Introduction to AdS/CFT Correspondence
Sitthichai Pinkanjanarod
Chulalongkorn University
quazact@gmail.com
February 3rd , 2020
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
1 / 19
Outlines
1
Motivation for AdS/CFT
Holographic Principle
Large Nc gauge theory as a string theory
2
String Theory
String Theory
Physics of D-branes
3
AdS/CFT Correspondence
Global stucture of AdS5 Spacetime
Conformal Field Theory
AdS/CFT Correspondence
4
Evidence for AdS/CFT
5
Summary
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
2 / 19
Motivation for AdS/CFT
Holographic Principle
Temperature of black hole
The motivation for the holographic
principle originated from study of
black hole (BH) [’t Hooft 93, Susskind 94]
Hawking radiation
= pair creation + BH pulling
the pair particles apart
one particle falls into,
the other escapes
the emitted particles provides
a temperature of the BH
(Hawking Temperature).
Source: S&T: Gregg Dindermann
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
3 / 19
Motivation for AdS/CFT
Holographic Principle
Black hole thermodynamics
Temperature of the BH is given by
TBH =
~
1
=
β
8πGN m
(1)
While it’s entropy is[Bekenstein, Hawking]
SBH =
AH
4~GN
(2)
where AH means area of horizon of the BH
First clue of AdS/CFT:
SBH suggests that a black hole in (d+1)-dimension can be described by
the usual statistical system in d-dimension (1 spatial dimension lower).
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
4 / 19
Motivation for AdS/CFT
Large Nc gauge theory as a string theory
Large Nc gauge theory
QCD: SU(3) gauge theory + Dirac field(fermions)
in the IR: baryons, mesons etc.→ strongly coupled (problematic)
[’t Hooft 74] replacing SU(3) → SU(Nc ) ?
Theory of Nc2 scaler fields
LSU(N)
1
1 4
1
µ
= − 2 Tr ∂µ Φ∂ Φ + Φ
g
2
4
then promotes Φ(x) → Φ(x)ab a Nc × Nc matrix
1
1 a b c d
1
a
µ a
LSU(N) = − 2 Tr (∂µ Φb )(∂ Φb ) + Φb Φc Φd Φa
g
2
4
(3)
(4)
where Φad → Uba Φbc (U † )cd ⇒ global U(Nc ) sym.
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
5 / 19
Motivation for AdS/CFT
Large Nc gauge theory as a string theory
Topological description of the large Nc gauge theory
2 N and N → ∞
Let λ ≡ gYM
c
c
< Φab Φcd >∝ δda δbc −
1 a b
N δc δd
≈ δda δbc
Feynmann → double lines
diagrams
diagrams
described topologically by associating
2
gYM
= λ/Nc , for each edge E
Nc /λ, for each edge V
Nc , for each face F
Image adapted from: M. Natsuume,
Amplitude for vacuum diagram becomes
A∼(
AdS/CFT duality user guide
λ E Nc V F
) ( ) Nc = λE −V Ncχ
Nc
λ
(5)
Note: χ = F + V − E = 2 − 2h is topological invariant
where h is a number of handle(genus).
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
6 / 19
Motivation for AdS/CFT
Large Nc gauge theory as a string theory
Large Nc gauge theory as a string theory
The double lines diagrams are topologies of 2-D
oriented surfaces.
In Nc gauge th., Σ(n-points scattering diagrams) is
now a partition function
lnZlarge Nc
=
gauge
∞
X
Nc2−2h−n fn(h) (λ)
(6)
h=1
Comparing to the partition function of n-closed
oriented string scattering
Image adapted from: M. Natsuume,
lnZpertb.
string
=
∞
X
gsn−2+2h fn(h) (λ)
(7)
AdS/CFT duality user guide
h=1
Second clue of AdS/CFT:
Zlarge Nc
Sitthichai Pinkanjanarod (CU)
gauge
= Zpertb.
closed oriented string
Introduction to AdS/CFT
February 3rd , 2020
7 / 19
String Theory
String Theory
The Need for String Theory
1st clue + 2nd clue ⇒ we need 5-D gravitational theory
(from string theory) dual to the large Nc gauge theory
String theory could provide the requirement above.
idea of string theory: a point particle → a string
their action can be represented by
SNG
1
=
2πα0
Z
√
d 2 −h
(8)
Σ
where induced metric hab = Gµν ∂a X µ ∂b X ν (the string world sheet Σ)
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
8 / 19
String Theory
String Theory
Open/Closed String Theory
A closed string has 2 modes
L-moving mode and R-moving mode
corresponding to spin-2 particle i.e. graviton
An open string has 2 d.o.f. or 2 polarizations
It represents U(1) gauge field
e.g photon, vector bosons, gluons
We are only interested in type II string theory where AdS/CFT duality is
well studied
no free open string in the theory
open strings exist only when bounded with D-branes
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
9 / 19
String Theory
Physics of D-branes
Dp-branes configuration
D-branes or Dp-branes = extended objects that is intrinsic to type II string
theory, defined by
dim 0 1 2 ... p p + 1 ... d − 1
B.C. N N N ... N
D
...
D
Closed strings interact with Nc Dp-branes
break open with endpoints on the
Dp-branes and vice versa.
Only closed strings can travel between
D-branes in bulk.
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
10 / 19
String Theory
Physics of D-branes
Dynamics on Dp-branes
Purturbative action for a D-brane is given by
Z
q
SDp = −Tp d p+1 x −|Gαβ + 2πα0 Fαβ |
(9)
where Gαβ = ∂α X µ ∂β X ν is induced metric on the Dp-brane and Fαβ is SU(N)
field strength tensor.
1st order perturbation tells us, there are
U(1) gauge fields on the Dp-brane.
for Nc of Dp-branes, there are U(Nc ) gauge
fields on them.
bosonic spectrum on a closed string:
dilaton(Φ), graviton(gµν ),
Kalb-Ramond field(Bµν ),
p-form field C (p)
bosonic spectrum on Dp-brane:
vector gauge boson (Aaµ ),
adjoint scalar field (Φa )
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
Bosonic spectrum on Dp-branes
February 3rd , 2020
11 / 19
String Theory
Physics of D-branes
D3-branes at near horizon limit
Consider a spacetime metric around D3-brane in SUGRA limit,
ds 2 = f (r )−1/2 (−dt 2 + d~x 2 ) + f (r )1/2 (dr 2 + r 2 dΩ25 )
where f (r ) = 1 +
R4
r4
(10)
and R 4 = N π42 GN T3
T3 is a tension between open strings on the D3-brane.
The tension T3 causes the spacetime metric curved.
As r → ∞ the spacetime is flat.
As r → 0 the spacetime turns into AdS5 × S 5 .
At near Horizon limit, the spacetime metric becomes
ds 2 =
R2
r2
(−dt 2 + d~x 2 ) + 2 dr 2 + R 2 dΩ25
2
r
|R
{z
} | {z5 }
AdS5
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
(11)
S
February 3rd , 2020
12 / 19
AdS/CFT Correspondence
Global stucture of AdS5 Spacetime
Global stucture of AdS5 Spacetime
A global AdS5 can be described by hyperboloid
−X02 − X52 + X12 + X22 + X32 + X42 = −L2
(12)
Symmetry group on the global AdS5 is SO(2, 4) isometry group,
where symmetry group on S 5 is SO(6) ' SU(4)
The local AdS5 metric is
−
ds 2
= −dX02 − dX52 + dX12 + dX22 + dX32 + dX42
L2
(13)
Under Poincare global coordinate transformation the metric becomes
ds 2
dr 2
2
2
2
=
r
(−dt
+
d~
x
)
+
L2
r2
(14)
which is similar to AdS5 metric in (11)
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
13 / 19
AdS/CFT Correspondence
Conformal Field Theory
CFT: N = 4 SYM Theory in (1+3)D
A part of string theory of interest of the center of mass of Nc D3-brane
L=−
1
1
1
Tr( F µν Fµν + (Dµ Φi )(D µ Φi ) + [Φi , Φj ]2 ) + fermionic (15)
2
4
2
gYM
Properties of the theory
conformally invariant because of the vanishing of β-function.
xµ → xµ + δxµ where δxµ = aµ + ωµν xν + λxµ + (bµ x 2 − 2xµ b · x)
Pµ , Lorentz trf. Jµν , dilation D, special conformal trf Kµ
Define anti sym. (6 × 6) matrix J¯MN where
K +P
K −P
J¯µν = Jµν , J¯µ4 = µ µ , J¯µ5 = µ µ , J¯45 = D
2
2
One finde that [J¯MN , J¯PQ ] is the algebra of the SO(2, 4)
invariant under N = 4 R-symmetry group SU(4) ∼ SO(6)
invariant under S-duality group SL(2, Z)
implies strong-weak duality τ → − τ1 with
2
2
coupling constant τ = 4πi/gYM
→ 4πi/gYM
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
14 / 19
AdS/CFT Correspondence
AdS/CFT Correspondence
AdS/CFT Correspondence
The AdS/CFT Correspondence
The superstring theory of type IIB on AdS5 × S 5 is equivalent (or dual to)
N = 4 SYM gauge theory in (1+3) dimensions
Large Nc gauge theory is equivalent to string theory
belongs to the holographic principle
Matching of symmetries
Symmetry
AdS5 × S 5
SO(2, 4) Isometry group of AdS5
SO(6)
Isometry of S 5 : SO(6)
Sitthichai Pinkanjanarod (CU)
4D N = 4 SYM th.
Conformal group
R-symmetry:SO(6)R
Introduction to AdS/CFT
February 3rd , 2020
15 / 19
AdS/CFT Correspondence
AdS/CFT Correspondence
AdS/CFT Dictionary[Witten, Gubser et al.]
Matching of parameters
N = 4 SYM th. ⇐⇒ AdS5 × S 5
2
gYM
=
4πgs
2 N
4 /α02
λ ≡ gYM
=
R
c
π 4 /(2Nc2 )
=
GN /R 8
Matching of correlation functions
N
= 4 SYM th.
⇐⇒
AdS5 × S 5
R 4
he d xφ0 (x)O(x) iCFT
=
Zstring φ(x, r )|r →∞ = φ0 (x)
Matching of the spectrums
N = 4 SYM th.
⇐⇒
AdS5 × S 5
Lagrangian of scalr op.: O(x) ⇐⇒
dilaton:Φ(x, r )
a
SO(6) vector operator: Jµ (x)
⇐⇒ vector gauge field:Aaµ (x, r )
stress tensor operator: Tµν (x) ⇐⇒
graviton:hµν (x, r )
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
16 / 19
Evidence for AdS/CFT
Evidence for AdS/CFT
First successful example is shear viscosity of strongly couple fluid
e.g. quark gluon plasma (QGP)
perturbative calculation does not work
AdS/CFT calculation of shear viscosity
[Kovtun, Son, and Starinets 2005]
computed by retarded Green function GRxy ,xy
associated with the correlator hT x,y , T x,y i
T x,y is holographically dual to hx,y
Shear viscosity of holographic fluid is
η
s
=
1
4π
lower bound for the most perfect fluid next to ideal fluid
Experiment at RHIC and LHC, shear viscosity of QGP is
η
s
'
2.5
4π
close and same order of magnitude → good agreement
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
17 / 19
Summary
Summary
Holographic principle tells us that BH in (d+1) dim can be described
by d dim statistical theory
Large Nc gauge th. can be described by weakly coupled string th.
Feynmann diagrams on gauge theory can be described by sum of
topology of 2-D surface similar to that of perturbative string theory
Type IIB string th.
closed strings (free gravitons) interact with Dp-brane and break open
open strings have their ends on the D-branes represents gauge bosons
Spacetime at near horizon limit of D3-branes turns into AdS5 × S 5
with a conformal boundary as r → ∞ where gauge th. live in.
4D N = 4 SYM gauge th.is dual to SUGRA on AdS5 × S 5
parameters between the 2 th. match
symmetries match: SO(2, 4) ⊗ SO(6)
matching partition functions
Shear viscosity of QGP provides real world example of AdS/CFT.
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
18 / 19
Summary
References
O. Aharony, and et al, Large N Field Theories, String Theory and Gravity,
Phys.Rept.323:183-386,2000
H. Liu, 8.821 String Theory and Holographic Duality, Fall 2014,
Massachusetts Institute of Technology: MIT OpenCourseWare,
https://ocw.mit.edu.
M. Natsuume, AdS/CFT Duality User Guide, Lecture Notes in Physics,
volume 903, Springer, 2015
M. Ammon and J. Erdmenger, Gauge/Gravity Foundation and Applications,
Cambridge University Press, 2015
K. Papadodimas, Introduction to AdS/CFT, XVIII European Workshop on
String Theory, http://www.physics.ntua.gr/corfu2012
H. Ngo, Introduction to the AdS/CFT correspondence, IMPRS/GK Young
Scientist Workshop at Ringberg Castle, Session: Gauge/Gravity Dualities,
2008
Thank you for your attention
Sitthichai Pinkanjanarod (CU)
Introduction to AdS/CFT
February 3rd , 2020
19 / 19