M. Dimitrijevic - Twisted Symmetry and Noncommutative Field Theory

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 Dimitrijevic

University of Belgrade, Faculty of Physics,Belgrade, Serbia

with L. Jonke, B. Nikolic and V. RadovanovicJHEP 0712, 059 (2007); JHEP 0904, 108 (2009); Phys. Rev. D, 81 (2010); Phys. Rev. D,

83 (2011); 1107.3475[hep-th].

M. Dimitrijevic, University of Belgrade – p.1

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).


Noncommutative spaces from a twist

Noncommutative space Ax, generated by xµ coordinates

µ = 0, 1, . . . n such that:

[xµ, xν ] = Θµν(x). (1)

It is an associative free algebra generated by xµ and divided by

the ideal generated by relations (1). Differential calculus, integral,

symmetries ["Munich" group around 1990 and from then on] can

be discussed, but. . .

⋆-product geometry: represent Ax on the space of commuting

coordinates, but keep track of the deformation

Ax 7→ A⋆x

f(x) 7→ f(x) and f(x)g(x) 7→ f ⋆ g(x).


MW ⋆-product

f ⋆ g (x) =

∞∑

n=0

( i

2

)n 1

n!θρ1σ1 . . . θρnσn

(

∂ρ1. . . ∂ρn

f(x))(

∂σ1. . . ∂σn

g(x))

= f · g +i

2θρσ(∂ρf) · (∂σg) + O(θ2). (2)

Associative, noncommutative; c. conjugation: (f ⋆ g)∗ = g∗ ⋆ f∗.

Special example: xµ ⋆ xν = xµxν +i

2θµν ,

[xµ ⋆, xν ] = iθµν . (3)

The ⋆-product (2) enabled: construction of quantum field theoriesand analysis of their renormalizability properties, construction ofNC Standard Model and the analysis of its phenomenologicalconsequences,. . . ["Munich" group around 2000 and from thenon].


Twist formalism

• Motivation 1: Product (2) can be viewed as coming from an

Abelian twist given by

F = e−i2θρσ∂ρ⊗∂σ (4)

asf ⋆ g = µ

F−1f ⊗ g

= µ

ei2θρσ∂ρ⊗∂σf ⊗ g

= f · g +i

2θρσ(∂ρf) · (∂σg) + O(θ2). (5)

• 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)?


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 Ug ⊗ Ug

-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 = fα ⊗ fα and F−1 = fα ⊗ fα.

["Munich" group around 2005 and from then on].


Action ofF

• F applied to Ug: twisted Hopf algebra UgF

[ta, tb] = ifabctc, ∆F (ta) = F∆(ta)F−1,

ε(ta) = 0, SF = fαS(fα)S(ta)S(fβ)fβ . (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)

= (fαf)(fαg) = f ⋆ g. (8)

• F applied to the symmetry of theory

δclg (φ1φ2) = (δcl

g φ1)φ2 + φ1(δclg φ2)

F ,F−1

→ deformed

Leibniz rule.


• F applied to Ω (exterior algebra of forms): Ω⋆

wedge product: ω1 ∧ ω2 = ω1 ⊗ ω2 − ω2 ⊗ ω1

⇓

⋆-wedge product: ω1 ∧⋆ ω2 = (fαω1) ∧ (fαω2).

• Differential calculus is classical: d : A⋆x → Ω⋆.

d2 = 0, d(f ⋆ g) = df ⋆ g + f ⋆ dg,

df = (∂µf)dxµ = (∂⋆µf) ⋆ dxµ. (9)

• Integral of a maximal form (d1 + d2 = dim(M)) is graded

cyclic:∫

ω1 ∧⋆ ω2 = (−1)d1d2

∫

ω2 ∧⋆ ω1. (10)


Comments I: Deformations by twist

1. Moyal-Weyl twist F = e−i2θµν∂µ⊗∂ν , θµν = −θνµ ∈ 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, [δ⋆ω, ∂ρ] = ω µ

ρ ∂µ, [δ⋆ω, δ⋆

ω′ ] = δ⋆[ω,ω′],

∆(δ⋆ω) = δ⋆

ω ⊗ 1 + 1 ⊗ δ⋆ω +

i

2θρσ

“

ωλρ∂λ ⊗ ∂σ + ∂ρ ⊗ ωλ

σ∂λ

”

.

• θ-deformed gravity: Aschieri et al. (Class. Quant. Grav. 22, 2005 and 23,2006).

[δ⋆ξ , δ⋆

η ] = δ⋆[ξ,η],

∆(δ⋆ξ ) = δ⋆

ξ ⊗ 1 + 1 ⊗ δ⋆ξ −

i

2θρσ

“

δ⋆(∂ρξ) ⊗ ∂σ + ∂ρ ⊗ δ⋆

(∂σξ)

”

+ . . . .

• 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).

[δ⋆α, δ⋆

β ] = δ⋆−i[α,β], α = αata,

∆(δ⋆α) = δ⋆

α ⊗ 1 + 1 ⊗ δ⋆α −

i

2θρσ

“

δ⋆(∂ρα) ⊗ ∂σ + ∂ρ ⊗ δ⋆

(∂σα)

”

+ . . . .


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), Dimitrijevic et al. (JHEP 0712 2007), . . .

F1 = e1

2Cαβ∂α⊗∂β+ 1

2C

αβ∂α

⊗∂β

, F2 = e1

2CαβDα⊗Dβ , . . .

Cαβ = Cβα∈ C, Dα = ∂α + iσm

ααθα∂m

3. Twist with commuting vector fields

F = e−i2

θabXa⊗Xb , Xa = Xµa ∂µ, [Xa, Xb] = 0, θab = const.

• dynamical NC: vector fields Xa are dynamical, global Lorentzsymmetry is preserved: Aschieri et al. (Lett. Math. Phys. 85 2008).

• deformed gravity: cosmological and black hole solutions, coupledto 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),. . .


Example I:κ-Minkowski space-time

Defined by:

[x0, xj ] = iaxj , [xi, xj ] = 0, (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

F = e−i2θabXa⊗Xb = e−

ia2

(∂0⊗xj∂j−xj∂j⊗∂0), (12)

with X1 = ∂0, X2 = xj∂j , [X1,X2] = 0 and θab = aǫab.


Action of the twist (11):

• ⋆-product of functions

f ⋆ g = µF−1 f ⊗ g

= f · g +ia

2xj

(

(∂0f)∂jg − (∂jf)∂0g)

+ O(a2). (13)

• [x0 ⋆, xj ] = iaxj and [xi ⋆, xj ] = 0 .

• Differential calculus

df = (∂⋆µ) ⋆ dxµ, ∂⋆

0 = ∂0, ∂⋆j = e−

i2a∂0∂j ,

f ⋆ dx0 = dx0 ⋆ f, f ⋆ dxj = dxj ⋆ eia∂0f,

dxµ ∧⋆ dxν = dxµ ∧ dxν , d4x = dx0 ∧ · · · ∧ dx3.

• Integral:∫

ω1 ∧⋆ ω2 = (−1)d1d2∫

ω2 ∧⋆ ω1,

with d1 + d2 = 4.


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 ),

with ∗F 0 = 12ǫµναβF 0αβdxµ ∧ dxν and F = dA − A ∧⋆ A.

The obvious choice ∗F = 12ǫµναβFαβ ⋆ dxµ ∧⋆ dxν does not lead

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:

S = −1

4

∫

2F0j ⋆ e−ia∂0X0j + Fij ⋆ e−2ia∂0Xij

⋆ d4x, (14)

with Xnj = Fnj −aAn ⋆Fnj and Xjk = F jk +aAn ⋆F jk. Analysis

of EOM, dispersion relations, . . . .


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?


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:

F1 = e12Cαβ∂α⊗∂β+ 1

2C

αβ∂α⊗∂β

, (15)

F2 = e12CαβDα⊗Dβ , (16)

with Cαβ1,2 = Cβα

1,2 ∈ C, ∂α = ∂∂θα , Dα = ∂α + iσm

ααθα∂m.

An obvious difference: F1 is hermitean and F2 is not hermitean

under the usual c.conjugation.


Twist F1 leads to:

• ⋆-product of superfields

F ⋆ G = µF−1

1 F ⊗ G

= F · G −1

2(−1)|F |Cαβ(∂αF ) · (∂βG)

−1

2(−1)|F |Cαβ(∂αF )(∂βG) + O(C2). (17)

where |F | = 1 if F is odd and |F | = 0 if F is even.

• θα ⋆, θβ = Cαβ , θα⋆, θβ = Cαβ , [xm ⋆, xn] = 0.

• Deformed Leibniz rule ⇒ twisted SUSY transformations

δ⋆ξ (F ⋆ G) = (δξF ) ⋆ G + F ⋆ (δξG) (18)

+i

2Cαβ

(

ξγσmαγ(∂mF ) ⋆ (∂βG) + (∂αF ) ⋆ ξγσm

βγ(∂mG))

−i

2Cαβ

(

ξασmαγεγα(∂mF ) ⋆ (∂βG) + (∂αF ) ⋆ ξασm

αγεγβ(∂mG))

.


• Chirality is broken; if Φ is chiral Φ ⋆ Φ is not chiral!

Project out chiral, antichiral and transverse components ofΦ ⋆ Φ and Φ ⋆ Φ ⋆ Φ using the projectors P1, P2 and PT

P1 =1

16

D2D2

, P2 =

1

16

D2D2

, f(x)

1

g(x) = f(x)

Z

d4y G(x − y)g(y).

• Deformed Wess-Zumino action

S =

∫

d4x

Φ+ ⋆ Φ∣

∣

∣

θθθθ(19)

+(m

2P2

(

Φ ⋆ Φ)

∣

∣

∣

θθ+

λ

3P2

(

Φ ⋆ P2

(

Φ ⋆ Φ)

)∣

∣

∣

θθ+ c.c.

)

• A minimal deformation of the commutative WZ action, good

commutative limit, it is non-local.


• 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.


Twist F2 leads to:

• ⋆-product of superfields

F ⋆ G = µF−1

1 F ⊗ G

= F · G −1

2(−1)|F |Cαβ(DαF ) · (DβG) + O(C2). (20)

where |F | = 1 if F is odd and |F | = 0 if F is even.

• θα ⋆, θβ = Cαβ , θα⋆, θβ = 0,

[xm ⋆, xn] = −Cαβ(σmnε)αβ θθ.

• Since Qα,Dβ = Qα,Dβ = 0 Leibniz rule for SUSYtransformations is undeformed.

δ⋆ξ (F ⋆ G) = (δξF ) ⋆ G + F ⋆ (δξG) (21)

• Chirality is broken again. Method of projectors. . .


• Deformed Wess-Zumino action

S =

∫

d4x

Φ+ ⋆ Φ∣

∣

∣

θθθθ+

[m

2

(

P2(Φ ⋆ Φ)∣

∣

∣

θθ+ 2a1P1(Φ ⋆ Φ)

∣

∣

∣

θθ

)

+λ

3

(

P2(P2(Φ ⋆ Φ) ⋆ Φ)∣

∣

∣

θθ+ 3a2P1(P2(Φ ⋆ Φ) ⋆ Φ)

∣

∣

∣

θθ

+2a3(P1(Φ ⋆ Φ) ⋆ Φ)∣

∣

∣

θθθθ+3a4P1(Φ ⋆ Φ) ⋆ Φ+

∣

∣

∣

θθ

+3a5C2P2(Φ ⋆ Φ) ⋆ Φ+

∣

∣

∣

θθθθ

)

+ c.c.]

. (22)

• A non-minimal deformation of the commutative WZ action,

good commutative limit, it is local.


• 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

divergences appearing in Γ(3)1

-divergences in Γ(4)1 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

renormalization, divergent parts of Γ(4)1 and higher-point

functions are zero.

• Non-minimal deformation and undeformed SUSY render a

renormalizable model. A more general conclusion?


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.


Education

M. Dimitrijevic - Twisted Symmetry and Noncommutative Field Theory