Large N Tensor Models
G. Tarnopolsky
Talk at PCTS workshop
‘’New Developments in Conformal Field Theory
AwayAbove
From Two Dimensions’’
Princeton University
March 7, 2017
Talk mostly based on
• Igor Klebanov, GT
“Uncolored Random
Tensors, Melon
Diagrams, and the
SYK models,”
arXiv:1611.08915
• These slides were
prepared in
collaboration with
Igor Klebanov.
Plan
1. Sachdev-Ye-Kitaev model and Uncolored Tensor model
and melonic Feynman diagrams
2. Tensor Models as Discretized 3-Geometries
3. Why melons dominate at large N in Tensor Models
4. 2pt function and 4pt function and spectrum of the twoparticle operators
5. Other Large N models
Sachdev-Ye-Kitaev Model
• Majorana fermions
•
Uncolored Tensor Model
• Majorana fermions
are Gaussian random
• No disorder
• Has O(N) symmetry after
averaging over disorder
• Has O(N)a x O(N)b x O(N)c symmetry
χ a1 b1 c1
c1
a1
b1
b2
Sachdev, Ye ‘93,
Georges, Parcollet, Sachdev’01
Kitaev ‘15
χ a1 b2 c2
c2
χ a2 b2 c1
a2
χ a2 b1 c2
Carrozza, Tanasa’15
Klebanov, GT’16
Melonic Graphs
• In the SYK and Uncolored Tensor Model only melon
graphs survive in the large N limit.
• Remarkably, these graphs may be summed explicitly,
so the “melonic” large N limit is exactly solvable!
• The dual structures of melon graphs are trees
• Here is the list of vacuum graphs up to 6
vertices Kleinert, Schulte-Frohlinde
• Only 4 out of these 27 graphs are melonic.
• The number of melonic graphs with p vertices
grows as cp Bonzom, Gurau, Riello, Rivasseau’11
Discretized 3-Geometries
• The 3-geometry interpretation
emerges directly as we
associate each 3-index tensor
with a face of a tetrahedron
• Wick contractions glue
a pair of triangles in
a special orientation:
red to red, blue to blue,
green to green.
χ a1 b2 c2
χ a2 b1 c2
c2
b1
a1
χ a1 b1 c1
b2
c1
a2
χ a2 b2 c1
Ambjorn, Durhuus, Jonsson’91
Sasakura’91
M. Gross ’91
• Feynman graphs tell us how to glue tetrahedra
• The original ‘91 models involved 3-index tensors transforming
under a single U(N) or O(N) group. Their large N limit is hard
to analyze.
Ambjorn, Durhuus, Jonsson’91
Sasakura’91
M. Gross ’91
• A new “melonic” large N limit in models
with multiple O(N) symmetries was discovered
’10
by Gurau, Rivasseau, … only in 2010! Gurau
Gurau, Rivasseau ’11
Bonzom, Gurau, Riello, Rivasseau ’11
• Recently Witten proposed to promote
d=0 models to d=1 Quantum Mechanics of Majorana fermions
Why melons dominate at large N
• For a 3-tensor with distinguishable indices the
propagator has index structure
• It may be represented graphically by 3 colored
a
a
b
b
wires c
c
• An interaction with O(N)a x O(N)b x O(N)c
a bc
symmetry
1 1 1
a1
b2
c2
c1
b2
a2
c2b1a2
Melonic Stranded Graphs
• The Feynman graphs of the quartic field theory
may be resolved in terms of the colored wires
(triple lines)
Non-Melonic Graphs
• Most Feynman graphs in the quartic field
theory are not melonic are therefore
subdominant in the new large N limit, e.g.
• Scales as
Large N Scaling
• Feynman diagram scales as
• ‘’Forgetting ” one color we get a double-line graph.
• The number of loops in a double-line graph is
is the Euler characteristic,
is the number of edges,
is the number of vertices,
• If we erase the blue lines we get
• Adding up such formulas, we find
• The total number of loops is
• The genus of a graph is
• Since
, maximal number of loops :
• Max diagram scales as
•
must be held fixed in the large N limit.
2-point function
• The two-point function obeys the SchwingerDyson equation like in SYK model
...
...
• Neglecting the bare term in IR we find
Sachdev, Ye’ 93
Georges, Parcollet, Sachdev ’01
Kitaev ‘15
Polchinski, Rosenhaus ’16
Maldacena, Stanford ’16
Jevicki, Suzuki, Yoon ’16
4-point function
• Four point function
t1
t3
...
t2
t4
t1
t3
...
...
t2
t4
• If we denote by
the ladder with n rungs
Kitaev ‘15
Polchinski, Rosenhaus ’16
Maldacena, Stanford ’16
Jevicki, Suzuki, Yoon ’16
Spectrum of two-particle operators
• For the two-particle operators
we derive S-D equation
χ
O2n
χ
Gross, Rosenhaus ‘16
• Assume that in IR 3pt function has conformal form
where
and is anomalous dimension of
• Then S-D equation, which determines is
• The 3pt function with an arbitrary
is an eigenvector of the kernel
• To compute
integral
we use basic conformal
Symanzik ‘72
• We find
• Scaling dimensions determined by
• Graphical solution for
• The first solution is h=2; dual to gravity.
• The higher scaling dimensions are
approaching
Maldacena, Stanford ’16
Gross, Rosenhaus ’16
Complex SYK Model
• Complex fermions
•
Complex Tensor Model
• Complex fermions
are Gaussian random
• No disorder
• Has U(N) symmetry after
averaging over disorder
• Has U(N)a x U(N)b x O(N)c symmetry
χ †a1 b1 c1
c1
a1
b1
b2
Sachdev ’15
Davison, Fu, Gu, Georges, Jensen, Sachdev ‘16
χ a1 b2 c2
c2
χ †a2 b2 c1
a2
χ a2 b1 c2
Klebanov, GT’16
Spectrum of two-particle operators
• S-D equation in IR for two-particle operators
including
• The integral equation also admits symmetric solutions
• Calculating the integrals we get
• Graphical solution for
• The first solution is h=1 corresponding to
U(1) charge
• The additional scaling dimensions
approach
Gross-Rosenhaus Model
q=4, f=4
• Majorana fermions
•
Gurau-Witten Model
• Majorana fermions
are Gaussian random
• No disorder
• Has O(N) x O(N) x O(N) x O(N)
symmetry
• Has O(N)a x O(N)b x O(N)c x O(N)d
x O(N)e x O(N)f symmetry
χ 1ade
e
b
Gross, Rosenhaus’ 16
χ 0abc
χ 2f be
d
a
c
f
χ 3f dc
Gurau ‘10
Witten’16
Gross-Rosenhaus Model
q=4, f=2
• Majorana fermions
•
Tensor Model
• Majorana fermions
are Gaussian random
• No disorder
• Has O(N) x O(N) symmetry
• Has O(N)a1 x O(N)a2 x O(N)b x O(N)c
symmetry
χ 0a b c
1 1 1
b1
c1
a
c2
Gross, Rosenhaus’ 16
χ 0a1 b2 c2
b2
χ 1a2 b1 c2
a2
χ 1a2 b2 c1
Bosonic Tensor Model in General d
• Action with a potential that is not positive
definite
• Schwinger-Dyson equation for 2pt function
Patashinsky, Pokrovsky ‘64
• Has solution
Spectrum of two-particle spin zero
operators
• Schwinger-Dyson equation
• Assume that in IR 3pt function is conformal
• We find
• Spectrum in d=1 again includes scaling
dimension h=2
• However, the leading solution is complex,
which suggests that the large N CFT is
unstable Giombi, Klebanov, GT (work in progress)
• Roughly, it corresponds to the operator
• The dual scalar field in AdS violates the
Breitenlohner-Freedman bound.
Conclusions
• Early exploration of the Random Tensor Models was to
study discretized quantum geometries in d>2.
• For models with multiple symmetry groups, like O(N)6 or
O(N)3 new solvable large N limits were later discovered,
dominated by the melonic Feynman graphs.
• Tensor models in 0+1 dimensions are similar to the SYK
models, quantum mechanics with random coupling.
• Gauging the Tensor QM removes the non-singlet states.
• Leaves a rich spectrum of gauge invariant operators
including the h=2 operator dual to gravity.
Thank you for your attention!