eigenvalues of ρ ) characterize the von-Neumann
tot
entropy [5].
2
According to AdS/CFT correspondence, any physical quantity of d+1 dimensional CFT theory can
be gained from the dual d + 2 dimensional anti de-Sitter space (AdS ). This is also the case with
d+2
entanglement entropy. In [2] and [3], it is clamed that entanglement entropy of d dimensional spacelike
submanifold A in d+1 dimensional CFT theory is given by the following formula:
Area(γ )
A
S(A)= . (10)
4G(d+2)
N
where Area(γ ) denotes the area of the surface γ , and G(d+2) is the Newton constant in the d + 2
A A N
dimensionalantide-Sitter space. The ddimensionalsurface γ is defined as the surfacewith minimal area
A
whose boundary coincides with the boundary of submanifold A.
The conjecture (10) is mathematically provedin two dimensionalCFT and in generaldimensionalcase
a good explanation is given in [6].
In addition, [7] shows a numerical evidence to prove that the holographic entanglement entropy (10)
follows the strong subadditivity, and [8] gives a mathematical proof of it.
2.2 The strong subadditivity of Wilson loops
From (1) and (2), the strong subadditivity of entanglement entropy (9) is translated into that of Wilson
loop
W W W W , (11)
∂A ∂B ∂(A∪B) ∂(A∩B)
h ih i ≤ h ih i
where W is a Wilson loop around the boundary of region A. To determine regions A and B for Wilson
∂A
loops, we only consider the case where Wilson loops are in spacelike two-dimensional flat plane.
Ifthelefthandsideandtherighthandsideof(4)arenotreal,thentheinequalitybecomesmeaningless.
However, when the system is invariant under charge conjugation A AT, the value of a Wilson loop
µ → − µ
in Euclidian space is real because when subjected to the charge conjugation we have
ePiHdxµigAµ ePiHdxµi(−gATµ) = ePiHdxµigAµ ∗. (12)
h i→h i h i
We also note that if Wilson loops follow only the area law and the perimeter law,
W =exp( K S(A) K L(∂A)), (13)
∂A 1 2
h i − −
where K and K are constants and S(A) is the area of A and L(A) is the length of the perimeter, the
1 2
expectation value of Wilson loops follows the equality,
W W = W W . (14)
∂A ∂B ∂(A∪B) ∂(A∩B)
h ih i h ih i
This is because
S(A)+S(B) = S(A B)+S(A B) (15)
∪ ∩
L(A)+L(B) = L(A B)+L(A B). (16)
∪ ∩
Therefore, the subadditivity comes from other factors than the area and the perimeter law factor. One
interesting example of the equality (14) is in the large N pure Yang-Mills lattice QCD . There, loop
2
equations are easily solved [9][10], and nonintersecting Wilson loops are calculated to
λa2 A/a2
W(C) = 1 (λ<1) (17)
(cid:18) − 2 (cid:19)
1 A/a2
W(C) = (λ>1), (18)
(cid:18)2λa2(cid:19)
where a is the lattice spacing. Therefore,Wilsonloops follow the pure arealaw both in weakcoupling and
in strong coupling regions. Hence, from the argument above, Wilson loops satisfy the equality (14).
In the next section, we will show three important applications of the strong subadditivity for Wilson
loops.
3
3 Application of the strong subadditivity for Wilson loops
3.1 Cusp anomalous dimensions
In this subsection, we consider the renormalizationof Wilson loops in four-dimensional Yang-Mills theory.
When a Wilson loop has cusps whose angles are θ respectively, the renormalization of Wilson loops is
k
multiplicatively renormalizable [11][12],such that:
k
Wren(M,C) =Z (M,C) Z (M,θ ) Wnon ren(C) , (19)
per cusp k
h i h i
Y
where M is a renormalization scale, and Z is renormalization constant which comes from perimeter,
per
Z (M,θ ) is an additional renormalization constant which comes from cusps, and Wren is finite when
cusp k
expressed via the renormalized charge.
By solving the Callan-Symanzik equation for Wilson loop [13] we have
Z (M,γ)=g (M)−Γcusp(γ)/Cβ, (20)
cusp R
where Γ (γ) is an anomalous dimension of Z (γ) and called a ”cusp anomalous dimension”, g (M)
cusp cusp R
is the renormalized coupling constant, and C is the coefficient of the β function:
β
11 2
C = N N . (21)
β c f
3 − 3
Now we apply the strong subadditivity of Wnon ren(C) to two Wilson loops whose cusp angles are
h i
a+b and b+c respectively (Fig.1).
a b
b
c
c
Figure 1: Two Wilson loops with cusp. One’s Figure 2: When all crossing points of two loops
angle is a+b and the other’s is b+c. are like this, the strong subadditivity gives a
stronger condition than (22).
Up to the secondorder perturbation Γ <0. So when M 1 and g (M) 1, Z−1 (M,γ) is much
cusp ≫ R ≪ cusp
greaterthan1. Z shouldcanceleachotheronbothsidesoftheinequality,asdivergenceofentanglement
per
entropy derived from perimeters cancel each other out. Then we have
Γ (a+b+c)+Γ (b) Γ (a+b)+Γ (b+c) (22)
cusp cusp cusp cusp
≤
This leads to the convexity
d2
Γ (θ) 0. (23)
d2θ cusp ≤
Up to the second order perturbation [13] we have
N2 1
Γ(2) (θ)=4 c − (θcotθ 1) (<0) (24)
cusp 2N −
c
4
for SU(N ) gauge theory. Then we can directly check the convexity of Γ(2) (θ) as
c cusp
d2 N2 1 2
Γ(2) (θ)=4 c − (θcotθ 1) <0. (25)
d2θ cusp 2Nc − sin2(θ)
Conversely,(22)derivesthestrongsubadditivitywhentwoloopshaveacrossingpointlikeFig.1,because
Z are main divergence parts of loops with cusps. (22) saturates when a =0 or b=0. Therefore when
cusp
all crossing points of two loops are like Fig.2 (i.e. when two loops don’t cross but just touch),1 the strong
subadditivity gives a nontrivial condition except (22).2
3.2 Quark potential
We considerrectangularWilsonloops shownin Fig.3 whereshortsides ofthe rectangleare a+bandb+c,
and long sides of them are all T.
a b c
T
Figure 3: Rectangular Wilson loops
The value of Wilson loops W(R,T) has a physical meaning as a quark potential V(R):
V(R)= lim lnW(R,T). (26)
−T→∞
The strong subadditivity of these Wilson loops is
V(a+b+c)+V(b) V(a+b)+V(b+c) (27)
≤
This is equivalent with the convexity of quark potential
d2
V(R) 0. (28)
d2R ≤
Asshownin[14][15],onecanalsoderivetheconvexityofcuspanomalousdimensionsandtheconvexity
of quark potential from Bachas inequality, which we will consider in the next section.
3.3 Inequality of F inserted Wilson loop
µν
LetusconsiderthreeloopsC,C+δC (x)andC+δC (y),whereδC (x)andδC (y)areinfinitesimalloops
1 2 1 2
attached to a loop C at points x and y (x=y) respectively and are located outside of C (Fig.4).
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Now we can see
Int(C+δC ) Int(C+δC ) = Int(C+δC +δC ) (29)
1 2 1 2
∪
Int(C+δC ) Int(C+δC ) = Int(C), (30)
1 2
∩
1Agoodexampleisshowninthenextsubsection(Fig.3).
2AsimilarsituationalsooccursinthecaseofBachasinequality[14]
5
x y
C C+δ C1 (x) C+δ C2 (y)
Figure 4: Wilson loop C and its small deformed Wilson loops, C+δC (x) and C+δC (y).
1 2
where Int(C) means the interior of C. The strong subadditivity for C+δC and C+δC is
1 2
W(C+δC ) W(C+δC ) W(C+δC +δC ) W(C) . (31)
1 2 1 2
h ih i ≤ h ih i
Using area derivative, we can expand W(C+δC) as
2
δW(C) 1 δW(C)
W(C+δC)=W(C)+δσµν + δσµν +O((δσµν)3), (32)
δσµν(x) 2(cid:18) δσµν(x)(cid:19)
where δσµν denotes the area enclosed by δCµν.
The area derivative is given by inserting the field strength iF into Wilson loop:
µν
δ
δσµν(x)W(C)=trP(iFµν(x)eiHcdξαAα), (33)
therefore the inequality (31) can be rewritten as
0 W(C) trP((F δσ)x(F δσ)yeiHcdξµAµ) , (34)
≥h ih · · i
where (F δσ) denotes F (x)δσµν.
x µν
·
This inequality is fundamental for the strong subadditivity. Indeed all inequalities of the strong sub-
additivity of small-deformed Wilson loops are derived from (34). Firstly let us consider three loops C,
C+ δC (x ) and C+ δC (y ), where δC (x ) and δC (y ) are infinitesimal loops attaced to a loop C
i i i j j j j i i
atpoPintsxi andyj (xi anPdyj arealldifferentpoints)respectivelyandarelocatedoutsideofC. Thereason
why we don’t have to consider a case where δC (x ) or δC (y ) are inside C or a case where x and y are
i i j j i j
not all different points is@that in those cases by redefining C as (C+ δC (x )) ( C+ δC (y )) we
i i i ∩ j j
can regain the original situation. P P
Now the strong subadditivity is
W(C+ δC (x )) W(C + δC (y )) W(C) W(C + δC (x )+ δC (y )) . (35)
i i j j i i j j
h ih i≤h ih i
Xi X Xi X
Rewriting it by the operator form, (35) is
0≥ hW(C)ihtrP((F ·δσ)xi(F ·δσ)yjeiHcdξµAµ)i. (36)
Xi,j
We can obtain this inequality from the former inequality (34). Other kind of small deformed Wilson loops
are obtained by summing infinite number of δC . Therefore, the inequality (34) will be the most essential
i
inequality for the strong subadditivity of Wilson loops. In the next section, we will give a perturbative
proof of the strong subadditivity for general small-deformedWilson loops, which gives a proof of (34) as a
special case.
6
4 Verification of the strong subadditivity of Wilson loops
In this section, we verify the strong subadditivity of Wilson loops in three ways.
Firstly, we assume AdS/CFT conjecture (1) and from the nature of the minimal surface we prove the
inequality at λ 1. Secondly, using Bachas inequality, which specially-shaped Wilson loops satisfy, we
≫
prove the strong subadditivity of specially-shaped Wilson loops in all coupling regions. Thirdly, we give a
perturbative proof of the strong subadditivity for all small-deformed Wilson loops.
4.1 Verification from the minimal surface conjecture
Assuming AdS/CFT conjecture (10) in the strong coupling region, it is possible to prove the strong sub-
additivity (4). We use the same logic as the proof for the strong subadditivity of entanglement entropy
shown in [8].
Letm(A)andm(B)be the minimalsurfaceofregionAandB respectivelywhereA andB areinteriors
ofWilson loops. m(A) is divided by m(B). Letm(A) be outside piece of m(A) with respectto m(B) and
o
let m(A) be inside piece of m(A) with respect to m(B). We also define m(B) , m(B) in the same way
i o i
(Fig.5).
m(B)
m(A) o
o
m(A) m(B)
i i
A
B
Figure 5: region A, B and their minimal surfaces
Since m(A) m(B) is a surface whose boundary is A B, its area is bigger than or equalto the area
o o
∪ ∪
of the minimal surface m(A B) whose boundary is A B:
∪ ∪
m(A) +m(B) m(A B). (37)
o o
≥ ∪
And more since m(A) m(B) is a surface whose boundary is A B, its area is bigger than or equal to
i i
∪ ∩
the area of the minimal surface m(A B) whose boundary is A B:
∩ ∩
m(A) +m(B) m(A B) (38)
i i
≥ ∩
Therefore, we have an inequality
m(A)+m(B)=m(A )+m(A )+m(B )+m(B ) m(A B)+m(A B). (39)
o i o i
≥ ∪ ∩
The strong subadditivity can be derived from this.
Onenoteshouldbe mentionedforthis subsection. Here,we haveconsideredthe casewheretwoWilson
loops are on a same flat plane. However, when two loops are on a same curved non-intersecting surface,
we can also prove the strong subadditivity in the same way.
4.2 Verification from Bachas inequality
4.2.1 Review of Bachas inequality
Here we review Bachas inequality. We define θ as Parity transformation along x1 axis and region L L
0 +
L as
−
L xµ;x1 =0 (40)
0
≡ { }
7
L xµ;x1 >0 (41)
+
≡ { }
L xµ;x1 <0 . (42)
−
≡ { }
Nowlet C be openlines whichexistin L andlettheir boundariesinL . Nowwe define a functionf as
i + 0 ab
M
f = kiW(C ) , (43)
ab i ab
Xi
where W(C ) is a Wilson line operator of C , k is an arbitrary real number, and a b are gauge indices.
i ab i i
Then we have
tr fθf† = Z−1tr dU e−S0 dU f(U(b))e−S+ dU f(U(θb))†e−S−
b b b
h i Z Z Z
bY∈L0 b∈YL+ b∈YL−
2
= Z−1 dU e−S0(cid:12) dU f(U(b))e−S+(cid:12) 0 (44)
Z b (cid:12)Z b (cid:12) ≥
bY∈L0 (cid:12)(cid:12) b∈YL+ (cid:12)(cid:12)
(cid:12) (cid:12)
where S , S and S are respectively action(cid:12)s in L , L and L . Sub(cid:12)stituting (43) we have
0 + − 0 + −
tr fθf† = ki W(C ) kj, (45)
ij
h i h i
Xij
where W(C ) is the Wilson loop made by C and mirror image of C
ij i j
W(C ) =tr W(C )W( θC ) . (46)
ij i j
h i h − i
Therefore the inequality (44) means the quadratic form (46) is positive definite i.e. the determinant of the
matrix W(C ) is positive.
ij
h i
det W(C ) 0 (47)
ij
ij h i≥
This is Bachas inequality which was extended by Pobyltsa [14]. When M =2 (47) derives the original
inequality presented by Bachas [15],
W(C ) W(C ) W(C ) W(C ) (48)
11 22 12 21
h ih i≥h ih i
4.2.2 Bachas inequality and the strong subadditivity: first example
Now wewill seesome Bachasinequalities areequivalentto orarederivedfromthe strongsubadditivity for
some symmetric Wilson loops but in all coupling region.
Consider two open lines C and C which touch an axis X. Let
1 2
Int(C +X) Int(C +X), (49)
1 2
⊃
where Int(C +X) is the interior of C +X. We consider the case where each C ’s two end points A and
i i 1
B are at the same place as C ’s end points (Fig.6).
2
As can be seen from Fig.7,
D D =D , D D =D
12 21 11 12 21 22
∪ ∩
are satisfied where D is the interior of C .
ij ij
So now the original Bachas inequality
W(C ) W(C ) W(C ) W(C ) , (50)
11 22 12 21
h ih i≥h ih i
is equivalent to the strong subadditivity
W(∂(D D )) W(∂(D D )) W(∂D ) W(∂D ) . (51)
12 21 12 21 12 21
h ∩ ih ∪ i≥h ih i
This example includes examples shown in section 3.1 and section 3.2 [14][15].
8
X X
A A
C1 D12 D21
C2
B
B
Figure6: TwoopenlinesC andC andaxisX. Figure 7: Configuration of D and D .
1 2 12 21
The interior of C +X is outside of the interior
1
of C +X.
2
4.2.3 Bachas Inequality and the strong subadditivity: second example
Letus introducethree openlines C ,C andC whichtouchesX-axis. Nowwe imposefollowingconditions
1 2 3
to these lines. Firstly the interior of C +X is inside or outside of the interior of C +X, C +X:
1 2 3
Int(C +X) Int(C +X), Int(C +X) (52)
1 2 3
⊃
or
Int(C +X) Int(C +X), Int(C +X). (53)
1 2 3
⊂
Secondly C and C are symmetric with respect to the Y-axis which is perpendicular to X axis and C
2 3 1
is axisymmetric with respect to the Y-axis. Thirdly C , C and C share their two end points A and B
1 2 3
(Fig.8).3
X X
A C A
2
C1
C2 Y C1 Y
C
3
B C B
3
Figure 8: Three open lines: C ,C and C . The interior of C +X is inside or outside of the interior of
1 2 3 1
C +X, C +X. C and C are symmetric with respect to the Y-axis, C is axisymmetric with respect to
2 3 2 3 1
the Y-axis.
From their symmetry, we have
W(C ) = W(C )=W(C )=W(C ) (54)
12 21 13 31
3OriginallytheBachasequation forthisconfigurationwasconsideredin[14]
9
W(C ) = W(C ). (55)
22 33
Now let us define r,t,x,p as
r = W(C ) , x= W(C ) = W(C )
11 12 13
h i h i h i
t= W(C ) = W(C ) , p= W(C ) (56)
22 33 23
h i h i h i
Then Bachas inequalities (47) for these three paths lead to
det W(C ) = t=t 0 (57)
ij
i,j=2h i ≥
t p
det W(C ) = det =t2 p2 0 (58)
i,j=2,3h ij i (cid:18) p t (cid:19) − ≥
r x x
det W(Cij) = det x t p =(t p)[r(t+p) 2x2] 0 (59)
i,j=1,2,3h i − − ≥
x p t
 
(57)(58) leads to t p. Therefore if t=p, (59) results in
≥ 6
r(t+p) 2x2. (60)
≥
This inequality also holds if t=p, because in this case (60) is equivalent to another Bachas inequality
r x
det W(C ) =det =rt x2 0. (61)
i,j=1,2h ij i (cid:18) x t (cid:19) − ≥
On the other hand, one can also derive (60) by the strong subadditivity. If the interior of C +X is
1
inside of the interior of C +X, we obtain
2
D D =D , D D =D (62)
12 21 11 12 21 22
∩ ∪
D D =D , D D =D (63)
13 21 11 13 21 23
∩ ∪
(See Fig.9). If the interior of C +X is outside of the interior of C +X, we obtain
1 2
X
X
1 1
1 1
2 2 2
Y Y
3
Figure9: ThesetwofiguresshowtheconfigurationofC ,C ,C andtheirmirrorimageswhentheinterior
1 2 3
of C +X is inside of the interior of C +X and C +X. The figure on the left shows (62), while that on
1 2 3
the right shows (63).
D D =D , D D =D (64)
12 21 11 12 21 22
∪ ∩
D D =D , D D =D (65)
13 21 11 13 21 23
∪ ∩
(see Fig.10) .
10