1. health and fitness

Locality and Classical
Fields in AdS,
from the Bootstrap
Jared Kaplan
Johns Hopkins University
based on various work with Liam Fitzpatrick,
David Poland, David Simmons-Duffin,
Matt Walters, & Junpu Wang
Motivation
Understand the universal features !
of AdS quantum gravity directly from the CFT:
• Fock space? AdS Locality? Long-Range Forces?!
• Effective Field Theory description in AdS?!
• Classical Field Configurations in AdS?!
• Black Hole Microstates - Temperature, Blackness…
Which of these require further assumptions/restrictions!
beyond Conformal Symmetry and QM?
CFT
EikonalCentral
limit. In the
caseQuestion
of CFT2 the resu
forbut
This
Talk
been observed,
we will
see that the gen
t will be sufficient to study just a few prima
What can we say about the CFT Spectrum, OPE
to ascoefficients, and correlators assuming only that:
O1 , O2 , T
with Oi (x)Oi (0)
T
s included inAnswer:
the OPE’.
use
Quite aWe
bit, all
of it ! 1 , 2 , T t
with
an
AdS
Interpretation.
d ⌧i and ⌧T to refer to their twists ⌧ ⌘
on or ‘light’ operator, such as the stress t
!
!
AdS/CFT
Kinematics
Energies and Dimensions
in AdS/CFT
ig:AdSCylinderIntro
AdS
CFT
t
exp
t
R
⇥
Figure 1: This figure shows how the AdS cylinder in global coordinates corresponds to the
CFT in radial quantization. The time translation operator in the bulk of AdS is the Dilatation
operator in the CFT, so energies in AdS correspond to dimensions in the CFT.
HAdS = DCF T
for general scalar theories at tree-level and for ⇥4 theory at one-loop. Recently we [2] verified
the conjecture for n-pt amplitudes in general scalar theories at tree-level by showing that our
diagrammatic rules for the Mellin amplitude reduce to the usual Feynman rules in the flat space
limit. By setting up the appropriate scattering experiment [10, 11, 12, 13] in AdS and making
Representations of
Global Conformal Alg
The momentum and special conformal generators act as
raising and lowering operators wrt Dimension
[D, Pµ ] = Pµ
[D, Kµ ] =
Kµ
Irreducible reps built from primaries:
[Kµ , O(0)] = 0 or Kµ |
Oi
=0
Can derive a unitarity relation for ⌧ =
d
1 and ⌧` d 2
s
2
`
Conformal Symmetry
and Primary States
AdS
CoM of Ground State
CFT
CFT Primary State
Excited/Descendant
States
AdS
n,` (t, ⇢, ⌦)
Center of Mass !
for Excited State
CFT
2n
@ @µ1 · · · @µ` O |0i
Descendant of a!
Primary
Two Particle States
AdS
n,` (ti , ⇢i , ⌦i )
Two Particle State,!
CoM at Origin
CFT
O@ 2n @µ1 · · · @µ` O |0i
`Double-Trace’ Primary
!
!
Long-Distance
AdS Locality
is Completely
Universal
Two Notions of Locality
AdS has an intrinsic length scale:
2
ds =
cosh
2
✓

RAdS
◆
2
2
RAdS
dt + d + sinh
2
✓

RAdS
◆
d⌦2
Long-Distance Locality: Particles/Systems
decouple when separated by !D
RAdS
!
Micro-Locality: There exists a good Effective Field
1
Theory Description in AdS with ⇤EF T
RAdS
Formal Definition of
Long-Distance Locality?
c
=
=) 9
d
cd
=
=
Statement about structure of the Hilbert Space:
HAdS = HCF T ⇡ Fock Space
A Fock Space
at Large Separation
How to Define
Distant Furballs?
AdS
Geodesic separation between cat & dog:
 ⇠ RAdS log `
Are there two furball states at!
large angular momentum???
Two Furball Physics?
AdS
AdS Energy = CFT Dimension:
En` = Ec + Ed + 2n + ` + (n, `)
Anomalous dimension, (n, `),!
is a kind of `binding energy’.
Existence of states as ` ! 1 with vanishing!
(n, `) implies AdS Cluster Decomposition.
General Theorem
(Any CFT in d>2)
Consider OPE of any two scalar primary operators:
O1 (x)O2 (0) =
X
,`
c12,` O
,` (0)
For each n there exists a tower of operators
O
,`
with
=
1
+
2
+ 2n + ` + (n, `)
as ` ! 1, where the anomalous dimensions
(n, `) !
n
`⌧m
or
ne
⌧m 
from leading twist exchange, generically Tµ⌫
The Idea of the Proof:
A Scattering Analogy
Free propagation and massless exchange!
require large amplitude at large ` , e.g.
1
1
cos ✓
⇡
X
P` (cos ✓)
`!1
Completely analogous CFT phenomenon.!
Implies existence of large ` states.
Partial Wave Amplitudes -> Conformal Partial Waves
The Conformal Partial
Wave Decomposition
Insert
1, organize according to!conformal symmetry:
hO1 O2
X
↵
|↵ih↵| O3 O4 i
Since operators = states in the CFT, write 4-pt as
=
X
O↵
↵
A sum over exchange labeled by primary operators,!
magnitude given by product of 3-pt correlators.
O2
O2
Light-Cone OPE Limit
(Diagrams = Blocks)
The equivalent of a t-channel singularity!
in the CFT is the light-cone OPE limit:
hO1 O1 O2 O2 i = u
1+
2
2
⇣
1+u
⌧T
2
fT (v) + · · ·
giving a bootstrap equation
O21
O
O1
+
O1O2
+
1
T
X
O2
1
+
[TT]
n,`
O21
O
O2
O1
O1
OO
2 1
O21
O
X
X
+T · · · +
⇡ [TT]
n,`
n,`
O1
O2
O2
O1
n,`
`!1
O2
+ ·[O· · O ⇡
]
O2O1
1
2 n,`
O2
⌘
O1
X
`!1
O1
Example: long-range gravity is completely universal.
[O
!
!
Micro-Locality
for
Special CFTs
Micro-Locality
(or flat space locality)
Showed that All CFTs have a Fock Space spectrum!
at long-distances. Would need at short-distances too.
To make progress, need perturbative expansion.!
(Our definition is in terms of AdS EFT.)
Both are special features, invalid for most CFTs.
Micro-Locality
(or flat space locality)
Need a 3rd special feature. Counter-example:
1
2
L = (@ )
2
1 ⇥
2
N
4
+
4
RAdS (@
4
) + ···
⇤
To see what went wrong, look at flat space S-Matrix:
1
A= 2
N
+ R4 (s2 + t2 + u2 ) + R8 s4 + R12 S 6 + · · ·
Breaks down at the scale R , despite weak coupling.
EFTs well-approximated by polynomial Amplitudes.
Mellin Space
as AdS Momentum Space
Can define Mellin Amplitude for CFT correlator:
hO(x1 ) · · · O(x4 )i /
Z
dsdt ˜
M
(s,
t)u
(2⇡i)2
s
v
t
Mellin variables direct analog of Mandelstam s and t.
Mellin Amplitude is `momentum space’ for AdS.!
Asymptotic bounds —> EFT description in AdS.
!
!
Classical AdS
Fields and
`Eikonalization’
Multiple Exchanges and
Classical Fields
In any spacetime, effective classical fields!
emerge from the summation of infinite!
series of simple diagrams, such as:
T
T T T
GM
=) V (r) = d 2
r
O1
O1
Can get Schwarzschild by including interactions.
wn as conformal blocks, and the CFT bootstrap e
Inter-related
in a relevant context in [] and in many other rece
Questions:
the conformal partial waves associated with the e
Doesor
the‘eikonalized’
Bootstrap know
about
the multiple exchanges
mmed
into
an exponential
form. Di
that build up a classical field in AdS?!
been explored [] at high energy with fixed impact
!
f the
Eikonal
limit.
In
the
case
of
CFT
the
resumm
2
Why are Mellin amplitude asymptotics important?!
lready
been observed, but we will see that the gener
!
1
Can
perturbation
theory
further?!
oses
it we
willtake
be sufficient
to study
just
a few primary
`
!
refer to as
How far can we go with no assumption except:!
!
O1 , O2 , T
with Oi (x)Oi (0)
T
t’s again consider the anomalous dimensions of the double-trace o
again consider the anomalous dimensions of the double-trac
large `, which arise due to the minimal twist operators present in
rge `, which arise due to the minimal twist operators present
(x)O1 (0) and O2 (x)O2 (0). The resulting shift in scaling dimension c
)O1 (0) and O2 (x)O2 (0). The resulting shift in scaling dimensio
the approximate form
A Bootstrap Implication
e approximate form
O1
O2
O1
O2
O1
+
+
O2
X
(n, `) ⇡
n
,
⌧
` mn
⇡
`⌧
+ · ·`)· ⇡
(n,
O1
X
,
m
O2
[O1 O2 ]n,`
[TT]
T
1
`!1
ere ⌧m
exchanged operators.
Using th
O is the
O
OtwistO of these
O minimal
O
O
O
at
can expand
[O1 O2exchanged
]n,` conformal
blocks as Using
a per
e ⌧atm large
is the` we
minimal
twist the
of these
operators.
n,
at`),large
` weright-hand
can expand
the [O1 O2 ]n,` conformal
blocks
On the
side,
anomalous
dim: as a
n,`
n,`
1
2
`),
1
2
✓
g⌧,` (v, u) ⇡✓ 1 +
g⌧,` (v, u) ⇡
1+
n
2`⌧m
n
2`⌧m
1
2
ln v +
2
n
8`2⌧m
ln v +
2
ln v + · · ·
2
n
8`2⌧m
2
◆
1
2
g◆
⌧n ,` (v, u)
ln v + · · ·
(
g⌧n ,` (v, u)
discussed
above,
term this
in this
reproduces side?
the s-channel
How
canthe
wefirst
match
onseries
the left-hand
e identity, while the second term contains a logarithmic singularity
iscussed above, the first term in this series reproduces the s-chan
Not
because
Bootstrap
is `on-shell’.
atches that
of obvious
the minimal
twist conformal
blocks.
For the sake of s
dentity, while the second term contains a logarithmic singula
rem [] referred to above states that in the OPE A(x)B(0) of a
re exist new primaries [AB]n,` labeled by positive integers n, `
A +
B + 2n + ` + (n, `), where the anomalous dimension
rescribed power-law rate. This immediately implies the existe
Be Fruitful and
Multiply…
Fock Space Theorem implies existence of:
[O1 O2 ]n,` , [O1 T ]n,` , [T T ]n,` , · · · , [[O1 O1 ]n,` T ]n0 ,`0 , · · ·
So at large
spin,`.weApplying
are allowed
to ask about:
combinations
at large
the theorem
recursively
O1
+
O2
O1
+
T
O2
X
–[TT]3n,`–
n,`
O1
O2
O1
O2
+···
⇡
X
`
A Needed OPE Limit
We would like to use the OPE to deduce at large spin:
O1
O1
T
O1
!
T
O1
T
O1
?
!
O1
[T T ]n,`
O1
Without further assumptions, ill-defined!!!
!
Cross-channel OPE limits of Blocks do not exist.
A Needed OPE Limit
O1
O1
T
O1
!
T
O1
T
O1
?
!
O1
[T T ]n,`
O1
Need to expand a function such as
2 F1 (1, 1, 2, z)
Near
=
log(1
z
z)
z=1
One reason why crossing symmetry so non-trivial!
Needed OPE Limit,
Understood from Mellin
Toy example:
G(z) =
Z
i1
M( ) z
d
i1
Both OPE limits, z = 0 and z =
studied simultaneously.
1 can
be!
Behavior near
z=0
Behavior near
z = 1 determined by asymptotics.
determined by poles.
A Needed OPE Limit,
Understood via Mellin
O1
O1
T
O1
!
T
O1
T
O1
?
!
O1
[T T ]n,`
O1
Pathology due to bad asymptotic behavior.!
!
Result controlled by leading Mellin pole,!
which gives the universal behavior expected!
from the bootstrap equation.
For any CFT, Conformal
Blocks can be Summed
If Mellin Amplitude is exponentially bounded, we can
sum or `Eikonalize’ the conformal blocks:
O1
O2
O1
+
1
O2
O1
+
T
O2
X
+···
[TT]n,`
n,`
O1
O2
O1
O2
O1
O1
O2
⇡
X
O2
[O1 O2 ]n,`
`!1
O1
O2
AdS QFT aka 1/N Perturbation Theory is a special case.
For any CFT, Conformal
Blocks can be Summed
O1
O2
O1
+
1
O2
O1
+
T
O2
X
+···
[TT]n,`
n,`
O1
O2
O1
O2
O1
In the limit
e
O1
O2
⌧T ⌧
⇡
1,
X
O2
[O1 O2 ]n,`
`!1
O1
2
T T
c11 c22 gT (u,v)
Block `eikonalizes’. Beyond this limit, !
more general series.
O2
Future
Directions
and
Conclusions
Conclusions
• Long-Range AdS Locality Completely General!
• Eikonalization can be seen directly in the bootstrap!
• Relation between Eikonalization/Classical Fields
and Short-Range locality!
• Can begin to improve large spin expansion
Future Directions:
Physics of CFT Spectra
AdS Black
Holes
Eikonal
twist
`
Lightcone OPE
Numerical
Bootstrap
spin `
Need to develop and apply new techniques to
understand the full CFT spectrum.
!
!
Extra Slides
!
What is the Bootstrap?
• Conformal Symmetry!
• Unitarity!
structure constants of the schematic form
X
X
f12k f34k (. . .) =
f14k f23k (. . .) .
• Crossing Symmetry
k
(3.3)
k
The (. . .) factors are functions of coordinates xi , called conformal partial waves. They are
produced by acting on the two-point function of the exchanged primary field k with the
di↵erential operators C appearing in the OPE of two external primaries. Thus, they are also
fixed by conformal invariance in terms of the dimensions and spins of the involved fields.
What can we learn from the fundamental principles?
1
X
f12k
4
k
f34k
k
1
=
X
k
k
2
3
f14k
4
2
f23k
3
Figure 1: The conformal bootstrap condition = associativity of the operator algebra.
nformal invariance relates the OPE coefficients of all operators in the same irr
nformal multiplet, and this allows one to reduce the sum above to a sum over
educible multiplets, or “conformal blocks”. When this expansion is performed in
ur-point function, the contribution of each block is just a constant “conformal blo
ent” PO / c2O for the entire multiplet times a function of the xi ’s whose functio
pends only on
the spinthat
`O and
dimension
lowest-weight
(i.e. “primary”)
O of the
Recall
4-pt
correlators
can
be written
the multiplet:
Consider the 4-pt
CFT Correlators
A(u,
v)
X
1
h
(x
)
(x
)
(x
)
(x
)i
=
1
2
3
4
2 2 P g
h (x1 ) (x2 ) (x3 ) (x4 )i =
(u, v),
2 (x
2 13 x24 ) O ⌧O ,`O
(x x )
12 34
ere xij = xi
O
the conformal
cross-ratios
xwhere
`O , and
j , the twist of O is ⌧O ⌘
O
✓ 2 2 ◆
✓ 2 2 ◆
x12 x34
x14 x23
u=
,
v=
,
2 2
2 2
x24 x13
x24 x13
are
We can use elementary 2quantum mechanics!
to rewrite this in a different way...
Formulate
CFT Bootstrap
e conformally invariant cross-ratios. The functions g⌧O ,`O (u, v) are also usually
conformal blocks or conformal partial waves [32–35], and they are crucial elem
ients in the bootstrap program.
the above, we took the OPE of (x1 ) (x2 ) and (x3 ) (x4 ) inside the four-point f
ne can also take the OPE in the additional “channels” (x1 ) (x3 ) and (x2 )
structure
constants
schematic
formthe bootstrap equation is the constraint that the decom
(x
(xof2 )the(x
4 ) and
3 ), and
X
X
Crossing
symmetry
the Bootstrap
Equation:
f f (. . .) =
f gives
f (. . .) .
(3.3)
erent channels
matches:
X
The (. . .) factors are 1
functions of
coordinates x , called conformal partial waves.
1 They areX
produced by acting on the two-point P
function
the(u,
exchanged
with the
v) =primary field
PO g⌧O ,`O (v, u).
O g⌧Oof,`O
2 2
2 2
di↵erential operators
in the OPE of two external primaries.(x
Thus,
(x12 xC34appearing
)
14 xthey
23 )are alsoO
O
12k 34k
14k 23k
k
k
i
k
fixed by conformal invariance in terms of the dimensions and spins of the involved fields.
4
1
4 fact that by unitarity, the co
of the power1 of this constraint
follows
from
the
f14k
Unitarity:
X PO must
X
coefficients
all
be
non-negative
in
each
of
these
channels, because the
k
f12k
f34k =
k
en to be the squares of real OPE coefficients.
2
k
k
2
3
2
f23k
3
PO = f > 0
The Bootstrap and Large ` Operators
Figure 1: The conformal bootstrap condition = associativity of the operator algebra.
The dream of the conformal bootstrap is that the condition (3.3), when imposed on fourpoint functions of sufficiently many (all?) primary fields, should allow one to determine the
CFT data and thus solve the CFT. Of course, there are presumably many di↵erent CFTs,
and so one can expect some (discrete?) set of solutions. One of the criteria which will help
us to select the solution representing the 3D Ising model is the global symmetry group,
ugh some of the arguments below are technical, the idea behind them is very sim
analogy, consider the s-channel partial wave decomposition of a tree-level sc
ude with poles in both the s and t channels. The center of mass energy is sim
What’s so great about
Mellin Amplitudes?
• Mellin Space is `Momentum Space for CFTs’ !
• Use complex analysis to study CFTs.
Mellin Amplitudes always meromorphic!
• Mellin Amplitude Factorizes on CFT States!
• Algebraic Feynman Rules for AdS/CFT!
• In Flat Space Limit of AdS/CFT:
Mellin Amplitude becomes S-Matrix,
Bootstrap gives Optical Theorem,
meromorphy becomes analyticity
CFT