chern-simons-matter theories, high-spin gravity, and 3d ... · chern-simons-matter theories,...
TRANSCRIPT
Chern-Simons-Matter Theories, High-Spin Gravity,
and 3D Bosonization
Guy Gur-AriWeizmann Institute, Israel
The Galileo Galilei Institute, Florence, May 2013[Aharony, GA, Yacoby 2011, 2012] [GA, Yacoby 2012][Aharony, Giombi, GA, Maldacena, Yacoby 2012]
Vector Models in 3D
• Complex scalars
• Singlet sector of
• Single-trace primaries:
i = 1, . . . , N
s = 0, 1, 2, . . .Js = �†
i@ · · · @�i
U(N)
�s = s+ 1
L = @µ�†i@µ�
i +�6
N2(�†
i�i)3
• Singlet sector of
• Single-trace primaries:
Vector Models in 3DL = i�
µ@µ i
J0 = Js = �@ · · · @ �0 = 2
�s = s+ 1
s = 1, 2, . . .
U(N)
Outline
• 3D vector models ↔ high-spin gravity
• Matter + Chern-Simons interactions
• 3D bosonization
Vasiliev’s Theory
• Defined on AdS4
• Scalar, photon, graviton, ...
• Consistent, classical interacting theory
dx
A+ A ⇤ A =
1
4+ ei✓0B ⇤K
�dz2 + c.c.
dx
B + A ⇤B �B ⇤ ⇡(A) = 0
Holographic Duality
[Sezgin, Sundell 2001]
GN ⇠ 1
NJ0, Js ! B, A ✓0 = 0
✓0 = ⇡/2
Boundary scalars:
Boundary fermions:
• dimension:
• Currents not renormalized at large N
• Bulk: change of scalar boundary conditions
Critical Vector Model
Wilson-Fischer
[Klebanov, Polyakov 2002]
Free Scalars�L ⇠ (�†�)2
J0 �IR0 = d��0 = 2
Detailed Evidence
• Conformal theory in flat 3D space + high-spin symmetry ⇒ free theory[Maldacena, Zhiboedov 2011]
• All 3-point functions agree[Giombi, Yin 2010]
Motivation• Weak-weak duality
• Tensionless limit of string theory[Chang, Minwalla, Sharma, Yin 2012]
• Boundary scalars and fermions continuously connected
Bulk Suggests Bosonization
Boundary Scalars
BoundaryFermions
Boundary scalars / fermions are connected
0 ⇡/2
dx
A+ A ⇤ A =
1
4+ ei✓0B ⇤K
�dz2 + c.c.
(Parity)✓0
Bulk Suggests Bosonization
Free ScalarsJ0 = �†�
� = 1
Critical Scalars� = 2
Critical Fermions� = 1
J0 =
� = 2
Free Fermions
0 ⇡/2✓0 (/P)
✓0 (/P)
Matter +Chern-Simons
Interactions
Why Chern-Simons?
• symmetry already gauged
• No additional primary operators
Smatter +ik
4⇡
ZTr
✓A ^ dA+
2
3A3
◆
� =N
kN, k ! 1
U(N)
• Chern-Simons level k is quantized
• deformation marginal at large N
• Theory is conformal at large N
Conformal Symmetry
(�†�)3
[Giombi, Minwalla, Prakash, Trivedi, Wadia, Yin 2011][Aharony, GA, Yacoby 2011]
Smatter +ik
4⇡
ZTr
✓A ^ dA+
2
3A3
◆
Fate of the High-Spin Symmetry
Currents not renormalizedat large N
@ · hJsJs0i = 0
Symmetry is broken, not renormalized
@ · hJsJs0Js00i 6= 0
�� s = 3
J0
@ · Js ⇠ 0 +1
NJs�2J0 + · · ·
3D BosonizationFree Scalar
J0 = �†�
� = 1
Critical Scalar� = 2
Critical Fermion� = 1
J0 =
� = 2
Free Fermion
0 ⇡/2✓0 (/P)
✓0 (/P)
• General theory, high-spin symmetry ‘slightly’ broken:
• Broken Ward identities constrain correlators
• 3-point functions of single-trace operators fixed by two parameters
CFT Perspective
[Maldacena, Zhiboedov 2012]
hJJJi�=1
= N
"1
1 + �2
h·ibos. +
�2
1 + �2
h·ifer. +
�
1 + �2
h·iodd
#N , �
?
@ · Js ⇠ Js�2J0/N + · · ·
N , � $ N,� = N/k
Computing Exact Correlators
hJ0J0i = J0 J0 + · · ·
hTT i , hJ0J0J0i , · · ·
[Aharony, GA, Yacoby 2012] [GA, Yacoby 2012]
h��†i
h��†��†i
Degrees of FreedomhTT i
hTT ifree=
sin(⇡�)
⇡�
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
l
(� = N/k)
[Aharony, GA, Yacoby 2012] [GA, Yacoby 2012]
Exact Correlators
✓0 =⇡�
2Bulk coupling:
hJJJi�=1
= N
"1
1 + �2
h·ibos. +
�2
1 + �2
h·ifer. +
�
1 + �2
h·iodd
#
=
2N sin(⇡�)
⇡�
"cos
2
✓⇡�
2
◆h·i
bos. + sin
2
✓⇡�
2
◆h·i
fer.
+
sin (⇡�)
2
h·iodd
#
[Aharony, GA, Yacoby 2012] [GA, Yacoby 2012]
[Chang, Minwalla, Sharma, Yin 2012]
hJ0J0J0i, . . .(� = N/k)
• Comparing correlators gives relation
• Level-rank duality in terms of
Bosonization Duality
k0 = k � sign(k)N
Scalar + U(N)k0
Critical Fermion + U(k0)�N
(Scalars) (Fermions)N,� $ N,�
� = 0
� = 1 � = 0
� = 1
(� = N/k)
Thermal Free Energy• Thermal cycle can support non-trivial
holonomy
• Eigenvalues hold gauge-invariant info
[Aharony, Giombi, GA, Maldacena, Yacoby 2012]
IA 6= 0
NV T 2 � N2
Matter sectorAttractive potential
Pure Chern-Simons
F/T :
Non-Trivial Holonomy
[Aharony, Giombi, GA, Maldacena, Yacoby 2012]
T 2 ⇥ R
SCS ! k
2⇡
Zdt a b
Reduce on
b ⇠= b+ 2⇡ ) �a =2⇡
k
Eigenvalues are quantized & fermionic,Spread evenly: 2⇡
kN = 2⇡�
!
!" "!"# "#
⇡b =k
2⇡a
[Douglas, 94]
a, b =
I
⌃1,2
A =
✓. . .
◆
1,2
Thermal Free Energy
• Free energy computed by summing planar diagrams
• Consistent with bosonization
[Aharony, Giombi, GA, Maldacena, Yacoby 2012]
Ff = �NV2T 3
2⇡2i�
µ2
3Li2
��e�µ+⇡i�
�+
Z 1
µdy y Li2
��e�y+⇡i�
�� c.c.
�
(1� �)µ =1
⇡i
⇥Li2(�e�µ�⇡i�)� c.c.
⇤Thermal mass
Open Questions• Better understand bosonization
• 1/N
• Mapping of fundamental fields
• Better understand holographic duality
• Singlet sector implementation
• Pure Chern-Simons contributions[Banerjee, Hellerman, Maltz, Shenker 2012]
• Vector models + Chern-Simons are dual to Vasiliev’s theory with broken parity
• 3D bosonization
• Single-trace 3-point functions, thermal free energy
Summary
Thank You
Scalar + U(N)k0
Critical Fermion + U(k0)�N
� = 0
� = 1 � = 0
� = 1