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

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, Serbia
with L. Jonke, B. Nikoli´ and V. Radovanovi´
c c
JHEP 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
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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
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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
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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
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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
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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
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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
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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
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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
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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 0601
2006), 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
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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
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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
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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
2
to 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)
4
with X nj = F nj − aAn ⋆ F nj and X jk = F jk + aAn ⋆ F jk . Analysis
of EOM, dispersion relations, . . . .
M. Dimitrijevi´ , University of Belgrade – p.13
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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
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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.
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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 ˙
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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.
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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
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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
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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.
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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
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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
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