T O W A R D S W E A K C O U P L I N G I N H O L O G R A P H Y
I N S T I T U U T- L O R E N T Z F O R T H E O R E T I C A L P H Y S I C S L E I D E N U N I V E R S I T Y
O X F O R D, 6 . 3 . 2 0 1 7
S AŠO G R O Z D A N O V
W O R K I N C O L L A B O R AT I O N W I T H J . C A S A L D E R R E Y- S O L A N A , N . K A P L I S , N . P O O V U T T I K U L , A . S TA R I N E T S A N D W. V A N D E R S C H E E
O U T L I N E
• from holography to experiment
• coupling constant dependence and universality in hydrodynamics
• thermalisation and higher-energy spectrum
• heavy ion collisions
• conclusion and future directions
2
F R O M H O L O G R A P H Y T O E X P E R I M E N T
• some strongly coupled field theories have a dual gravitational description (AdS/CFT correspondence)
• originally a duality in type IIB string theory
• so far: universality at strong coupling
• challenges
• field content (no supersymmetry)
• away from infinite N
• away from infinite coupling
Z[field theory] = Z[string theory]
3
↵0 / 1/�1/2
� ⌘ g2YMN
T W O C L A S S E S O F T H E O R I E S• top-down dual of N=4 theory with ’t Hooft coupling corrections from
type IIB string theory
SIIB =1
2210
Zd
10x
p�g
✓R� 1
2(@�)2 � 1
4 · 5!F25 + �e
� 32�W + . . .
◆
� = ↵03⇣(3)/8 ↵0/L2 = ��1/2
W = C↵���Cµ��⌫C⇢�µ
↵ C⌫⇢�� +
1
2C↵���Cµ⌫��C
⇢�µ↵ C⌫
⇢��
4
• bottom-up curvature-squared theory with a special, non-perturbative case of Gauss-Bonnet gravity
SR2 =1
225
Zd
5x
p�g
⇥R� 2⇤+ L
2�↵1R
2 + ↵2Rµ⌫Rµ⌫ + ↵3Rµ⌫⇢�R
µ⌫⇢��⇤
SGB =1
225
Zd
5x
p�g
R� 2⇤+
�GB
2L
2�R
2 � 4Rµ⌫Rµ⌫ +Rµ⌫⇢�R
µ⌫⇢���
H Y D R O DY N A M I C S
H Y D R O DY N A M I C S
• QCD and quark-gluon plasma
• low-energy limit of QFTs (effective field theory)
• tensor structures (phenomenological gradient expansions) with transport coefficients (microscopic)
rµTµ⌫ = 0
Tµ⌫�u�, T, µ
�= ("+ P )uµu⌫ + Pgµ⌫ � ⌘�µ⌫ � ⇣r · u�µ⌫ + . . .
6
H Y D R O DY N A M I C S
• conformal (Weyl-covariant) hydrodynamics
• infinite-order asymptotic expansion
• classification of tensors beyond Navier-Stokes
first order: 2 (1 in CFT) - shear and bulk viscosities
second order: 15 (5 in CFT) - relaxation time, … [Israel-Stewart and extensions]
third order: 68 (20 in CFT) - [S. G., Kaplis, PRD 93 (2016) 6, 066012, arXiv:1507.02461]
Tµµ = 0
gµ⌫
! e�2!(x)gµ⌫ Tµ⌫ ! e6!(x) Tµ⌫
! =OHX
n=0
↵nkn+1Tµ⌫ =
1X
n=0
Tµ⌫(n)
7
H Y D R O DY N A M I C S• diffusion and sound dispersion relations in CFT
• loop corrections break analyticity of the gradient expansion (long-time tails), but are 1/N suppressed [Kovtun, Yaffe (2003)]
• entropy current, constraints on transport and new transport coefficients (anomalies, broken parity)
• non-relativistic hydrodynamics
• hydrodynamics from effective Schwinger-Keldysh field theory with dissipation [Nicolis, et. al.; S. G., Polonyi; Haehl, Loganayagam, Rangamani; de Boer, Heller, Pinzani-Fokeeva; Crossley, Glorioso, Liu]
shear: ! = �i⌘
"+ Pk2 � i
"⌘2⌧⇧
("+ P )
2 � 1
2
✓1"+ P
#k4 +O �
k5�
sound: ! = ±csk � i�ck2 ⌥ �c
2cs
��c � 2c2s⌧⇧
�k3 � i
"8
9
⌘2⌧⇧
("+ P )
2 � 1
3
✓1 + ✓2"+ P
#k4 +O �
k5�
8
H Y D R O DY N A M I C S F R O M H O L O G R A P H Y
• holography can compute microscopic transport coefficients [Policastro, Son, Starinets (2001)]
• low-energy limit of QFTs / low-energy gravitational perturbations in backgrounds with black holes
• Green’s functions and Kubo formulae
• quasi-normal modes [Kovtun, Starinets (2005)]
• fluid-gravity [Bhattacharyya, Hubeny, Minwalla, Rangamani (2007)]
⌦T
abR (0)
↵=G
abR (0)� 1
2
Zd
4xG
ab,cdRA (0, x)hcd(x) +
1
8
Zd
4xd
4yG
ab,cd,efRAA (0, x, y)hcd(x)hef (y) + . . .
! =1X
n=0
↵nkn+1
9
N = 4 AT I N F I N I T E C O U P L I N G• type IIB theory on S5 dual to N=4 supersymmetric Yang-Mills at
infinite ’t Hooft coupling and infinite Nc
• black brane
• use to find field theory stress-energy tensor to third order
S =1
225
Zd
5x
p�g
✓R+
12
L
2
◆
5 = 2⇡/Nc
ds
2 =r
20
u
��f(u)dt2 + dx
2 + dy
2 + dz
2�+
du
2
4u2f(u)
f(u) = 1� u2
T ab = "uaub + P�ab � ⌘�ab + ⌘⌧⇧
hD�abi +
1
3�ab (r · u)
�+
hRhabi � 2ucR
chabidud
i
+ �1�hac�
bic + �2�hac⌦
bic + �3⌦hac⌦
bic +20X
n=1
�(3)n On
10
N = 4 AT I N F I N I T E C O U P L I N G• transport coefficients [S. G., Kaplis, PRD 93 (2016) 6, 066012, arXiv:1507.02461]
⌘
s=
1
4⇡
�(3)1 + �(3)
2 + �(3)4 ⌘ �✓1 = �N2
c T
32⇡
�(3)3 + �(3)
5 + �(3)6 ⌘ �✓2 =
N2c T
384⇡
✓⇡2
12+ 18 ln 2� ln2 2� 22
◆
�(3)1 � �(3)
16 =N2
c T
16⇡
✓⇡2
12+ 4 ln 2� ln2 2
◆
�(3)17 =
N2c T
16⇡
✓⇡2
12+ 2 ln 2� ln2 2
◆
�(3)1
6+
4�(3)2
3+
4�(3)3
3+
5�(3)4
6+
5�(3)5
6+
4�(3)6
3� �(3)
7
2
+3�(3)
8
2+
�(3)9
2� 2�(3)
10
3� 11�(3)
11
6� �(3)
12
3+
�(3)13
6� �(3)
15 =N2
c T
648⇡
�15� 2⇡2 � 45 ln 2 + 24 ln2 2
�
⌘ =⇡
8N2
c T3
⌧⇧ =(2� ln 2)
2⇡T =
N2c T
2
8�1 =
N2c T
2
16�2 = �N2
c T2
8ln 2 �3 = 0
11
T O P- D O W N C O N S T R U C T I O N• type IIB action with ’t Hooft coupling corrections
• dimensional reduction
• black brane
SIIB =1
2210
Zd
10x
p�g
✓R� 1
2(@�)2 � 1
4 · 5!F25 + �e
� 32�W + . . .
◆
� = ↵03⇣(3)/8 ↵0/L2 = ��1/2
W = C↵���Cµ��⌫C⇢�µ
↵ C⌫⇢�� +
1
2C↵���Cµ⌫��C
⇢�µ↵ C⌫
⇢��
S =1
225
Zd
5x
p�g
✓R+
12
L
2+ �W
◆
ds
2 =r
20
u
��f(u)Ztdt
2 + dx
2 + dy
2 + dz
2�+ Zu
du
2
4u2f
f(u) = 1� u2
Zt = 1� 15��5u2 + 5u4 � 3u6
�Zu = 1 + 15�
�5u2 + 5u4 � 19u6
�
12
T O P- D O W N C O N S T R U C T I O N• N=4 transport coefficients to second order [S. G., Starinets, JHEP 1503 (2015)
007 arXiv:1412.5685]
⌘ =⇡
8N2
c T3 (1 + 135� + . . . )
⌧⇧ =(2� ln 2)
2⇡T+
375�
4⇡T+ . . .
=N2
c T2
8(1� 10� + . . . )
�1 =N2
c T2
16(1 + 350� + . . . )
�2 = �N2c T
2
16(2 ln 2 + 5 (97 + 54 ln 2) � + . . . )
�3 =25N2
c T2
2� + . . .
⌘
s=
1
4⇡
⇣1 + 15⇣(3)��3/2 + . . .
⌘
0
h̄
4πkB
η
s
g2Nc
[Kovtun, Son, Starinets (2005)]
13
B O T T O M - U P C O N S T R U C T I O N• curvature-squared theory [S. G., Starinets, JHEP 1503 (2015) 007 arXiv:1412.5685]
SR2 =1
225
Zd
5x
p�g
⇥R� 2⇤+ L
2�↵1R
2 + ↵2Rµ⌫Rµ⌫ + ↵3Rµ⌫⇢�R
µ⌫⇢��⇤
⌘ =r3+22
5
(1� 8 (5↵1 + ↵2)) +O(↵2i )
⌘⌧⇧ =r2+ (2� ln 2)
425
✓1� 26
3(5↵1 + ↵2)
◆� r2+ (23 + 5 ln 2)
1225
↵3 +O(↵2i )
=r2+22
5
✓1� 26
3(5↵1 + ↵2)
◆� 25r2+
625
↵3 +O(↵2i )
�1 =r2+42
5
✓1� 26
3(5↵1 + ↵2)
◆� r2+
1225
↵3 +O(↵2i )
�2 = �r2+ ln 2
225
✓1� 26
3(5↵1 + ↵2)
◆� r2+ (21 + 5 ln 2)
625
↵3 +O(↵2i )
�3 = �28r2+25
↵3 +O(↵2i )
14
B O T T O M - U P C O N S T R U C T I O N• Gauss-Bonnet theory [S. G., Starinets, Theor. Math. Phys. 182 (2015) 1, 61-73]
SGB =1
225
Zd
5x
p�g
R� 2⇤+
�GB
2L
2�R
2 � 4Rµ⌫Rµ⌫ +Rµ⌫⇢�R
µ⌫⇢���
⌘ = s�2/4⇡
⌧⇧ =
1
2⇡T
✓1
4
(1 + �)
✓5 + � � 2
�
◆� 1
2
log
2 (1 + �)
�
�◆
�1 =
⌘
2⇡T
(1 + �)
�3� 4� + 2�3
�
2�2
!
�2 = � ⌘
⇡T
✓�1
4
(1 + �)
✓1 + � � 2
�
◆+
1
2
log
2 (1 + �)
�
�◆
�3 = � ⌘
⇡T
(1 + �)
�3 + � � 4�2
�
�2
!
=
⌘
⇡T
(1 + �)
�2�2 � 1
�
2�2
!
✓1 =
⌘
8⇡2T 2��2�2
+ � � 1
�
� =p1� 4�GB
⌘
s=
1
4⇡(1� 4�GB)
15
L I M I T S O F T H E G A U S S - B O N N E T T H E O R Y
• exact spectrum in the extreme (anomalous) limit of
16
�GB = 1/4
Scalar: w = �i⇣4 + 2n1 �
p4� 3q2
⌘, w = �i
⇣4 + 2n2 +
p4� 3q2
⌘
Shear: w = �2i (1 + n1) , w = �2i (3 + n2)
Sound: w = �i⇣4 + 2n1 �
p4 + q2
⌘, w = �i
⇣4 + 2n2 +
p4 + q2
⌘
• in the extreme ``weak” limit there is a curvature singularity, which needs a stringy resolution
lim�GB!�1
SGB =�GBL
2
425
Zd
5x
p�g
R
2 � 4Rµ⌫Rµ⌫ +Rµ⌫⇢�R
µ⌫⇢� � 4⇤
�GBL2
�
ds
2 =p
��GB
2
4� r̃
2
L
2
r1�
r̃
4+
r̃
4dt
2 +L
2
r̃
2
q1� r̃4+
r̃4
dr̃
2 +r̃
2
L
2
�dx
2 + dy
2 + dz
2�3
5
�GB ! �1
• how can we interpret the extreme limits of the Gauss-Bonnet coupling?
C H A R G E D I F F U S I O N• construct the most general four-derivative action [S. G., Starinets (2016)
arXiv:1611.07053]
17
• make it such that the equations of motion are only second order
S =1
225
Zd
5x
p�g [R� 2⇤+ LGB ] +
Zd
5x
p�gLA
LA = �1
4Fµ⌫F
µ⌫ + ↵4RFµ⌫Fµ⌫ + ↵5R
µ⌫Fµ⇢F⇢
⌫ + ↵6Rµ⌫⇢�Fµ⌫F⇢� + ↵7 (Fµ⌫F
µ⌫)2
+ ↵8rµF⇢�rµF ⇢� + ↵9rµF⇢�r⇢Fµ� + ↵10rµFµ⌫r⇢F⇢⌫ + ↵11F
µ⌫F⌫⇢F⇢�F�µ
LA =� 1
4Fµ⌫F
µ⌫ + �1L2 (RFµ⌫F
µ⌫ � 4Rµ⌫Fµ⇢F⇢
⌫ +Rµ⌫⇢�Fµ⌫F⇢�)
+ �2L2 (Fµ⌫F
µ⌫)2 + �3L2Fµ⌫F⌫⇢F
⇢�F�µ
D =(1 + �GB)(1 + 2�)
⇣� +
p�2 � �2
GB
⌘
6(� � 1)h�⇣� +
p�2 � �2
GB
⌘� �2
GB
i(p(1� �2
GB) (�2 � �2
GB) ln
"�GB
1 +p1� �2
GB
#
��� � �2
GB
�ln
"�GB
� +p�2 � �2
GB
#)
• diffusion
�GB =p
1� 4�GB
� = 1 + 48�1
D 6= D = 0
D 6= D = 0
U N I V E R S A L I T Y• membrane paradigm (conserved current)
• first-order hydrodynamics [Kovtun, Policastro, Son, Starinets]
• second-order hydrodynamics [Haack, Yarom (2009); S. G., Starinets, JHEP 1503 (2015) 007 arXiv:1412.5685]
⌘
s=
1
4⇡
@rJ = 0
2⌘⌧⇧ � 4�1 � �2 = O ��2
�
2⌘⌧⇧ � 4�1 � �2 = O �↵2i
�
2⌘⌧⇧ � 4�1 � �2 = � ⌘
⇡T
(1� �GB)�1� �2
GB
�(3 + 2�GB)
�2GB
= �40�2GB⌘
⇡T+O �
�3GB
�
18
( M O R E ) U N I V E R S A L I T Y• non-renormalisation of anomalous conductivities
[Gursoy, Tarrio (2015); S. G., Poovuttikul, JHEP 1609 (2016) 046 arXiv:1603.08770]
• universal values
• bulk with arbitrarily high derivatives (gauge, diffeomorphism)
• proof to all orders in the coupling constant expansion
✓h�Jµih�Jµ
5 i
◆=
✓�JB �J!
�J5B �J5!
◆✓Bµ
!µ
◆
�J5B = �2�µ �JB = �2�µ5,
�J5! = µ25 + �µ2 + 2�(2⇡T )2 �J! = 2�µ5µ
S =
Zd
5x
p�g {L [Aa, Va, gab,�i] + LCS [Aa, Va, gab]}
LCS [Aa, Va, gab] = ✏abcdeAa
⇣3FA,bcFA,de + �FV,bcFV,de + �Rp
qbcRqpde
⌘
19
B E Y O N D H Y D R O DY N A M I C S
W E A K C O U P L I N G ( K I N E T I C T H E O R Y )
• coupling constant dependence of non-hydrodynamic transport [S. G., Kaplis, Starinets, JHEP 1607 (2016) 151 arXiv:1605.02173]
• instead of hydrodynamics, start with (weakly coupled) kinetic theory
• essential concept: quasi-particles
• Boltzmann equation
• close to equilibrium
• resulting equation
• solution
21
@F
@t+
pim
@F
@ri� @U(r)
@ri@F
@pi= C[F ]
F (t, r,p) = F0(r,p) [1 + '(t, r,p)]
@'
@t= �pi
m
@'
@ri+
@U(r)
@ri@'
@pi+ L0[']
'(t, r,p) = etL '0(r,p) =1
2⇡i
�+i1Z
��i1
est Rsds'0(r,p)
Rs = (sI � L)�1
W E A K C O U P L I N G ( K I N E T I C T H E O R Y )
• ansatz for homogeneous eq. distribution
• eigenvalue equation for lin. coll. operator
• hierarchy of relaxation times
22
'(t,p) = e�⌫th(p)
�⌫h = L0[h]
'(t,p) =X
n
Cne�⌫nthn(p)
dominant:
⌧R = 1/⌫min
T H E R M A L I S AT I O N ( R E L A X AT I O N )
• KGB equation
• kinetic theory predicts
• relaxation time bound [Sachdev]
• Ising model (BTZ)
⌘ = ⌧R s T
⌧R � C ~kBT
GR(!, q) =
C�⇡ �
2(�� 1) sin⇡�
�����✓�
2
+
i(! � q)
4⇡T
◆�
✓�
2
+
i(! + q)
4⇡T
◆����2
⇥"cosh
q
2T� cos⇡� cosh
!
2T+ i sin⇡� sinh
!
2T
#
! = ±q � i4⇡T
✓n+
�
2
◆
⌧R =1
2⇡�
~kBT
23
@F
@t+
pim
@F
@ri� @U(r)
@ri@F
@pi= �F � F0
⌧R
cute: � = 2 ) “⌘/s” = 1/4⇡
F U L L Q U A S I N O R M A L S P E C T R U M • weak/strong coupling (perturbative/holographic quasi-normal
mode calculations) [Hartnoll, Kumar (2005)]
qq-q-q
-i4πT-i4πT
-i8πT-i8πT
-i12πT-i12πT
-i16πT-i16πT
Reω
Imω
Reω
Imω
� ! 0 � ! 1
24
R E S U LT S : Q N M S T R U C T U R E
• different trends depending on
• poles become denser (branch cut)
• new poles on imaginary axis
x xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxxx
◆
◆
◆
ReωImω
x xx
x xx
x xx
x xx
x xx
x xx
x xx
x xx
x xx
x xx
x xx
xx
◆
◆
◆
ReωImω
⌘/s
⌘/s > ~/4⇡kB ⌘/s < ~/4⇡kB
``weak limit”
� ! 1�GB ! �1
``anomalous limit”�GB ! 1/4
25
Q N M S T R U C T U R E• shear channel QNMs in N=4
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◆
◆
◆
ReωImω
x xx
x xx
x xx
x xx
x xx
x xx
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◆
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26
Q N M S T R U C T U R E• shear channel QNMs in Gauss-Bonnet
x xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxxx
◆
◆
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ReωImω
x xx
x xx
x xx
x xx
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◆
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ReωImω
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K I N E T I C T H E O R Y R E S U LT• kinetic theory behaviour quickly approached
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ReωImω
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ReωImω
⌘/s ⇠ const ⌧RT
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H Y D R O DY N A M I C S• breakdown of hydrodynamics (diffusion) qc ⇠ �3/4
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H Y D R O DY N A M I C S• breakdown of hydrodynamics (diffusion)
⟩
⟩
qc ⇠ �3/4
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H Y D R O DY N A M I C S• breakdown of hydrodynamics (sound)
⟩
⟩
qc ⇠ �3/4
31
H Y D R O DY N A M I C S• breakdown of hydrodynamics due to new poles
• shear and sound correlators in Gauss-Bonnet [S. G., Starinets (2016) arXiv:
1611.07053]
32
Gtt,tt(!, q) =3p2⇡4T 4
(1 + �GB)3/225
�5q2 � 3!2
�(1� !/!g)� i�2
GB!q2/⇡T
(3!2 � q2) (1� !/!g) + i�2GB!q
2/⇡T
!
Gxz,xz
(!, q) =
p2⇡3T 3�2
GB
(1 + �GB)3/225
✓!2
i! � i!2/!g � �2GBq
2/4⇡T
◆
• dispersion relations
!1 = �i�2
GB
4⇡Tq2 !2 = !g + i
�2GB
4⇡Tq2
!1,2 = ± 1p3q � i
�2GB
6⇡Tq2 !3 = !g + i
�2GB
3⇡Tq2
!g = � 8⇡Ti
�GB (�GB + 2)� 3 + 2 ln⇣
2�GB+1
⌘ ⇡ �8⇡Ti
�2GB
• gap
Q U A S I - PA R T I C L E S ( T R A N S P O R T P E A K )
• quasi-particles appear in the spectrum [Casalderrey-Solana, S. G., Starinets (2017)]
• weak coupling (Boltzmann equation)
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• holographic results in N=4 theory (scalar channel)
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Q U A S I - PA R T I C L E S ( T R A N S P O R T P E A K )
• holographic results in a dual of the Gauss-Bonnet theory (scalar channel)
35
Q U A S I - PA R T I C L E S ( T R A N S P O R T P E A K )
Q U A S I - PA R T I C L E S I N N = 436
Q U A S I - PA R T I C L E S I N G A U S S - B O N N E T37
• quasi-particles appear also in other channels (shear)
38
• how do these peaks manifest themselves (if at all) in a weakly coupled field theory like QCD?
Q U A S I - PA R T I C L E S I N G A U S S - B O N N E T
H E A V Y I O N C O L L I S I O N S
H E A V Y I O N C O L L I S I O N S• R2 coupling constant corrections to shockwave collisions
[S. G., van der Schee (2016) arXiv:1610.08976]
• energy density along the longitudinal direction after collision: less stopping of narrow shocks (88% higher energy density on lightcone) and decreased energy density of wide shocks
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H E A V Y I O N C O L L I S I O N S• delayed hydrodynamisation
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thydThyd = {0.41� 0.52�GB,
0.43� 6.3�GB}
at �GB = �0.2 :
{25%, 290%}
H E A V Y I O N C O L L I S I O N S
• rapidity profile: start wider and smaller but later become comparable
42
H E A V Y I O N C O L L I S I O N S• change in entropy density: enhanced on lightcone, negative in plasma
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• total entropy: reduced at intermediate coupling
C O N C L U S I O N A N D F U T U R E D I R E C T I O N S
• various weakly coupled properties are recovered remarkably quickly
• the topic of this workshop: how precisely do they match onto those
computed from weakly coupled, perturbative physics?
• use interpolations between weakly coupled physics and coupling-
dependent holography to understand intermediate coupling
• can we see a signature of higher-QNMs (quasi-particles) in
experiments?
• a harder task for the future: understand 1/N corrections
44
T H A N K Y O U !