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

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

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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’.

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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 ofdeformed Hopf algebra. Use framework of deformation quantization using star-product and (formal) power series expansion. Noncommutative space-time lost, new kinematics/dynamics ineffective field theory.

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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.

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

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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].

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Star-product (a ≡ a0 = 1/κ) : ia jf 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 + 1dm+1 x := dx 0 ∧ dx 1 ∧ . . . dx m = dx 0 ∧ dx 1 ∧ . . . dx m = dm+1 x.

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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

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The field-strength tensor is a two-form given by 1 F = Fµν dx µ ∧ dx ν = dA − iA ∧ A 2or 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 AiOne can check that field-strength tensor transforms covariantly: δα F = i[Λ , F ]

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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 2does not work since it does not transform covariantly. Assume that∗F has the form 1 ∗F := µναβ X αβ dx µ ∧ dx ν , 2where 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 .

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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

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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

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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

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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

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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].

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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.