Developments in the Conformal Bootstrap

David Simmons-Duffin

CaltechJune 28, 2018

What’s New With Boo(tstrap)?

David Simmons-Duffin

CaltechJune 28, 2018

Outline
1 Introduction

2 Large-spin perturbation theory

3 The simplicity of dDisc

4 Analyticity in spin

5 Other developments

Conformal Blocks and Crossing Symmetry• Conformal block expansion

hφ(x1 )φ(x2 )φ(x3 )φ(x4 )iXX=
h0|φ(x1 )φ(x2 )|αihα|βi−1 hβ|φ(x3 )φ(x4 )|0i
O α,β=O,∂O,∂ 2 O=X

2fφφO
G∆O ,JO (xi )O• Conformal blocks fixed by symmetry

G∆,J (xi ) =G∆,J (z, z)
2∆2∆
x12 φ x34 φ,zz =x212 x234
x2 x2
, (1 − z)(1 − z) = 223 214
22x13 x24
x13 x24• Crossing symmetry

XO

2fφφO
G∆O ,JO (xi ) =X

2fφφO
G∆O ,JO (xi )|1↔3O• Unitarity: fφφO ∈ R and ∆ > ∆min (J)Information in the crossing equations• Algorithm to place bounds on ∆O ’s and fφφO ’s: [Rattazzi, Rychkov,Tonni, Vichi ’08]

• Numerical investigations over past decade: crossing+unitarity issurprisingly powerful• Example: 3d Ising model figure from [Kos, Poland, DSD, Vichi ’16]Δϵ1.41301.4129Monte Carlo
1.41265
1.412641.41263
1.41262

1.41281.41261
1.41260

1.41271.41260.518146 0.518148 0.518150 0.518152

Bootstrap

1.41250.51808 0.51810 0.51812 0.51814 0.51816 0.51818

ΔσChallenges• Achieve similar precision for other CFTs
• Map the space of CFTs
• Explain numerical results analytically, go beyond?
• Tom’s talk: crossing+unitarity encodes ANEC/conformal collider

bounds/causality. What else?
• What can we learn about AdS/CFT?

Lorentzian Inversion Formula [Caron-Huot ’17]

• Unifies many analytic bootstrap studies• New conceptual ideas (analyticity in spin)

This talk:• Use LIF to explore some physics of the crossing equation• Use LIF as a lens to understand some recent works• Discuss analyticity in spin

Harmonic Analysis [Dobrev, Mack, Petkova, Petrova, Todorov ’77]
hφ(x1 )φ(x2 )φ(x3 )φ(x4 )i =∞ I d +i∞X2d∆C(∆, J)Ψ∆,J (xi )d2πi
J=0 2 −i∞Ψ∆,J (xi ) = 12 (G∆,J (xi ) + Gd−∆,J (xi ))Recover conformal block expansion by deforming ∆ contour e.g. SYK model[Stanford, Maldacena ’16]

C(∆, J) ∼P

k−fφφOk fφφOk∆−∆kOrthogonality:
dx1 · · · dx4 Ψ∆,J (xi )Ψd−∆0 ,J 0 (xi )∝ δ(∆ − ∆0 )δJJ 0vol SO(d + 1, 1)
C(∆, J) = (hφφφφi, Ψd−∆,J ) “Euclidean inversion formula”Z(Ψ∆,J , Ψd−∆0 ,J 0 ) ==⇒Lorentzian Inversion formula [Caron-Huot ’17]κ∆+JC(∆, J) =4Z 1Z 1dzdzµ(z, z)GJ+d−1,∆−d+1 (z, z)
00× h[φ(x4 ), φ(x1 )][φ(x2 ), φ(x3 )]i2∆

2∆x12 φ x34 φ h[φ, φ][φ, φ]i = 2g(z, z) − g (z, z) − g  (z, z)(around z = 1)

≡ dDisc[g](z, z)• Simplifies large-spin perturbation theory• dDisc[g] can be much simpler than g• dDisc[g] is positive ( =⇒ ANEC+more, see Tom’s talk)• Analyticity in spinOutline
1 Introduction

2 Large-spin perturbation theory

3 The simplicity of dDisc4 Analyticity in spin

5 Other developments

Large-spin perturbation theory• Even in a nonperturbative theory, 1/J is a good expansion parameter• Dynamics purely from crossing symmetry• Explains much of the numerical data for 3d Ising (103/112 “stable”

operators) [Alday, Zhiboedov ’15] [DSD ’16]
τ[σσ]0 (h)
τ
1.04

1.03

1.02

1.01

1.00
10203040h

How to succeed at large-spin perturbation theory without
really trying• Set J large in the inversion formula!

GJ+d−1,∆−d+1 (z, z) ∼ zlarge J1J−∆2∼zJ+∆2

pushes integral towards z ∼ 1zz(1−z)(1−z)

 ∆φ(t-channel unit operator)

poles z0Z 1Z 1000z1 ∆φp1 ( ∆+Jp0 ( ∆+Jdz dz J−∆ J+∆ zz2 )2 )z 2 z 2=++ ...
(1−z)(1−z)∆−J∆−Jz z2 − ∆φ2 − ∆φ − 1• Correct! At large-spin, we have “double-twist” operators [Komargodski,Zhiboedov ’12] [Fitzpatrick, Kaplan, Poland, DSD ’12]

[φφ]n,J ∼ “φ∂ µ1 · · · ∂ µJ ∂ 2n φ”∆n,J = 2∆φ + 2n + J + γn (J)Corrections at large spin
• Include another t-channel block G∆0 ,J 0 (1 − z, 1 − z):C(∆, J) ⊃p(J)γ(J)δp(J)+2(∆ − J − 2∆φ )∆ − J − ∆φ

• Gives double-twist anomalous dimensions

γ(J) ∼

11+ τ 0 +1 + . . .
Jτ0J(τ 0 = ∆0 − J 0 )Agrees with all-orders asymptotic expansions in 1/J [Alday ’16] [DSD ’16]
• Cartoon:

φφO0φφAway from large J?
p(J)γ(J)δp(J)• Problem: every t-channel operator O 0 gives (∆−J−2∆)2 + ∆−J−∆φφ(poles of 6j symbol/crossing kernel for SO(d + 1, 1) [Gadde ’17]
[Hogervorst, van Rees ’17] [Sleight, Toronna ’18])• Correct answer [Fitzpatrick, Kaplan, Walters ’15], [Alday, Bissi ’16], [DSD ’16]:
ppγpγ 2p∆−J−2∆φ +γ = ∆−J−2∆φ + (∆−J−2∆φ )2 + (∆−J−2∆φ )3 + . . .

=+O0+X
J0[O 0 O 0 ]J 0+ ...

Need multi-twist operators [O1 O2 · · · On ]J1 ,...,Jn−1• Inversion integral is only guaranteed to work on principal series∆ = d2 + is. To move into physical region, need to (at least) resum
multi-twists.
• Organizing spectrum is difficult. Small parameters can help.

Outline
1 Introduction

2 Large-spin perturbation theory

3 The simplicity of dDisc

4 Analyticity in spin

5 Other developments

Small parameters and the simplicity of dDisc: large NConsider a large-N theory:11h· · ·i(1) + 4 h· · ·i(2) + . . .
2NN∆[φφ]n,J = 2∆φ + 2n + J + γ (1) (n, J) + γ (2) (n, J) + . . .
hφφφφi = hφφihφφi +dDisc[G∆,J (1 − z, 1 − z)] = sin2π2(∆ − J − 2∆φ ) G∆,J (1 − z, 1 − z)

• At O(N −2 ), only single-trace operators contribute to dDisc!

=⇒ tree-level double-trace data γ (1) (n, J) from single-trace data.
• At O(N −4 ), double-traces contribute as γ (1) (n, J)2=⇒ 1-loop double-trace data γ (2) (n, J) from (tree-level)2 .
Interpretation: g = 1 + iM. Disc[g] = M. dDisc[g] = Im(M).

Loops in the bulk from CFT!

• Scalar triangle diagram in AdS using large-spin perturbation theory,squaring tree-level results [Aharony, Alday, Bissi, Perlmutter ’16].
• Bulk loop corrections in N = 4 SYM at strong coupling [Alday, Bissi’17] [Aprile, Drummond, Heslop, Paul ’17] [Alday, Caron-Huot ’17] (using tree-level

data from old days, [Rastelli, Zhou ’17])• CFT version of unitarity-cut methods from amplitudes. Can

higher-loop techniques be adapted too? [Ye Yuan ’17]

Small parameters and the simplicity of dDisc• Wilson Fisher theory in 4 −  dimensions: only dDisc[G1 , Gφ2 ] arenonzero at O(3 ).
• Fixes anomalous dimensions γ[φφ] (J) up to O(3 ) from a single

application of the inversion formula.
• Explains some success of Mellin bootstrap for Wilson-Fisher[Gopakumar, Kaviraj, Sen, Sinha ’16]

• To get O(4 ), another amplitudes technique: transcendentality [Alday,Henriksson, van Loon ’18]



z−1
log2 z, Li2 (1 − z), . . . , Li3z
5 2233 310065155CT=1−
 − −−ζ 3 4 + . . .
Cfree324
8748
3779136 2916

dDisc[g (4) ] ⊂• Other theories with small parameters [Turiaci, Zhiboedov ’18], [Aharony, Alday,Bissi, Yacoby ’18]

• In 3d Ising, γ[σσ]0 < 0.036 are numerically small. Allows one toresum their effects [DSD ’16].

Outline
1 Introduction

2 Large-spin perturbation theory

3 The simplicity of dDisc

4 Analyticity in spin

5 Other developments

The OPE in Lorentzian signature
Consider a CFT correlation function hO1 · · · On i• Euclidean signature: all singularities described by the OPE

O1 (x1 )O2 (x2 ) =X∆k −∆1 −∆2
f12k x12Ok (x2 )k• Lorentzian signature: the OPE is valid when both operators act on

the vacuum [Mack ’76]
O1 O2 |Ωi =X

f12k Ok |Ωik

But it’s easy to find situations where x212 → 0 and the O1 × O2
OPE doesn’t work

The Regge limit• Position-space version of high-energy scattering. Operators 1 and 2

are highly-boosted relative to 3 and 4.
• O1 , O3 create excitations that scatter, measured by O2 , O4 .
• x212 → 0 but the O1 × O2 OPE is not validhΩ|T {O1 O2 O3 O4 }|Ωi = hΩ|O4 O1 O2 O3 |ΩiAnalyticity in spin: toy model [Caron-Huot ’17]
Consider an “amplitude” A(w) = A(eiθ ) such that
• A is bounded in the “Regge limit” w → ∞A(w) . wJ0asw→∞• A is analytic outside of w ∈ [1, ∞)• A has partial wave decomposition A(w) =P∞

J=0 a(J)w

JI

dw −Jw A(w)2πiw
Z ∞dw −Jw disc[A](w)=w
1wa(J) =“Euclidean inversion”“Lorentzian inversion”

Analyticity in spin
Z ∞a(J) =1dw −J
w disc[A](w)w

Can now write the amplitude in a way that manifests Regge behavior(Sommerfeld-Watson trick)IA(w) =dJ
Γ0a(J)wJ
1 − e−2πiJ

vs.

∞
X
J=0a(J)wJ

Conformal Regge Theory
1−J• In Regge regime G∆,J → G1−J,1−∆ ∼ (zz) 2 . Big at big J.
• Sommerfeld-Watson in CFT gives [Brower, Polchinski, Strassler, Tan ’06]
[Cornalba ’07] [Costa, Goncalves, Penedones ’12] [Kravchuk, DSD ’18]

Ihφφφφi ∼IdJd∆ C(∆, J)(G1−J,1−∆ (xi ) + . . .)2πi 1 − e−2πiJ• Many applications (see Tom’s talk) [Li, Meltzer, Poland ’17], [Costa, Hansen,Penedones ’17] [Meltzer, Perlmutter ’17] [Afkhami-Jeddi, Hartman, Kundu, Tajdini ’17]
[Gromov, Kazakov, Korchemsky, Sizov ’17]

• Caron-Huot’s formula justifies conformal Regge theory in a general

CFT. Derive it by deforming the Euclidean inversion contour toLorentzian space [Caron-Huot ’17] [Stanford, DSD, Witten ’17], like in our toymodel. Boundedness from [Maldacena, Shenker, Stanford ’15].
• Similar story for chaos in 1d and 2d [Maldacena, Stanford ’16], [Murugan,Stanford, Witten ’17], [Stanford, DSD, Witten ’17] ...

Light-ray operators [Kravchuk, DSD ’18] (see Kravchuk’s poster)
What does non-integer spin actually mean?
• We can’t analytically continue a local operator O µ1 ···µJ in spin.

Continuous spin operators kill the vacuum, local operators do not(Edward’s talk).
• Instead consider the integral over a null line

Z ∞L[OJ ] =dx− O−−···− (x− )“light-transform”−∞• L[OJ ] can be analytically continued in J =⇒ light-ray operator OJ• Construct OJ as a bilocal integral

ZOJ = Res∆

dd x1 dd x2 K∆,J (x1 , x2 )O1 (x1 )O2 (x2 )Generalized inversion formula [Kravchuk, DSD ’18] (see Kravchuk’s poster)• Res∆ C(∆, J) is a matrix element of OJ• Gives a new proof/generalization of Caron-Huot’s formula.

C(∆, J) = −12πiZ

dd x1 · · · dd x4hΩ|[O4 , O1 ][O2 , O3 ]|Ωifvol SO(d,2)×hO1 O2 L[O]i−1 hO4 O3 L[O]i−1
hL[O]L[O]i−1• Setting J = 2, Res∆=d gives ANEC from positivity of h[ψ, O][ψ, O]i• Other residues give all other OPE data

• The Regge limit is an expansion in light-ray operators.
Regge

hO1 · · · O4 i ∼IIdJd∆ C(∆, J) hO1 O2 L[O]ihO3 O4 L[O]i
2πi 1 − e−2πiJhL[O]L[O]i• The Reggeon/Pomeron are (families of) light-ray operators

A Riemann surface of CFT operators
Chew-Frautschi plot [Brower, Polchinski, Strassler, Tan ’06]
JL[T ]

L[O4 ] . . .

∆ − d2
N = 4 SYM at finite λ from integrability [Gromov, Levkovich-Maslyuk, Sizov ’15]

Outline
1 Introduction

2 Large-spin perturbation theory

3 The simplicity of dDisc

4 Analyticity in spin

5 Other developments

Other developments• More sophisticated numerical studies. hJJJJi in 3d [Dymarsky,Penedones, Trevisani, Vichi ’17], hT T T T i in 3d [Dymarsky, Kos, Kravchuk, Poland,
DSD ’17]. Many interesting supersymmetric computations(Madalena’s talk).
• To isolate theories and improve precision, need larger-scale problems

with multiple correlators. Better algorithms, bigger machines?
[Bootstrap collaboration] in progress• Analytic bootstrap bounds [Mazac ’16] [Mazac, Paulos ’18], using algorithmof [Rattazzi, Rychkov, Tonni, Vichi] analytically. Currently in 1d. Promising
direction, perhaps for numerical/analytical hybrid?

Bootstrap in new settings• Defects [Gaiotto, Paulos, Mazac, Lemos, Liendo, Meineri, Sarkar, Meneghelli, Mitev,Lauria, Trevisani, Gadde, Isachenkov, Linke, Schomerus, ...]• Finite temperature [Iliesiu, Koloǧlu, Mahajan, Perlmutter, DSD ’18] [Gobeil, Maloney,Ng, Wu ’18]

• Large charge [Hellerman, Kobayashi, Maeda, Watanabe, Monin, Pirtskhalava, Rattazzi,Seibold, Jafferis, Mukhametzhanov, Zhiboedov, ...]In defect and finite-temperature cases, one can derive Lorentzian
inversion formulas with many of the nice properties we’ve discussed.

Questions• Can we solve the complete large-spin dynamics of a CFT (andisolate what’s left over)?
• Can we input large-spin perturbation theory into numerics?
• Can we iterate the inversion formula?
• Can we compute higher loops in the bulk?
• What other operators exist besides light-ray/local operators? Whatphysics do they encode? Can we describe the bulk-point limit?
• Connections between analyticity in spin and information theory?