the analytic conformal bootstrap - strings 2017 › wp-content › uploads › 2017 › 06 ›...

The Analytic Conformal Bootstrap

Luis Fernando Alday

University of Oxford

Strings 2017 - Tel Aviv

Luis Fernando Alday The Analytic Conformal Bootstrap

What will this talk be about?

Conformal Field Theories in D > 2 dimensions

Very relevant for Physics:

Asymptotic regimes of QFT.Condensed matter/ description of critical points, ...

Interesting interplay with Mathematics

Representation theory, Langlands program,...

Many important theories are conformal:

4d N = 4 SYM.3d Ising model.Critical O(N) models,...

Ubiquitous in dualities in string and gauge theory.


CFTD-difficulties

Studying CFT in D > 2 is not so easy...

In general CFTs do not have a Lagrangian description...

In a Lagrangian theory we can use Feynman diagrams:

A(g) = A(0) + gA(1) + · · ·

But generic CFTs don’t have a small coupling constant!

We don’t have the luxuries of D = 2:

No Virasoro Algebra 7No finite number of primaries 7Representation theory is less powerful 7

In spite of all this, progress can be made!


Conformal bootstrap

Conformal bootstrap: resort to consistency conditions!

Conformal symmetryProperties of the OPEUnitarityCrossing symmetry

This idea was successfully applied to 2d in the eighties! [Ferrara,

Gatto, Grillo; Belavin, Polyakov, Zamolodchikov]

25 years later it was finally implemented in D > 2! [Rattazzi, Rychkov,

Tonni, Vichi ’08]

Starting point of an impressive numerical revolution! [Poland’s review

talk Strings 2015 and still going on]


Recent analytic approaches to CFT

Today: Analytic results for CFTs from consistency conditions!

`

�� `

Numerical Bootstrap

Bulk Point Singularity Regge

Limit

Light Cone Bootstrap

spin

twist

Great progress by many people! [L.F.A, Maldacena; Fitzpatrick, Kaplan, Poland,

Simmons-Duffin; Komargodski, Zhiboedov; Bissi; Cornalba, Costa, Penedones; Kaviraj, Sen,

Sinha; Lukowski, Walters, Wang, Vos, Aharony, Perlmutter, Li, Kulaxizi, Parnachev, , ...]


Recent analytic approaches to CFT

Other approaches

. Large central charge limit [Heemskerk, Penedones, Polchinski, Sully;

Fitzpatrick, Kaplan; L.F.A, Bissi, Lukowski; Rastelli, Zhou,...]

. Solvable protected subsectors [Dolan, Osborn; Beem, Lemos, Liendo,

Peelaers, Rastelli, van Rees; Chester, Lee, Pufu, Yacoby, ...]

. Crossing in Mellin space [Gopakumar, Kaviraj, Sen, Sinha; Rastelli, Zhou,

L.F.A., Bissi, Lukowski] [Talk by Rastelli]

. Large global charge limit [Hellerman, Orlando, Reffert, Watanabe,

Alvarez-Gaume, Loukas, Monin, Pirtskhalava, Rattazzi, Seibold,...]

. ANEC from bootstrap [Maldacena, Shenker, Stanford, Hartman, Jain, Kundu,

Hofman, Li, Meltzer, Poland, Rejon-Barrera,...]

. Great progress in 2d [Talks by Kaplan and Yin]


Analytic conformal bootstrap

Our plan is to start with the light-cone bootstrap and expand to allother regions!

Which kind of analytic results will you get today?

Analytic bootstrap

Results for operators with spin in a generic CFT!

ODT∼ ϕ∂µ1 · · · ∂µ`ϕ, O

ST∼ Trϕ∂µ1 · · · ∂µ`ϕ

Study their scaling dimension ∆ for large values of the spin `:

∆(`) = `+ c0 +c1

`+

c2

`2+ · · · , double-twist

∆(`) = d0 log `+ `+ d1 +c2

`+ · · · , single-trace

We will obtain analytic results to all orders in 1/` resorting only toconsistency conditions.Valid for vast families of CFTs!


CFT - Symmetries

Conformal algebra:

Scale transformations → dilatation D

Poincare Algebra: Pµ and Mµν

Special conformal transformations: Kµ

[D,Pµ] = Pµ, [D,Kµ] = −Kµ, [Kµ,Pν ] = ηµνD − iMµν

Specific CFTs may have extra symmetries but we will keep thediscussion very general.


CFT - Ingredients

Main ingredient:

Conformal Primary local operators: O∆,`

(x), [Kµ,O(0)] = 0

@@@I

��

Dimension Lorentz spin

In addition we have descendants Pµk ...Pµ1O∆,`= ∂µk ...∂µ1O∆,`

.

Operators form an algebra (OPE)

Oi (x)Oj(0) =∑

k∈prim.

Cijk |x |∆k−∆i−∆j

Ok(0) +xµ∂µOk(0) + · · ·︸︷︷︸all fixed

CFT data: The set ∆i and Cijk characterizes the CFT.


CFT - Basics

Main observable:

Correlation functions of primary operators

〈O1(x1)...On(xn)〉

〈Oi (x1)Oj(x2)〉 =δij

|x12|2∆i

〈Oi (1)Oj(2)Ok(3)〉 =Cijk

|x12|∆1+∆2−∆3 |x13|∆1+∆3−∆2 |x23|∆2+∆3−∆1

Four-point function of identical operators:

〈O(x1)O(x2)O(x3)O(x4)〉 =G(u, v)

x2∆O12 x2∆O

34

where u =x2

12x234

x213x

224, v =

x214x

223

x213x

224


Four-point function - properties

Conformal partial wave decomposition

OPE: O ×O =∑

i Oi + descendants

C∆,` C∆,`

2 3

1 4

O∆,`∑∆,`

〈OOOO〉 =

G(u, v) = 1 +∑∆,`

C 2∆,`G∆,`(u, v)

��>

6Z

ZZZ}

Identity operator Conformal primaries Conformal blocks

Conformal blocks: For a given primary, take into account thecontribution from all its descendants. Fully fixed function!


Conformal bootstrap

Crossing symmetry

v∆OG(u, v) =︸︷︷︸x1↔x3

u∆OG(v , u)

C∆,` C∆,`

2 3

1 4

O∆,`∑∆,`

=∑∆,`

C∆,`

C∆,`

O∆,`

2 3

1 4

A remarkable...but hard equation!

v∆O

1 +∑∆,`

C 2∆,`G∆,`(u, v)

︸︷︷︸

Easy to expand around u = 0, v = 1

= u∆O

1 +∑∆,`

C 2∆,`G∆,`(v , u)

︸︷︷︸

Easy to expand around u = 1, v = 0


Numerical vs Analytic bootstrap

Study this equation in different regions, u = zz̄ , v = (1− z)(1− z̄)

z̄

z1

1

In the Euclidean regime z̄ = z∗.

We can study crossing aroundu = v = 1

4

Starting point of the numericalbootstrap.

�

��

In the Lorentzian regime z , z̄ are independent real variables and wecan consider u, v → 0.

Starting point of the analytic (light-cone) bootstrap!


Analytic bootstrap

Analytic bootstrap

Why is this a good idea?

v∆O

1 +∑∆,`

C 2∆,`G∆,`(u, v)

= u∆O

1 +∑∆,`

C 2∆,`G∆,`(v , u)

Direct channel ⇔ Crossed channel

Very complicated interplay between l.h.s. and r.h.s. ... but:

Operators with large spin

DT operators with large spin ⇔ Identity operatorST operators with large spin ⇔ ST operators with large spin


Conformal blocks - technicalities

Eigenfunctions of a Casimir operator

CG∆,`(u, v) = J2G∆,`(u, v)

where J2 = (`+ ∆)(`+ ∆− 1) ∼ `2

Small u limit:

G∆,`(u, v) ∼ uτ/2f collτ,` (v), τ = ∆− `

We will introduce the notation

G∆,`(u, v) ≡ uτ/2fτ,`(u, v)

Small v limit:fτ,`(u, v) ∼ log v


Overlays

Consider the v � 1 limit of the crossing equation: C 2∆,` → aτ,`

v∆O

1 +∑τ,`

aτ,`uτ/2fτ,`(u, v)

= u∆O

1 +∑τ,`

aτ,`vτ/2fτ,`(v , u)

⇓

1 +∑τ,`

aτ,`uτ/2fτ,`(u, v) =

u∆O

v∆O+ subleading terms︸︷︷︸

rest of operators sorted by twist

The r.h.s. is divergent as v → 0.Each term on the l.h.s. diverges as fτ,`(u, v) ∼ log v .In order to reproduce the divergence on the right, we need infiniteoperators, with large spin and whose twist approaches τ = 2∆O(actually τn = 2∆O + 2n)


Overlays

Consider the v � 1 limit of the crossing equation:

v∆O

1 +∑τ,`


= u∆O

1 +∑τ,`

aτ,`vτ/2fτ,`(v , u)

⇓

1 +∑τ,`


u∆O

v∆O+ subleading terms︸︷︷︸

rest of operators sorted by twist

The r.h.s. is divergent as v → 0.

Each term on the l.h.s. diverges as fτ,`(u, v) ∼ log v .

In order to reproduce the divergence on the right, we need infiniteoperators, with large spin and whose twist approaches τ = 2∆O(actually τn = 2∆O + 2n)


Example: Generalised free fields

Simplest solution: Large N CFTs - Generalised free fields

G(0)(u, v) = 1 +(uv

)∆O+ u∆O

Intermediate ops: Double twist operators: O�n∂µ1 · · · ∂µ`Oτn,` = 2∆O + 2n

an,` = a(0)n,`

Their OPE coefficients are such that the divergence of a singleconformal block (∼ log v) , as v → 0, is enhanced!

1 +∑τ,`

a(0)τn,`

uτn/2fτn,`(u, v) = 1 +(uv

)∆O︸︷︷︸+u∆O

6

But this divergence is quite universal!


Large spin sector

Additivity property [Fitzpatrick, Kaplan, Poland, Simmons-Duffin; Komargodski, Zhiboedov]

In any CFT with O in the spectrum, crossing symmetry implies theexistence of double twist operators with arbitrarily large spin and

τn,` = 2∆O + 2n +O(

1

`

)an,` = a

(0)n,`

(1 +O

(1

`

))

All CFTs have a large spin sector, for which the operators become”free”!

Can we do perturbations around large spin? YES!


General picture

We would like to exploit the following idea

∑τ,`


u∆O

v∆O+ · · ·

Behaviour at large spin ⇔ Enhanced divergences as v → 0

The presence of the identity on the r.h.s. already led to aremarkable result!

Let’s take this to the next level!

We will focus in corrections to GFF.


New ingredient: Twist conformal blocks

GFF has accumulation points in the twist at

τn = 2∆O + 2n

It is convenient to define ”twist” conformal blocks, thecontribution to the 4pt function from a given twist.

∑`

a(0)τ,`u

τ/2fτ,`(u, v) ≡ Hτ (u, v)

And twist conformal blocks with insertions:

∑`

a(0)τ,`

J2muτ/2fτ,`(u, v) ≡ H(m)

τ (u, v)


Twist conformal blocks

Properties

. Correlator decomposition

G(0)(u, v) =∑n

H(0)τn (u, v)

. Recurrence relation

CH(m+1)τ (u, v) = H(m)

τ (u, v)

. Prescribed behaviour at small u and small v .

H(m)τ (u, v) ∼ uτ/2, small u

H(m)τ (u, v) ∼ v−∆O+m, small v

The functions H(m)τ (u, v) can be systematically constructed!


Corrections to GFF

Let us compute 1/N corrections to large N CFTs/GFF!

G(u, v) = G(0)(u, v) +1

N2G(1)(u, v) + · · ·

Two Sources of corrections

1 Double twist operators will acquire corrections:

τn,` = 2∆O + 2n +1

N2γn,` + · · ·

an,` = a(0)n,` +

1

N2a

(1)n,` + · · ·

2 We can also have new intermediate operators at order 1/N2.

Which corrections are consistent with crossing symmetry

G(1)(u, v) =(uv

)∆O G(1)(v , u)


Corrections to GFF

Conformal partial wave expansion

G(u, v) =∑τ,`


Plug τn,` = 2∆O + 2n + 1N2γn,` + · · · and expand in 1

N2 :

G(1)(u, v) =∑n,`

a(0)τ,`u

∆O+n γn,`2

fτ,`(u, v) log u + · · ·

Assume γn,` admits an expansion in 1/J:

γn,` = 2bn,1J2

+ 2bn,2J4

+ · · ·

⇓

G(1)(u, v) =∑n,m

bn,mH(m)τn (u, v) log u + · · ·


Corrections to GFF

Plugging this into the crossing equation...

∑n,m

bn,mH(m)n (u, v) =

(uv

)∆O G(1)(v , u)

∣∣∣∣log u

Now both sides can be expanded around small v !

Matching divergences we can fix all coefficients bn,m. Hence γn,` toall orders in 1/`!

Crossing equation has become algebraic!


Corrections to GFF

∑n,m

bn,mH(m)n (u, v) =

(uv

)∆O G(1)(v , u)

∣∣∣∣log u

Case 1: No new operators at order 1/N2

The r.h.s has no divergences as v → 0.

All H(m)n (u, v) on the l.h.s. must be absent.

γn,` vanishes to all orders in 1/`!

While truncated in the spin are allowed! [Heemskerk, Penedones, Polchinski,

Sully]

Simplest solution, d = 2,∆ = 2

γn,` =

{3

3+2n ` = 0

0 ` 6= 0


Corrections to GFF

Case 2: New single-trace operators at order 1/N2, e.g. O itself:

O ×O = 1 + [O,O]n,` +1

N2O

Now the situation is more interesting:∑n,m

bn,mH(m)τn (u, v) ∼

(uv

)∆Ov∆O/2fO(v , u)

∣∣∣∣log u

Reproducing the divergences as v → 0 fixes all bn,m, and henceγn,` to all orders in 1/`!

e.g. ∆ = 2

γn,` = − 2

(`+ n + 2)(`+ n + 1)

This result was not known for generic ∆!


Wider perspective on CFT

The additivity property we have seen is a particular example of a moregeneral one

If two operators Oτ1 ,Oτ2 of twists τ1 and τ2 are part of the spectrumthen there is a tower of operators [Oτ1 ,Oτ2 ]n,` of twist

τ[Oτ1 ,Oτ2 ]n,` = τ1 + τ2 + 2n +O(

1

`

)

This additivity in the twist property should make you happy andsad at the same time!

The spectrum of generic CFTs is hard!

. O is part of the spectrum.

. [O,O]n,` is also part of the spectrum.

. And [[O,O]n1,`1 , [O,O]n2,`2 ]n3,`3too, and so on!


Large spin expansions for non-perturbative CFTs

In non-perturbative CFTs the spectrum is very rich. Hard to apply ouridea

∑τ,`

aτ,`uτ/2fτ,`(u, v) = Rich spectrum in the crossed channel

Behaviour at large spin ⇔ complicated divergences as v → 0

However

γ` = − c0

`τε+ · · · ⇔ Operator Oε in the crossed channel

Having control on the low twist spectrum of the theory,e.g. throughthe numerical bootstrap, we can compute the leading terms in the1/` expansion of the anomalous dimensions of DT operators.


Large spin expansions for non-perturbative CFTs

3d Ising Model

Spin operator σ: ∆σ = 0.518151 = 1/2 + γσ.

Higher spin-operators σ∂µ1 · · · ∂µ`σ, ∆` = 1 + γ`

One predicts [Work with Zhiboedov]

γ` ∼ 2γσ −0.0027

`− 0.0926

`1.4126− 0.0024

`2.4126+ · · ·

Next corrections are hard to control but small for large spin.

` = 4 is already large!

Two important comments

Powerful interplay with the numerical bootstrap, taken to the nextlevel by [Simmons-Duffin]!

The convergence properties of these series was stablished in[Caron-Huot] where it was also understood why they work so well!


3d Ising - Comparison with numerical results

468 12 20 400.00

0.01

0.02

0.03

0.04 γ`

`


Large spin perturbation theory

If the CFT has a small parameter we are better of, as thisparameter further organises the problem.

The range of applicability of our method is very vast [Work with Aharony,

Bissi, Perlmutter, Zhiboedov]

. Weakly coupled gauge theories.

. Theories with weakly broken higher spin symmetry.

. 1/N4 corrections to GFF.

Each one interesting on its own right! The series in 1/` can beresummed in all examples!


Weakly coupled conformal gauge theories

Single trace operators with spin

O` ∼ Trϕ∂µ1 · · · ∂µ`ϕ

Highly non-trivial fact:

∆− ` = f (g) log `+ · · ·

Get this from crossing

Study the following correlator in a weakly coupled gauge theory

〈OOOO〉, O = Trϕ2

The O` are the operators with leading twist in the theory

τ` = ∆` − ` = ∆O + γ` ← Small in pert. theory

Focus on their contribution by taking the small u, v limit!


Weakly coupled conformal gauge theories

u−∆O∑`

a`u∆O/2+γ`/2f `coll(v) = v−∆O

∑`

a`v∆O/2+γ`/2f `coll(u)

ST operators with large spin map to themselves!

Match divergences on both sides → constraints on γ`

Logarithmic behaviour?

General ansatz: γ` = f log `+ a2 log2 `+ a3 log3 `+ · · ·Crossing symmetry: γ` = f log `+��

�XXXXa2 log2 ` +��XXXXa3 log3 ` + · · ·

Higher order corrections? At one loop crossing is very powerful!

γ` = g log `+ c0 +c1

`+

c2

`2+

c3

`3+ · · ·︸︷︷︸

all fixed!

→ g S1(`)


CFT with weakly broken HS symmetry

CFT with HS symmetry

The following is part of the spectrumFundamental scalar field ϕ

∂µ∂µϕ = 0→ ∆ϕ =

d − 2

2

Tower of HS conserved currents J(s) = ϕ∂µ1 · · · ∂µsϕ,

conservation→ ∆s = d − 2 + s

Weakly broken HS symmetry

The fundamental field ϕ and HS currents J(s) acquire ananomalous dimensions:

∆ϕ =d − 2

2+ gγϕ + · · ·

∆s = d − 2 + s + gγs + · · ·

Which anomalous dimensions are consistent with crossing symmetry?Luis Fernando Alday The Analytic Conformal Bootstrap

CFT with weakly broken HS symmetry

Crossing for the correlator 〈ϕϕϕϕ〉 fixes the spectrum!

e.g. Single scalar in d = 4

γϕ2 = g + · · · , γϕ =1

12g2 + · · · , γ` = 2γϕ

(1− 6

`(`+ 1)

)+ · · ·

We studied several models. Again crossing symmetry fixes thespectrum!

Wilson-Fisher models around d = 4 X

Large N critical O(N) models in 2 < d < 4 X

Cubic models around d = 6 X


1/N4 corrections to GFF

AdS/CFT

Large N CFT in D-dimensions(GFF + corrections)

⇔ Gravitational theory inAdSD+1

1N2 expansion in CFT ↔ loops in AdS/powers of GN .

+ + + + +…G = { {{

N0 1/N2 1/N4

Diagrams in AdS are hard to compute...Use crossing for the CFT!


1/N4 corrections to GFF

Loops in AdS are a largely unexplored subject.

We can approach this by studying 1/N4 corrections to GFF!

τn,` = 2∆O + 2n +1

N2γ

(1)n,` +

1

N4γ

(2)n,` + · · ·

Consistency conditions fix γ(2)n,` from the leading order solution!

Even 1/N4 corrections to N = 4 SYM at strong coupling!equivalent to loop corrections to supergravity on AdS5 × S5.

∆0,2 = 6− 4

N2− 45

N4+ · · ·

First non-protected quantity ever computed to this order inAdS/CFT !


Conclusions

Generic CFTs have a large spin sector which becomes essentiallyfree and we have shown how to perform a perturbation around thatsector.

The method applies to vast families of CFTs.

It provides an alternative to Feynman diagram computations.

It provides an on-shell gauge invariant way to study weakly coupledgauge theories.

It allows to learn about loops in AdS and quantum gravity.

It allows connections with the numerical bootstrap.

This is just the beginning!!


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