Introduction to the AdS/CFT correspondence

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Introduction to the AdS/CFT
correspondence
Dmitri Bykov
Steklov Mathematical Institute, Moscow
Samara, 27 August 2012

2/22
The main concept
Quantum field theory in R D can be described
by gravity in AdS D 1 Maldacena, 1997
Especially strong quantitative evidence in the
conformal case, however we will also describe
the non-conformal setup
The extra dimension has the interpretation of
RG-scale in QFT

2/22
The main concept
Quantum field theory in R D can be described
by gravity in AdS D 1 Maldacena, 1997
Especially strong quantitative evidence in the
conformal case, however we will also describe
the non-conformal setup
The extra dimension has the interpretation of
RG-scale in QFT

2/22
The main concept
Quantum field theory in R D can be described
by gravity in AdS D 1 Maldacena, 1997
Especially strong quantitative evidence in the
conformal case, however we will also describe
the non-conformal setup
The extra dimension has the interpretation of
RG-scale in QFT

2/22
The main concept
Quantum field theory in R D can be described
by gravity in AdS D 1 Maldacena, 1997
Especially strong quantitative evidence in the
conformal case, however we will also describe
the non-conformal setup
The extra dimension has the interpretation of
RG-scale in QFT

3/22
The AdS space
A non-compact manifold with constant
negative curvature and Lorentzian signature
Can be viewed as a quotient space (coset)
AdS D 1 SOp2,Dq
SOp1,Dq
For the moment let us set D 2 and
consider AdS 3
The simplest model: a hyperboloid
¡Y 2 ¡1 ¡ Y 2
0 Y 2
1 Y 2
2 ¡R2 inside R 2,2 :
ds 2 ¡dY¡1 2 ¡ dY 0 2 dY 1 2 dY 2
2

3/22
The AdS space
A non-compact manifold with constant
negative curvature and Lorentzian signature
Can be viewed as a quotient space (coset)
AdS D 1 SOp2,Dq
SOp1,Dq
For the moment let us set D 2 and
consider AdS 3
The simplest model: a hyperboloid
¡Y 2 ¡1 ¡ Y 2
0 Y 2
1 Y 2
2 ¡R2 inside R 2,2 :
ds 2 ¡dY¡1 2 ¡ dY 0 2 dY 1 2 dY 2
2

3/22
The AdS space
A non-compact manifold with constant
negative curvature and Lorentzian signature
Can be viewed as a quotient space (coset)
AdS D 1 SOp2,Dq
SOp1,Dq
For the moment let us set D 2 and
consider AdS 3
The simplest model: a hyperboloid
¡Y 2 ¡1 ¡ Y 2
0 Y 2
1 Y 2
2 ¡R2 inside R 2,2 :
ds 2 ¡dY¡1 2 ¡ dY 0 2 dY 1 2 dY 2
2

3/22
The AdS space
A non-compact manifold with constant
negative curvature and Lorentzian signature
Can be viewed as a quotient space (coset)
AdS D 1 SOp2,Dq
SOp1,Dq
For the moment let us set D 2 and
consider AdS 3
The simplest model: a hyperboloid
¡Y 2 ¡1 ¡ Y 2
0 Y 2
1 Y 2
2 ¡R2 inside R 2,2 :
ds 2 ¡dY¡1 2 ¡ dY 0 2 dY 1 2 dY 2
2

3/22
The AdS space
A non-compact manifold with constant
negative curvature and Lorentzian signature
Can be viewed as a quotient space (coset)
AdS D 1 SOp2,Dq
SOp1,Dq
For the moment let us set D 2 and
consider AdS 3
The simplest model: a hyperboloid
¡Y 2 ¡1 ¡ Y 2
0 Y 2
1 Y 2
2 ¡R2 inside R 2,2 :
ds 2 ¡dY¡1 2 ¡ dY 0 2 dY 1 2 dY 2
2

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The AdS space. 2.
The global coordinates
Y ¡1 cosh prq cos ptq, Y 0 cosh prq sin ptq,
Y 1 sinh prq cos pφq, Y 2 sinh prq sin pφq
ds 2 ¡ cosh prq 2 dt 2 dr 2 sinh prq 2 dφ 2 ,
r P r0, 8q, t P p¡8, 8q, φ P r0, 2πq
A conformal rescaling ds 2 cosh prq 2 d˜s 2
and change of coordinates
w 2 arctan pe r q ¡ π produces a metric
2
d˜s 2 ¡dt 2 dw 2 sin pwq 2 dφ 2 where
w P r0, π 2 s ñ Disc

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The AdS space. 2.
The global coordinates
Y ¡1 cosh prq cos ptq, Y 0 cosh prq sin ptq,
Y 1 sinh prq cos pφq, Y 2 sinh prq sin pφq
ds 2 ¡ cosh prq 2 dt 2 dr 2 sinh prq 2 dφ 2 ,
r P r0, 8q, t P p¡8, 8q, φ P r0, 2πq
A conformal rescaling ds 2 cosh prq 2 d˜s 2
and change of coordinates
w 2 arctan pe r q ¡ π produces a metric
2
d˜s 2 ¡dt 2 dw 2 sin pwq 2 dφ 2 where
w P r0, π 2 s ñ Disc

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The AdS space. 2.
The global coordinates
Y ¡1 cosh prq cos ptq, Y 0 cosh prq sin ptq,
Y 1 sinh prq cos pφq, Y 2 sinh prq sin pφq
ds 2 ¡ cosh prq 2 dt 2 dr 2 sinh prq 2 dφ 2 ,
r P r0, 8q, t P p¡8, 8q, φ P r0, 2πq
A conformal rescaling ds 2 cosh prq 2 d˜s 2
and change of coordinates
w 2 arctan pe r q ¡ π produces a metric
2
d˜s 2 ¡dt 2 dw 2 sin pwq 2 dφ 2 where
w P r0, π 2 s ñ Disc

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The AdS space. 3.
Hence we arrive at the Penrose diagram of
AdS — a cylinder:

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The AdS space. 3.
Hence we arrive at the Penrose diagram of
AdS — a cylinder:

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The AdS space. 4.
AdS has a boundary, i.e. a hypersurface
reflecting light rays: r 8
Massive geodesics are confined inside AdS

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The AdS space. 4.
AdS has a boundary, i.e. a hypersurface
reflecting light rays: r 8
Massive geodesics are confined inside AdS

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The AdS space. 4.
AdS has a boundary, i.e. a hypersurface
reflecting light rays: r 8
Massive geodesics are confined inside AdS

7/22
Partition functions
The correspondence can be made more precise
Gubser, Klebanov, Polyakov; Witten, 1998
To every field ϕ in the bulk of AdS there
corresponds a local operator Opxq on the
boundary
For example, g µν ô T µν , A µ ô J µ
Moreover, xe i ³ d 4 x φ 0 pxq Opxq y Z bulk pφ 0 q
— the partition function of the bulk theory
with the condition φ Ñ φ 0 at the boundary
i
»
d 4 x A µ B pxq J µ pxq is gauge-invariant!

7/22
Partition functions
The correspondence can be made more precise
Gubser, Klebanov, Polyakov; Witten, 1998
To every field ϕ in the bulk of AdS there
corresponds a local operator Opxq on the
boundary
For example, g µν ô T µν , A µ ô J µ
Moreover, xe i ³ d 4 x φ 0 pxq Opxq y Z bulk pφ 0 q
— the partition function of the bulk theory
with the condition φ Ñ φ 0 at the boundary
i
»
d 4 x A µ B pxq J µ pxq is gauge-invariant!

7/22
Partition functions
The correspondence can be made more precise
Gubser, Klebanov, Polyakov; Witten, 1998
To every field ϕ in the bulk of AdS there
corresponds a local operator Opxq on the
boundary
For example, g µν ô T µν , A µ ô J µ
Moreover, xe i ³ d 4 x φ 0 pxq Opxq y Z bulk pφ 0 q
— the partition function of the bulk theory
with the condition φ Ñ φ 0 at the boundary
i
»
d 4 x A µ B pxq J µ pxq is gauge-invariant!

7/22
Partition functions
The correspondence can be made more precise
Gubser, Klebanov, Polyakov; Witten, 1998
To every field ϕ in the bulk of AdS there
corresponds a local operator Opxq on the
boundary
For example, g µν ô T µν , A µ ô J µ
Moreover, xe i ³ d 4 x φ 0 pxq Opxq y Z bulk pφ 0 q
— the partition function of the bulk theory
with the condition φ Ñ φ 0 at the boundary
i
»
d 4 x A µ B pxq J µ pxq is gauge-invariant!

7/22
Partition functions
The correspondence can be made more precise
Gubser, Klebanov, Polyakov; Witten, 1998
To every field ϕ in the bulk of AdS there
corresponds a local operator Opxq on the
boundary
For example, g µν ô T µν , A µ ô J µ
Moreover, xe i ³ d 4 x φ 0 pxq Opxq y Z bulk pφ 0 q
— the partition function of the bulk theory
with the condition φ Ñ φ 0 at the boundary
i
»
d 4 x A µ B pxq J µ pxq is gauge-invariant!

7/22
Partition functions
The correspondence can be made more precise
Gubser, Klebanov, Polyakov; Witten, 1998
To every field ϕ in the bulk of AdS there
corresponds a local operator Opxq on the
boundary
For example, g µν ô T µν , A µ ô J µ
Moreover, xe i ³ d 4 x φ 0 pxq Opxq y Z bulk pφ 0 q
— the partition function of the bulk theory
with the condition φ Ñ φ 0 at the boundary
i
»
d 4 x A µ B pxq J µ pxq is gauge-invariant!

Partition functions. 2.
Imagine a scalar field ϕ in the bulk of AdS
with the action
S
»
d 5 y ? ¡g p∇ϕq 2
m 2 ϕ 2¨ obeying
a wave equation ¡△ϕ m 2 ϕ 0, where
△ ? 1
¡g
B µ p ? ¡g g µν B ν ¤q is the Laplacian.
We want to solve it with a prescribed
boundary value ϕ| boundary
ϕ 0
8/22

Partition functions. 2.
Imagine a scalar field ϕ in the bulk of AdS
with the action
S
»
d 5 y ? ¡g p∇ϕq 2
m 2 ϕ 2¨ obeying
a wave equation ¡△ϕ m 2 ϕ 0, where
△ ? 1
¡g
B µ p ? ¡g g µν B ν ¤q is the Laplacian.
We want to solve it with a prescribed
boundary value ϕ| boundary
ϕ 0
8/22

Partition functions. 2.
Imagine a scalar field ϕ in the bulk of AdS
with the action
S
»
d 5 y ? ¡g p∇ϕq 2
m 2 ϕ 2¨ obeying
a wave equation ¡△ϕ m 2 ϕ 0, where
△ ? 1
¡g
B µ p ? ¡g g µν B ν ¤q is the Laplacian.
We want to solve it with a prescribed
boundary value ϕ| boundary
ϕ 0
8/22

9/22
Partition functions. 3.
Use Poincare coordinates:
ds 2 ¡dt2 d⃗x 2 d¡1 dz2
z 2
. The boundary
is at z 0. The equation takes the form
(after the ansatz ϕ e i k¤x f pzq)
f 11 1 ¡ d
z f 1 ¡ pk 2 pmRq 2
q f 0
z 2
What kind of boundary conditions can one
impose?

9/22
Partition functions. 3.
Use Poincare coordinates:
ds 2 ¡dt2 d⃗x 2 d¡1 dz2
z 2
. The boundary
is at z 0. The equation takes the form
(after the ansatz ϕ e i k¤x f pzq)
f 11 1 ¡ d
z f 1 ¡ pk 2 pmRq 2
q f 0
z 2
What kind of boundary conditions can one
impose?

9/22
Partition functions. 3.
Use Poincare coordinates:
ds 2 ¡dt2 d⃗x 2 d¡1 dz2
z 2
. The boundary
is at z 0. The equation takes the form
(after the ansatz ϕ e i k¤x f pzq)
f 11 1 ¡ d
z f 1 ¡ pk 2 pmRq 2
q f 0
z 2
What kind of boundary conditions can one
impose?

10/22
Partition functions. 4.
In the limit z Ñ 0 the equation becomes
homogeneous in z:
z Ñ 0 : f 11 1 ¡ d
z
f 1 ¡ pmRq2 f 0.
z 2
Therefore the solution is a power function
f z ∆
Plugging the ansatz into the equation and
solving for ∆, one obtains
dd 2
Ƭ d 2
¨
4
pmRq 2 .

10/22
Partition functions. 4.
In the limit z Ñ 0 the equation becomes
homogeneous in z:
z Ñ 0 : f 11 1 ¡ d
z
f 1 ¡ pmRq2 f 0.
z 2
Therefore the solution is a power function
f z ∆
Plugging the ansatz into the equation and
solving for ∆, one obtains
dd 2
Ƭ d 2
¨
4
pmRq 2 .

10/22
Partition functions. 4.
In the limit z Ñ 0 the equation becomes
homogeneous in z:
z Ñ 0 : f 11 1 ¡ d
z
f 1 ¡ pmRq2 f 0.
z 2
Therefore the solution is a power function
f z ∆
Plugging the ansatz into the equation and
solving for ∆, one obtains
dd 2
Ƭ d 2
¨
4
pmRq 2 .

Partition functions. 5.
Therefore we impose the condition
ϕpx, zq| zÑ0
Ñ z ∆¡ ϕ 0 pxq
To solve the equation with this b.c. one needs
to find a function Kp⃗x, ⃗y, zq — the
boundary-to-boundary propagator — with the
property Kp⃗x, ⃗y, zq| zÑ0
Ñ z ∆¡ δp⃗x ¡ ⃗yq .
Then the desired solution may be written as
ϕp⃗x, zq
»
d 4 y Kp⃗x, ⃗y, zq ϕ 0 pyq
Check that this function is
z ¨∆
K
z 2 p⃗x ¡ ⃗yq 2
11/22

Partition functions. 5.
Therefore we impose the condition
ϕpx, zq| zÑ0
Ñ z ∆¡ ϕ 0 pxq
To solve the equation with this b.c. one needs
to find a function Kp⃗x, ⃗y, zq — the
boundary-to-boundary propagator — with the
property Kp⃗x, ⃗y, zq| zÑ0
Ñ z ∆¡ δp⃗x ¡ ⃗yq .
Then the desired solution may be written as
ϕp⃗x, zq
»
d 4 y Kp⃗x, ⃗y, zq ϕ 0 pyq
Check that this function is
z ¨∆
K
z 2 p⃗x ¡ ⃗yq 2
11/22

Partition functions. 5.
Therefore we impose the condition
ϕpx, zq| zÑ0
Ñ z ∆¡ ϕ 0 pxq
To solve the equation with this b.c. one needs
to find a function Kp⃗x, ⃗y, zq — the
boundary-to-boundary propagator — with the
property Kp⃗x, ⃗y, zq| zÑ0
Ñ z ∆¡ δp⃗x ¡ ⃗yq .
Then the desired solution may be written as
ϕp⃗x, zq
»
d 4 y Kp⃗x, ⃗y, zq ϕ 0 pyq
Check that this function is
z ¨∆
K
z 2 p⃗x ¡ ⃗yq 2
11/22

Partition functions. 5.
Therefore we impose the condition
ϕpx, zq| zÑ0
Ñ z ∆¡ ϕ 0 pxq
To solve the equation with this b.c. one needs
to find a function Kp⃗x, ⃗y, zq — the
boundary-to-boundary propagator — with the
property Kp⃗x, ⃗y, zq| zÑ0
Ñ z ∆¡ δp⃗x ¡ ⃗yq .
Then the desired solution may be written as
ϕp⃗x, zq
»
d 4 y Kp⃗x, ⃗y, zq ϕ 0 pyq
Check that this function is
z ¨∆
K
z 2 p⃗x ¡ ⃗yq 2
11/22

12/22
Partition functions. 6.
To compute the path integral for the scalar
field ϕ with the prescribed b.c. we calculate
the value of the action S on the solution:
S
»
»
d D x B µ p ? ¡g g µν φB ν φq
¡ d D x 1 Bϕ
ɛD¡1ϕ Bz
³ | zɛ
For z Ñ 0 we have the asymptotics
ϕ Ñ z ∆ ¡ ϕ 0 pxq z ∆ d 4 y ϕ 0pyq
2∆ , so
|x¡y|
S div.terms ¡
»
d 4 x d 4 y ϕ 0pxq ϕ 0 pyq
|x ¡ y| 2∆

12/22
Partition functions. 6.
To compute the path integral for the scalar
field ϕ with the prescribed b.c. we calculate
the value of the action S on the solution:
S
»
»
d D x B µ p ? ¡g g µν φB ν φq
¡ d D x 1 Bϕ
ɛD¡1ϕ Bz
³ | zɛ
For z Ñ 0 we have the asymptotics
ϕ Ñ z ∆ ¡ ϕ 0 pxq z ∆ d 4 y ϕ 0pyq
2∆ , so
|x¡y|
S div.terms ¡
»
d 4 x d 4 y ϕ 0pxq ϕ 0 pyq
|x ¡ y| 2∆

12/22
Partition functions. 6.
To compute the path integral for the scalar
field ϕ with the prescribed b.c. we calculate
the value of the action S on the solution:
S
»
»
d D x B µ p ? ¡g g µν φB ν φq
¡ d D x 1 Bϕ
ɛD¡1ϕ Bz
³ | zɛ
For z Ñ 0 we have the asymptotics
ϕ Ñ z ∆ ¡ ϕ 0 pxq z ∆ d 4 y ϕ 0pyq
2∆ , so
|x¡y|
S div.terms ¡
»
d 4 x d 4 y ϕ 0pxq ϕ 0 pyq
|x ¡ y| 2∆

13/22
A confining potential
Let us assume that the AdS warp factor 1 z 2 is
cut-off at some finite value z 0 of the
coordinate z

13/22
A confining potential
Let us assume that the AdS warp factor 1 z 2 is
cut-off at some finite value z 0 of the
coordinate z

14/22
A confining potential. 2.
The solution of the equation
f 11 1 ¡ d
z f 1 ¡ pk 2 pmRq 2
q f 0
z 2
may be written as
f pk, zq e i k¤x z d{2 K ν p|k| zq, ν ∆ ¡ ∆ ¡
2
The boundary condition f pk, z 0 q 0 leads
to the quantization of k 2 n λ n — the
spectrum of masses of the mesons

14/22
A confining potential. 2.
The solution of the equation
f 11 1 ¡ d
z f 1 ¡ pk 2 pmRq 2
q f 0
z 2
may be written as
f pk, zq e i k¤x z d{2 K ν p|k| zq, ν ∆ ¡ ∆ ¡
2
The boundary condition f pk, z 0 q 0 leads
to the quantization of k 2 n λ n — the
spectrum of masses of the mesons

14/22
A confining potential. 2.
The solution of the equation
f 11 1 ¡ d
z f 1 ¡ pk 2 pmRq 2
q f 0
z 2
may be written as
f pk, zq e i k¤x z d{2 K ν p|k| zq, ν ∆ ¡ ∆ ¡
2
The boundary condition f pk, z 0 q 0 leads
to the quantization of k 2 n λ n — the
spectrum of masses of the mesons

14/22
A confining potential. 2.
The solution of the equation
f 11 1 ¡ d
z f 1 ¡ pk 2 pmRq 2
q f 0
z 2
may be written as
f pk, zq e i k¤x z d{2 K ν p|k| zq, ν ∆ ¡ ∆ ¡
2
The boundary condition f pk, z 0 q 0 leads
to the quantization of k 2 n λ n — the
spectrum of masses of the mesons

15/22
The Wilson loop
µ An important observable in a QFT is the
dt A µ 9x qy,
Wilson loop xtr pP exp i ³
which depends on the contour C in spacetime
AdS/CFT provides a method for calculating it
Maldacena, 1998
At large ’t Hooft coupling λ one should find a
minimal area surface ending on the Wilson
loop at the boundary
C

15/22
The Wilson loop
µ An important observable in a QFT is the
dt A µ 9x qy,
Wilson loop xtr pP exp i ³
which depends on the contour C in spacetime
AdS/CFT provides a method for calculating it
Maldacena, 1998
At large ’t Hooft coupling λ one should find a
minimal area surface ending on the Wilson
loop at the boundary
C

15/22
The Wilson loop
µ An important observable in a QFT is the
dt A µ 9x qy,
Wilson loop xtr pP exp i ³
which depends on the contour C in spacetime
AdS/CFT provides a method for calculating it
Maldacena, 1998
At large ’t Hooft coupling λ one should find a
minimal area surface ending on the Wilson
loop at the boundary
C

15/22
The Wilson loop
µ An important observable in a QFT is the
dt A µ 9x qy,
Wilson loop xtr pP exp i ³
which depends on the contour C in spacetime
AdS/CFT provides a method for calculating it
Maldacena, 1998
At large ’t Hooft coupling λ one should find a
minimal area surface ending on the Wilson
loop at the boundary
C

The quark-antiquark potential
Let us calculate the Wilson loop depicted
above at strong coupling
The area is A
»
dσ dτ ? det h with h
the induced metric: h ab BXM BX M
Bσ a Bσ b
AdS 3 with coordinates pX, T, Zq. We set
X σ, T τ, Z ZpX q, therefore
A T
2»
L
dX
d
Z 12 1
Z 4 16/22
¡ L 2

The quark-antiquark potential
Let us calculate the Wilson loop depicted
above at strong coupling
The area is A
»
dσ dτ ? det h with h
the induced metric: h ab BXM BX M
Bσ a Bσ b
AdS 3 with coordinates pX, T, Zq. We set
X σ, T τ, Z ZpX q, therefore
A T
2»
L
dX
d
Z 12 1
Z 4 16/22
¡ L 2

The quark-antiquark potential
Let us calculate the Wilson loop depicted
above at strong coupling
The area is A
»
dσ dτ ? det h with h
the induced metric: h ab BXM BX M
Bσ a Bσ b
AdS 3 with coordinates pX, T, Zq. We set
X σ, T τ, Z ZpX q, therefore
A T
2»
L
dX
d
Z 12 1
Z 4 16/22
¡ L 2

The quark-antiquark potential
Let us calculate the Wilson loop depicted
above at strong coupling
The area is A
»
dσ dτ ? det h with h
the induced metric: h ab BXM BX M
Bσ a Bσ b
AdS 3 with coordinates pX, T, Zq. We set
X σ, T τ, Z ZpX q, therefore
A T
2»
L
dX
d
Z 12 1
Z 4 16/22
¡ L 2

17/22
The quark-antiquark potential. 2.
Check that the solution has the form
X ¨
» z m
z
z m¨2
dz
b1 ¡
z
z m¨4
The regularized area is
A 2T
Z
» z m
ɛ
dz
z 2b1 z
¨4 T1 ¡ ɛ ¡
z m
?
2π π
Γp 1 4 q2 z m
¤ ¤ ¤
The first term corresponds to mass
renormalization
The second term is the (Coulomb) potential

17/22
The quark-antiquark potential. 2.
Check that the solution has the form
X ¨
» z m
z
z m¨2
dz
b1 ¡
z
z m¨4
The regularized area is
A 2T
Z
» z m
ɛ
dz
z 2b1 z
¨4 T1 ¡ ɛ ¡
z m
?
2π π
Γp 1 4 q2 z m
¤ ¤ ¤
The first term corresponds to mass
renormalization
The second term is the (Coulomb) potential

17/22
The quark-antiquark potential. 2.
Check that the solution has the form
X ¨
» z m
z
z m¨2
dz
b1 ¡
z
z m¨4
The regularized area is
A 2T
Z
» z m
ɛ
dz
z 2b1 z
¨4 T1 ¡ ɛ ¡
z m
?
2π π
Γp 1 4 q2 z m
¤ ¤ ¤
The first term corresponds to mass
renormalization
The second term is the (Coulomb) potential

17/22
The quark-antiquark potential. 2.
Check that the solution has the form
X ¨
» z m
z
z m¨2
dz
b1 ¡
z
z m¨4
The regularized area is
A 2T
Z
» z m
ɛ
dz
z 2b1 z
¨4 T1 ¡ ɛ ¡
z m
?
2π π
Γp 1 4 q2 z m
¤ ¤ ¤
The first term corresponds to mass
renormalization
The second term is the (Coulomb) potential

17/22
The quark-antiquark potential. 2.
Check that the solution has the form
X ¨
» z m
z
z m¨2
dz
b1 ¡
z
z m¨4
The regularized area is
A 2T
Z
» z m
ɛ
dz
z 2b1 z
¨4 T1 ¡ ɛ ¡
z m
?
2π π
Γp 1 4 q2 z m
¤ ¤ ¤
The first term corresponds to mass
renormalization
The second term is the (Coulomb) potential

18/22
The advent of mesons
What will change if we impose a cut-off z 0 on
the AdS warp factor, as before?
The surface will ‘flatten’ at z z 0 .
At a large separation L the arcs will be
negligible, and the area will be proportional to
the area of the rectangle, i.e. xW y e ¡ T L
z
0
2
The area law and a linear potential!

18/22
The advent of mesons
What will change if we impose a cut-off z 0 on
the AdS warp factor, as before?
The surface will ‘flatten’ at z z 0 .
At a large separation L the arcs will be
negligible, and the area will be proportional to
the area of the rectangle, i.e. xW y e ¡ T L
z
0
2
The area law and a linear potential!

18/22
The advent of mesons
What will change if we impose a cut-off z 0 on
the AdS warp factor, as before?
The surface will ‘flatten’ at z z 0 .
At a large separation L the arcs will be
negligible, and the area will be proportional to
the area of the rectangle, i.e. xW y e ¡ T L
z
0
2
The area law and a linear potential!

18/22
The advent of mesons
What will change if we impose a cut-off z 0 on
the AdS warp factor, as before?
The surface will ‘flatten’ at z z 0 .
At a large separation L the arcs will be
negligible, and the area will be proportional to
the area of the rectangle, i.e. xW y e ¡ T L
z
0
2
The area law and a linear potential!

18/22
The advent of mesons
What will change if we impose a cut-off z 0 on
the AdS warp factor, as before?
The surface will ‘flatten’ at z z 0 .
At a large separation L the arcs will be
negligible, and the area will be proportional to
the area of the rectangle, i.e. xW y e ¡ T L
z
0
2
The area law and a linear potential!

19/22
AdS 5 ¢ S 5 vs. N 4 SYM
N 4 SYM — the maximally
supersymmetric conformal QFT in D 4
A vector multiplet + 3 chiral multiplets (in
N 1 superspace)
P SU p2, 2|4q isometries of AdS 5 ¢ S 5
The bosonic part is SOp2, 4q ¢ SU p4q — the
conformal symmetry and the R-symmetry
(rotating the scalars, for example)
λ gY 2 M N — the ‘t Hooft coupling,
g Y M Ñ 0, N Ñ 8

19/22
AdS 5 ¢ S 5 vs. N 4 SYM
N 4 SYM — the maximally
supersymmetric conformal QFT in D 4
A vector multiplet + 3 chiral multiplets (in
N 1 superspace)
P SU p2, 2|4q isometries of AdS 5 ¢ S 5
The bosonic part is SOp2, 4q ¢ SU p4q — the
conformal symmetry and the R-symmetry
(rotating the scalars, for example)
λ gY 2 M N — the ‘t Hooft coupling,
g Y M Ñ 0, N Ñ 8

19/22
AdS 5 ¢ S 5 vs. N 4 SYM
N 4 SYM — the maximally
supersymmetric conformal QFT in D 4
A vector multiplet + 3 chiral multiplets (in
N 1 superspace)
P SU p2, 2|4q isometries of AdS 5 ¢ S 5
The bosonic part is SOp2, 4q ¢ SU p4q — the
conformal symmetry and the R-symmetry
(rotating the scalars, for example)
λ gY 2 M N — the ‘t Hooft coupling,
g Y M Ñ 0, N Ñ 8

19/22
AdS 5 ¢ S 5 vs. N 4 SYM
N 4 SYM — the maximally
supersymmetric conformal QFT in D 4
A vector multiplet + 3 chiral multiplets (in
N 1 superspace)
P SU p2, 2|4q isometries of AdS 5 ¢ S 5
The bosonic part is SOp2, 4q ¢ SU p4q — the
conformal symmetry and the R-symmetry
(rotating the scalars, for example)
λ gY 2 M N — the ‘t Hooft coupling,
g Y M Ñ 0, N Ñ 8

19/22
AdS 5 ¢ S 5 vs. N 4 SYM
N 4 SYM — the maximally
supersymmetric conformal QFT in D 4
A vector multiplet + 3 chiral multiplets (in
N 1 superspace)
P SU p2, 2|4q isometries of AdS 5 ¢ S 5
The bosonic part is SOp2, 4q ¢ SU p4q — the
conformal symmetry and the R-symmetry
(rotating the scalars, for example)
λ gY 2 M N — the ‘t Hooft coupling,
g Y M Ñ 0, N Ñ 8

19/22
AdS 5 ¢ S 5 vs. N 4 SYM
N 4 SYM — the maximally
supersymmetric conformal QFT in D 4
A vector multiplet + 3 chiral multiplets (in
N 1 superspace)
P SU p2, 2|4q isometries of AdS 5 ¢ S 5
The bosonic part is SOp2, 4q ¢ SU p4q — the
conformal symmetry and the R-symmetry
(rotating the scalars, for example)
λ gY 2 M N — the ‘t Hooft coupling,
g Y M Ñ 0, N Ñ 8

20/22
AdS 5 ¢ S 5 vs. N 4 SYM
The string coupling g st 1 is small,
N
therefore the string is free
»
ñ Described by the σ-model
?
λ
S d 2 σ γ ab B a X M G MN B b X N
2π
At strong coupling λ Ñ 8 may be treated
semiclassically
Turns out to be integrable; the spectrum of
states may be computed exactly! 2002—now

20/22
AdS 5 ¢ S 5 vs. N 4 SYM
The string coupling g st 1 is small,
N
therefore the string is free
»
ñ Described by the σ-model
?
λ
S d 2 σ γ ab B a X M G MN B b X N
2π
At strong coupling λ Ñ 8 may be treated
semiclassically
Turns out to be integrable; the spectrum of
states may be computed exactly! 2002—now

20/22
AdS 5 ¢ S 5 vs. N 4 SYM
The string coupling g st 1 is small,
N
therefore the string is free
»
ñ Described by the σ-model
?
λ
S d 2 σ γ ab B a X M G MN B b X N
2π
At strong coupling λ Ñ 8 may be treated
semiclassically
Turns out to be integrable; the spectrum of
states may be computed exactly! 2002—now

20/22
AdS 5 ¢ S 5 vs. N 4 SYM
The string coupling g st 1 is small,
N
therefore the string is free
»
ñ Described by the σ-model
?
λ
S d 2 σ γ ab B a X M G MN B b X N
2π
At strong coupling λ Ñ 8 may be treated
semiclassically
Turns out to be integrable; the spectrum of
states may be computed exactly! 2002—now

20/22
AdS 5 ¢ S 5 vs. N 4 SYM
The string coupling g st 1 is small,
N
therefore the string is free
»
ñ Described by the σ-model
?
λ
S d 2 σ γ ab B a X M G MN B b X N
2π
At strong coupling λ Ñ 8 may be treated
semiclassically
Turns out to be integrable; the spectrum of
states may be computed exactly! 2002—now

21/22
Questions / Answers
We have discussed the following topics:
Properties of the AdS space
Semiclassical calculation of the scalar field
partition function
The holographic calculation of Wilson loops
The AdS 5 ¢ S 5 string theory

21/22
Questions / Answers
We have discussed the following topics:
Properties of the AdS space
Semiclassical calculation of the scalar field
partition function
The holographic calculation of Wilson loops
The AdS 5 ¢ S 5 string theory

21/22
Questions / Answers
We have discussed the following topics:
Properties of the AdS space
Semiclassical calculation of the scalar field
partition function
The holographic calculation of Wilson loops
The AdS 5 ¢ S 5 string theory

21/22
Questions / Answers
We have discussed the following topics:
Properties of the AdS space
Semiclassical calculation of the scalar field
partition function
The holographic calculation of Wilson loops
The AdS 5 ¢ S 5 string theory

Thank you!
22/22