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

c 2011 Till Bargheer

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]

AdS/CFT dual to strings on AdS4 × S7/Zk Planar limit: N → ∞ (gauge group U(N) × U(N))

• Similar to N = 4 SYM

→ Integrability Planar spectrum appears integrable! [Minahan

Zarembo][Zwiebel][ Minahan Schulgin,Zarembo]

Integrability Typically fully constrains spectrum & dynamics Usually in two-dimensional models

1 1 3 3 2 2

=

1 1 3 3 2 2

Here d > 2 → Interesting! Motivation: Does integrability permit complete “solution” of the model? Leitmotiv: Understand the integrable structure. Study similarities & differences between d = 3/d = 4.

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Scattering Amplitudes

Status N = 4 SYM:

◮ Integrability for spectrum ◮ Integrability for amplitudes

N = 6 ABJM:

◮ Integrability for spectrum ◮ Integrability for amplitudes ?

AN =6

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= ?

Very little is known about amplitudes in N = 6 ABJM. → Need to set up framework and compute some amplitudes

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Superfield & Superamplitudes

Superfield Very advantageous in N = 4 SYM! ΦN =4 = G + ηAψA + 1

2ηAηBφAB + 1 6εABCDηAηBηC ¯

ψD +

1 24εABCDηAηBηCηD ¯

G Can do similar in N = 6 ABJM: Fields: Scalars φA, ¯ φA, fermions ψA, ¯ ψA [TB, Loebbert

Meneghelli ]

ΦN =6 = φ4 + ηAψA + 1

2εABCηAηBφC + 1 6εABCηAηBηCψ4

¯ ΦN =6 = ¯ ψ4 + ηA ¯ φA + 1

2εABCηAηB ¯

ψC + 1

6εABCηAηBηC ¯

φ4 Fermionic u(3) spinor ηA u(3): part of su(4) R-symmetry Superamplitudes Individual n-point amplitudes combine into superamplitude

An

Φ Φ Φ Φ ¯ Φ ¯ Φ ¯ Φ ¯ Φ

c 2011 Till Bargheer

An = An(η1, . . . , ηn) “Bookkeeping” variables ηA

k enumerate components

Superfields Φ and ¯ Φ alternate

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Superfield & Superamplitudes

Superfield Very advantageous in N = 4 SYM! ΦN =4 = G + ηAψA + 1

2ηAηBφAB + 1 6εABCDηAηBηC ¯

ψD +

1 24εABCDηAηBηCηD ¯

G Can do similar in N = 6 ABJM: Fields: Scalars φA, ¯ φA, fermions ψA, ¯ ψA [TB, Loebbert

Meneghelli ]

ΦN =6 = φ4 + ηAψA + 1

2εABCηAηBφC + 1 6εABCηAηBηCψ4

¯ ΦN =6 = ¯ ψ4 + ηA ¯ φA + 1

2εABCηAηB ¯

ψC + 1

6εABCηAηBηC ¯

φ4 Fermionic u(3) spinor ηA u(3): part of su(4) R-symmetry Superamplitudes Individual n-point amplitudes combine into superamplitude

An

Φ Φ Φ Φ ¯ Φ ¯ Φ ¯ Φ ¯ Φ

c 2011 Till Bargheer

An = An(η1, . . . , ηn) “Bookkeeping” variables ηA

k enumerate components

Superfields Φ and ¯ Φ alternate

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Symmetries for Amplitudes

Superconformal Symmetry Symmetry algebra: osp(6|4) (Lorentz, conformal, and supersymmetry) External superstates parametrized by Λk = (λa

k, ηA k ), pµ k = σµ abλa kλb k

λa, ηA

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Singleton representation: Generators J ∈ osp(6|4) Pab = λaλb, La

b = λa ∂

∂λb , Qa

A = λa ∂

∂ηA , . . . Amplitudes: An = An(Λ1, . . . , Λn) Tree-Level: JAn =

n

• k=1

An

k J

c 2011 Till Bargheer

R-Symmetry Superfield ΦN =6 = φ4 + ηAψA + 1

2εABCηAηBφC + 1 6εABCηAηBηCψ4

Only u(3) covariant: RA

B = ηA

∂ ∂ηB − 1

2δA B,

RAB = ηAηB, RAB = ∂ ∂ηA ∂ ∂ηB . Trace RC

C = ηC∂/∂ηC − 3/2 = 0, hence An ∼ (η)3n/2

Different than in N = 4 SYM!

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Symmetries for Amplitudes

Superconformal Symmetry Symmetry algebra: osp(6|4) (Lorentz, conformal, and supersymmetry) External superstates parametrized by Λk = (λa

k, ηA k ), pµ k = σµ abλa kλb k

λa, ηA

c 2011 Till Bargheer

Singleton representation: Generators J ∈ osp(6|4) Pab = λaλb, La

b = λa ∂

∂λb , Qa

A = λa ∂

∂ηA , . . . Amplitudes: An = An(Λ1, . . . , Λn) Tree-Level: JAn =

n

• k=1

An

k J

c 2011 Till Bargheer

R-Symmetry Superfield ΦN =6 = φ4 + ηAψA + 1

2εABCηAηBφC + 1 6εABCηAηBηCψ4

Only u(3) covariant: RA

B = ηA

∂ ∂ηB − 1

2δA B,

RAB = ηAηB, RAB = ∂ ∂ηA ∂ ∂ηB . Trace RC

C = ηC∂/∂ηC − 3/2 = 0, hence An ∼ (η)3n/2

Different than in N = 4 SYM!

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R-Symmetry Trace

N = 4 SYM An = (η)8AMHV

n,2

+ (η)12ANMHV

n,3

+ . . . + (η)4n−8AMHV

n,n−2

# particles n helicity h

. . . . . . . . . . . . . . . A4,2 A5,2 A5,3 A6,2 A6,3 A6,4 A7,2 A7,3 A7,4 A7,5

simple complicated

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N = 6 ABJM An ∼ (η)3n/2

# A4 A6 A8 . . .

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Amplitudes A4

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and A6

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computed at tree level [Agarwal, Beisert

McLoughlin ][TB, Loebbert Meneghelli ] 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 K D P

• K
• D
• P

K(2) D(2) P(2) . . . . . . . . .

• sp(6|4)

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◮ Infinite-dimensional symmetry algebra Y(osp(6|4)) ◮ Infinitely many levels J = J(0),

J = J(1), J(2), . . .

◮ Level one: Bilocal form

• Jα An = fα

βγ n

• j,k=1

j<k

An

Jβ Jγ j k 1 n

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Yangian Symmetry

Invariance

• P = 1

2

• j<k
• L[jPk] + D[jPk] − Q[jQk]
• J

A4

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= 0,

• J

A6

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= 0 Atree

4

and Atree

6

invariant under P Invariance under all J follows from [J, J] ∼ J [TB, Loebbert

Meneghelli ]

Serre Relations [ Jα, [ Jβ, Jγ]] + [ Jβ, [ Jγ, Jα]] + [ Jγ, [ Jα, Jβ]] ∼ fαρ

λfβσ µfγτ νf ρστ{Jλ, Jµ, Jν}

Sufficient to show: 0 = {JJJ}

• X

J ∼ spinor representation T ij ∼ [γi, γj] Representation theory ⇒ {JJJ}

• X = 0

[TB, Loebbert

Meneghelli ]

Invariance under consistent Yangian symmetry algebra

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Recent Developments

Dual superconformal symmetry x1 x2 x3 x4 x5 p1 p2 p3 p4 p5

c 2011 Till Bargheer

λa

j λb j = xab j − xab j+1

λa

j ηA j = θaA j

− θaA

j+1

ηA

j ηB j = yAB j

− yAB

j+1

Superconformal symmetry acting on x, θ [ Huang

Lipstein][Gang, Huang, Koh Lee, Lipstein ]

On-Shell Recursion Relations λj → + 1

2(z + 1 z )λj + i 2(z − 1 z )λk

λk → − i

2(z − 1 z )λj + 1 2(z + 1 z )λk

AL AR

j k

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[Gang, Huang, Koh

Lee, Lipstein ]

Graßmannian formula L2k(Λ) =

• dν(t)

M1 · · · Mk δk(k+1)/2(t · tT) δ2k|3k(t · Λ) [Lee]

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Summary and Outlook

Summary ★ On-shell superfield formalism for N = 6 ABJM amplitudes ★ Explicit tree-level amplitudes for four and six points ★ Invariance under Yangian symmetry Yangian ˜ P K D= ˜ D P

• K
• D
• P= ˜

K K(2) D(2) P(2) . . . . . . . . .

• rdinary

dual

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★ Dual superconformal symmetry ★ Conjectured Graßmannian functional Outlook ? Self-T-Duality of dual strings on AdS4 × CP3? [

Adam Dekel, Oz][ Grassi Sorokin, Wulff][ Adam Dekel, Oz][Bakhmatov][Dekel Oz ]

? Wilson Loop / Amplitude / Correlator Duality? [

Bianchi, Leoni, Mauri Penati, Ratti, Santambrogio]

? Loop-level Yangian / dual symmetry?

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