silviu s. pufu, princeton universitysilviu pufu (princeton university) 10-26-2018 5 / 32 in 11d, we...

M-theory S-Matrix from 3d SCFT

Silviu S. Pufu, Princeton University

Based on:

arXiv:1711.07343 with N. Agmon and S. Chester

arXiv:1804.00949 with S. Chester and X. Yin

arXiv:1808.10554 with D. Binder and S. Chester

arXiv:1406.4814, arXiv:1412.0334 with S. Chester, J. Lee, and R. Yacoby

arXiv:1610.00740 with M. Dedushenko and R. Yacoby

Trieste, October 17, 2018

Silviu Pufu (Princeton University) 10-26-2018 1 / 32

Motivation

Learn about (reconstruct?) gravity / string theory / M-theory fromCFT using AdS/CFT.

Work toward a constructive proof of AdS/CFT.

Most well-established examples:

4d SU(N) N = 4 SYM at large N and large ’t Hooft coupling / typeIIB strings on AdS5 × S5

3d U(N)k × U(N)−k ABJM theory at large N / M-theory onAdS4 × S7/Zk .

Both have maximal SUSY (for ABJM only when k = 1 or 2).

I’ll focus on the ABJM example in the case k = 1.

Motivation

Learn about (reconstruct?) gravity / string theory / M-theory fromCFT using AdS/CFT.

Work toward a constructive proof of AdS/CFT.

Most well-established examples:

4d SU(N) N = 4 SYM at large N and large ’t Hooft coupling / typeIIB strings on AdS5 × S5

3d U(N)k × U(N)−k ABJM theory at large N / M-theory onAdS4 × S7/Zk .

Both have maximal SUSY (for ABJM only when k = 1 or 2).

I’ll focus on the ABJM example in the case k = 1.

Motivation

Last 10 years: progress in QFT calculations

using supersymmetric localization;

using conformal bootstrap in CFTs.

Example: using SUSic loc., the S3 partition function of ABJMtheory can be written as a 2N-dim’l integral [Kapustin, Willett, Yaakov ’09]

∫dNλdNµ

∏i<j 4 sinh2(λi − λj) sinh2(µi − µj)∏

i,j cosh2(λi − µj)ei k

∑i (λ

2i −µ

2i ) .

Expand at large N to find F = − log Z = π√

23 k1/2N3/2 + O(N1/2).

N3/2 scaling matches # of d.o.f.’s on N coincident M2-branes ascomputed using 11d SUGRA.

Subleading corrections contain info beyond 11d SUGRA. Whatexactly can we learn from them??

Motivation

∫dNλdNµ

∑i (λ

2i −µ

2i ) .

23 k1/2N3/2 + O(N1/2).

Motivation

∫dNλdNµ

∑i (λ

2i −µ

2i ) .

23 k1/2N3/2 + O(N1/2).

Motivation

∫dNλdNµ

∑i (λ

2i −µ

2i ) .

23 k1/2N3/2 + O(N1/2).

M-theory S-matrix

This talk: Reconstruct M-theory S-matrix perturbatively at smallmomentum.

scatter gravitons and superpartners in 11d.

Equivalently, reconstruct the derivative expansion of the M-theoryeffective action. Schematically,

∫d11x

√g[R + Riem4 + · · ·+ (SUSic completion)

Restrict momenta to be in 4 out of the 11 dimensions.

M-theory S-matrix

∫d11x

M-theory S-matrix

∫d11x

In 11d, we can scatter: gravitons, gravitini, 3-form gauge particles.Momenta within 4d =⇒ can use 4d N = 8 language. We scatter:

graviton (1);gravitinos (8);gravi-photons (28);gravi-photinos (56);scalars (70 = 35 + 35)

At leading order in small momentum (i.e. p2), scatteringamplitudes are those in N = 8 SUGRA at tree level. Examples:

ASUGRA, tree(h−h−h+h+) =〈12〉4[34]4

ASUGRA, tree(S1S1S2S2) =tus,

where s = (p1 + p2)2, t = (p1 + p4)2, u = (p1 + p3)2.Amplitudes depend on the particles being scattered, but they’re allrelated by SUSY. (See Elvang & Huang’s book.)

Momentum expansion

Momentum expansion takes a universal form (independent of thetype of particle):

A = ASUGRA, tree

(1 + `6pfR4(s, t) + `9pf1-loop(s, t)

+ `12p fD6R4(s, t) + `14

p fD8R4(s, t) + · · ·).

fD2nR4 = symmetric polyn in s, t ,u of degree n + 3

Known from type II string theory + SUSY [Green, Tseytlin, Gutperle,

Vanhove, Russo, Pioline, . . . ] :

fR4(s, t) =stu

3 · 27 , fD6R4(s, t) =(stu)2

15 · 215 .

`10p fD4R4 allowed by SUSY, but known to vanish.

This talk: Reproduce fR4 and fD4R4 = 0 from ABJM theory.Silviu Pufu (Princeton University) 10-26-2018 6 / 32

Flat space limit of CFT correlators

Idea: Flat space scattering amplitudes can be obtained as limit ofCFT correlators [Polchinski ’99; Susskind ’99; Giddings ’99; Penedones ’10;

Fitzpatrick, Kaplan ’11] .

For a CFT3 operator φ(x) with ∆φ = 1,

〈φ(x1)φ(x2)φ(x3)φ(x4)〉conn =1

x212x2

34g(U,V )

with U =x2

12x234

x213x2

24, V =

x214x2

13x224

, go to Mellin space

g(U,V ) =

∫ds dt

(4πi)2 U t/2V (u−2)/2Γ2(

1− s2

1− t2

1− u2

)M(s, t)

where s + t + u = 4.

From the large s, t limit of M(s, t) one can extract 4d scatteringamplitude A(s, t) [Penedones ’10; Fitzpatrick, Kaplan ’11] .

Idea: Flat space scattering amplitudes can be obtained as limit ofCFT correlators [Polchinski ’99; Susskind ’99; Giddings ’99; Penedones ’10;

Fitzpatrick, Kaplan ’11] .

For a CFT3 operator φ(x) with ∆φ = 1,

〈φ(x1)φ(x2)φ(x3)φ(x4)〉conn =1

x212x2

34g(U,V )

with U =x2

12x234

x213x2

24, V =

x214x2

13x224

, go to Mellin space

g(U,V ) =

∫ds dt

(4πi)2 U t/2V (u−2)/2Γ2(

1− s2

1− t2

1− u2

)M(s, t)

where s + t + u = 4.

From the large s, t limit of M(s, t) one can extract 4d scatteringamplitude A(s, t) [Penedones ’10; Fitzpatrick, Kaplan ’11] .

If L is the radius of AdS (and L/2 is the radius of S7), then

A(s, t) = limL→∞

∫ c+i∞

c−i∞dα eαα−1/2M

(− L2

4αs,− L2

Expect M(s, t) to have a series expansion in `p/L ∝ N−1/6.

At leach order in `p/L, it is only the large s, t behavior of M(s, t)that contributes to A(s, t).

In order for A(s, t) to have an expansion in `p times momentum,we need

M =∞∑

(function that grows as k th power of s, t ,u)

Instead of 1/N or `p/L I will use 1/cT , where 〈TµνTρσ〉 ∝ cT .Silviu Pufu (Princeton University) 10-26-2018 8 / 32

If L is the radius of AdS (and L/2 is the radius of S7), then

A(s, t) = limL→∞

∫ c+i∞

c−i∞dα eαα−1/2M

(− L2

4αs,− L2

Expect M(s, t) to have a series expansion in `p/L ∝ N−1/6.

At leach order in `p/L, it is only the large s, t behavior of M(s, t)that contributes to A(s, t).

In order for A(s, t) to have an expansion in `p times momentum,we need

M =∞∑

(function that grows as k th power of s, t ,u)

Instead of 1/N or `p/L I will use 1/cT , where 〈TµνTρσ〉 ∝ cT .Silviu Pufu (Princeton University) 10-26-2018 8 / 32

To obtain scattering amplitude of graviton + superpartners inM-theory, look at stress tensor multiplet in k = 1 ABJM theory:

Stress tensor multiplet of any N = 8 SCFT (R-symm is so(8)):

focus on this −→

dimension spin so(8)R couples to1 0 35c scalars

3/2 1/2 56v gravi-photinos2 0 35s pseudo-scalars2 1 28 gravi-photons

5/2 3/2 8v gravitinos3 2 1 graviton

Easier to look at scalars than at operators with spin.

There are 35 ∆ = 1 scalars SIJ (traceless symmetric) and 35∆ = 2 pseudo-scalars PAB (traceless symmetric).

Task: find the Mellin amplitudes MSSSS (6 fns), MSSPP (3 fns),MPPPP (6 fns) in the 1/cT expansion, and then take flat space limit.

Ward identity

〈SSSS〉 = 1x2

12x234× 6 functions Si(U,V ), i = 1, . . .6

〈SSPP〉 = 1x2

12x434× 3 functions Ri(U,V ), i = 1, . . .3

〈PPPP〉 = 1x4

12x434× 6 functions Pi(U,V ), i = 1, . . .6

The Si(U,V ) obey differential relations [Dolan, Gallot, Sokatchev ’04] .Example:

∂US4(U,V ) =1US4(U,V ) +

(1U− ∂U − ∂V

)S2(U,V ) +

+ (U − 1)∂U + V∂V

)S3(U,V ) ,

∂VS4(U,V ) = −1

2VS4(U,V )−

(1− U∂U + (U − 1)∂V )S2(U,V )− (∂U + ∂V )S3(U,V ) .

One can derive differential relations relating Ri and Pi to Si [Binder,Chester, SSP ’18] . (Pretty hard!) Example:

R1(U,V ) =14

(U2 − 4U

)∂U +

(4 + U − 2U2 + 7UV − 4V 2

+ 2U(2V − U + 2)(U∂2U + (U + V − 1)∂U∂V + V∂2

]S1(U,V ) .

Ward identity

〈SSSS〉 = 1x2

12x234× 6 functions Si(U,V ), i = 1, . . .6

〈SSPP〉 = 1x2

12x434× 3 functions Ri(U,V ), i = 1, . . .3

〈PPPP〉 = 1x4

12x434× 6 functions Pi(U,V ), i = 1, . . .6

The Si(U,V ) obey differential relations [Dolan, Gallot, Sokatchev ’04] .Example:

∂US4(U,V ) =1US4(U,V ) +

(1U− ∂U − ∂V

)S2(U,V ) +

+ (U − 1)∂U + V∂V

)S3(U,V ) ,

∂VS4(U,V ) = −1

2VS4(U,V )−

(1− U∂U + (U − 1)∂V )S2(U,V )− (∂U + ∂V )S3(U,V ) .

One can derive differential relations relating Ri and Pi to Si [Binder,Chester, SSP ’18] . (Pretty hard!) Example:

R1(U,V ) =14

(U2 − 4U

)∂U +

(4 + U − 2U2 + 7UV − 4V 2

+ 2U(2V − U + 2)(U∂2U + (U + V − 1)∂U∂V + V∂2

]S1(U,V ) .

CFT 4-pt correlators at large N

In general, one cannot compute 4-pt functions using SUSiclocalization.

However, the requirementsThe Mellin amplitudes obey SUSY Ward identities;

The Mellin amplitudes are consistent with crossing symmetry;

At order 1/c79 + 2

9 nT , the Mellin amplitude grows at most as the nth

power of s, t ,u;

The Mellin amplitudes have the analytic properties appropriate fortree-level Witten diagrams

determine MSSSS, MSSPP , MPPPP up to a few constants whosenumber depends on n.

Number of solutions:

degree in s, t , u 1 2 3 4 5 6 7 . . .

11D vertex R R4 D4R4 D6R4 . . .

scaling c−1T c

− 53

T (0×)c− 19

− 73

T# of params 1 2 3 4 . . .

(degree 1 in [Zhou ’18] ); degree ≥ 2 in [Chester, SSP, Yin ’18] .)

The number of solutions matches number of solutions to the Wardidentity for the flat space scattering amplitudes.So:

To determine M(s, t) to order 1/cT we should compute one CFTquantity.

To determine M(s, t) to order 1/c5/3T we should compute two CFT

quantities. Etc.

Luckily, we can compute three CFT quantities (one being cT ) andrelate them to M(s, t).

degree in s, t , u 1 2 3 4 5 6 7 . . .

scaling c−1T c

− 53

T (0×)c− 19

− 73

T# of params 1 2 3 4 . . .

quantities. Etc.

degree in s, t , u 1 2 3 4 5 6 7 . . .

scaling c−1T c

− 53

T (0×)c− 19

− 73

T# of params 1 2 3 4 . . .

quantities. Etc.

degree in s, t , u 1 2 3 4 5 6 7 . . .

scaling c−1T c

− 53

T (0×)c− 19

− 73

T# of params 1 2 3 4 . . .

quantities. Etc.

What to compute

What to compute: mass deformed S3 partition functionZ (m1,m2) = e−F (m1,m2) as a function of two masses m1, m2.

How to compute: use SUSic localization to write Z (m1,m2) as anintegral [Kapustin, Willett, Yaakov ’09] ; then use statistical physicstechniques to extract F (m1,m2) to all orders in the 1/Nexpansion! [Marino, Putrov ’11; Nosaka ’15]

The following quantities

∂2F∂m2

∣∣∣∣m1=m2=0

,∂4F∂m4

∣∣∣∣m1=m2=0

,∂4F

∂m21∂m2

∣∣∣∣m1=m2=0

can be related to 〈SSSS〉, 〈SSPP〉, and 〈PPPP〉 and thus can beused to determine M(s, t) up to order 1/c19/9

Mass-deformed S3 partition function

ABJM theory at k = 1 (or 2) has so(8)R R-symmetry.

As an N = 2 SCFT, it has so(2)R R-symmetry as well as su(4)flavor symmetry.

So there exists an su(4) flavor current multiplet (conserved currentjµ, scalar J with ∆ = 1, pseudoscalar K with ∆ = 2, fermions).

Can couple it to an su(4) background vector multiplet (Aµ,D, σ,fermions): ∫

d3x tr(

Aµjµ + DJ + Kσ + fermions).

On S3, the following background preserves SUSY:

Aµ(x) = 0 , D(x) =imr, σ(x) = m , fermions = 0 .

The deformation∫

d3x tr m(

ir J + K

)is a real mass deformation.

Two mass parameters

The Cartan of su(4) is 3-dimensional, so there are 3 independentmass parameters.

However, expanded to order m4, the S3 free energy takes the form

F (m) = f0 + f2 tr m2 + f3 tr m3 + f4,1(tr m2)2 + f4,2 tr m4 + O(m5) .

Up to order m4, we don’t lose any info by considering only 2 massparameters (instead of 3):

m = diagm1,−m1,m2,−m2 .

Localization result

Using supersymmetric localization [Kapustin, Willett, Yaakov ’09] :

ZS3(m1,m2) =

∫dNλdNµ

eik∑

i (λ2i −µ

2i )∏

i<j sinh2(λi − λj) sinh2(µi − µj)∏i,j cosh(λi − µj + m1

2 ) cosh(λi − µj + m22 )

Small N: can evaluate integral exactly.

Large N: rewrite ZS3(m) as the partition function of non-interactingFermi gas of N particles with [Marino, Putrov ’11; Nosaka ’15]

U(x) = log(2 cosh x)−m1x , T (p) = log(2 cosh p)−m2p .

Resummed perturbative expansion [Nosaka ’15] :

ZS3(m) ∼ Ai (f1(m1,m2)N − f2(m1,m2))

for some known functions f1 and f2. (log Z ∝ N3/2)

Localization result

Using supersymmetric localization [Kapustin, Willett, Yaakov ’09] :

ZS3(m1,m2) =

∫dNλdNµ

eik∑

i (λ2i −µ

2i )∏

i<j sinh2(λi − λj) sinh2(µi − µj)∏i,j cosh(λi − µj + m1

2 ) cosh(λi − µj + m22 )

Small N: can evaluate integral exactly.

Large N: rewrite ZS3(m) as the partition function of non-interactingFermi gas of N particles with [Marino, Putrov ’11; Nosaka ’15]

U(x) = log(2 cosh x)−m1x , T (p) = log(2 cosh p)−m2p .

Resummed perturbative expansion [Nosaka ’15] :

ZS3(m) ∼ Ai (f1(m1,m2)N − f2(m1,m2))

for some known functions f1 and f2. (log Z ∝ N3/2)

Integrated correlators

Relate mass derivatives to 〈SSSS〉, 〈SSPP〉, 〈PPPP〉.

Recall that the mass deformations are

2∑i=1

∫d3x

( irJi + Ki

)The ∆ = 1 operators Ji are linear combinations of the 35c scalarsSIJ .

The ∆ = 2 operators Ki are linear combinations of the 35spseudoscalars PIJ .

∂2F∂m2

i= −

⟨(∫ ir Ji +

∫Ki)2⟩

= integrated 2-pt fn.

∂4F∂m2

i ∂m2j

= −⟨(∫ i

r Ji +∫

Ki)2 (∫ i

r Jj +∫

Kj)2⟩

= integrated 4-pt fn.

Integrated 2-pt function

Because S and P belong to the same multiplet as the stresstensor, 〈TµνTρσ〉 ∝ cT implies 〈SS〉 ∝ cT and 〈PP〉 ∝ cT .

Then 〈JJ〉 ∝ cT and 〈KK 〉 ∝ cT , and

∂2F∂m2

32cT .

But what does this have to do with the 4-pt function?

(Super)Conformal block decomposition

〈SSSS〉 =1

x212x2

∑multipletsM

λ2MgM(U,V )

Then, in a normalization where λ2id = 1,

λ2stress =

〈SSTµν〉〈SSTρσ〉〈TµνTρσ〉

=256cT

Because S and P belong to the same multiplet as the stresstensor, 〈TµνTρσ〉 ∝ cT implies 〈SS〉 ∝ cT and 〈PP〉 ∝ cT .

Then 〈JJ〉 ∝ cT and 〈KK 〉 ∝ cT , and

∂2F∂m2

32cT .

But what does this have to do with the 4-pt function?

(Super)Conformal block decomposition

〈SSSS〉 =1

x212x2

∑multipletsM

λ2MgM(U,V )

Then, in a normalization where λ2id = 1,

λ2stress =

〈SSTµν〉〈SSTρσ〉〈TµνTρσ〉

=256cT

Four mass derivatives

Expressing the four mass derivatives in terms of cT :

− 1c2

∂4F∂m4

+(3π)4/3

210/31

+(· · · )c2

T− (18π2)1/3 1

+ · · ·

− 1c2

∂4F∂m2

1∂m22

= −π2

+5π4/3

210/332/31

+(· · · )c2

T− 4(2π2)1/3

310/31

+ · · ·

Impose these constraints on the 4-pt functions 〈SSSS〉, 〈SSPP〉,〈PPPP〉 using

∂4F∂m2

i ∂m2j

⟨(∫irJi +

)2(∫ irJj +

If 〈AABB〉 = 1x

2∆A12 x

2∆B34

G(U,V ) in flat space, then the integrated

correlator on S3 (with metric ds2 = Ω−2(~x)d~x2) is⟨(∫A)2(∫

B)2⟩

∫ 4∏i=1

d3~xi(Ω(~x1)Ω(~x2))∆A−3(Ω(~x3)Ω(~x4))∆B−3

x2∆A12 x2∆B

with Ω(~x) = 1 + x2

This non-conformal integral can be written as⟨(∫A)2(∫

B)2⟩∝∫

dU dV D3−∆A,3−∆A,3−∆B ,3−∆B (U,V )G(U,V )

U∆A.

where D function is the (Euclidean) AdS contact Witten diagramfor the 4-pt function of ops of dim 3−∆A, 3−∆A, 3−∆B, 3−∆B.

If 〈AABB〉 = 1x

2∆A12 x

2∆B34

G(U,V ) in flat space, then the integrated

correlator on S3 (with metric ds2 = Ω−2(~x)d~x2) is⟨(∫A)2(∫

B)2⟩

∫ 4∏i=1

d3~xi(Ω(~x1)Ω(~x2))∆A−3(Ω(~x3)Ω(~x4))∆B−3

x2∆A12 x2∆B

with Ω(~x) = 1 + x2

This non-conformal integral can be written as⟨(∫A)2(∫

B)2⟩∝∫

dU dV D3−∆A,3−∆A,3−∆B ,3−∆B (U,V )G(U,V )

U∆A.

where D function is the (Euclidean) AdS contact Witten diagramfor the 4-pt function of ops of dim 3−∆A, 3−∆A, 3−∆B, 3−∆B.

D functionNow for the 4 pt functions, we normally consider both contact andexchange diagrams:

They respectively yield the following contributions:

where ,J(X , X ) is the spin J bulk to bulk propagator.

Heng-Yu Chen National Taiwan University Spinning Witten Diagrams: From Geodesic to Mellin

= Dr1,r2,r3,r4(~xi) =∫ dz0dd~z

∏4i=1 Gri

B∂(z0, ~z;~xi)

GriB∂(z0, ~z;~xi) =

z20 + (~z − ~xi)2

The D function is defined as

Dr1,r2,r3,r4 (U,V ) =x

12∑4

i=1 ri−r413 x r2

x12∑4

i=1 ri−r1−r414 x

12∑4

i=1 ri−r3−r434

i=1 Γ(ri )

πd2 Γ

(−d+

∑4i=1 ri

)Dr1,r2,r3,r4 (xi )

The reason why D functions appear in the integrated 4-ptfunctions on S3 is that SO(4,1)/SO(4) = H4.

Summary of computation

Superconformal Ward id + asymptotic growth in Mellin space +crossing symmetry + analytic structure of Mellin tree amplitudes=⇒ determine Mellin amplitudes MSSSS, MSSPP , MPPPP in 1/cTexpansion up to a few undetermined constants at each order

SUSic localization =⇒ ZS3(m1,m2) =⇒ F (m1,m2) in 1/cTexpansion =⇒ 2nd and 4th derivatives of F (m1,m2) in 1/cTexpansion =⇒ conditions on integrated correlators 〈SSSS〉,〈SSPP〉, 〈PPPP〉.

Combine above =⇒ determine tree-level MSSSS, MSSPP , MPPPP upto 1/c19/9

T =⇒ take flat space limit to determine scatteringamplitudes in 11d.

Precision test of AdS/CFT

The flat space limit implies fR4(s, t) = stu3·27 and fD4R4 = 0, as

expected.

This is a precision test of AdS/CFT beyond supergravity!!

Precision test of AdS/CFT

The flat space limit implies fR4(s, t) = stu3·27 and fD4R4 = 0, as

expected.

This is a precision test of AdS/CFT beyond supergravity!!

Beyond fR4 and fD4R4?

Can one go beyond reconstructing fR4 and fD4R4?

More SUSic localization results for ABJM theory are available,e.g. partition function on a squashed sphere.

It is likely that one can also determine fD6R4 in a similar way.

Another approach: conformal bootstrap.

Generally, we obtain bounds on various quantities.

If the bounds are saturated, then we can solve for the CFT data.

Exact OPE coefficients

As already mentioned, in the superconformal block decomposition

〈SSSS〉 =1

x212x2

∑multipletsM

λ2MgM(U,V )

λ2stress = 256/cT can be computed to all orders in 1/N because cT

is proportional to ∂2F∂m2

∣∣∣m1=m2=0

There is another OPE coefficient λ2(B,2) of a 1/4-BPS multiplet

“(B,2)” that can be related to ∂4F∂m4

∣∣∣m1=m2=0

and hence can be

computed to all orders in 1/N.

Exact OPE coefficients

As already mentioned, in the superconformal block decomposition

〈SSSS〉 =1

x212x2

∑multipletsM

λ2MgM(U,V )

λ2stress = 256/cT can be computed to all orders in 1/N because cT

is proportional to ∂2F∂m2

∣∣∣m1=m2=0

There is another OPE coefficient λ2(B,2) of a 1/4-BPS multiplet

“(B,2)” that can be related to ∂4F∂m4

∣∣∣m1=m2=0

and hence can be

computed to all orders in 1/N.

Topological sector

3d N = 4 SCFTs have a 1d topological sector [Beem, Lemos, Liendo,

Peelaers, Rastelli, van Rees ’13; Chester, Lee, SSP, Yacoby ’14; Dedushenko, SSP,

Yacoby ’16] defined on a line (0,0, x) in R3.

〈O1(x1) . . .On(xn)〉 depends only on the ordering of xi on the line.

Ops in 1d are 3d 1/2-BPS operators O(~x) placed at ~x = (0,0, x)and contracted with x-dependent R-symmetry polarizations.

The operators O(x) are in the cohomology of a superchargeQ = “Q + S′′ cohomology s.t. translations in x are Q-exact.

The topological sector is defined either on a line in flat space or ona great circle of S3.

In ABJM, construct 1d operators Sα(x) from SIJ , α = 1,2,3. Their2-pt function depends only on cT ; their 4-pt function depends onlyon cT and λ2

(B,2).Silviu Pufu (Princeton University) 10-26-2018 26 / 32

Topological sector

From ZS3(m1) to OPE coefficients

One can argue ZS3(m1) = ZS1(m1), and so derivatives of the ZS3

w.r.t. m1 corresponds to integrated correlators in the 1d theory.

From 2 derivatives of ZS3 w.r.t. m1 we can extract cT .

From 4 derivatives of ZS3 w.r.t. m1 we can extract λ2(B,2).

So the (resummed) perturbative expansion of cT , λ2B,2 can be

written in terms of derivatives of the Airy function!

Eliminating N gives

λ2B,2 =

323− 1024(4π2 − 15)

9π21cT

+ 40960(

) 13 1

+ · · ·

(Tangent: For 2d bulk dual of the 1d topological sector of ABJMtheory, see [Mezei, SSP, Wang ’17] . The 1d theory is exactly solvable,and its 2d bulk dual is 2d YM.)

Eliminating N gives

λ2B,2 =

323− 1024(4π2 − 15)

9π21cT

+ 40960(

) 13 1

+ · · ·

Eliminating N gives

λ2B,2 =

323− 1024(4π2 − 15)

9π21cT

+ 40960(

) 13 1

+ · · ·

Known N = 8 SCFTs

A few families of N = 8 SCFTs:A1,A2

k −k

With holographic duals:

ABJMN,1: U(N)1 × U(N)−1 ←→ AdS4 × S7.

ABJMN,2: U(N)2 × U(N)−2 ←→ AdS4 × S7/Z2.

ABJN,2: U(N)2 × U(N + 1)−2 ←→ AdS4 × S7/Z2.

Without known holographic duals:

BLGk : SU(2)k × SU(2)−k .

Bootstrap bounds [Agmon, Chester, SSP ’17]

Bounds from conformal bootstrap applying to all N = 8 SCFTs.

ABJMN,1int

ABJMN,2

ABJM1,1

0.2 0.4 0.6 0.8 1.0

λ(B,2)2

SUGRA (leading large cT ) saturates bootstrap bounds.Conjecture: ABJMN,1 or ABJMN,2 or ABJN,2 saturate bound at allN in the limit of infinite precision.

λ2B,2 =

323− 1024(4π2 − 15)

9π21cT

+ 40960(

29π8k2

) 13 1

+ · · · .Silviu Pufu (Princeton University) 10-26-2018 29 / 32

Bootstrap bounds [Agmon, Chester, SSP ’17]

Bounds from conformal bootstrap applying to all N = 8 SCFTs.

ABJMN,1int

ABJMN,2

ABJM1,1

0.2 0.4 0.6 0.8 1.0

λ(B,2)2

SUGRA (leading large cT ) saturates bootstrap bounds.Conjecture: ABJMN,1 or ABJMN,2 or ABJN,2 saturate bound at allN in the limit of infinite precision.

λ2B,2 =

323− 1024(4π2 − 15)

9π21cT

+ 40960(

29π8k2

) 13 1

+ · · · .Silviu Pufu (Princeton University) 10-26-2018 29 / 32

Bound saturation =⇒ read off CFT data

On the boundary of the bootstrap bounds, the solution to crossingshould be unique =⇒ can find 〈SIJSKLSMNSPQ〉 and solve for thespectrum !! [Agmon, Chester, SSP ’17]

lowest spin 0

lowest spin 2

lowest spin 4

0.1 0.2 0.3 0.4 0.5 0.6 0.716/cT

Red lines are leading SUGRA tree level results [Zhou ’17; Chester ’18] .Lowest operators have the form SIJ∂µ1 · · · ∂µ`SIJ .

λ2(A,2)j

and λ2(A,+)j

from extremal functional

Semishort (A,2)j and (A,+)j OPE coefficients for low spin j in terms of16cT

from extremal functional:

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

λ(A,2) j2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

λ(A,+) j2

Red line is tree level SUGRA result [Chester ’18] .

λ2(A,+)j

appears close to linear in 16/cT .

More precision needed.Silviu Pufu (Princeton University) 10-26-2018 31 / 32

λ2(A,2)j

and λ2(A,+)j

from extremal functional

Semishort (A,2)j and (A,+)j OPE coefficients for low spin j in terms of16cT

from extremal functional:

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

λ(A,2) j2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

λ(A,+) j2

Red line is tree level SUGRA result [Chester ’18] .

λ2(A,+)j

appears close to linear in 16/cT .

More precision needed.Silviu Pufu (Princeton University) 10-26-2018 31 / 32

Conclusion

A combination of techniques (supersymmetric localization, SUSYWard identities, Mellin space) can be used to recover gravitonscattering amplitudes (at small momentum) from ABJM theory.

We can reproduce the fR4(s, t) = stu3·27 term in the flat space

4-graviton scattering amplitude and show that fD4R4 = 0.

Bootstrap bounds are almost saturated by N = 8 SCFTs withholographic duals.

For the future:

Generalize to other dimensions, other 4-point function, lessSUSY. (See also [Chester, Perlmutter ’18] in 6d.)

Study other SCFTs from which one can compute scatteringamplitudes of gauge bosons on branes. (?)

Loops in AdS.Silviu Pufu (Princeton University) 10-26-2018 32 / 32

Conclusion

A combination of techniques (supersymmetric localization, SUSYWard identities, Mellin space) can be used to recover gravitonscattering amplitudes (at small momentum) from ABJM theory.

We can reproduce the fR4(s, t) = stu3·27 term in the flat space

4-graviton scattering amplitude and show that fD4R4 = 0.

Bootstrap bounds are almost saturated by N = 8 SCFTs withholographic duals.

For the future:

Generalize to other dimensions, other 4-point function, lessSUSY. (See also [Chester, Perlmutter ’18] in 6d.)

Study other SCFTs from which one can compute scatteringamplitudes of gauge bosons on branes. (?)

Loops in AdS.Silviu Pufu (Princeton University) 10-26-2018 32 / 32

silviu s. pufu, princeton universitysilviu pufu (princeton university) 10-26-2018 5 / 32 in 11d, we...

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