Yangian Symmetry of N=4 SYM

Transcription

1 Yangian Symmetry of N=4 SYM Aleksander Garus Joint work with N. Beisert and M. Rosso Zakopane,

2 Table of contents 1. Introduction and Motivation* 2. Yangian Symmetry 3. Correlation Functions in Pictures 4. Conclusions & Outlook *Shortened due to overlap with Jan Plefka s lecture

3 N=4 SYM Maximally supersymmetric gauge theory in 4D 1 gauge field A µ, 4 spinors Ψ, 4 conjugate spinors Ψ, _ 6 scalars φm psu(2,2 4) - symmetry algebra of the action

4 Integrability in N=4 SYM Spectra of local operators } Increased Wilson loops symmetry algebra: Ya ng ia n Scattering amplitudes

5 Yangian Algebra Infinitely many levels of generators. Level 0 - original psu(2,2 4): [J a,j b } = fabj c c Level 1 - first higher level Yangian generators: [J a, Ĵb} = fabĵc c Coproduct (action on multiparticle states) (J a )=id J a + J a id

6 Yangian Algebra Infinitely many levels of generators. Level 0 - original psu(2,2 4): [J a,j b } = f c abj c Level 1 - first higher level Yangian generators: [J a, Ĵb} = f c abĵc Coproduct (action on multiparticle states) (Ĵa) =id Ĵa + Ĵa id + fa bc J b J c

7 Motivation (for advanced) We know how Yangian generators act on spin chains scattering amplitudes Wilson loops

8 Motivation (for advanced) We know how Yangian generators act on spin chains scattering amplitudes (also mentioned in Jacob Bourjaily s talk) See Jan Plefka s lecture for details! Wilson loops

9 Yangian Action on Action S = Z d 4 xt r(...) Due to the trace, the action is cyclic: Tr(ABC) =Tr(CAB) However: Ĵ(ABC) 6= Ĵ(CAB)

10 Yangian Action on Action S = Z d 4 xt r(...) Due to the trace, the action is cyclic: Tr(ABC) =Tr(CAB) However: Ĵ(ABC) 6= Ĵ(CAB) Incompatible!

11 Symmetry of the EoMs (they are no longer cyclic) Simplest level-1 generator: ( ˆP µ )=D^P µ + L µ i ^ P 4 Easiest equation of motion: µ Qȧ ^ Q a [D, a] ig m ab [ m, b ] =0

12 Symmetry of the EoMs (they are no longer cyclic) Simplest level-1 generator: ( ˆP µ )=D^P µ + L µ i ^ P 4 Easiest equation of motion: µ Qȧ ^ Q a [D, a] ig m ab [ m, b ] =0

13 Symmetry of the EoMs (they are no longer cyclic) Simplest level-1 generator: ( ˆP µ )=D^P µ + L µ i ^ P 4 - Easiest equation of motion: µ Qȧ ^ Q a [D, a] ig m ab [ m, b ] =0 etc

14 Symmetry of the EoMs (they are no longer cyclic) Introducing the single field action: ˆP µ (D )= 1 2 g2 µ { m, m} we obtain schematically: ˆP (Dirac) =0

15 Symmetry of the EoMs (they are no longer cyclic) Introducing the single field action: ˆP µ (D )= 1 2 g2 µ { m, m} we obtain schematically:

16 Symmetry of the EoMs (they are no longer cyclic) Introducing the single field action: ˆP µ (D )= 1 2 g2 µ { m, m} we obtain schematically: Similarly for all the other equations of motion!

17 Off-shell? For level-0 symmetry generators we have: 0=JS = Z (JZ a ) S Z a Varying again w. r. t. to a field: Vanishes on-shell J S Z c + (JZ a) Z c S Z a =0 Off-shell equality regarding behavior of eoms under J.

18 Off-shell! Generalize to level-1 generators: Ĵ k S Z c = (Ĵ k Z a ) Z c S Z a + f k mn (J m Z a ) Za ^ S Z b Z c (J n Z b ) Vanishes on shell Holds off-shell!

19 Off-shell! Generalize to level-1 generators: ˆ J k S c = ( J ˆk a) Vanishes on shell c S a + f k mn (J m a) Holds off-shell! a ^ S b c (J n b) Doesn t hold for non-integrable field theories!

20 Brief recap Yangian algebra appears in many observables in N=4 SYM. At the first level, the Yangian generators act bilocally (here - in color space). Cannot act with them on the action, but they are an on-shell symmetry of equations of motion. We put forward an off-shell equality we believe equivalent to the invariance of the action.

21 Correlation Functions (tree level) J If JS =0 and leaves the PI measure invariant, the correlation functions are also invariant (Slavnov-Taylor / Ward identity): J<Z 1 (x 1 )Z 2 (x 2 )...Z n (x n ) >= NX i=1 <Z 1 (x 1 )...(JZ i (x i ))...Z n (x n ) >= 0

22 Ars Gratia Artis* Introduce graphical notation: *Art for art s sake

23 Example: Propagator Invariance Level-0 generator J acting on a propagator: Level-0 magic formula J S Z c + (JZ a) Z c S Z a =0 at 0-th order:

24 Invariance under level-1 Draw level-1 magic formula Draw action of Yangian generator on the correlation function Local action cancels easily - tricky part is the bilocal part of the coproduct Level-0 is a symmetry, we can move it around Dual coxeter number of psu(2,2 4) vanishes: f a bcf bc d = h a b =0

25 Final recap Yangian of psu(2,2 4) - correct symmetry algebra of planar N=4 SYM. Invariance of equations of motion under level-1. Cannot act on the action, but exists a way to circumvent it. Gauge-fixing (BRST) - compatible. Implies invariance of the correlation functions.

26 Final recap Yangian of psu(2,2 4) - correct symmetry algebra of planar N=4 SYM. Invariance of equations of motion under level-1. Cannot act on the action, but exists a way to circumvent it. Gauge-fixing (BRST) - compatible. Implies invariance of the correlation functions. Holds for ABJM too!

27 To investigate Anomalies Algebraic relations for Yangian generators Proper Ward identities

28 Thanks for your attention!