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
Silviu Pufu (Princeton University) 10-26-2018 1 / 32
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 AdS 5 × S 5 3d U ( N ) k × U ( N ) − k ABJM theory at large N / M-theory on AdS 4 × S 7 / Z k . 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
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 AdS 5 × S 5 3d U ( N ) k × U ( N ) − k ABJM theory at large N / M-theory on AdS 4 × S 7 / Z k . 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
Motivation Last 10 years: progress in QFT calculations using supersymmetric localization; using conformal bootstrap in CFTs. Example: using SUSic loc., the S 3 partition function of ABJM theory can be written as a 2 N -dim’l integral [Kapustin, Willett, Yaakov ’09] i < j 4 sinh 2 ( λ i − λ j ) sinh 2 ( µ i − µ j ) � � i ) . e i k i ( λ 2 i − µ 2 d N λ d N µ � Z = π i , j cosh 2 ( λ i − µ j ) � √ 3 k 1 / 2 N 3 / 2 + O ( N 1 / 2 ) . Expand at large N to find F = − log Z = π 2 N 3 / 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
Motivation Last 10 years: progress in QFT calculations using supersymmetric localization; using conformal bootstrap in CFTs. Example: using SUSic loc., the S 3 partition function of ABJM theory can be written as a 2 N -dim’l integral [Kapustin, Willett, Yaakov ’09] i < j 4 sinh 2 ( λ i − λ j ) sinh 2 ( µ i − µ j ) � � i ) . e i k i ( λ 2 i − µ 2 d N λ d N µ � Z = π i , j cosh 2 ( λ i − µ j ) � √ 3 k 1 / 2 N 3 / 2 + O ( N 1 / 2 ) . Expand at large N to find F = − log Z = π 2 N 3 / 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
Motivation Last 10 years: progress in QFT calculations using supersymmetric localization; using conformal bootstrap in CFTs. Example: using SUSic loc., the S 3 partition function of ABJM theory can be written as a 2 N -dim’l integral [Kapustin, Willett, Yaakov ’09] i < j 4 sinh 2 ( λ i − λ j ) sinh 2 ( µ i − µ j ) � � i ) . e i k i ( λ 2 i − µ 2 d N λ d N µ � Z = π i , j cosh 2 ( λ i − µ j ) � √ 3 k 1 / 2 N 3 / 2 + O ( N 1 / 2 ) . Expand at large N to find F = − log Z = π 2 N 3 / 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
Motivation Last 10 years: progress in QFT calculations using supersymmetric localization; using conformal bootstrap in CFTs. Example: using SUSic loc., the S 3 partition function of ABJM theory can be written as a 2 N -dim’l integral [Kapustin, Willett, Yaakov ’09] i < j 4 sinh 2 ( λ i − λ j ) sinh 2 ( µ i − µ j ) � � i ) . e i k i ( λ 2 i − µ 2 d N λ d N µ � Z = π i , j cosh 2 ( λ i − µ j ) � √ 3 k 1 / 2 N 3 / 2 + O ( N 1 / 2 ) . Expand at large N to find F = − log Z = π 2 N 3 / 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
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, d 11 x √ g � � R + Riem 4 + · · · + (SUSic completion) � S = . Restrict momenta to be in 4 out of the 11 dimensions. Silviu Pufu (Princeton University) 10-26-2018 4 / 32
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, d 11 x √ g � � R + Riem 4 + · · · + (SUSic completion) � S = . Restrict momenta to be in 4 out of the 11 dimensions. Silviu Pufu (Princeton University) 10-26-2018 4 / 32
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, d 11 x √ g � � R + Riem 4 + · · · + (SUSic completion) � S = . Restrict momenta to be in 4 out of the 11 dimensions. Silviu Pufu (Princeton University) 10-26-2018 4 / 32
In 11d, we can scatter: gravitons, gravitini, 3-form gauge particles. ⇒ can use 4d N = 8 language. We scatter: Momenta within 4d = graviton (1); gravitinos (8); gravi-photons (28); gravi-photinos (56); scalars (70 = 35 + 35) At leading order in small momentum (i.e. p 2 ), scattering amplitudes are those in N = 8 SUGRA at tree level. Examples: A SUGRA, tree ( h − h − h + h + ) = � 12 � 4 [ 34 ] 4 , stu A SUGRA, tree ( S 1 S 1 S 2 S 2 ) = tu s , where s = ( p 1 + p 2 ) 2 , t = ( p 1 + p 4 ) 2 , u = ( p 1 + p 3 ) 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
In 11d, we can scatter: gravitons, gravitini, 3-form gauge particles. ⇒ can use 4d N = 8 language. We scatter: Momenta within 4d = graviton (1); gravitinos (8); gravi-photons (28); gravi-photinos (56); scalars (70 = 35 + 35) At leading order in small momentum (i.e. p 2 ), scattering amplitudes are those in N = 8 SUGRA at tree level. Examples: A SUGRA, tree ( h − h − h + h + ) = � 12 � 4 [ 34 ] 4 , stu A SUGRA, tree ( S 1 S 1 S 2 S 2 ) = tu s , where s = ( p 1 + p 2 ) 2 , t = ( p 1 + p 4 ) 2 , u = ( p 1 + p 3 ) 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
In 11d, we can scatter: gravitons, gravitini, 3-form gauge particles. ⇒ can use 4d N = 8 language. We scatter: Momenta within 4d = graviton (1); gravitinos (8); gravi-photons (28); gravi-photinos (56); scalars (70 = 35 + 35) At leading order in small momentum (i.e. p 2 ), scattering amplitudes are those in N = 8 SUGRA at tree level. Examples: A SUGRA, tree ( h − h − h + h + ) = � 12 � 4 [ 34 ] 4 , stu A SUGRA, tree ( S 1 S 1 S 2 S 2 ) = tu s , where s = ( p 1 + p 2 ) 2 , t = ( p 1 + p 4 ) 2 , u = ( p 1 + p 3 ) 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
In 11d, we can scatter: gravitons, gravitini, 3-form gauge particles. ⇒ can use 4d N = 8 language. We scatter: Momenta within 4d = graviton (1); gravitinos (8); gravi-photons (28); gravi-photinos (56); scalars (70 = 35 + 35) At leading order in small momentum (i.e. p 2 ), scattering amplitudes are those in N = 8 SUGRA at tree level. Examples: A SUGRA, tree ( h − h − h + h + ) = � 12 � 4 [ 34 ] 4 , stu A SUGRA, tree ( S 1 S 1 S 2 S 2 ) = tu s , where s = ( p 1 + p 2 ) 2 , t = ( p 1 + p 4 ) 2 , u = ( p 1 + p 3 ) 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
Momentum expansion Momentum expansion takes a universal form (independent of the type of particle): � 1 + ℓ 6 p f R 4 ( s , t ) + ℓ 9 A = A SUGRA, tree p f 1-loop ( s , t ) � + ℓ 12 p f D 6 R 4 ( s , t ) + ℓ 14 p f D 8 R 4 ( s , t ) + · · · . f D 2 n 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, . . . ] : f D 6 R 4 ( s , t ) = ( stu ) 2 stu f R 4 ( s , t ) = 3 · 2 7 , 15 · 2 15 . ℓ 10 p f D 4 R 4 allowed by SUSY, but known to vanish. This talk: Reproduce f R 4 and f D 4 R 4 = 0 from ABJM theory. Silviu Pufu (Princeton University) 10-26-2018 6 / 32
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