M-theory S-Matrix from 6d CFT
Shai M. Chester
Princeton University
Based on arXiv:1805.00892 with E. Perlmutter

Shai Chester (Princeton University)

3-2-2018

1/6

M-theory S-Matrix
M-theory is a quantum theory of interacting supergravitons in 11d
with no dimensionless coupling.
Graviton S-matrix in small momentum (`11  1) expansion:
D2m R4R+=∞
X+ Loops

m=0= A(s, t, u)+Exchange= ℓ911 AR (s, t, u)+Exchange= ℓ15+2m
AD2m R4 (s, t, u)
11

Protected terms from type IIA string theory + duality [Green,Tseytlin] :(stu)2stu
AR 4 = AR 3·2
AD 2 R 4 = AD 4 R 4 = 0 , AD 6 R 4 = AR 15·2
7 ,15 .
Goal: Find all tree level terms AD 2m R 4 for m > 3 using AdS/CFT.
Shai Chester (Princeton University)

3-2-2018

2/6

M-theory S-Matrix
M-theory is a quantum theory of interacting supergravitons in 11d
with no dimensionless coupling.
Graviton S-matrix in small momentum (`11  1) expansion:
D2m R4R+=∞
X+ Loops

m=0= A(s, t, u)+Exchange= ℓ911 AR (s, t, u)+Exchange= ℓ15+2mAD2m R4 (s, t, u)
11

Protected terms from type IIA string theory + duality [Green,Tseytlin] :(stu)2stu
AR 4 = AR 3·2
AD 2 R 4 = AD 4 R 4 = 0 , AD 6 R 4 = AR 15·2
7 ,15 .
Goal: Find all tree level terms AD 2m R 4 for m > 3 using AdS/CFT.
Shai Chester (Princeton University)

3-2-2018

2/6

M-theory S-Matrix
M-theory is a quantum theory of interacting supergravitons in 11d
with no dimensionless coupling.
Graviton S-matrix in small momentum (`11  1) expansion:
D2m R4R+=∞
X+ Loops

m=0= A(s, t, u)+Exchange= ℓ911 AR (s, t, u)+Exchange= ℓ15+2m
AD2m R4 (s, t, u)
11

Protected terms from type IIA string theory + duality [Green,Tseytlin] :(stu)2stu
AR 4 = AR 3·2
AD 2 R 4 = AD 4 R 4 = 0 , AD 6 R 4 = AR 15·2
7 ,15 .
Goal: Find all tree level terms AD 2m R 4 for m > 3 using AdS/CFT.
Shai Chester (Princeton University)

3-2-2018

2/6

M-theory S-Matrix
M-theory is a quantum theory of interacting supergravitons in 11d
with no dimensionless coupling.
Graviton S-matrix in small momentum (`11  1) expansion:
D2m R4R+=∞
X+ Loops

m=0= A(s, t, u)+Exchange= ℓ911 AR (s, t, u)+Exchange= ℓ15+2m
AD2m R4 (s, t, u)
11

Protected terms from type IIA string theory + duality [Green,Tseytlin] :(stu)2stu
AR 4 = AR 3·2
AD 2 R 4 = AD 4 R 4 = 0 , AD 6 R 4 = AR 15·2
7 ,15 .
Goal: Find all tree level terms AD 2m R 4 for m > 3 using AdS/CFT.
Shai Chester (Princeton University)

3-2-2018

2/6

AdS/CFT for M-theory
M-theory contains two non-perturbative dynamical objects: M2branes and M5 branes.
We study (2, 0) AN−1 6d SCFT that describes stack of N M5
branes, and is dual at large N to M-theory on AdS7 × S 4 .
2k x 2k hO 1 O 2 O 3 O 4 i of k -th lowest
We compute: Gk (U, V ) = x1234
k k k kdimension half-BPS operators in CFT6 in large N expansion.
x2 x2x2 x2

13 24

13 24

U ≡ x122 x342 , V ≡ x142 x232 are conformal cross ratios.
Gk dual to correlator of k -th lowest KK modes of M-theory onAdS7 × S 4 .
Flat space limit of Gk gives 11d S-matrix A|7d with momenta
restricted to 7d.
Shai Chester (Princeton University)

3-2-2018

3/6

AdS/CFT for M-theory
M-theory contains two non-perturbative dynamical objects: M2
branes and M5 branes.
We study (2, 0) AN−1 6d SCFT that describes stack of N M5
branes, and is dual at large N to M-theory on AdS7 × S 4 .
2k x 2k hO 1 O 2 O 3 O 4 i of k -th lowest
We compute: Gk (U, V ) = x1234k k k kdimension half-BPS operators in CFT6 in large N expansion.
x2 x2x2 x2

13 24

13 24U ≡ x122 x342 , V ≡ x142 x232 are conformal cross ratios.
Gk dual to correlator of k -th lowest KK modes of M-theory on
AdS7 × S 4 .
Flat space limit of Gk gives 11d S-matrix A|7d with momenta
restricted to 7d.
Shai Chester (Princeton University)

3-2-2018

3/6

AdS/CFT for M-theory
M-theory contains two non-perturbative dynamical objects: M2branes and M5 branes.
We study (2, 0) AN−1 6d SCFT that describes stack of N M5
branes, and is dual at large N to M-theory on AdS7 × S 4 .
2k x 2k hO 1 O 2 O 3 O 4 i of k -th lowest
We compute: Gk (U, V ) = x1234
k k k kdimension half-BPS operators in CFT6 in large N expansion.
x2 x2x2 x2

13 24

13 24

U ≡ x122 x342 , V ≡ x142 x232 are conformal cross ratios.
Gk dual to correlator of k -th lowest KK modes of M-theory onAdS7 × S 4 .
Flat space limit of Gk gives 11d S-matrix A|7d with momentarestricted to 7d.
Shai Chester (Princeton University)

3-2-2018

3/6

AdS/CFT for M-theory
M-theory contains two non-perturbative dynamical objects: M2branes and M5 branes.
We study (2, 0) AN−1 6d SCFT that describes stack of N M5
branes, and is dual at large N to M-theory on AdS7 × S 4 .
2k x 2k hO 1 O 2 O 3 O 4 i of k -th lowest
We compute: Gk (U, V ) = x12
34
k k k kdimension half-BPS operators in CFT6 in large N expansion.
x2 x2x2 x2

13 24

13 24

U ≡ x122 x342 , V ≡ x142 x232 are conformal cross ratios.
Gk dual to correlator of k -th lowest KK modes of M-theory onAdS7 × S 4 .
Flat space limit of Gk gives 11d S-matrix A|7d with momenta
restricted to 7d.
Shai Chester (Princeton University)

3-2-2018

3/6

AdS/CFT for M-theory
M-theory contains two non-perturbative dynamical objects: M2
branes and M5 branes.
We study (2, 0) AN−1 6d SCFT that describes stack of N M5
branes, and is dual at large N to M-theory on AdS7 × S 4 .
2k x 2k hO 1 O 2 O 3 O 4 i of k -th lowest
We compute: Gk (U, V ) = x1234k k k kdimension half-BPS operators in CFT6 in large N expansion.
x2 x2x2 x2

13 24

13 24

U ≡ x122 x342 , V ≡ x142 x232 are conformal cross ratios.
Gk dual to correlator of k -th lowest KK modes of M-theory onAdS7 × S 4 .
Flat space limit of Gk gives 11d S-matrix A|7d with momenta
restricted to 7d.
Shai Chester (Princeton University)

3-2-2018

3/6

AdS/CFT for M-theory
M-theory contains two non-perturbative dynamical objects: M2branes and M5 branes.
We study (2, 0) AN−1 6d SCFT that describes stack of N M5
branes, and is dual at large N to M-theory on AdS7 × S 4 .
2k x 2k hO 1 O 2 O 3 O 4 i of k -th lowest
We compute: Gk (U, V ) = x1234
k k k kdimension half-BPS operators in CFT6 in large N expansion.
x2 x2x2 x2

13 24

13 24

U ≡ x122 x342 , V ≡ x142 x232 are conformal cross ratios.
Gk dual to correlator of k -th lowest KK modes of M-theory onAdS7 × S 4 .
Flat space limit of Gk gives 11d S-matrix A|7d with momenta
restricted to 7d.
Shai Chester (Princeton University)

3-2-2018

3/6

Tree Level Half-BPS Four Point functions: Constraints
We use Mellin transform Gk (U, V ) → Mk (s, t, u) [Mack, Penedones] ,where s, t, u are like Mandelstam variables in Mellin space.
For N large, let Mktree (s, t, u) ≡P∞(p)degree in s, t, u → ∞, and Mk(p)
p=1 Mk (s, t, u), where p isfixed in terms of CFT6 data by:

1

Crossing symmetry.

2

Superconformal Ward identities [Dolan, Gallot, Sokatchev; Rastelli, Zhou] .

3Poles in s, t, u correspond to dimensions of operators inOk × Ok , for tree level only allow poles for half-BPS operators.
(p)Flat space limit of Mk (s, t, u) gives 2p derivative contribution to(1)

A|7d (s, t, u) [Penedones] , e.g. Mk
Shai Chester (Princeton University)→ AR |7d .
3-2-2018

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Tree Level Half-BPS Four Point functions: Constraints
We use Mellin transform Gk (U, V ) → Mk (s, t, u) [Mack, Penedones] ,where s, t, u are like Mandelstam variables in Mellin space.
For N large, let Mktree (s, t, u) ≡P∞(p)degree in s, t, u → ∞, and Mk(p)
p=1 Mk (s, t, u), where p isfixed in terms of CFT6 data by:

1

Crossing symmetry.

2

Superconformal Ward identities [Dolan, Gallot, Sokatchev; Rastelli, Zhou] .

3Poles in s, t, u correspond to dimensions of operators inOk × Ok , for tree level only allow poles for half-BPS operators.
(p)Flat space limit of Mk (s, t, u) gives 2p derivative contribution to(1)A|7d (s, t, u) [Penedones] , e.g. Mk
Shai Chester (Princeton University)→ AR |7d .
3-2-2018

4/6

Tree Level Half-BPS Four Point functions: Constraints
We use Mellin transform Gk (U, V ) → Mk (s, t, u) [Mack, Penedones] ,where s, t, u are like Mandelstam variables in Mellin space.
For N large, let Mktree (s, t, u) ≡P∞(p)degree in s, t, u → ∞, and Mk(p)p=1 Mk (s, t, u), where p isfixed in terms of CFT6 data by:1Crossing symmetry.

2

Superconformal Ward identities [Dolan, Gallot, Sokatchev; Rastelli, Zhou] .

3Poles in s, t, u correspond to dimensions of operators inOk × Ok , for tree level only allow poles for half-BPS operators.
(p)Flat space limit of Mk (s, t, u) gives 2p derivative contribution to(1)A|7d (s, t, u) [Penedones] , e.g. Mk
Shai Chester (Princeton University)→ AR |7d .
3-2-2018

4/6

Tree Level Half-BPS Four Point functions: Constraints
We use Mellin transform Gk (U, V ) → Mk (s, t, u) [Mack, Penedones] ,where s, t, u are like Mandelstam variables in Mellin space.
For N large, let Mktree (s, t, u) ≡P∞(p)degree in s, t, u → ∞, and Mk(p)
p=1 Mk (s, t, u), where p isfixed in terms of CFT6 data by:

1

Crossing symmetry.

2

Superconformal Ward identities [Dolan, Gallot, Sokatchev; Rastelli, Zhou] .

3Poles in s, t, u correspond to dimensions of operators inOk × Ok , for tree level only allow poles for half-BPS operators.
(p)Flat space limit of Mk (s, t, u) gives 2p derivative contribution to(1)A|7d (s, t, u) [Penedones] , e.g. Mk
Shai Chester (Princeton University)→ AR |7d .
3-2-2018

4/6

Tree Level Half-BPS Four Point functions: Constraints
We use Mellin transform Gk (U, V ) → Mk (s, t, u) [Mack, Penedones] ,where s, t, u are like Mandelstam variables in Mellin space.
For N large, let Mktree (s, t, u) ≡P∞(p)degree in s, t, u → ∞, and Mk(p)
p=1 Mk (s, t, u), where p isfixed in terms of CFT6 data by:

1

Crossing symmetry.

2

Superconformal Ward identities [Dolan, Gallot, Sokatchev; Rastelli, Zhou] .

3Poles in s, t, u correspond to dimensions of operators inOk × Ok , for tree level only allow poles for half-BPS operators.
(p)Flat space limit of Mk (s, t, u) gives 2p derivative contribution to(1)A|7d (s, t, u) [Penedones] , e.g. Mk
Shai Chester (Princeton University)→ AR |7d .
3-2-2018

4/6

Tree Level Half-BPS Four Point functions: Constraints
We use Mellin transform Gk (U, V ) → Mk (s, t, u) [Mack, Penedones] ,where s, t, u are like Mandelstam variables in Mellin space.
For N large, let Mktree (s, t, u) ≡P∞(p)degree in s, t, u → ∞, and Mk(p)
p=1 Mk (s, t, u), where p isfixed in terms of CFT6 data by:

1

Crossing symmetry.

2

Superconformal Ward identities [Dolan, Gallot, Sokatchev; Rastelli, Zhou] .

3Poles in s, t, u correspond to dimensions of operators inOk × Ok , for tree level only allow poles for half-BPS operators.
(p)Flat space limit of Mk (s, t, u) gives 2p derivative contribution to(1)A|7d (s, t, u) [Penedones] , e.g. Mk
Shai Chester (Princeton University)→ AR |7d .
3-2-2018

4/6

Tree Level Half-BPS Four Point functions: Solutions(1)

Mk fixed by central charge c1T ≈ N −3 [Rastelli, Zhou] ⇒ AR |7d is
proportional to gravitational coupling κ2 ≈ N −3 as expected.
(p)

No Mk

for p = 2, 3 ⇒ no AR 2 |7d or AR 3 |7d [SMC, Perlmutter] .

(p)

Mk for 4 ≤ p < 10, which gives AD 2p−8 R 4 |7d in flat space limit,fixed by small set of CFT6 OPE coefficients [SMC, Perlmutter] .
(p)

Mk for p ≥ 10 has same N scaling as loop terms, so require loop
Mellin amplitudes to fix unambiguously.
(4)

Mk fixed by half-BPS OPE coefficient λ2BPS [SMC, Perlmutter] thatcan be computed exactly [Beem, Rastelli, van Rees] .
(4)Flat space limit of Mk
Shai Chester (Princeton University)

correctly reproduces the known AR 4 |7d !
3-2-2018

5/6

Tree Level Half-BPS Four Point functions: Solutions(1)

Mk fixed by central charge c1T ≈ N −3 [Rastelli, Zhou] ⇒ AR |7d is
proportional to gravitational coupling κ2 ≈ N −3 as expected.
(p)

No Mkfor p = 2, 3 ⇒ no AR 2 |7d or AR 3 |7d [SMC, Perlmutter] .

(p)

Mk for 4 ≤ p < 10, which gives AD 2p−8 R 4 |7d in flat space limit,
fixed by small set of CFT6 OPE coefficients [SMC, Perlmutter] .
(p)

Mk for p ≥ 10 has same N scaling as loop terms, so require loop
Mellin amplitudes to fix unambiguously.
(4)

Mk fixed by half-BPS OPE coefficient λ2BPS [SMC, Perlmutter] thatcan be computed exactly [Beem, Rastelli, van Rees] .
(4)Flat space limit of Mk
Shai Chester (Princeton University)

correctly reproduces the known AR 4 |7d !
3-2-2018

5/6

Tree Level Half-BPS Four Point functions: Solutions(1)

Mk fixed by central charge c1T ≈ N −3 [Rastelli, Zhou] ⇒ AR |7d is
proportional to gravitational coupling κ2 ≈ N −3 as expected.
(p)

No Mk

for p = 2, 3 ⇒ no AR 2 |7d or AR 3 |7d [SMC, Perlmutter] .

(p)

Mk for 4 ≤ p < 10, which gives AD 2p−8 R 4 |7d in flat space limit,
fixed by small set of CFT6 OPE coefficients [SMC, Perlmutter] .
(p)

Mk for p ≥ 10 has same N scaling as loop terms, so require loop
Mellin amplitudes to fix unambiguously.
(4)

Mk fixed by half-BPS OPE coefficient λ2BPS [SMC, Perlmutter] thatcan be computed exactly [Beem, Rastelli, van Rees] .
(4)Flat space limit of Mk
Shai Chester (Princeton University)

correctly reproduces the known AR 4 |7d !
3-2-2018

5/6

Tree Level Half-BPS Four Point functions: Solutions(1)

Mk fixed by central charge c1T ≈ N −3 [Rastelli, Zhou] ⇒ AR |7d is
proportional to gravitational coupling κ2 ≈ N −3 as expected.
(p)

No Mk

for p = 2, 3 ⇒ no AR 2 |7d or AR 3 |7d [SMC, Perlmutter] .

(p)

Mk for 4 ≤ p < 10, which gives AD 2p−8 R 4 |7d in flat space limit,fixed by small set of CFT6 OPE coefficients [SMC, Perlmutter] .
(p)

Mk for p ≥ 10 has same N scaling as loop terms, so require loop
Mellin amplitudes to fix unambiguously.
(4)

Mk fixed by half-BPS OPE coefficient λ2BPS [SMC, Perlmutter] thatcan be computed exactly [Beem, Rastelli, van Rees] .
(4)Flat space limit of Mk
Shai Chester (Princeton University)

correctly reproduces the known AR 4 |7d !
3-2-2018

5/6

Tree Level Half-BPS Four Point functions: Solutions(1)

Mk fixed by central charge c1T ≈ N −3 [Rastelli, Zhou] ⇒ AR |7d is
proportional to gravitational coupling κ2 ≈ N −3 as expected.
(p)

No Mk

for p = 2, 3 ⇒ no AR 2 |7d or AR 3 |7d [SMC, Perlmutter] .

(p)

Mk for 4 ≤ p < 10, which gives AD 2p−8 R 4 |7d in flat space limit,
fixed by small set of CFT6 OPE coefficients [SMC, Perlmutter] .
(p)

Mk for p ≥ 10 has same N scaling as loop terms, so require loop
Mellin amplitudes to fix unambiguously.
(4)

Mk fixed by half-BPS OPE coefficient λ2BPS [SMC, Perlmutter] thatcan be computed exactly [Beem, Rastelli, van Rees] .
(4)Flat space limit of Mk
Shai Chester (Princeton University)

correctly reproduces the known AR 4 |7d !
3-2-2018

5/6

Tree Level Half-BPS Four Point functions: Solutions(1)

Mk fixed by central charge c1T ≈ N −3 [Rastelli, Zhou] ⇒ AR |7d is
proportional to gravitational coupling κ2 ≈ N −3 as expected.
(p)

No Mkfor p = 2, 3 ⇒ no AR 2 |7d or AR 3 |7d [SMC, Perlmutter] .

(p)

Mk for 4 ≤ p < 10, which gives AD 2p−8 R 4 |7d in flat space limit,
fixed by small set of CFT6 OPE coefficients [SMC, Perlmutter] .
(p)

Mk for p ≥ 10 has same N scaling as loop terms, so require loop
Mellin amplitudes to fix unambiguously.
(4)

Mk fixed by half-BPS OPE coefficient λ2BPS [SMC, Perlmutter] thatcan be computed exactly [Beem, Rastelli, van Rees] .
(4)Flat space limit of Mk
Shai Chester (Princeton University)

correctly reproduces the known AR 4 |7d !
3-2-2018

5/6

Conclusion
Results:
Tree level D 2m R 4 contributions to 11d M-theory S-matrix for m < 6in terms of CFT6 data.
Known half-BPS CFT6 data precisely reproduces R 4 contribution.
Future Directions:
Derive loop Mellin amplitudes ⇒ loop 11d S-matrix terms.
6d numerical bootstrap [Beem, Lemos, Rastelli, van Rees] to fix CFT6 data
⇒ 11d S-matrix coefficients.
Apply method to AdSd+1 /CFTd for other d.
See Silviu Pufu’s talk tomorrow for d = 3 case [SCM, Pufu, Yin] .

See my poster for more details!
Shai Chester (Princeton University)

3-2-2018

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