The SEENET-MTP Workshop BW2011 Particle Physics from TeV to Plank Scale 28 August – 1 September 2011, Donji Milanovac, Serbia

1. A twisted look on kappa-Minkowski:
U(1) gauge theory
Larisa Jonke
Rudjer Boˇkovi´ Institute, Zagreb
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Based on:
M. Dimitrijevi´ and L. Jonke, arXiv:1107.3475[hep-th],
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M. Dimitrijevi´, L. Jonke and L. M¨ller, JHEP 0509 (2005) 068.
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Donji Milanovac, August 2011

2. Overview
Introduction
Kappa-Minkowski via twist
U(1) gauge theory
Conclusions & Outlook

3. Introduction
Classical concepts of space and time are expected to break down at
the Planck scale due to the interplay between gravity and quantum
mechanics.
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’.

4. Describe manifold using C ∗ algebra of functions on manifold,
deform commutative C ∗ algebra into noncommutative
algebra, and forget about manifold.
Noncommutative geometry ∼ noncommutative algebra.
Deform Hopf algebra of symmetry Lie algebra.
Noncommutative space-time defined through representations of
deformed Hopf algebra.
Use framework of deformation quantization using star-product
and (formal) power series expansion.
Noncommutative space-time lost, new kinematics/dynamics in
effective field theory.

5. Here: κ-Minkowski space-time:
i j
[ˆ0 , x j ] =
x ˆ x , [ˆi , x j ] = 0.
ˆ x ˆ
κ
Dimensionful deformation of the global Poincar´ group, the
e
κ-Poincar´ group [Lukierski, Nowicki, Ruegg, ’92].
e
An arena for formulating new physical concepts: Double
Special Relativity [Amelino-Camelia ’02], The principle of relative
locality [Amelino-Camelia, Freidel, Kowalski-Glikman, Smolin, ’11]
Potentially interesting phenomenology.

6. We are interested in the construction of gauge field theory on
κ-Minkowski as a step towards extracting observable consequences
of underlying noncommutative structure.
Existing results [Dimitrijevi´, Jonke, M¨ller, ’05] consistent, but with
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ambiguities.
No geometric formulation of gauge theory.
Use twist formalism [Drinfel’d ’85].

7. Kappa-Minkowski via twist
Deformation of symmetry Lie algebra g by an bidifferential
operator F acting on symmetry Hopf algebra.
Cannot express κ-Poincar´ by twist, we choose twist to
e
reproduce κ-Minkowski commutation relations and to obtain
hermitean star product.
i ia
F = exp − θab Xa ⊗ Xb = exp − (∂0 ⊗ x j ∂j − x j ∂j ⊗ ∂0 )
2 2
Abelian twist, vector fields X1 = ∂0 and X2 = x j ∂j commute.
The vector field X2 not in universal enveloping algebra of
Poincar´ algebra, we enlarge it to get twisted igl(1, 4) [Borowiec,
e
Pachol, ’09].

8. Star-product (a ≡ a0 = 1/κ) :
ia j
f g = µ{F −1 f ⊗ g } = f · g + x (∂0 f )∂j g − (∂j f )∂0 g + O(a2 )
2
Differential calculus:
df = (∂µ f )dx µ = (∂µ f ) dx µ
dx µ ∧ dx ν = −dx ν ∧ dx µ
f dx j = dx j e ia∂n f
i
∂j = e − 2 a∂n ∂j , ∂j (f g ) = (∂j f ) e −ia∂n g + f (∂j g )
Integral:
ω1 ∧ ω2 = (−1)d1 +d2 ω2 ∧ ω1 , d1 + d2 = m + 1
dm+1 x := dx 0 ∧ dx 1 ∧ . . . dx m = dx 0 ∧ dx 1 ∧ . . . dx m = dm+1 x.

9. U(1) gauge theory
The covariant derivative Dψ is defined
Dψ = dψ − iA ψ = Dµ ψ dx µ
D0 = ∂0 ψ − iA0 ψ, Dj = ∂j ψ − iAj e −ia∂0 ψ
where the noncommutative connection is
A = Aµ dx µ
The transformation law of the covaraint derivative
δα Dψ = iΛα Dψ
defines the transformation law of the noncommutative connection.
It is given by
δα A = dα + i[Λα , A]
or in the components
δα A0 = ∂n Λα + i[Λα , A0 ]
δα Aj = ∂j Λα + iΛα Aj − iAj e −ia∂n Λα

10. The field-strength tensor is a two-form given by
1
F = Fµν dx µ ∧ dx ν = dA − iA ∧ A
2
or in components
F0j = ∂0 Aj − ∂j A0 − iA0 Aj + iAj e −ia∂0 A0
Fij = ∂i Aj − ∂j Ai − iAi e −ia∂0 Aj + iAj e −ia∂0 Ai
One can check that field-strength tensor transforms covariantly:
δα F = i[Λ , F ]

11. The noncommutative gauge field action
S∝ F ∧ (∗F )
where ∗F is the noncommutative Hodge dual. The obvious guess
1 αβ
∗F = µναβ F dx µ ∧ dx ν
2
does not work since it does not transform covariantly. Assume that
∗F has the form
1
∗F := µναβ X αβ dx µ ∧ dx ν ,
2
where X αβ components are determined demanding
δα (∗F ) = i[Λα , ∗F ]
Up to first order we obtain
X 0j = F 0j − aA0 F 0j ,
X jk = F jk + aA0 F jk .

12. Action
Gauge fields
Sg ∝ F ∧ (∗F )
1
Sg = − 2F0j e −ia∂0 X 0j + Fij e −2ia∂0 X ij d4 x.
4
Fermions
Sm ∝ (Dψ)B ¯
ψA − ψB (Dψ)A ∧ (V ∧ V ∧ V γ5 )BA ,
V = Vµ dx µ = Vµ γa dx µ = δµ γa dx µ = γµ dx µ ,
a a
After tracing over spinor indices
1 ¯ ¯
Sm = ψ (iγ µ Dµ − m)ψ − (iDµ ψγ µ + mψ) ψ d4 x.
2

13. Seiberg-Witten map
We construct the SW map relating noncommutative and
commutative degrees of freedom from the consistency relation for
gauge transformations:
(δα δβ − δβ δα )ψ(x) = δ−i[α,β] ψ
and assuming the noncommutative gauge transformations are
induced by commutative ones:
δα ψ = iΛα ψ(x),
δα A = dΛα + i[Λα , A],
These relations are solved order by order in deformation parameter
a. The solutions for the fields have free parameters, e.g.
1 ρσ ρσ
ψ = ψ 0 − Cλ x λ A0 (∂σ ψ 0 ) + id1 Cλ x λ Fρσ ψ 0 + d2 aD0 ψ 0 .
ρ
0 0
2

14. Expanded action
The action expanded up to first order in a, expanding -product
and using SW map.
(1) 1 1 ρσ
Sg = − d4 x Fµν F 0µν − Cλ x λ F 0µν Fµν Fρσ +
0 0 0
4 2
ρσ
+2Cλ x λ F 0µν Fµρ Fνσ
0 0
(1) 1 ¯ a
Sm = d4 x ψ 0 iγ µ Dµ ψ 0 − mψ 0 + γ j D0 Dj0 ψ 0 +
0 0
2 2
i ρσ
+ Cλ x λ γ µ Fρµ (Dσ ψ 0 ) −
0
2
0 ¯ a 0 i ρσ 0
− iDµ ψ γ µ + mψ 0 − D0 Dj ψ γ j + Cλ x λ Dσ ψ γ µ Fρµ ψ 0
0
2 2

15. Equation of motion
fermions:
a i ρσ
iγ µ (Dµ − m)ψ 0 + γ j D0 Dj0 ψ 0 + Cλ x λ γ µ Fρµ (Dσ ψ 0 ) = 0
0 0 0
2 2
gauge field:
a α ρσ
∂µ F 0αµ + δ0 F 0µν Fµν + 2aF 0αµ F0µ − Cλ x λ ∂µ (Fρ Fσ )+
0 0 0µ 0α
4
¯ i ρσ 0
+ Fµσ (∂ρ F 0µα ) = ψ 0 γ α ψ 0 + Cλ x λ Dσ ψ γ α (Dρ ψ)0 +
0
2
¯ 0 ia α ¯ 0
+ia ψ 0 γ α (D0 ψ)0 − D0 ψ γ α ψ 0 + δ0 ψ 0 γ 0 (D0 ψ)0 − D0 ψ γ 0 ψ 0
2
conserved U(1) current up to first order in a
¯ a 0 ¯ ia ¯
j 0 = ψ 0 γ 0 ψ 0 − x j Fjσ ψ 0 γ σ ψ 0 − ψ 0 γ j Dj0 ψ 0 ,
2 2
k
j =ψ ¯0 γ k ψ 0 + a x k F 0 ψ 0 γ σ ψ 0 + ia D 0 ψ 0 γ k ψ 0 .
¯
0σ
2 2 0

16. Conclusions
In the twist formalism we have
Unique four-dimensional differential calculus.
Integral with trace property, no need to introduce additional
measure function.
Constructed the action for the gauge and matter fields in a
geometric way, as an integral of a maximal form.
No ambiguities coming from the Seiberg-Witten map in the
action expanded up to the first order in the deformation parameter.
In the twist formalism we do not have κ-Poincar´ symmetry:
e
Use five-dimensional differential calculus [Sitarz ’95].
Use nonassociative differential algebra [Beggs, Majid ’06].

17. Outlook
Generically, the noncommutative geometrical structure prevents
decoupling of translation and gauge symmetries. Here we see it in
construction 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 induced
gravity coupling to the remaining SU(N).
Check models with larger gauge group.
Geometric interpretation of x-dependent terms in action.