A 2011 Till Bargheer c Mostly based on [1003.6120] with Florian - PowerPoint PPT Presentation

Scattering Amplitudes in N = 6 Super Chern–Simons Theory and Yangian Symmetry Till Bargheer Uppsala University A � 2011 Till Bargheer c Mostly based on [1003.6120] with Florian Loebbert and Carlo Meneghelli March 25, 2011 Niels Bohr International Academy 27th Nordic Network Meeting on Strings, Fields and Branes

Introduction & Motivation N = 6 Super Chern–Simons (ABJM) � Superconformal theory in d = 3 dimensions [ Aharony, Bergman Jafferis, Maldacena ] Similar to N = 4 SYM AdS/CFT dual to strings on AdS 4 × S 7 / Z k → Integrability Planar limit: N → ∞ (gauge group U( N ) × U( N ) ) Planar spectrum appears integrable! [ Minahan Zarembo ][ Zwiebel ][ Schulgin,Zarembo ] Minahan Integrability 3 2 1 3 2 1 Typically fully constrains spectrum & dynamics Usually in two-dimensional models = Here d > 2 → Interesting! 1 2 3 1 2 3 Motivation: Does integrability permit complete “solution” of the model? Leitmotiv: Understand the integrable structure. Study similarities & differences between d = 3 / d = 4 . 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 1 / 9

Scattering Amplitudes Status N = 4 SYM: ◮ Integrability for spectrum � ◮ Integrability for amplitudes � N = 6 ABJM: ◮ Integrability for spectrum � ◮ Integrability for amplitudes ? A N =6 = ? � 2011 Till Bargheer c Very little is known about amplitudes in N = 6 ABJM. → Need to set up framework and compute some amplitudes 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 2 / 9

Superfield & Superamplitudes Superfield Very advantageous in N = 4 SYM! 6 ε ABCD η A η B η C ¯ ψ D + 24 ε ABCD η A η B η C η D ¯ Φ N =4 = G + η A ψ A + 1 2 η A η B φ AB + 1 1 G Can do similar in N = 6 ABJM: Fields: Scalars φ A , ¯ φ A , fermions ψ A , ¯ ψ A [ TB, Loebbert Meneghelli ] Φ N =6 = φ 4 + η A ψ A + 1 2 ε ABC η A η B φ C + 1 6 ε ABC η A η B η C ψ 4 ψ 4 + η A ¯ 2 ε ABC η A η B ¯ ψ C + 1 6 ε ABC η A η B η C ¯ Φ N =6 = ¯ ¯ φ A + 1 φ 4 Fermionic u (3) spinor η A u (3) : part of su (4) R-symmetry ¯ Φ Φ Φ Superamplitudes ¯ Φ A n Individual n -point amplitudes combine into superamplitude ¯ Φ Φ A n = A n ( η 1 , . . . , η n ) Φ ¯ Φ “Bookkeeping” variables η A k enumerate components � 2011 Till Bargheer c Superfields Φ and ¯ Φ alternate 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 3 / 9

Superfield & Superamplitudes Superfield Very advantageous in N = 4 SYM! 6 ε ABCD η A η B η C ¯ ψ D + 24 ε ABCD η A η B η C η D ¯ Φ N =4 = G + η A ψ A + 1 2 η A η B φ AB + 1 1 G Can do similar in N = 6 ABJM: Fields: Scalars φ A , ¯ φ A , fermions ψ A , ¯ ψ A [ TB, Loebbert Meneghelli ] Φ N =6 = φ 4 + η A ψ A + 1 2 ε ABC η A η B φ C + 1 6 ε ABC η A η B η C ψ 4 ψ 4 + η A ¯ 2 ε ABC η A η B ¯ ψ C + 1 6 ε ABC η A η B η C ¯ Φ N =6 = ¯ ¯ φ A + 1 φ 4 Fermionic u (3) spinor η A u (3) : part of su (4) R-symmetry ¯ Φ Φ Φ Superamplitudes ¯ Φ A n Individual n -point amplitudes combine into superamplitude ¯ Φ Φ A n = A n ( η 1 , . . . , η n ) Φ ¯ Φ “Bookkeeping” variables η A k enumerate components � 2011 Till Bargheer c Superfields Φ and ¯ Φ alternate 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 3 / 9

Symmetries for Amplitudes Superconformal Symmetry Symmetry algebra: osp (6 | 4) (Lorentz, conformal, and supersymmetry) λ a , η A k ) , p µ k = σ µ External superstates parametrized by Λ k = ( λ a k , η A ab λ a k λ b k Singleton representation: Generators J ∈ osp (6 | 4) � 2011 Till Bargheer c b = λ a ∂ A = λ a ∂ P ab = λ a λ b , L a Q a ∂λ b , ∂η A , . . . Amplitudes: A n = A n ( Λ 1 , . . . , Λ n ) � n k J A n Tree-Level: J A n = k =1 � 2011 Till Bargheer c R-Symmetry Φ N =6 = φ 4 + η A ψ A + 1 2 ε ABC η A η B φ C + 1 6 ε ABC η A η B η C ψ 4 Superfield ∂ ∂ ∂ R A B = η A ∂η B − 1 2 δ A R AB = η A η B , Only u (3) covariant: B , R AB = ∂η B . ∂η A C = η C ∂/∂η C − 3 / 2 � = 0 , hence A n ∼ ( η ) 3 n/ 2 Trace R C Different than in N = 4 SYM! 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 4 / 9

Symmetries for Amplitudes Superconformal Symmetry Symmetry algebra: osp (6 | 4) (Lorentz, conformal, and supersymmetry) λ a , η A k ) , p µ k = σ µ External superstates parametrized by Λ k = ( λ a k , η A ab λ a k λ b k Singleton representation: Generators J ∈ osp (6 | 4) � 2011 Till Bargheer c b = λ a ∂ A = λ a ∂ P ab = λ a λ b , L a Q a ∂λ b , ∂η A , . . . Amplitudes: A n = A n ( Λ 1 , . . . , Λ n ) � n k J A n Tree-Level: J A n = k =1 � 2011 Till Bargheer c R-Symmetry Φ N =6 = φ 4 + η A ψ A + 1 2 ε ABC η A η B φ C + 1 6 ε ABC η A η B η C ψ 4 Superfield ∂ ∂ ∂ R A B = η A ∂η B − 1 2 δ A R AB = η A η B , Only u (3) covariant: B , R AB = ∂η B . ∂η A C = η C ∂/∂η C − 3 / 2 � = 0 , hence A n ∼ ( η ) 3 n/ 2 Trace R C Different than in N = 4 SYM! 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 4 / 9

R-Symmetry Trace N = 4 SYM N = 6 ABJM + . . . + ( η ) 4 n − 8 A MHV A n ∼ ( η ) 3 n/ 2 A n = ( η ) 8 A MHV + ( η ) 12 A NMHV n, 2 n, 3 n,n − 2 # # particles n . . . . . . . . . . . . . . . . . . A 7 , 2 A 7 , 3 A 7 , 4 A 7 , 5 A 8 A 6 , 2 A 6 , 3 A 6 , 4 simple A 6 complicated A 5 , 2 A 5 , 3 helicity h A 4 A 4 , 2 � 2011 Till Bargheer c � 2011 Till Bargheer c A 4 A 6 [ Agarwal, Beisert Amplitudes and computed at tree level McLoughlin ][ TB, Loebbert Meneghelli ] � 2011 Till Bargheer c � 2011 Till Bargheer c 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 5 / 9

Yangian Symmetry Integrability for Amplitudes? States = tensor products of osp (6 | 4) (tree-level amplitudes) Integrability ⇔ Yangian symmetry [ Drinfel’d 1985 ] Yangian ◮ Infinite-dimensional symmetry algebra Y( osp (6 | 4)) . . . . . . . . . ◮ Infinitely many levels J = J (0) , � J = J (1) , J (2) , . . . ◮ Level one: Bilocal form K (2) D (2) P (2) � � � P K D � n j � J β βγ A n J α A n = f α J γ P K D k j,k =1 1 j<k n osp (6 | 4) � 2011 Till Bargheer c � 2011 Till Bargheer c 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 6 / 9

Yangian Symmetry Invariance � � � � � � P = 1 A 4 A 6 L [ j P k ] + D [ j P k ] − Q [ j Q k ] = 0 , = 0 J J 2 j<k invariant under � A tree and A tree P � 4 6 � 2011 Till Bargheer c � 2011 Till Bargheer c Invariance under all � J follows from [ J , � J ] ∼ � [ TB, Loebbert J � Meneghelli ] Serre Relations [ � J α , [ � J β , J γ ]] + [ � J β , [ � J γ , J α ]] + [ � J γ , [ � λ f βσ µ f γτ ν f ρστ { J λ , J µ , J ν } J α , J β ]] ∼ f αρ � � Sufficient to show: 0 = { JJJ } X J ∼ spinor representation T ij ∼ [ γ i , γ j ] � � Representation theory ⇒ { JJJ } [ TB, Loebbert X = 0 � Meneghelli ] Invariance under consistent Yangian symmetry algebra 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 7 / 9

Recent Developments Dual superconformal symmetry x 4 λ a j λ b j = x ab j − x ab p 4 x 5 j +1 p 3 Superconformal symmetry λ a j η A j = θ aA − θ aA x 3 p 5 j j +1 [ Huang Lipstein ][ Gang, Huang, Koh acting on x , θ Lee, Lipstein ] p 2 η A j η B j = y AB − y AB j j +1 p 1 x 1 x 2 � 2011 Till Bargheer c On-Shell Recursion Relations λ j → + 1 2 ( z + 1 z ) λ j + i 2 ( z − 1 z ) λ k A L A R [ Gang, Huang, Koh Lee, Lipstein ] λ k → − i 2 ( z − 1 z ) λ j + 1 2 ( z + 1 z ) λ k j k � 2011 Till Bargheer c Graßmannian formula � d ν ( t ) M 1 · · · M k δ k ( k +1) / 2 ( t · t T ) δ 2 k | 3 k ( t · Λ ) L 2 k ( Λ ) = [ Lee ] 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 8 / 9

Summary and Outlook Yangian . . . . . . . . . Summary ★ On-shell superfield formalism for N = 6 ABJM amplitudes P (2) K (2) D (2) ★ Explicit tree-level amplitudes for four and six points ★ Invariance under Yangian symmetry P = ˜ � � � K K D ★ Dual superconformal symmetry ★ Conjectured Graßmannian functional D = ˜ P K D ordinary dual ˜ P c � 2011 Till Bargheer Outlook ? Self-T-Duality of dual strings on AdS 4 × C P 3 ? [ Dekel, Oz ][ Adam Sorokin, Wulff ][ Grassi Dekel, Oz ][ Bakhmatov ][ Dekel Adam Oz ] Bianchi, Leoni, Mauri ? Wilson Loop / Amplitude / Correlator Duality? [ Penati, Ratti, Santambrogio ] ? Loop-level Yangian / dual symmetry? 25.03.11 Till Bargheer: N = 6 Super Chern–Simons Amplitudes and Yangian Symmetry 9 / 9