A twisted look on kappa-Minkowski:U(1) gauge theory

Larisa JonkeRudjer Boskovic Institute, Zagreb

Based on:M. Dimitrijevic and L. Jonke, arXiv:1107.3475[hep-th],

M. Dimitrijevic, L. Jonke and L. Moller, JHEP 0509 (2005) 068.

Donji Milanovac, August 2011

Overview

Introduction

Kappa-Minkowski via twist

U(1) gauge theory

Conclusions & Outlook

Introduction

Classical concepts of space and time are expected to break down atthe Planck scale due to the interplay between gravity and quantummechanics.

Q: How to modify space-time structure?

Direct geometrical construction (e.g. triangulation models).

Non-local fundamental observables (strings, loops).

Deform algebra of functions on ’noncommutative space-time’.

I Describe manifold using C ∗ algebra of functions on manifold,deform commutative C ∗ algebra into noncommutativealgebra, and forget about manifold.

Noncommutative geometry ∼ noncommutative algebra.

I Deform Hopf algebra of symmetry Lie algebra.

Noncommutative space-time defined through representations ofdeformed Hopf algebra.

I Use framework of deformation quantization using star-productand (formal) power series expansion.

Noncommutative space-time lost, new kinematics/dynamics ineffective field theory.

Here: κ-Minkowski space-time:

[x0, x j ] =i

κx j , [x i , x j ] = 0.

I Dimensionful deformation of the global Poincare group, theκ-Poincare group [Lukierski, Nowicki, Ruegg, ’92].

I An arena for formulating new physical concepts: DoubleSpecial Relativity [Amelino-Camelia ’02], The principle of relativelocality [Amelino-Camelia, Freidel, Kowalski-Glikman, Smolin, ’11]

Potentially interesting phenomenology.

We are interested in the construction of gauge field theory onκ-Minkowski as a step towards extracting observable consequencesof underlying noncommutative structure.

I Existing results [Dimitrijevic, Jonke, Moller, ’05] consistent, but withambiguities.

I No geometric formulation of gauge theory.

Use twist formalism [Drinfel’d ’85].

Kappa-Minkowski via twist

Deformation of symmetry Lie algebra g by an bidifferentialoperator F acting on symmetry Hopf algebra.

Cannot express κ-Poincare by twist, we choose twist toreproduce κ-Minkowski commutation relations and to obtainhermitean star product.

F = exp

{− i

2θabXa ⊗ Xb

}= exp

{− ia

2(∂0 ⊗ x j∂j − x j∂j ⊗ ∂0)

} Abelian twist, vector fields X1 = ∂0 and X2 = x j∂j commute.

The vector field X2 not in universal enveloping algebra ofPoincare algebra, we enlarge it to get twisted igl(1, 4) [Borowiec,

Pachol, ’09].

I Star-product (a ≡ a0 = 1/κ) :

f ? g = µ{F−1 f ⊗ g} = f · g +ia

2x j((∂0f )∂j g − (∂j f )∂0g

)+O(a2)

I Differential calculus:

df = (∂µf )dxµ = (∂?µf ) ? dxµ

dxµ ∧? dxν = −dxν ∧? dxµ

f ? dx j = dx j ? e ia∂n f

∂?j = e−i2

a∂n∂j , ∂?j (f ? g) = (∂?j f ) ? e−ia∂n g + f ? (∂?j g)

I Integral:∫ω1 ∧? ω2 = (−1)d1+d2

∫ω2 ∧? ω1, d1 + d2 = m + 1

dm+1? x := dx0 ∧? dx1 ∧? . . . dxm = dx0 ∧ dx1 ∧ . . . dxm = dm+1x .

U(1) gauge theoryThe covariant derivative Dψ is defined

Dψ = dψ − iA ? ψ = D?µψ ? dxµ

D?0 = ∂?0ψ − iA0 ? ψ, D?

j = ∂?j ψ − iAj ? e−ia∂0ψ

where the noncommutative connection is

A = Aµ ? dxµ

The transformation law of the covaraint derivative

δ?αDψ = iΛα ? Dψ

defines the transformation law of the noncommutative connection.It is given by

δ?αA = dα + i [Λα ?, A]

or in the components

δ?αA0 = ∂nΛα + i [Λα ?, A0]

δ?αAj = ∂?j Λα + iΛα ? Aj − iAj ? e−ia∂n Λα

The field-strength tensor is a two-form given by

F =1

2Fµν ? dxµ ∧? dxν = dA− iA ∧? A

or in components

F0j = ∂?0Aj − ∂?j A0 − iA0 ? Aj + iAj ? e−ia∂0A0

Fij = ∂?i Aj − ∂?j Ai − iAi ? e−ia∂0Aj + iAj ? e−ia∂0Ai

One can check that field-strength tensor transforms covariantly:

δ?αF = i [Λ ?, F ]

The noncommutative gauge field action

S ∝∫

F ∧? (∗F )

where ∗F is the noncommutative Hodge dual. The obvious guess

∗F =1

2εµναβFαβ ? dxµ ∧? dxν

does not work since it does not transform covariantly. Assume that∗F has the form

∗F :=1

2εµναβXαβ ? dxµ ∧? dxν ,

where Xαβ components are determined demanding

δ?α(∗F ) = i [Λα ?, ∗F ]

Up to first order we obtain

X 0j = F 0j − aA0 ? F 0j ,

X jk = F jk + aA0 ? F jk .

Action

Gauge fields

Sg ∝∫

F ∧? (∗F )

Sg = −1

4

∫ {2F0j ? e−ia∂0X 0j + Fij ? e−2ia∂0X ij

}? d4x .

Fermions

Sm ∝∫ (

(Dψ)B ? ψA − ψB ? (Dψ)A

)∧? (V ∧? V ∧? V γ5)BA,

V = Vµ ? dxµ = V aµγa ? dxµ = δa

µγa ? dxµ = γµdxµ,

After tracing over spinor indices

Sm =1

2

∫ (ψ ? (iγµD?

µ −m)ψ − (iD?µψγ

µ + mψ) ? ψ)? d4x .

Seiberg-Witten map

We construct the SW map relating noncommutative andcommutative degrees of freedom from the consistency relation forgauge transformations:

(δ?αδ?β − δ?βδ?α)ψ(x) = δ?−i [α,β]ψ

and assuming the noncommutative gauge transformations areinduced by commutative ones:

δ?αψ = iΛα ? ψ(x),

δ?αA = dΛα + i [Λα ?, A],

These relations are solved order by order in deformation parametera. The solutions for the fields have free parameters, e.g.

ψ = ψ0 − 1

2C ρσλ xλA0

ρ(∂σψ0) + id1C ρσ

λ xλF 0ρσψ

0 + d2aD00ψ

0.

Expanded action

The action expanded up to first order in a, expanding ?-productand using SW map.

S(1)g = −1

4

∫d4x{

F 0µνF 0µν − 1

2C ρσλ xλF 0µνF 0

µνF 0ρσ +

+2C ρσλ xλF 0µνF 0

µρF 0νσ

}S

(1)m =

1

2

∫d4x{ψ0(

iγµD0µψ

0 −mψ0 +a

2γj D0

0 D0j ψ

0 +

+i

2C ρσλ xλγµF 0

ρµ(Dσψ0))−

−(

iDµψ0γµ + mψ0 − a

2D0Djψ

0γj +

i

2C ρσλ xλDσψ

0γµF 0

ρµ

)ψ0}

Equation of motionI fermions:

iγµ(D0µ −m)ψ0 +

a

2γj D0

0 D0j ψ

0 +i

2C ρσλ xλγµF 0

ρµ(Dσψ0) = 0

I gauge field:

∂µF 0αµ +a

4δα0 F 0µνF 0

µν + 2aF 0αµF 00µ − C ρσ

λ xλ(∂µ(F 0µ

ρ F 0ασ )+

+ F 0µσ(∂ρF 0µα)

)= ψ0γαψ0 +

i

2C ρσλ xλDσψ

0γα(Dρψ)0 +

+ia(ψ0γα(D0ψ)0 − D0ψ

0γαψ0

)+

ia

2δα0(ψ0γ0(D0ψ)0 − D0ψ

0γ0ψ0

)I conserved U(1) current up to first order in a

j0 = ψ0γ0ψ0 − a

2x j F 0

jσψ0γσψ0 − ia

2ψ0γj D0

j ψ0,

jk = ψ0γkψ0 +a

2xk F 0

0σψ0γσψ0 +

ia

2D0

0ψ0γkψ0.

Conclusions

In the twist formalism we have

I Unique four-dimensional differential calculus.

I Integral with trace property, no need to introduce additionalmeasure function.

Constructed the action for the gauge and matter fields in ageometric way, as an integral of a maximal form. No ambiguities coming from the Seiberg-Witten map in theaction expanded up to the first order in the deformation parameter.

In the twist formalism we do not have κ-Poincare symmetry:

I Use five-dimensional differential calculus [Sitarz ’95].

I Use nonassociative differential algebra [Beggs, Majid ’06].

Outlook

Generically, the noncommutative geometrical structure preventsdecoupling of translation and gauge symmetries. Here we see it inconstruction of the Hodge dual, a relation which introduces(geo)metric degrees of freedom in U(1) gauge theory.

In the framework of Yang-Mills type matrix models [Steinacker ’10],U(1) part of general U(N) gauge group is interpreted as inducedgravity coupling to the remaining SU(N).

I Check models with larger gauge group.

I Geometric interpretation of x-dependent terms in action.