Gravity in three dimensions as a noncommutative gauge theory George Manolakos ESI workshop “MM for NCG and string theory” July 10, 2018 Joint work with: A. Chatzistavrakidis, L. Jonke, D. Jurman, P. Manousselis, G. Zoupanos NTUA [arXiv:1802.07550]
Table of contents 1. Gravity as gauge theory in 1+2 (and 1+3) dimensions 2. How to build gauge theories on noncommutative (nc) spaces 3. 3-d noncommutative spaces based on SU(2) and SU(1,1) 4. Gravity as gauge theory on 3-d nc spaces (Euclidean - Lorentzian)
Gravity in three dimensions as a gauge theory The algebra Witten ’88 ◮ 3-d Gravity: gauge theory of iso (1 , 2) (Poincar´ e - isometry of M 3 ) ◮ Presence of Λ: dS or AdS algebras, i.e. so (1 , 3) , so (2 , 2) ◮ Corresponding generators: P a , J ab , a = 1 , 2 , 3 (translations, LT) ◮ Satisfy the following CRs: [ J ab , J cd ] = 4 η [ a [ c J d ] b ] , [ P a , J bc ] = 2 η a [ b P c ] , [ P a , P b ] = Λ J ab ◮ CRs valid in arbitrary dim; particularly in 3-d: [ J a , J b ] = ǫ abc J c , [ P a , J b ] = ǫ abc P c , [ P a , P b ] = Λ ǫ abc J c ◮ After the redefinition: J a = 1 2 ǫ abc J bc
The gauging procedure ◮ Intro of a gauge field for each generator: e a µ , ω a µ (transl, LT) ◮ The Lie-valued 1-form gauge connection is: A µ = e a µ ( x ) P a + ω a µ ( x ) J a ◮ Transforms in the adjoint rep, according to the rule: δA µ = ∂ µ ǫ + [ A µ , ǫ ] ◮ The gauge transformation parameter is expanded as: ǫ = ξ a ( x ) P a + λ a ( x ) J a ◮ Combining the above → transformations of the fields: µ = ∂ µ ξ a − ǫ abc ( ξ b ω µc + λ b e µc ) δe a µ = ∂ µ λ a − ǫ abc ( λ b ω µc + Λ ξ b e µc ) δω a
Curvatures and action ◮ Curvatures of the fields are given by: R µν ( A ) = 2 ∂ [ µ A ν ] + [ A µ , A ν ] ◮ Tensor R µν is also Lie-valued: a a R µν ( A ) = T µν P a + R µν J a ◮ Combining the above → curvatures of the fields: a ν ] + 2 ǫ abc ω [ µb e ν ] c a T = 2 ∂ [ µ e µν a ν ] + ǫ abc ( ω µb ω νc + Λ e µb e νc ) a = 2 ∂ [ µ ω R µν ◮ The Chern-Simons action functional of the Poincar´ e, dS or AdS algebra is found to be identical to the 3-d E-H action: � 1 ǫ µνρ ( e a µ ( ∂ ν ω ρa − ∂ ρ ω νa ) + ǫ abc e a µ ω b ν ω c ρ + Λ 3 ǫ abc e c µ e b ν e c S CS = ρ ) ≡ S EH 16 πG 3-d gravity is a Chern-Simons gauge theory.
Remarks on 4-d gravity Utiyama ’56 , Kibble ’61 MacDowell-Mansouri ’77 Kibble-Stelle ’85 ◮ Vielbein formulation of GR: Gauging Poincar´ e algebra iso (1 , 3) ◮ Comprises ten generators: P a , J ab , a = 1 , . . . 4 (transl, LT) ◮ Satisfy the aforementioned CRs (for Λ = 0) ◮ Gauging in the same way leading to field transformations ◮ Curvatures are obtained accordingly ◮ Dynamics follow from the E-H action: � S EH 4 = 1 d 4 xǫ µνρσ ǫ abcd e a µ e b cd ν R ρσ 2 ◮ Form of Einstein action: A 2 ( dA + A 2 ) ◮ Such action does not exist in gauge theories ◮ In that sense, gravity cannot be considered as gauge theory.
Gauge theories on noncommutative spaces ◮ Employ the nc type of matrix geometries Ishibashi-Kawai-Kitazawa-Tsuchiya ’97 ◮ Operators X µ ∈ A satisfy the CR: [ X µ , X ν ] = iθ µν , θ µν arbitrary ◮ Lie-type nc: [ X µ , X ν ] = iC ρ µν X ρ ◮ Natural intro of nc gauge theories through covariant nc coordinates : X µ = X µ + A µ Madore-Schraml-Schupp-Wess ’00 ◮ Obeys a covariant gauge transformation rule: δ X µ = i [ ǫ, X µ ] ◮ A µ transforms in analogy with the gauge connection: δA µ = − i [ X µ , ǫ ] + i [ ǫ, A µ ] , ( ǫ - the gauge parameter) ◮ Definition of a (Lie-type) nc covariant field strength tensor : ρ F µν = [ X µ , X ν ] − iC µν X ρ
Non-Abelian case ◮ Gauge theory could be Abelian or non-Abelian: ◮ Abelian if ǫ is a function in A ◮ Non-Abelian if ǫ is matrix valued (Mat( A )) ⊲ In non-Abelian case, where are the gauge fields valued? ◮ Let us consider the CR of two elements of an algebra: [ ǫ, A ] = [ ǫ A T A , A B T B ] = 1 2 { ǫ A , A B } [ T A , T B ]+1 2[ ǫ A , A B ] { T A , T B } ◮ Not possible to restrict to a matrix algebra: last term neither vanishes in nc nor is an algebra element ◮ There are two options to overpass the difficulty: ◮ Consider the universal enveloping algebra ◮ Extend the generators and/or fix the rep so that the anticommutators close ⊲ We employ the second option
3-d fuzzy spaces based on SU(2) and SU(1,1) The Euclidean case ◮ Euclidean case: 3-d fuzzy space based on SU (2) ◮ Fuzzy sphere, S 2 F : Matrix approximation of ordinary sphere, S 2 Hoppe ’82, Madore ’92 For higher-dim S F see: Kimura ’02, Dolan - O’Connor ’03, Sperling - Steinacker ’17 ◮ S 2 defined by coordinates of R 3 modulo � 3 a =1 x a x a = r 2 ◮ S 2 F defined by three rescaled angular momentum operators, X i = λJ i , J i the Lie algebra generators in a UIR of SU (2). The X i s satisfy: � 3 X i X i = λ 2 j ( j + 1) := r 2 , λ ∈ R , 2 j ∈ N [ X i , X j ] = iλǫ ijk X k , i =1 ◮ Allowing X i to live in reducible rep: obtain the nc R 3 λ , viewed as direct sum F with all possible radii (determined by 2 j ) - a discrete foliation of R 3 by of S 2 multiple S 2 Hammou-Lagraa-Sheikh Jabbari ’02 F Vitale-Wallet ’13, Vitale ’14
The fuzzy space R 3 λ λ : Foliation of R 3 by R 3 fuzzy spheres (onion-like Matrix (coordinate) of R 3 λ as a block diagonal form - construction) each block is a fuzzy sphere
The Lorentzian case ◮ In analogy: Lorentzian case: 3-d fuzzy space based on SU (1 , 1) Grosse - Pre ˇ s najder ’93 Jurman-Steinacker ’14 ◮ Fuzzy hyperboloids, dS 2 F , defined by three rescaled operators, X i = λJ i , (in appropriate irreps) satisfying: � η ij X i X j = λ 2 j ( j − 1) , k [ X i , X j ] = iλC ij X k , i,j ◮ where C k are the structure constants of su (1 , 1) ij ◮ Difference to previous case: Non-compact group, i.e. no finite-dim UIRs but infinite-dim ◮ Again, letting X i live in (infinite-dim) reducible reps: Block diagonal form - each block being a dS 2 F ◮ 3-d Minkowski spacetime foliated with leaves being dS 2 F of different radii
Gravity as gauge theory on 3-d fuzzy spaces The Lorentzian case Aschieri-Castellani ’09 ◮ Consideration of the foliated M 3 with Λ > 0 ◮ Natural symmetry of the space: SO (1 , 3) ( SO (4) for the Eucl.) Kov´ a ˇ c ik - Presnajder ’13 ◮ Consider the corresponding spin group: SO (1 , 3) ∼ = Spin (1 , 3) = SL (2 , C ) ◮ Anticommutators do not close → Fix at spinor rep generated by: � AB = 1 2 γ AB = 1 4[ γ A , γ B ] , A = 1 , . . . 4 ◮ Satisfying the CRs and aCRs: [ γ AB , γ CD ] = 8 η [ A [ C γ D ] B ] , { γ AB , γ CD } = 4 η C [ B η A ] D 1 l+2 iǫ ABCD γ 5 ◮ Inclusion of γ 5 and identity in the algebra → extension of SL (2 , C ) to GL (2 , C ) with set of generators: { γ AB , γ 5 , i 1 l }
SO(3) notation γ a ≡ ǫ abc γ bc , with a = 1 , 2 , 3 ◮ In SO (3) notation: γ a 4 ≡ γ a and ˜ ◮ The CRs and aCRs are now written: γ b ] = − 4 ǫ abc γ c , [ γ a , γ b ] = ǫ abc ˜ γ c , [ γ 5 , γ AB ] = 0 γ a , ˜ γ b ] = − 4 ǫ abc ˜ [˜ γ c , [ γ a , ˜ γ a , ˜ γ b } = − 8 η ab 1 γ b } = 4 iδ b { ˜ l , { γ a , ˜ a γ 5 , { γ a , γ b } = 2 η ab 1 l , { γ 5 , γ a } = i ˜ γ 5 , γ a } = − 4 iγ a γ a , { ˜ ◮ Proceed with the gauging of GL (2 , C ) ◮ Determine the covariant coordinate: X µ = X µ + A µ µ ( X a ) ⊗ T i the gl (2 , C )-valued gauge connection A µ = A i ◮ Gauge connection expands on the generators as: l + ˜ A µ = e a µ ( X ) ⊗ γ a + ω a µ ( X ) ⊗ ˜ γ a + A µ ( X ) ⊗ i 1 A µ ( X ) ⊗ γ 5 See also: Nair ’03,’06, Abe - Nair ’03 ◮ Gauge parameter, ǫ , expands similarly: ǫ = ξ a ( X ) ⊗ γ a + λ a ( X ) ⊗ ˜ γ a + ǫ 0 ( X ) ⊗ i 1 l + ˜ ǫ 0 ( X ) ⊗ γ 5
Kinematics ◮ Covariant transf rule: δ X µ = [ ǫ, X µ ] → transf of the gauge fields: µ = − i [ X µ + A µ , ξ a ] − 2 { ξ b , ω µc } ǫ abc − 2 { λ b , e µc } ǫ abc + i [ ǫ 0 , e a µ ] − 2 i [ λ a , ˜ δe a A µ ] − 2 i [˜ ǫ 0 , ω a µ ] 2 { ξ b , e µc } ǫ abc − 2 { λ b , ω µc } ǫ abc + i [ ǫ 0 , ω a µ = − i [ X µ + A µ , λ a ] + 1 µ ] + i 2 [ ξ a , ˜ A µ ] + i δω a 2 [˜ ǫ 0 , e a µ ] ǫ 0 , ˜ δA µ = − i [ X µ + A µ , ǫ 0 ] − i [ ξ a , e a µ ] + 4 i [ λ a , ω a µ ] − i [˜ A µ ] µ ] + i [ ǫ 0 , ˜ δ � ǫ 0 ] + 2 i [ ξ a , ω a µ ] + 2 i [ λ a , e a A µ = − i [ X µ + A µ , ˜ A µ ] ◮ Commutative limit: Y-M and gravity fields disentangle and inner derivation becomes [ X µ , f ] → − i∂ µ f : µ = − ∂ µ ξ a − 4 ξ b ω µc ǫ abc − 4 λ b e µc ǫ abc δe a µ = − ∂ µ λ a + ξ b e µc ǫ abc − 4 λ b ω µc ǫ abc δω a γ a → − 4 J a , 4 λ a → λ a , ◮ After the redefinitions: γ a → 2 i Λ P a , ˜ √ √ ξ a 2 i Λ → − ξ a , e a 2 i e a Λ µ , ω a µ → − 1 4 ω a µ → µ → 3-d gravity √
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