[PPT] - M-theory S-Matrix from 3d SCFT Silviu S. Pufu, Princeton University PowerPoint Presentation

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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 Also: 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

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

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Motivation

Learn about (reconstruct?) gravity / string theory / M-theory from CFT 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 / type IIB strings on AdS5 × S5 3d U(N)k × U(N)−k ABJM theory at large N / M-theory on AdS4 × 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.

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

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Motivation

Learn about (reconstruct?) gravity / string theory / M-theory from CFT 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 / type IIB strings on AdS5 × S5 3d U(N)k × U(N)−k ABJM theory at large N / M-theory on AdS4 × 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.

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

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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 ABJM theory can be written as a 2N-dim’l integral [Kapustin, Willett, Yaakov ’09] Z =

dNλ dNµ
i<j 4 sinh2(λi − λj) sinh2(µi − µj)
i,j cosh2(λi − µj)

ei k

π

i(λ2

i −µ2 i ) .

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

√ 2 3 k1/2N3/2 + O(N1/2).

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

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

Silviu Pufu (Princeton University) 10-26-2018 3 / 32

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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 ABJM theory can be written as a 2N-dim’l integral [Kapustin, Willett, Yaakov ’09] Z =

dNλ dNµ
i<j 4 sinh2(λi − λj) sinh2(µi − µj)
i,j cosh2(λi − µj)

ei k

π

i(λ2

i −µ2 i ) .

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

√ 2 3 k1/2N3/2 + O(N1/2).

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

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

Silviu Pufu (Princeton University) 10-26-2018 3 / 32

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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 ABJM theory can be written as a 2N-dim’l integral [Kapustin, Willett, Yaakov ’09] Z =

dNλ dNµ
i<j 4 sinh2(λi − λj) sinh2(µi − µj)
i,j cosh2(λi − µj)

ei k

π

i(λ2

i −µ2 i ) .

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

√ 2 3 k1/2N3/2 + O(N1/2).

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

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

Silviu Pufu (Princeton University) 10-26-2018 3 / 32

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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 ABJM theory can be written as a 2N-dim’l integral [Kapustin, Willett, Yaakov ’09] Z =

dNλ dNµ
i<j 4 sinh2(λi − λj) sinh2(µi − µj)
i,j cosh2(λi − µj)

ei k

π

i(λ2

i −µ2 i ) .

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

√ 2 3 k1/2N3/2 + O(N1/2).

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

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

Silviu Pufu (Princeton University) 10-26-2018 3 / 32

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M-theory S-matrix

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

scatter gravitons and superpartners in 11d.

Equivalently, reconstruct the derivative expansion of the M-theory effective action. Schematically, S =

d11x √g
R + Riem4 + · · · + (SUSic completion)
.

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

Silviu Pufu (Princeton University) 10-26-2018 4 / 32

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M-theory S-matrix

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

scatter gravitons and superpartners in 11d.

Equivalently, reconstruct the derivative expansion of the M-theory effective action. Schematically, S =

d11x √g
R + Riem4 + · · · + (SUSic completion)
.

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

Silviu Pufu (Princeton University) 10-26-2018 4 / 32

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M-theory S-matrix

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

scatter gravitons and superpartners in 11d.

Equivalently, reconstruct the derivative expansion of the M-theory effective action. Schematically, S =

d11x √g
R + Riem4 + · · · + (SUSic completion)
.

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

Silviu Pufu (Princeton University) 10-26-2018 4 / 32

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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), scattering amplitudes are those in N = 8 SUGRA at tree level. Examples: ASUGRA, tree(h−h−h+h+) = 124[34]4 stu , ASUGRA, tree(S1S1S2S2) = tu s , where s = (p1 + p2)2, t = (p1 + p4)2, u = (p1 + p3)2. Amplitudes depend on the particles being scattered, but they’re all related by SUSY. (See Elvang & Huang’s book.)

Silviu Pufu (Princeton University) 10-26-2018 5 / 32

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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), scattering amplitudes are those in N = 8 SUGRA at tree level. Examples: ASUGRA, tree(h−h−h+h+) = 124[34]4 stu , ASUGRA, tree(S1S1S2S2) = tu s , where s = (p1 + p2)2, t = (p1 + p4)2, u = (p1 + p3)2. Amplitudes depend on the particles being scattered, but they’re all related by SUSY. (See Elvang & Huang’s book.)

Silviu Pufu (Princeton University) 10-26-2018 5 / 32

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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), scattering amplitudes are those in N = 8 SUGRA at tree level. Examples: ASUGRA, tree(h−h−h+h+) = 124[34]4 stu , ASUGRA, tree(S1S1S2S2) = tu s , where s = (p1 + p2)2, t = (p1 + p4)2, u = (p1 + p3)2. Amplitudes depend on the particles being scattered, but they’re all related by SUSY. (See Elvang & Huang’s book.)

Silviu Pufu (Princeton University) 10-26-2018 5 / 32

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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), scattering amplitudes are those in N = 8 SUGRA at tree level. Examples: ASUGRA, tree(h−h−h+h+) = 124[34]4 stu , ASUGRA, tree(S1S1S2S2) = tu s , where s = (p1 + p2)2, t = (p1 + p4)2, u = (p1 + p3)2. Amplitudes depend on the particles being scattered, but they’re all related by SUSY. (See Elvang & Huang’s book.)

Silviu Pufu (Princeton University) 10-26-2018 5 / 32

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Momentum expansion

Momentum expansion takes a universal form (independent of the type of particle): A = ASUGRA, tree

1 + ℓ6

pfR4(s, t) + ℓ9 pf1-loop(s, t)

+ ℓ12

p fD6R4(s, t) + ℓ14 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 . ℓ10

p 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

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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, φ(x1)φ(x2)φ(x3)φ(x4)conn = 1 x2

12x2 34

g(U, V) with U =

x2

12x2 34

x2

13x2 24 , V =

x2

14x2 23

x2

13x2 24 , go to Mellin space

g(U, V) =

ds dt

(4πi)2 Ut/2V (u−2)/2Γ2 1 − s 2

Γ2
1 − t

2

Γ2

1 − u 2

M(s, t)

where s + t + u = 4. 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) 10-26-2018 7 / 32

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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, φ(x1)φ(x2)φ(x3)φ(x4)conn = 1 x2

12x2 34

g(U, V) with U =

x2

12x2 34

x2

13x2 24 , V =

x2

14x2 23

x2

13x2 24 , go to Mellin space

g(U, V) =

ds dt

(4πi)2 Ut/2V (u−2)/2Γ2 1 − s 2

Γ2
1 − t

2

Γ2

1 − u 2

M(s, t)

where s + t + u = 4. 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) 10-26-2018 7 / 32

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Flat space limit of CFT correlators

If L is the radius of AdS (and L/2 is the radius of S7), then A(s, t) = lim

L→∞

NL7 ℓ9

p

c+i∞

c−i∞

dα eαα−1/2M

− L2

4αs, − L2 4αt

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

At leach order in ℓp/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 ℓp times momentum, we need M =

∞

k=1

ℓp L 7+2k (function that grows as kth power of s, t, u) Instead of 1/N or ℓp/L I will use 1/cT, where TµνTρσ ∝ cT.

Silviu Pufu (Princeton University) 10-26-2018 8 / 32

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Flat space limit of CFT correlators

If L is the radius of AdS (and L/2 is the radius of S7), then A(s, t) = lim

L→∞

NL7 ℓ9

p

c+i∞

c−i∞

dα eαα−1/2M

− L2

4αs, − L2 4αt

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

At leach order in ℓp/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 ℓp times momentum, we need M =

∞

k=1

ℓp L 7+2k (function that grows as kth power of s, t, u) Instead of 1/N or ℓp/L I will use 1/cT, where TµνTρσ ∝ cT.

Silviu Pufu (Princeton University) 10-26-2018 8 / 32

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To obtain scattering amplitude of graviton + superpartners in M-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 − → focus on this − → dimension spin so(8)R couples to 1 35c scalars 3/2 1/2 56v gravi-photinos 2 35s pseudo-scalars 2 1 28 gravi-photons 5/2 3/2 8v gravitinos 3 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.

Silviu Pufu (Princeton University) 10-26-2018 9 / 32

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Ward identity

SSSS =

1 x2

12x2 34 × 6 functions Si(U, V), i = 1, . . . 6

SSPP =

1 x2

12x4 34 × 3 functions Ri(U, V), i = 1, . . . 3

PPPP =

1 x4

12x4 34 × 6 functions Pi(U, V), i = 1, . . . 6

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

∂US4(U, V) = 1 U S4(U, V) + 1 U − ∂U − ∂V

S2(U, V) +

1 U + (U − 1)∂U + V∂V

S3(U, V) ,

∂V S4(U, V) = − 1 2V S4(U, V) − 1 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) = 1 4

4 +
U2 − 4U
∂U +
4 + U − 2U2 + 7UV − 4V 2

∂V + 2U(2V − U + 2)(U∂2

U + (U + V − 1)∂U∂V + V∂2 V )

S1(U, V) .

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

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Ward identity

SSSS =

1 x2

12x2 34 × 6 functions Si(U, V), i = 1, . . . 6

SSPP =

1 x2

12x4 34 × 3 functions Ri(U, V), i = 1, . . . 3

PPPP =

1 x4

12x4 34 × 6 functions Pi(U, V), i = 1, . . . 6

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

∂US4(U, V) = 1 U S4(U, V) + 1 U − ∂U − ∂V

S2(U, V) +

1 U + (U − 1)∂U + V∂V

S3(U, V) ,

∂V S4(U, V) = − 1 2V S4(U, V) − 1 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) = 1 4

4 +
U2 − 4U
∂U +
4 + U − 2U2 + 7UV − 4V 2

∂V + 2U(2V − U + 2)(U∂2

U + (U + V − 1)∂U∂V + V∂2 V )

S1(U, V) .

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

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CFT 4-pt correlators at large N

In general, one cannot compute 4-pt functions using SUSic localization. However, the requirements

The Mellin amplitudes obey SUSY Ward identities; The Mellin amplitudes are consistent with crossing symmetry; At order 1/c

7 9 + 2 9 n

T

, the Mellin amplitude grows at most as the nth power of s, t, u; The Mellin amplitudes have the analytic properties appropriate for tree-level Witten diagrams

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

Silviu Pufu (Princeton University) 10-26-2018 11 / 32

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Number of solutions: degree in s, t, u 1 2 3 4 5 6 7 . . . 11D vertex R R4 D4R4 D6R4 . . . scaling c−1

T

c

− 5

3

T

(0×)c

− 19

9

T

c

− 7

3

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 Ward identity for the flat space scattering amplitudes. So: To determine M(s, t) to order 1/cT we should compute one CFT quantity. To determine M(s, t) to order 1/c5/3

T

we should compute two CFT

quantities. Etc.

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

Silviu Pufu (Princeton University) 10-26-2018 12 / 32

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Number of solutions: degree in s, t, u 1 2 3 4 5 6 7 . . . 11D vertex R R4 D4R4 D6R4 . . . scaling c−1

T

c

− 5

3

T

(0×)c

− 19

9

T

c

− 7

3

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 Ward identity for the flat space scattering amplitudes. So: To determine M(s, t) to order 1/cT we should compute one CFT quantity. To determine M(s, t) to order 1/c5/3

T

we should compute two CFT

quantities. Etc.

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

Silviu Pufu (Princeton University) 10-26-2018 12 / 32

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Number of solutions: degree in s, t, u 1 2 3 4 5 6 7 . . . 11D vertex R R4 D4R4 D6R4 . . . scaling c−1

T

c

− 5

3

T

(0×)c

− 19

9

T

c

− 7

3

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 Ward identity for the flat space scattering amplitudes. So: To determine M(s, t) to order 1/cT we should compute one CFT quantity. To determine M(s, t) to order 1/c5/3

T

we should compute two CFT

quantities. Etc.

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

Silviu Pufu (Princeton University) 10-26-2018 12 / 32

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Number of solutions: degree in s, t, u 1 2 3 4 5 6 7 . . . 11D vertex R R4 D4R4 D6R4 . . . scaling c−1

T

c

− 5

3

T

(0×)c

− 19

9

T

c

− 7

3

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 Ward identity for the flat space scattering amplitudes. So: To determine M(s, t) to order 1/cT we should compute one CFT quantity. To determine M(s, t) to order 1/c5/3

T

we should compute two CFT

quantities. Etc.

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

Silviu Pufu (Princeton University) 10-26-2018 12 / 32

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What to compute

What to compute: mass deformed S3 partition function Z(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 an integral [Kapustin, Willett, Yaakov ’09] ; then use statistical physics techniques to extract F(m1, m2) to all orders in the 1/N expansion! [Marino, Putrov ’11; Nosaka ’15] The following quantities ∂2F ∂m2

1

m1=m2=0

, ∂4F ∂m4

1

m1=m2=0

, ∂4F ∂m2

1∂m2 2

m1=m2=0

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

T

.

Silviu Pufu (Princeton University) 10-26-2018 13 / 32

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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 current jµ, 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) = im r , σ(x) = m , fermions = 0 . The deformation

d3x tr m
i

r J + K

is a real mass deformation.

Silviu Pufu (Princeton University) 10-26-2018 14 / 32

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Two mass parameters

The Cartan of su(4) is 3-dimensional, so there are 3 independent mass 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 mass parameters (instead of 3): m = diag{m1, −m1, m2, −m2} .

Silviu Pufu (Princeton University) 10-26-2018 15 / 32

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Localization result

Using supersymmetric localization [Kapustin, Willett, Yaakov ’09] : ZS3(m1, m2) =

dNλ dNµ

eik

i(λ2 i −µ2 i )

i<j sinh2(λi − λj) sinh2(µi − µj)

i,j cosh(λi − µj + m1

2 ) cosh(λi − µj + m2 2 )

Small N: can evaluate integral exactly. Large N: rewrite ZS3(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) − 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)

Silviu Pufu (Princeton University) 10-26-2018 16 / 32

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Localization result

Using supersymmetric localization [Kapustin, Willett, Yaakov ’09] : ZS3(m1, m2) =

dNλ dNµ

eik

i(λ2 i −µ2 i )

i<j sinh2(λi − λj) sinh2(µi − µj)

i,j cosh(λi − µj + m1

2 ) cosh(λi − µj + m2 2 )

Small N: can evaluate integral exactly. Large N: rewrite ZS3(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) − 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)

Silviu Pufu (Princeton University) 10-26-2018 16 / 32

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Integrated correlators

Relate mass derivatives to SSSS, SSPP, PPPP. Recall that the mass deformations are

2

i=1

mi

d3x

i r Ji + Ki

The ∆ = 1 operators Ji are linear combinations of the 35c scalars

SIJ. The ∆ = 2 operators Ki are linear combinations of the 35s pseudoscalars PIJ.

∂2F ∂m2

i = −

i

r Ji +

Ki

2 = integrated 2-pt fn.

∂4F ∂m2

i ∂m2 j = −

i

r Ji +

Ki

2 i

r Jj +

Kj

2 = integrated 4-pt fn.

Silviu Pufu (Princeton University) 10-26-2018 17 / 32

SLIDE 35

Integrated 2-pt function

Because S and P belong to the same multiplet as the stress tensor, TµνTρσ ∝ cT implies SS ∝ cT and PP ∝ cT. Then JJ ∝ cT and KK ∝ cT, and ∂2F ∂m2

i

= π2 32cT . But what does this have to do with the 4-pt function? (Super)Conformal block decomposition SSSS = 1 x2

12x2 34

multiplets M

λ2

MgM(U, V)

Then, in a normalization where λ2

id = 1,

λ2

stress = SSTµνSSTρσ

TµνTρσ = 256 cT .

Silviu Pufu (Princeton University) 10-26-2018 18 / 32

SLIDE 36

Integrated 2-pt function

Because S and P belong to the same multiplet as the stress tensor, TµνTρσ ∝ cT implies SS ∝ cT and PP ∝ cT. Then JJ ∝ cT and KK ∝ cT, and ∂2F ∂m2

i

= π2 32cT . But what does this have to do with the 4-pt function? (Super)Conformal block decomposition SSSS = 1 x2

12x2 34

multiplets M

λ2

MgM(U, V)

Then, in a normalization where λ2

id = 1,

λ2

stress = SSTµνSSTρσ

TµνTρσ = 256 cT .

Silviu Pufu (Princeton University) 10-26-2018 18 / 32

SLIDE 37

Four mass derivatives

Expressing the four mass derivatives in terms of cT: − 1 c2

T

∂4F ∂m4

1

= 3π2 64 1 cT + (3π)4/3 210/3 1 c5/3

T

+ (· · · ) c2

T

− (18π2)1/3 1 c7/3

T

+ · · · − 1 c2

T

∂4F ∂m2

1∂m2 2

= −π2 64 1 cT + 5π4/3 210/332/3 1 c5/3

T

+ (· · · ) c2

T

− 4(2π2)1/3 310/3 1 c7/3

T

+ · · · Impose these constraints on the 4-pt functions SSSS, SSPP, PPPP using ∂4F ∂m2

i ∂m2 j

= − i r Ji +

Ki

2 i r Jj +

Kj

2

Silviu Pufu (Princeton University) 10-26-2018 19 / 32

SLIDE 38

Integrated 4-pt function

If AABB =

1 x

2∆A 12

x

2∆B 34

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∆A

12 x2∆B 34

G with Ω( x) = 1 + x2

4r 2 .

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 diagram for the 4-pt function of ops of dim 3 − ∆A, 3 − ∆A, 3 − ∆B, 3 − ∆B.

Silviu Pufu (Princeton University) 10-26-2018 20 / 32

SLIDE 39

Integrated 4-pt function

If AABB =

1 x

2∆A 12

x

2∆B 34

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∆A

12 x2∆B 34

G with Ω( x) = 1 + x2

4r 2 .

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 diagram for the 4-pt function of ops of dim 3 − ∆A, 3 − ∆A, 3 − ∆B, 3 − ∆B.

Silviu Pufu (Princeton University) 10-26-2018 20 / 32

SLIDE 40

¯ D function

= Dr1,r2,r3,r4( xi) = dz0dd

z zd+1

4

i=1 Gri B∂(z0,

z; xi) Gri

B∂(z0,

z; xi) =

z0

z2

0 + (

z − xi)2 ri The ¯ D function is defined as

¯ Dr1,r2,r3,r4(U, V) = x

1 2

4

i=1 ri −r4

13

xr2

24

x

1 2

4

i=1 ri −r1−r4

14

x

1 2

4

i=1 ri −r3−r4

34

2 4

i=1 Γ(ri)

π

d 2 Γ

−d+4

i=1 ri

2

Dr1,r2,r3,r4(xi)

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

Silviu Pufu (Princeton University) 10-26-2018 21 / 32

SLIDE 41

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/cT expansion up to a few undetermined constants at each order SUSic localization = ⇒ ZS3(m1, m2) = ⇒ F(m1, m2) in 1/cT expansion = ⇒ 2nd and 4th derivatives of F(m1, m2) in 1/cT expansion = ⇒ conditions on integrated correlators SSSS, SSPP, PPPP. Combine above = ⇒ determine tree-level MSSSS, MSSPP, MPPPP up to 1/c19/9

T

= ⇒ take flat space limit to determine scattering amplitudes in 11d.

Silviu Pufu (Princeton University) 10-26-2018 22 / 32

SLIDE 42

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/cT expansion up to a few undetermined constants at each order SUSic localization = ⇒ ZS3(m1, m2) = ⇒ F(m1, m2) in 1/cT expansion = ⇒ 2nd and 4th derivatives of F(m1, m2) in 1/cT expansion = ⇒ conditions on integrated correlators SSSS, SSPP, PPPP. Combine above = ⇒ determine tree-level MSSSS, MSSPP, MPPPP up to 1/c19/9

T

= ⇒ take flat space limit to determine scattering amplitudes in 11d.

Silviu Pufu (Princeton University) 10-26-2018 22 / 32

SLIDE 43

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/cT expansion up to a few undetermined constants at each order SUSic localization = ⇒ ZS3(m1, m2) = ⇒ F(m1, m2) in 1/cT expansion = ⇒ 2nd and 4th derivatives of F(m1, m2) in 1/cT expansion = ⇒ conditions on integrated correlators SSSS, SSPP, PPPP. Combine above = ⇒ determine tree-level MSSSS, MSSPP, MPPPP up to 1/c19/9

T

= ⇒ take flat space limit to determine scattering amplitudes in 11d.

Silviu Pufu (Princeton University) 10-26-2018 22 / 32

SLIDE 44

Precision test of AdS/CFT

The flat space limit implies fR4(s, t) = stu

3·27 and fD4R4 = 0, as

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

Silviu Pufu (Princeton University) 10-26-2018 23 / 32

SLIDE 45

Precision test of AdS/CFT

The flat space limit implies fR4(s, t) = stu

3·27 and fD4R4 = 0, as

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

Silviu Pufu (Princeton University) 10-26-2018 23 / 32

SLIDE 46

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.

Silviu Pufu (Princeton University) 10-26-2018 24 / 32

SLIDE 47

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.

Silviu Pufu (Princeton University) 10-26-2018 24 / 32

SLIDE 48

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.

Silviu Pufu (Princeton University) 10-26-2018 24 / 32

SLIDE 49

Exact OPE coefficients

As already mentioned, in the superconformal block decomposition SSSS = 1 x2

12x2 34

multiplets M

λ2

MgM(U, V)

λ2

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

is proportional to ∂2F

∂m2

1

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

1

m1=m2=0 and hence can be

computed to all orders in 1/N.

Silviu Pufu (Princeton University) 10-26-2018 25 / 32

SLIDE 50

Exact OPE coefficients

As already mentioned, in the superconformal block decomposition SSSS = 1 x2

12x2 34

multiplets M

λ2

MgM(U, V)

λ2

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

is proportional to ∂2F

∂m2

1

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

1

m1=m2=0 and hence can be

computed to all orders in 1/N.

Silviu Pufu (Princeton University) 10-26-2018 25 / 32

SLIDE 51

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 supercharge Q = “Q + S′′ cohomology s.t. translations in x are Q-exact. The topological sector is defined either on a line in flat space or on a great circle of S3. In ABJM, construct 1d operators Sα(x) from SIJ, α = 1, 2, 3. Their 2-pt function depends only on cT; their 4-pt function depends only

n cT and λ2

(B,2).

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

SLIDE 52

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 supercharge Q = “Q + S′′ cohomology s.t. translations in x are Q-exact. The topological sector is defined either on a line in flat space or on a great circle of S3. In ABJM, construct 1d operators Sα(x) from SIJ, α = 1, 2, 3. Their 2-pt function depends only on cT; their 4-pt function depends only

n cT and λ2

(B,2).

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

SLIDE 53

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 supercharge Q = “Q + S′′ cohomology s.t. translations in x are Q-exact. The topological sector is defined either on a line in flat space or on a great circle of S3. In ABJM, construct 1d operators Sα(x) from SIJ, α = 1, 2, 3. Their 2-pt function depends only on cT; their 4-pt function depends only

n cT and λ2

(B,2).

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

SLIDE 54

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 supercharge Q = “Q + S′′ cohomology s.t. translations in x are Q-exact. The topological sector is defined either on a line in flat space or on a great circle of S3. In ABJM, construct 1d operators Sα(x) from SIJ, α = 1, 2, 3. Their 2-pt function depends only on cT; their 4-pt function depends only

n cT and λ2

(B,2).

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

SLIDE 55

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, λ2

B,2 can be

written in terms of derivatives of the Airy function! Eliminating N gives λ2

B,2 = 32

3 − 1024(4π2 − 15) 9π2 1 cT + 40960 2 9π8 1

3

1 c5/3

T

+ · · · (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) 10-26-2018 27 / 32

SLIDE 56

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, λ2

B,2 can be

written in terms of derivatives of the Airy function! Eliminating N gives λ2

B,2 = 32

3 − 1024(4π2 − 15) 9π2 1 cT + 40960 2 9π8 1

3

1 c5/3

T

+ · · · (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) 10-26-2018 27 / 32

SLIDE 57

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, λ2

B,2 can be

written in terms of derivatives of the Airy function! Eliminating N gives λ2

B,2 = 32

3 − 1024(4π2 − 15) 9π2 1 cT + 40960 2 9π8 1

3

1 c5/3

T

+ · · · (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) 10-26-2018 27 / 32

SLIDE 58

Known N = 8 SCFTs

A few families of N = 8 SCFTs: A1, A2 B1, B2 k −k G1 G2 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.

Silviu Pufu (Princeton University) 10-26-2018 28 / 32

SLIDE 59

Bootstrap bounds [Agmon, Chester, SSP ’17]

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

BLGk ABJMN,1

int

ABJN ABJMN,2 ABJM1,1

0.2 0.4 0.6 0.8 1.0

16 cT

2 4 6 8 10

λ(B,2)

2

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. 32 1024(4π2 − 15) 1

2

1

3

1

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

SLIDE 60

Bootstrap bounds [Agmon, Chester, SSP ’17]

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

BLGk ABJMN,1

int

ABJN ABJMN,2 ABJM1,1

0.2 0.4 0.6 0.8 1.0

16 cT

2 4 6 8 10

λ(B,2)

2

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. 32 1024(4π2 − 15) 1

2

1

3

1

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

SLIDE 61

Bound saturation = ⇒ read off CFT data

On the boundary of the bootstrap bounds, the solution to crossing should be unique = ⇒ can find SIJSKLSMNSPQ and solve for the spectrum !! [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 0.7 16/cT 1 2 3 4 5 6

Δ

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

Silviu Pufu (Princeton University) 10-26-2018 30 / 32

SLIDE 62

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:

j = 1 j = 3 j = 5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

16 cT

5 10 15 20 25

λ(A,2)j

2 j = 0 j = 2 j = 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

16 cT

5 10 15 20 25

λ(A,+)j

2

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

SLIDE 63

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:

j = 1 j = 3 j = 5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

16 cT

5 10 15 20 25

λ(A,2)j

2 j = 0 j = 2 j = 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

16 cT

5 10 15 20 25

λ(A,+)j

2

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

SLIDE 64

Conclusion

A combination of techniques (supersymmetric localization, SUSY Ward identities, Mellin space) can be used to recover graviton scattering amplitudes (at small momentum) from ABJM theory. We can reproduce the fR4(s, t) = stu

3·27 term in the flat space

4-graviton scattering amplitude and show that fD4R4 = 0. 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 also [Chester, Perlmutter ’18] in 6d.)

Study other SCFTs from which one can compute scattering amplitudes of gauge bosons on branes. (?) Loops in AdS.

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

SLIDE 65

Conclusion

A combination of techniques (supersymmetric localization, SUSY Ward identities, Mellin space) can be used to recover graviton scattering amplitudes (at small momentum) from ABJM theory. We can reproduce the fR4(s, t) = stu

3·27 term in the flat space

4-graviton scattering amplitude and show that fD4R4 = 0. 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 also [Chester, Perlmutter ’18] in 6d.)

Study other SCFTs from which one can compute scattering amplitudes of gauge bosons on branes. (?) Loops in AdS.

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