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# M. Dimitrijevic - Twisted Symmetry and Noncommutative Field Theory

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The SEENET-MTP Workshop JW2011
Scientific and Human Legacy of Julius Wess
27-28 August 2011, Donji Milanovac, Serbia

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### M. Dimitrijevic - Twisted Symmetry and Noncommutative Field Theory

1. 1. Balkan Summer Institute 2011, Workshop JW2011: Scientiﬁc and Human Legacy of Julius Wess, 27th -28th August 2011, Donji Milanovac, Serbia Twisted Symmetry and Noncommutative Field Theory Marija Dimitrijevi´ c University of Belgrade, Faculty of Physics, Belgrade, Serbiawith L. Jonke, B. Nikoli´ and V. Radovanovi´ c cJHEP 0712, 059 (2007); JHEP 0904, 108 (2009); Phys. Rev. D, 81 (2010); Phys. Rev. D,83 (2011); 1107.3475[hep-th]. M. Dimitrijevi´ , University of Belgrade – p.1 c
2. 2. Outline • Noncommutative spaces from a twist -motivation -deﬁnition -examples • Example I: κ-Mikowski space-time • Example II: Non(anti)commutative SUSY Notation: "Munich" group: students, postdocs, collaborators of Julius Wess during the time he was in Munich (1990-2005) and Hamburg (2005-2007). M. Dimitrijevi´ , University of Belgrade – p.2 c
3. 3. Noncommutative spaces from a twist ˆˆ Noncommutative space Ax , generated by xµ coordinates ˆ µ = 0, 1, . . . n such that: [ˆµ , xν ] = Θµν (ˆ). x ˆ x (1) It is an associative free algebra generated by xµ and divided by ˆ the ideal generated by relations (1). Differential calculus, integral, symmetries ["Munich" group around 1990 and from then on] can be discussed, but. . . ˆˆ ⋆-product geometry: represent Ax on the space of commuting coordinates, but keep track of the deformation ˆˆ Ax → Ax⋆ ˆx f (ˆ) → f (x) and ˆx g x f (ˆ)ˆ(ˆ) → f ⋆ g(x). M. Dimitrijevi´ , University of Belgrade – p.3 c
4. 4. MW ⋆-product ∞ i n 1 ρ1 σ1 f ⋆ g (x) = θ . . . θρn σn n=0 2 n! ∂ρ1 . . . ∂ρn f (x) ∂σ1 . . . ∂σn g(x) i = f · g + θ ρσ (∂ρ f ) · (∂σ g) + O(θ 2 ). (2) 2 Associative, noncommutative; c. conjugation: (f ⋆ g)∗ = g ∗ ⋆ f ∗ . µ ν µ ν i µν Special example: x ⋆ x = x x + θ , 2 [xµ ⋆ xν ] = iθµν . , (3) The ⋆-product (2) enabled: construction of quantum ﬁeld theories and analysis of their renormalizability properties, construction of NC Standard Model and the analysis of its phenomenological consequences,. . . ["Munich" group around 2000 and from then on]. M. Dimitrijevi´ , University of Belgrade – p.4 c
5. 5. Twist formalism • Motivation 1: Product (2) can be viewed as coming from an Abelian twist given by i − 2 θρσ ∂ρ ⊗∂σ F =e (4) as f ⋆ g = µ F −1 f ⊗ g i ρσ 2θ ∂ρ ⊗∂σ = µ e f ⊗g i = f · g + θ ρσ (∂ρ f ) · (∂σ g) + O(θ 2 ). (5) 2 • Motivation 2: Deformation [xµ ⋆ xν ] = iθµν breaks the , classical Lorentz symmetry. Is there a deformation of Lorentz symmetry such that it is a symmetry of (3)? M. Dimitrijevi´ , University of Belgrade – p.5 c
6. 6. Basic idea Consider ﬁrst a deformation (twist) of a classical symmetry algebra g (Lorentz, SUSY, gauge,. . . ). Then deform the space-time itself. A twist F (introduced by Drinfel’d in 1983-1985) is: -an element of U g ⊗ U g -invertible -fulﬁlls the cocycle condition (ensures the associativity of the ⋆-product) F ⊗ 1(∆ ⊗ id)F = 1 ⊗ F (id ⊗ ∆)F . (6) -additionally: F = 1 ⊗ 1 + O(h); h-deformation parameter. Notation: F = f α ⊗ fα and F −1 = ¯α ⊗ ¯α . f f ["Munich" group around 2005 and from then on]. M. Dimitrijevi´ , University of Belgrade – p.6 c
7. 7. Action of F • F applied to U g: twisted Hopf algebra U gF [ta , tb ] = if abc tc ,∆F (ta ) = F∆(ta )F −1 , ε(ta ) = 0, SF = f α S(fα )S(ta )S(¯β )¯β . f f (7) • F applied to Ax (algebra of smooth functions on M): A⋆ x pointwise multiplication: µ(f ⊗ g) = f · g ⇓ ⋆-multiplication: µ⋆ (f ⊗ g) ≡ µ ◦ F −1 (f ⊗ g) = (¯α f )(¯α g) = f ⋆ g. f f (8) • F applied to the symmetry of theory cl cl cl F ,F −1 δg (φ1 φ2 ) = (δg φ1 )φ2 + φ1 (δg φ2 ) → deformed Leibniz rule. M. Dimitrijevi´ , University of Belgrade – p.7 c
8. 8. • F applied to Ω (exterior algebra of forms): Ω⋆ wedge product: ω1 ∧ ω2 = ω1 ⊗ ω2 − ω2 ⊗ ω1 ⇓ ⋆-wedge product: ω1 ∧⋆ ω2 = (¯α ω1 ) ∧ (¯α ω2 ). f f• Differential calculus is classical: d : A⋆ → Ω⋆ . x d2 = 0, d(f ⋆ g) = df ⋆ g + f ⋆ dg, df = (∂µ f )dxµ = (∂µ f ) ⋆ dxµ . ⋆ (9)• Integral of a maximal form (d1 + d2 = dim(M)) is graded cyclic: ω1 ∧⋆ ω2 = (−1)d1 d2 ω2 ∧⋆ ω1 . (10) M. Dimitrijevi´ , University of Belgrade – p.8 c
9. 9. Comments I: Deformations by twist i µν 1. Moyal-Weyl twist F = e− 2 θ ∂µ ⊗∂ν , θ µν = −θ νµ ∈ R: • θ-deformed Poincaré symmetry: Chaichian et al. (Phys. Lett. B604 2004), Wess (hep-th/0408080), Koch et al. (Nucl. Phys. B717 2005). [∂µ , ∂ν ] = 0, ⋆ [δω , ∂ρ ] = ωρµ ∂µ , ⋆ ⋆ ⋆ [δω , δω ′ ] = δ[ω,ω ′ ] , ⋆ ⋆ ⋆ i ρσ “ λ λ ” ∆(δω ) = δω ⊗ 1 + 1 ⊗ δω + θ ω ρ ∂λ ⊗ ∂σ + ∂ρ ⊗ ω σ ∂λ . 2 • θ-deformed gravity: Aschieri et al. (Class. Quant. Grav. 22, 2005 and 23, 2006). ⋆ ⋆ ⋆ [δξ , δη ] = δ[ξ,η] , ⋆ ⋆ ⋆ i ρσ “ ⋆ ⋆ ” ∆(δξ ) = δξ ⊗1+1⊗ δξ − θ δ(∂ρ ξ) ⊗ ∂σ + ∂ρ ⊗ δ(∂σ ξ) + . . . . 2 • θ-deformed gauge theory: Aschieri et al. (Lett. Math. Phys. 78 2006), Vassilevich (Mod. Phys. Lett. A 21 2006), Giller et al. (Phys. Lett. B655, 2007). ⋆ ⋆ ⋆ [δα , δβ ] = δ−i[α,β] , α = αa t a , ⋆ ⋆ ⋆ i ρσ “ ⋆ ⋆ ” ∆(δα ) = δα ⊗1+1⊗ δα − θ δ(∂ρ α) ⊗ ∂σ + ∂ρ ⊗ δ(∂σ α) + . . . . 2 M. Dimitrijevi´ , University of Belgrade – p.9 c
10. 10. 2. Twisted supersymmetry: ´ Kosinski et al. (J. Phys. A 27 1994), Kobayashi et al.(Int. J. Mod. Phys. A 20 2005), Zupnik (Phys. Lett. B 627 2005), Ihl et al. (JHEP 06012006), Dimitrijevi´ et al. (JHEP 0712 2007), . . . c 1 C αβ ∂ ⊗∂ + 1 C ¯˙ ¯˙ ¯ ˙ ∂ α ⊗∂ β 1 αβ F1 = e2 α β 2 αβ˙ , F2 = e 2 C Dα ⊗Dβ ,... ¯˙ C αβ = C βα ∈ C, Dα = ∂α + iσ mα θα ∂m α˙3. Twist with commuting vector ﬁelds i ab X ⊗X F = e− 2 θ a b, Xa = Xa ∂µ , [Xa , Xb ] = 0, θab = const. µ • dynamical NC: vector ﬁelds Xa are dynamical, global Lorentz symmetry is preserved: Aschieri et al. (Lett. Math. Phys. 85 2008). • deformed gravity: cosmological and black hole solutions, coupled to fermions, deformed supergravity: Schupp et al. (0906.2724[hep-th]), Ohl et al. (JHEP 0901, 2009), Aschieri et al. (JHEP 0906 2009; JHEP 0906 2009). • κ-Minkowski: Meljanac et al. (Eur. Phys. J. C 53 2008), Borowiec et al. (Phys. Rev. D 79 2009),. . . M. Dimitrijevi´ , University of Belgrade – p.10 c
11. 11. Example I: κ-Minkowski space-time Deﬁned by: [ˆ0 , xj ] = iaˆj , x ˆ x [ˆi , xj ] = 0, x ˆ (11) with a = 1/κ and i, j = 1, 2, 3. Interesting phenomenological consequences: modiﬁed Lorentz symmetry, modiﬁed dispersion relations, DSR theories, . . . ⋆-product approach ["Munich" group 2002-2005] has problems with: non-unique derivatives, diferential calculus, non-cyclic integral. ⇒ Difﬁcult to do ﬁeld theory. . . Suggestion: apply the twist formalism with i − 2 θab Xa ⊗Xb − ia (∂0 ⊗xj ∂j −xj ∂j ⊗∂0 ) F =e =e 2 , (12) with X1 = ∂0 , X2 = xj ∂j , [X1 , X2 ] = 0 and θab = aǫab . M. Dimitrijevi´ , University of Belgrade – p.11 c
12. 12. Action of the twist (11): • ⋆-product of functions f ⋆g = µ{F −1 f ⊗ g} ia = f · g + xj (∂0 f )∂j g − (∂j f )∂0 g + O(a2 ). (13) 2 • [x0 ⋆ xj ] = iaxj and [xi ⋆ xj ] = 0 . , , • Differential calculus i ⋆ df = (∂µ ) ⋆ dxµ , ⋆ ∂0 = ∂0 , ∂j = e− 2 a∂0 ∂j , ⋆ f ⋆ dx0 = dx0 ⋆ f, f ⋆ dxj = dxj ⋆ eia∂0 f, dxµ ∧⋆ dxν = dxµ ∧ dxν , d4 x = dx0 ∧ · · · ∧ dx3 . • Integral: ω1 ∧⋆ ω2 = (−1)d1 d2 ω2 ∧⋆ ω1 , with d1 + d2 = 4. M. Dimitrijevi´ , University of Belgrade – p.12 c
13. 13. Enough ingredients to construct scalar and spinor ﬁeld theories.For a gauge theory a ⋆-Hodge dual is needed: S= F 0 ∧ (∗F 0 ) → S = F ∧⋆ (∗F ), 1 with ∗F 0 = 2 ǫµναβ F 0αβ dxµ ∧ dxν and F = dA − A ∧⋆ A.The obvious choice ∗F = 1 ǫµναβ F αβ ⋆ dxµ ∧⋆ dxν does not lead 2to a gauge invariant action.Using the Seiberg-Witten map we were able to construct the⋆-Hodge dual up to ﬁrst order in a. The invariant action is: 1 S=− 2F0j ⋆ e−ia∂0 X 0j + Fij ⋆ e−2ia∂0 X ij ⋆ d4 x, (14) 4with X nj = F nj − aAn ⋆ F nj and X jk = F jk + aAn ⋆ F jk . Analysisof EOM, dispersion relations, . . . . M. Dimitrijevi´ , University of Belgrade – p.13 c
14. 14. Comments II • Advantages of twist formalism: -mathematically well deﬁned -differential calculus -cyclic integral. • Disadvantages: -Hodge dual is difﬁcult to generalize -global Poinaceré symmetry iso(1, 3) is replaced by global inhomogenious general linear symmetry igl(1, 3) -problem of conserved charges. • Possibilities: -natural basis -new deﬁnition of ⋆-Hodge dual -twisted gauge symmetry? M. Dimitrijevi´ , University of Belgrade – p.14 c
15. 15. Example II: Twisted SUSY Non(anti)commutative ﬁeld theories: from 2003 intensively, ["Munich" group around 2006 and from then on]. Different types of deformation of superspace, Wess-Zumino and Yang-Mills models, their renormalizability properties,. . . For an ilustration let us compare two different twists: 1 αβ 1 ¯ ¯˙ ¯˙ C ∂α ⊗∂β + 2 Cαβ ∂ α ⊗∂ β ˙ F1 = e 2 ˙ , (15) 1 αβ C Dα ⊗Dβ F2 = e 2 , (16) αβ βα with C1,2 = C1,2 ∈ C, ∂α = ∂ ¯˙ Dα = ∂α + iσ m α θα ∂m . ∂θα , α˙ An obvious difference: F1 is hermitean and F2 is not hermitean under the usual c.conjugation. M. Dimitrijevi´ , University of Belgrade – p.15 c
16. 16. Twist F1 leads to: • ⋆-product of superﬁelds −1 F ⋆ G = µ{F1 F ⊗ G} 1 = F · G − (−1)|F | C αβ (∂α F ) · (∂β G) 2 1 ¯ ˙ ¯˙ ¯˙ − (−1)|F | Cαβ (∂ α F )(∂ β G) + O(C 2 ). ˙ (17) 2 where |F | = 1 if F is odd and |F | = 0 if F is even. ¯˙ , ¯ ˙ ¯ ˙ • {θα ⋆ θβ } = C αβ , {θα ⋆ θβ } = Cαβ , [xm ⋆ xn ] = 0. , , ˙ • Deformed Leibniz rule ⇒ twisted SUSY transformations ⋆ δξ (F ⋆ G) = (δξ F ) ⋆ G + F ⋆ (δξ G) (18) i αβ ¯γ m ¯˙ + C ξ ˙ σ αγ (∂m F ) ⋆ (∂β G) + (∂α F ) ⋆ ξ γ σ mγ (∂m G) ˙ β˙ 2 i ¯ α m γα ˙ ˙ ˙ ¯˙ ˙ ˙ ¯β G) + (∂ α F ) ⋆ ξ α σ m εγ β (∂m G) . − Cαβ ξ σαγ ε (∂m F ) ⋆ (∂ ˙ ˙ αγ ˙ 2 ˙ M. Dimitrijevi´ , University of Belgrade – p.16 c
17. 17. • Chirality is broken; if Φ is chiral Φ ⋆ Φ is not chiral! Project out chiral, antichiral and transverse components of Φ ⋆ Φ and Φ ⋆ Φ ⋆ Φ using the projectors P1 , P2 and PT ¯ 1 D2 D2 ¯ 1 D2 D2 1 Z P1 = , P2 = , f (x) g(x) = f (x) d4 y G(x − y)g(y). 16 16• Deformed Wess-Zumino action S = d4 x Φ+ ⋆ Φ (19) ¯¯ θθ θθ m λ + P2 Φ ⋆ Φ + P2 Φ ⋆ P2 Φ ⋆ Φ + c.c. 2 θθ 3 θθ• A minimal deformation of the commutative WZ action, good commutative limit, it is non-local. M. Dimitrijevi´ , University of Belgrade – p.17 c
18. 18. • One-loop renormalizability using supergraph technique and the background ﬁeld method: -no tadpole -the divergences in two-point function cannot be removed ⇒ The model is NOT renormalizable.• What can be done: -add new terms to absorb divergences ⇒ non-minimal deformation -understand better the interplay between twisted symmetry and renormalizability. M. Dimitrijevi´ , University of Belgrade – p.18 c
19. 19. Twist F2 leads to: • ⋆-product of superﬁelds −1 F ⋆ G = µ{F1 F ⊗ G} 1 = F · G − (−1)|F | C αβ (Dα F ) · (Dβ G) + O(C 2 ). (20) 2 where |F | = 1 if F is odd and |F | = 0 if F is even. • {θα ⋆ θβ } = C αβ , , ¯˙ , ¯ ˙ {θα ⋆ θβ } = 0, [xm ¯¯ ⋆ xn ] = −C αβ (σ mn ε) θ θ. , αβ ¯˙ • Since {Qα , Dβ } = {Qα , Dβ } = 0 Leibniz rule for SUSY transformations is undeformed. ⋆ δξ (F ⋆ G) = (δξ F ) ⋆ G + F ⋆ (δξ G) (21) • Chirality is broken again. Method of projectors. . . M. Dimitrijevi´ , University of Belgrade – p.19 c
20. 20. • Deformed Wess-Zumino action 4 + m S = d x Φ ⋆Φ + P2 (Φ ⋆ Φ) + 2a1 P1 (Φ ⋆ Φ) ¯θ θθ θ ¯ 2 θθ ¯¯ θθ λ + P2 (P2 (Φ ⋆ Φ) ⋆ Φ) + 3a2 P1 (P2 (Φ ⋆ Φ) ⋆ Φ) 3 θθ ¯¯ θθ +2a3 (P1 (Φ ⋆ Φ) ⋆ Φ) +3a4 P1 (Φ ⋆ Φ) ⋆ Φ+ ¯¯ θθ θθ ¯¯ θθ ¯ +3a5 C 2 P2 (Φ ⋆ Φ) ⋆ Φ+ + c.c. . (22) ¯¯ θθ θθ• A non-minimal deformation of the commutative WZ action, good commutative limit, it is local. M. Dimitrijevi´ , University of Belgrade – p.20 c
21. 21. • One-loop renormalizability using supergraph technique and the background ﬁeld method: -no tadpole, no mass renormaization -a4 and a5 -terms in the action (20) required to absorb the (3) divergences appearing in Γ1 (4) -divergences in Γ1 cannot be absorbed ⇒ The general model IS NOT renormalizable.• HOWEVER: There is a special choice: a2 = a3 = a4 = 0 when the model is renormalizable! Almost all commutative SUSY results remain valid: no tadpole, no mass (4) renormalization, divergent parts of Γ1 and higher-point functions are zero.• Non-minimal deformation and undeformed SUSY render a renormalizable model. A more general conclusion? M. Dimitrijevi´ , University of Belgrade – p.21 c
22. 22. Summary • NC spaces can be deﬁned via twist. • Mathematically well deﬁned, good control of deformed symmetries, differential calculus, integral. • More loose ends: Generalization of Noether theorem, conserved charges, Hodge dual,. . . Better understanding of renormalizability versus twisted symmetries also needed. M. Dimitrijevi´ , University of Belgrade – p.22 c