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
Also:
arXiv:1406.4814, arXiv:1412.0334 with S. Chester, J. Lee, and R. Yacoby
arXiv:1610.00740 with M. Dedushenko and R. Yacoby

OIST, June 26, 2018
Silviu Pufu (Princeton University)

6-26-20181 / 20Silviu Pufu (Princeton University)

6-26-20181 / 20Motivation
Learn about gravity / string theory / M-theory from CFT.
3d maximally supersymmetric (N = 8) CFTs w/ gravity duals:
explicit Lagrangians; no marginal coupling; SUSY.
Most well-understood example:
M-theory on AdS4 × S 7 ⇐⇒ U(N)k × U(N)−k ABJM theory at CS
level k = 1.
Last 10 years: progress in QFT calculations
using supersymmetric localization;using conformal bootstrap in CFTs.

What do these calculations tell us about M-theory?
Example: S 3 partition function of ABJM theory can be written as
an N-dim’l integral. What info about M-theory does it contain?
Silviu Pufu (Princeton University)

6-26-20182 / 20Motivation
Learn about gravity / string theory / M-theory from CFT.
3d maximally supersymmetric (N = 8) CFTs w/ gravity duals:
explicit Lagrangians; no marginal coupling; SUSY.
Most well-understood example:
M-theory on AdS4 × S 7 ⇐⇒ U(N)k × U(N)−k ABJM theory at CS
level k = 1.
Last 10 years: progress in QFT calculations
using supersymmetric localization;using conformal bootstrap in CFTs.

What do these calculations tell us about M-theory?
Example: S 3 partition function of ABJM theory can be written as
an N-dim’l integral. What info about M-theory does it contain?
Silviu Pufu (Princeton University)

6-26-20182 / 20Motivation
Learn about gravity / string theory / M-theory from CFT.
3d maximally supersymmetric (N = 8) CFTs w/ gravity duals:
explicit Lagrangians; no marginal coupling; SUSY.
Most well-understood example:
M-theory on AdS4 × S 7 ⇐⇒ U(N)k × U(N)−k ABJM theory at CS
level k = 1.
Last 10 years: progress in QFT calculations
using supersymmetric localization;using conformal bootstrap in CFTs.

What do these calculations tell us about M-theory?
Example: S 3 partition function of ABJM theory can be written as
an N-dim’l integral. What info about M-theory does it contain?
Silviu Pufu (Princeton University)

6-26-20182 / 20Motivation
Learn about gravity / string theory / M-theory from CFT.
3d maximally supersymmetric (N = 8) CFTs w/ gravity duals:
explicit Lagrangians; no marginal coupling; SUSY.
Most well-understood example:
M-theory on AdS4 × S 7 ⇐⇒ U(N)k × U(N)−k ABJM theory at CS
level k = 1.
Last 10 years: progress in QFT calculations
using supersymmetric localization;using conformal bootstrap in CFTs.

What do these calculations tell us about M-theory?
Example: S 3 partition function of ABJM theory can be written as
an N-dim’l integral. What info about M-theory does it contain?
Silviu Pufu (Princeton University)

6-26-20182 / 20M-theory S-matrix

This talk: Reconstruct M-theory S-matrix perturbatively at small
momentum (scatter gravitons and superpartners).
Equivalently, reconstruct the derivative expansion of the M-theory
effective action. Schematically,Zi√ h
S = d 11 x g R + Riem4 + · · · + (SUSic completion) .
Restrict momenta to be in 4 out of the 11 dimensions.

Silviu Pufu (Princeton University)

6-26-20183 / 20M-theory S-matrix

This talk: Reconstruct M-theory S-matrix perturbatively at small
momentum (scatter gravitons and superpartners).
Equivalently, reconstruct the derivative expansion of the M-theory
effective action. Schematically,Zi√ h
S = d 11 x g R + Riem4 + · · · + (SUSic completion) .
Restrict momenta to be in 4 out of the 11 dimensions.

Silviu Pufu (Princeton University)

6-26-20183 / 20M-theory S-matrix

This talk: Reconstruct M-theory S-matrix perturbatively at small
momentum (scatter gravitons and superpartners).
Equivalently, reconstruct the derivative expansion of the M-theory
effective action. Schematically,Zi√ h
S = d 11 x g R + Riem4 + · · · + (SUSic completion) .
Restrict momenta to be in 4 out of the 11 dimensions.

Silviu Pufu (Princeton University)

6-26-20183 / 20From the 4d point of view, we can scatter:
graviton (1);gravitinos (8);gravi-photons (28);
gravi-photinos (56);
scalars (70 = 35 + 35)

At leading order in small momentum (i.e. momentum squared),scattering amplitudes are those in N = 8 SUGRA at tree level.
Amplitude depends on the type of particle, e.g.
h12i4 [34]4,stutu
ASUGRA, tree (S1 S1 S2 S2 ) =
,
s
etc.

ASUGRA, tree (h− h− h+ h+ ) =but they’re all related by SUSY. (See Elvang & Huang’s book.)
Silviu Pufu (Princeton University)

6-26-2018

4 / 20From the 4d point of view, we can scatter:
graviton (1);gravitinos (8);gravi-photons (28);
gravi-photinos (56);
scalars (70 = 35 + 35)

At leading order in small momentum (i.e. momentum squared),scattering amplitudes are those in N = 8 SUGRA at tree level.
Amplitude depends on the type of particle, e.g.
h12i4 [34]4,stutu
ASUGRA, tree (S1 S1 S2 S2 ) =
,
s
etc.

ASUGRA, tree (h− h− h+ h+ ) =but they’re all related by SUSY. (See Elvang & Huang’s book.)
Silviu Pufu (Princeton University)

6-26-2018

4 / 20Momentum expansion
Momentum expansion takes a universal form (independent of thetype of particle):
A = ASUGRA, tree 1 + `6p fR 4 (s, t) + `9p f1-loop (s, t)1412+ `p fD 6 R 4 (s, t) + `p fD 8 R 4 (s, t) + · · · .
fD 2n R 4 = symmetric polyn in s, t, u of degree n + 3
Known from type II string theory + SUSY [Green, Tseytlin, Gutperle,
Vanhove, Russo, Pioline, . . . ] :stu(stu)2,f.
6 R 4 (s, t) =D
3 · 2715 · 215`10
p fD 4 R 4 allowed by SUSY, but known to vanish.
fR 4 (s, t) =This talk: Reproduce fR 4 from 3d SCFT.
Silviu Pufu (Princeton University)

6-26-2018

5 / 20Flat space limit of CFT correlators
Idea: Flat space scattering amplitudes can be obtained as limit of
CFT correlators [Polchinski ’99; Susskind ’99; Giddings ’99; Penedones ’10;
Fitzpatrick, Kaplan ’11] .
For a CFT3 operator φ(x) with ∆φ = 1,hφ(x1 )φ(x2 )φ(x3 )φ(x4 )iconn =

1
2 x2
x12
34

g(U, V )go to Mellin space

 
Z
s 2
ds dt t/2 (u−2)/2 2 
t
u
2
1
−
g(U, V ) =
U
V
Γ
Γ
1
−
1
−
M(s, t)Γ222(4πi)2
x2 x2

x2 x2

13 24

13 24

14 23
34
where s + t + u = 4 and U = x12
2 x2 , V = x2 x2 .

From the large s, t limit of M(s, t) one can extract 4d scattering
amplitude A(s, t) [Penedones ’10; Fitzpatrick, Kaplan ’11] .
Silviu Pufu (Princeton University)

6-26-2018

6 / 20Flat space limit of CFT correlators
Idea: Flat space scattering amplitudes can be obtained as limit of
CFT correlators [Polchinski ’99; Susskind ’99; Giddings ’99; Penedones ’10;
Fitzpatrick, Kaplan ’11] .
For a CFT3 operator φ(x) with ∆φ = 1,hφ(x1 )φ(x2 )φ(x3 )φ(x4 )iconn =

1
2 x2
x12
34

g(U, V )go to Mellin space

 
Z
s 2
ds dt t/2 (u−2)/2 2 
t
u
2
1
−
g(U, V ) =
U
V
Γ
Γ
1
−
1
−
M(s, t)Γ222(4πi)2
x2 x2

x2 x2

13 24

13 24

14 23
34
where s + t + u = 4 and U = x12
2 x2 , V = x2 x2 .

From the large s, t limit of M(s, t) one can extract 4d scattering
amplitude A(s, t) [Penedones ’10; Fitzpatrick, Kaplan ’11] .
Silviu Pufu (Princeton University)

6-26-2018

6 / 20To obtain scattering amplitude of graviton + superpartners in
M-theory, look at stress tensor multiplet in ABJM theory (ABJM
theory is a 3d N = 8 SCFT, and so it has so(8)R R-symmetry):
focus on this −→dimension
1
3/2
2
2
5/2
3

spin
0
1/2
0
1
3/2
2

so(8)R
35c
56v
35s
28
8v
1

couples toscalars
gravi-photinos
pseudo-scalars
gravi-photons
gravitinos
graviton

Task: find the Mellin amplitude M(s, t) corresponding tohSIJ SKL SMN SPQ i by solving superconformal Ward identity [Dolan,−1/3 ∝ c −2/9 .
Gallot, Sokatchev ’04] order by order in `2p ∝NTHere, hTµν Tρσ i ∝ cT ∝ N 3/2 .
Silviu Pufu (Princeton University)

6-26-20187 / 20Require: 1) at order `2k
p , M(s, t) should not grow faster than(k + 1)st power of s, t, u;2) right analytic properties to correspond to a bulk tree-level
Witten diagram.
Number of such solutions to Ward identity is:degree in s, t, u
11D vertex

1R

scaling# of params

cT−1
1

2

3

4
R4
−5

cT 3
2

5

6
D4R4
− 19

(0×)cT 9
3

7
D6R4

...
...

−7

cT 3
4

...

(degree 1 in [Zhou ’18] ); degree ≥ 2 in [Chester, SSP, Yin ’18] .)So:
To determine M(s, t) to order 1/cT we should compute one CFT
quantity.
5/3

To determine M(s, t) to order 1/cT
quantities.
Silviu Pufu (Princeton University)

we should compute two CFT
6-26-20188 / 20Require: 1) at order `2k
p , M(s, t) should not grow faster than(k + 1)st power of s, t, u;2) right analytic properties to correspond to a bulk tree-level
Witten diagram.
Number of such solutions to Ward identity is:degree in s, t, u
11D vertex

1R

scaling# of params

cT−1
1

2

3

4
R4
−5

cT 3
2

5

6
D4R4
− 19

(0×)cT 9
3

7
D6R4

...
...

−7

cT 3
4

...

(degree 1 in [Zhou ’18] ); degree ≥ 2 in [Chester, SSP, Yin ’18] .)So:
To determine M(s, t) to order 1/cT we should compute one CFT
quantity.
5/3

To determine M(s, t) to order 1/cT
quantities.
Silviu Pufu (Princeton University)

we should compute two CFT
6-26-20188 / 20CFT quantities
The CFT quantities can be, for instance, squared OPE coefficients
appearing in the superconformal block decomposition.
Schematically,X
1
λ2M GM (U, V ) .
hSIJ SKL SMN SPQ i = 2 2x12 x34 N = 8 supermultiplets M(M is a superconformal multiplet appearing in the S × S OPE.)
Tµν Ward identity gives λ2stress = 256/cT . Using SUSY tricks, one
can compute [Agmon, Chester, SSP ’17] :
1

2 3 1
32 1024(4π 2 − 15) 1+
40960
λ2B,2 =
−
+ ···
5/3
3
cT
9π 2
9π 8
cT
where “stress” is the stress tensor multiplet, and “(B, 2)” is a
1/4-BPS mutliplet appearing in the OPE S × S.
Silviu Pufu (Princeton University)

6-26-2018

9 / 20Precision test of AdS/CFT

Using these two expressions, we determined M(s, t) to order
5/3
1/cT .
stu
The flat space limit implies fR 4 (s, t) = 3·2
7 , as expected.

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

Silviu Pufu (Princeton University)

6-26-201810 / 20

Precision test of AdS/CFT

Using these two expressions, we determined M(s, t) to order
5/3
1/cT .
stu
The flat space limit implies fR 4 (s, t) = 3·2
7 , as expected.

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

Silviu Pufu (Princeton University)

6-26-201810 / 20

Precision test of AdS/CFT

Using these two expressions, we determined M(s, t) to order
5/3
1/cT .
stu
The flat space limit implies fR 4 (s, t) = 3·2
7 , as expected.

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

Silviu Pufu (Princeton University)

6-26-201810 / 20

OPE coefficients from SUSic localization

It is hard to calculate correlation functions at separated points
using SUSic localization. See however [Gerkchovitz, Gomis, Ishtiaque,Karasik, Komargodski, SSP ’16; Dedushenko, SSP, Yacoby ’16] .
How were cT and λ2(B,2) computed?
From derivatives of the S 3 partition function with respect to an
N = 4-preserving mass parameter m, which can be computed
using supersymmetric localization.
(For cT , see also [Chester, Lee, SSP, Yacoby ’14] for another method
based on [Closset, Dumitrescu, Festuccia, Komargodski, Seiberg ’12] .)Silviu Pufu (Princeton University)

6-26-201811 / 20

OPE coefficients from SUSic localization

It is hard to calculate correlation functions at separated points
using SUSic localization. See however [Gerkchovitz, Gomis, Ishtiaque,Karasik, Komargodski, SSP ’16; Dedushenko, SSP, Yacoby ’16] .
How were cT and λ2(B,2) computed?
From derivatives of the S 3 partition function with respect to an
N = 4-preserving mass parameter m, which can be computed
using supersymmetric localization.
(For cT , see also [Chester, Lee, SSP, Yacoby ’14] for another method
based on [Closset, Dumitrescu, Festuccia, Komargodski, Seiberg ’12] .)Silviu Pufu (Princeton University)

6-26-201811 / 20

Mass-deformed S 3 partition function
For an N = 4-preserving mass deformation of ABJM theory,
ZS 3 (m) is [Kapustin, Willett, Yaakov ’09] :QZP
sinh2 (λi − λj ) sinh2 (µi − µj )NNik i (λ2i −µ2i ) Q i<j
ZS 3 (m) = d λ d µ ei,j cosh(λi − µj + m) cosh(λi − µj )Small N: can evaluate integral exactly.
Large N: rewrite ZS 3 (m) as the partition function of non-interacting
Fermi gas of N particles with [Marino, Putrov ’11; Nosaka ’15]
U(x) = log(2 cosh x) − mx ,T (p) = log(2 cosh p) .

Resummed perturbative expansion [Nosaka ’15] :ZS 3 (m) ∼ Ai (f1 (m)N − f2 (m))
for some known functions f1 (m) and f2 (m). (log Z ∝ N 3/2 )Silviu Pufu (Princeton University)

6-26-201812 / 20

Mass-deformed S 3 partition function
For an N = 4-preserving mass deformation of ABJM theory,
ZS 3 (m) is [Kapustin, Willett, Yaakov ’09] :QZP
sinh2 (λi − λj ) sinh2 (µi − µj )NNik i (λ2i −µ2i ) Q i<j
ZS 3 (m) = d λ d µ ei,j cosh(λi − µj + m) cosh(λi − µj )Small N: can evaluate integral exactly.
Large N: rewrite ZS 3 (m) as the partition function of non-interacting
Fermi gas of N particles with [Marino, Putrov ’11; Nosaka ’15]
U(x) = log(2 cosh x) − mx ,T (p) = log(2 cosh p) .

Resummed perturbative expansion [Nosaka ’15] :ZS 3 (m) ∼ Ai (f1 (m)N − f2 (m))
for some known functions f1 (m) and f2 (m). (log Z ∝ N 3/2 )Silviu Pufu (Princeton University)

6-26-201812 / 20

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 .

hO1 (x1 ) . . . On (xn )i 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 supercharge
Q = “Q + S 00 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 S 3 .
In ABJM, construct 1d operators Sα (x) from SIJ , α = 1, 2, 3. Their
2-pt function depends on cT ; their 4-pt function depends on cT
and λ2(B,2) .
Silviu Pufu (Princeton University)

6-26-201813 / 20

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 .

hO1 (x1 ) . . . On (xn )i 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 supercharge
Q = “Q + S 00 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 S 3 .
In ABJM, construct 1d operators Sα (x) from SIJ , α = 1, 2, 3. Their
2-pt function depends on cT ; their 4-pt function depends on cT
and λ2(B,2) .
Silviu Pufu (Princeton University)

6-26-201813 / 20

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 .

hO1 (x1 ) . . . On (xn )i 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 supercharge
Q = “Q + S 00 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 S 3 .
In ABJM, construct 1d operators Sα (x) from SIJ , α = 1, 2, 3. Their
2-pt function depends on cT ; their 4-pt function depends on cT
and λ2(B,2) .
Silviu Pufu (Princeton University)

6-26-201813 / 20

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 .

hO1 (x1 ) . . . On (xn )i 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 supercharge
Q = “Q + S 00 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 S 3 .
In ABJM, construct 1d operators Sα (x) from SIJ , α = 1, 2, 3. Their
2-pt function depends on cT ; their 4-pt function depends on cT
and λ2(B,2) .
Silviu Pufu (Princeton University)

6-26-201813 / 20

From ZS 3 (m) to OPE coefficients
One can show ZS 3 (m) = ZS 1 (m), and so derivatives of the ZS 3
w.r.t. m corresponds to integrated correlators in the 1d theory.
From 2 derivatives of ZS 3 w.r.t. m we can extract cT .
From 4 derivatives of ZS 3 w.r.t. m 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
32 1024(4π 2 − 15) 1
λ2B,2 =−
+ 40960
3
cT
9π 2



2
9π 8

1
3

1
5/3
cT

+ ···

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

6-26-201814 / 20

From ZS 3 (m) to OPE coefficients
One can show ZS 3 (m) = ZS 1 (m), and so derivatives of the ZS 3
w.r.t. m corresponds to integrated correlators in the 1d theory.
From 2 derivatives of ZS 3 w.r.t. m we can extract cT .
From 4 derivatives of ZS 3 w.r.t. m 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
32 1024(4π 2 − 15) 1
λ2B,2 =−
+ 40960
3
cT
9π 2



2
9π 8

1
3

1
5/3
cT

+ ···

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

6-26-201814 / 20

From ZS 3 (m) to OPE coefficients
One can show ZS 3 (m) = ZS 1 (m), and so derivatives of the ZS 3
w.r.t. m corresponds to integrated correlators in the 1d theory.
From 2 derivatives of ZS 3 w.r.t. m we can extract cT .
From 4 derivatives of ZS 3 w.r.t. m 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
32 1024(4π 2 − 15) 1
λ2B,2 =−
+ 40960
3
cT
9π 2



2
9π 8

1
3

1
5/3
cT

+ ···

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

6-26-201814 / 20

Beyond fR 4 ?
Can one go beyond reconstructing fR 4 ?
More SUSic localization results for ABJM theory are available: ZS 3
as a function of three real mass parameters; partition function ona squashed sphere, etc.
Cannot use the 1d topological sector in this case, but it is very likelythat this extra data will show fD4 R 4 = 0 and maybe even determine
fD6 R 4 . (Work in progress with D. Binder and S. Chester.)

Another approach: conformal bootstrap.
Generally, we obtain bounds on various quantities.
If the bounds are saturated, then we can solve for the CFT data.

Silviu Pufu (Princeton University)

6-26-201815 / 20

Beyond fR 4 ?
Can one go beyond reconstructing fR 4 ?
More SUSic localization results for ABJM theory are available: ZS 3
as a function of three real mass parameters; partition function ona squashed sphere, etc.
Cannot use the 1d topological sector in this case, but it is very likelythat this extra data will show fD4 R 4 = 0 and maybe even determine
fD6 R 4 . (Work in progress with D. Binder and S. Chester.)

Another approach: conformal bootstrap.
Generally, we obtain bounds on various quantities.
If the bounds are saturated, then we can solve for the CFT data.

Silviu Pufu (Princeton University)

6-26-201815 / 20

Beyond fR 4 ?
Can one go beyond reconstructing fR 4 ?
More SUSic localization results for ABJM theory are available: ZS 3
as a function of three real mass parameters; partition function ona squashed sphere, etc.
Cannot use the 1d topological sector in this case, but it is very likelythat this extra data will show fD4 R 4 = 0 and maybe even determine
fD6 R 4 . (Work in progress with D. Binder and S. Chester.)

Another approach: conformal bootstrap.
Generally, we obtain bounds on various quantities.
If the bounds are saturated, then we can solve for the CFT data.

Silviu Pufu (Princeton University)

6-26-201815 / 20

Known N = 8 SCFTs
A few families of N = 8 SCFTs:
A1 , A2
G1

G2

k

−k
B1 , B2

With holographic duals:
ABJMN,1 : U(N)1 × U(N)−1←→

AdS4 × S 7 .

ABJMN,2 : U(N)2 × U(N)−2←→

AdS4 × S 7 /Z2 .

ABJN,2 : U(N)2 × U(N + 1)−2←→

AdS4 × S 7 /Z2 .

Without known holographic duals:
BLGk : SU(2)k × SU(2)−k .
Silviu Pufu (Princeton University)

6-26-201816 / 20

Bootstrap bounds [Agmon, Chester, SSP ’17]
Bounds from conformal bootstrap applying to all N = 8 SCFTs.
λ2(B,2)
10

BLGk

8

ABJMint
N,1
ABJMN,2

6

ABJN

4

ABJM1,1

2

0.2

0.4

0.6

0.8

1.0

16
cT

SUGRA (leading large cT ) saturates bootstrap bounds.
Conjecture: ABJMN,1 or ABJMN,2 or ABJN,2 saturate bound at all
N in the limit of infinite precision.
Silviu Pufu (Princeton University) 1 6-26-2018 17 / 20

Bootstrap bounds [Agmon, Chester, SSP ’17]
Bounds from conformal bootstrap applying to all N = 8 SCFTs.
λ2(B,2)
10

BLGk

8

ABJMint
N,1
ABJMN,2

6

ABJN

4

ABJM1,1

2

0.2

0.4

0.6

0.8

1.0

16
cT

SUGRA (leading large cT ) saturates bootstrap bounds.
Conjecture: ABJMN,1 or ABJMN,2 or ABJN,2 saturate bound at all
N in the limit of infinite precision.
Silviu Pufu (Princeton University) 1 6-26-2018 17 / 20

Bound saturation =⇒ read off CFT data
On the boundary of the bootstrap bounds, the solution to crossing
should be unique =⇒ can find hSIJ SKL SMN SPQ i and solve for the
spectrum !! [Agmon, Chester, SSP ’17]Δ
lowest spin 4654

lowest spin 2

32

lowest spin 0

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

16/cT

Red lines are leading SUGRA tree level results [Zhou ’17; Chester ’18] .
Lowest operators have the form SIJ ∂µ1 · · · ∂µ` S IJ .
Silviu Pufu (Princeton University)

6-26-201818 / 20

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 of
16
cT from extremal functional:
λ2(A,2) j

λ2(A,+) j

25

25
j=5

j=4

20

20
j=3

15
10

10

j=1

5

0.0

j=2

15

j=0

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

16
cT

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

16
cT

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)

6-26-201819 / 20

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 of
16
cT from extremal functional:
λ2(A,2) j

λ2(A,+) j

25

25
j=5

j=4

20

20
j=3

15
10

10

j=1

5

0.0

j=2

15

j=0

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

16
cT

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

16
cT

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)

6-26-201819 / 20

Conclusion
Can compute OPE coefficients in N = 8 SCFTs with Lagrangian
descriptions using supersymmetric localization.
stu
For ABJM theory, we can reproduce the fR 4 (s, t) = 3·2
7 term in theflat space 4-graviton scattering amplitude.

Bootstrap bounds are almost saturated by N = 8 SCFTs with
holographic duals.
For the future:
Generalize to other dimensions, other 4-point function, less
SUSY. (See [Chester, Perlmutter ’18] on 6d as well as Shai Chester’stalk & poster.)Study other SCFTs from which one can compute scattering
amplitudes of gauge bosons on branes. (?)
Loops in AdS.
Silviu Pufu (Princeton University)

6-26-201820 / 20

Conclusion
Can compute OPE coefficients in N = 8 SCFTs with Lagrangian
descriptions using supersymmetric localization.
stu
For ABJM theory, we can reproduce the fR 4 (s, t) = 3·2
7 term in theflat space 4-graviton scattering amplitude.

Bootstrap bounds are almost saturated by N = 8 SCFTs with
holographic duals.
For the future:
Generalize to other dimensions, other 4-point function, less
SUSY. (See [Chester, Perlmutter ’18] on 6d as well as Shai Chester’stalk & poster.)Study other SCFTs from which one can compute scattering
amplitudes of gauge bosons on branes. (?)
Loops in AdS.
Silviu Pufu (Princeton University)

6-26-201820 / 20