Noncommutative OSp ( 4 | 2 ) SUGRA canin 1 Dragoljub Go 1Faculty of - PowerPoint PPT Presentation

Noncommutative OSp ( 4 | 2 ) SUGRA canin 1 Dragoljub Goˇ 1Faculty of Physics, University of Belgrade, Studentski Trg 12-16, 11000 Belgrade, Serbia D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of N = 2 AdS 4 SUGRA” arXiv:1909.01069 . 13 September 2019, MPHYS10 Belgrade D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14

Deformation quantization 1 Deformation quantization (phase space quantum mechanics). Classical system ( M , ω, H ) is deformed by imposing noncommutative (NC) geometry on its phase space; ⋆ -product deformation of commutative algebra C ∞ ( M ) . 2 NC Field Theory - field theory on NC-deformed space-time. Introduce an abstract algebra of coordinates x µ , ˆ x ν ] = iC µν (ˆ [ˆ x ) . Canonical (or θ -constant) deformation , x ν ] = i θ µν ∼ Λ 2 x µ , ˆ [ˆ NC , with constant deformation parameters θ µν = − θ νµ . For canonical noncommutativity , we use the Moyal ⋆ -product, i 2 θ µν ∂ ∂ ∂ y ν f ( x ) g ( y ) | y → x . (ˆ ∂ x µ f ⋆ ˆ g )( x ) = e The leading term is the commutative point-wise multiplication, and the higher order terms represent (non-classical) NC corrections. D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14

NC gauge field theory C Let { T A } satisfy some Lie algebra relations [ T A , T B ] = if AB T C . Closure rule holds [ δ ǫ 1 , δ ǫ 2 ] = δ − i [ ǫ 1 ,ǫ 2 ] . If NC gauge parameter ˆ Λ is supposed to be Lie algebra-valued, ˆ Λ( x ) = ˆ Λ A ( x ) T A , then, for some generic NC field ˆ Ψ from the fund. rep. 2 ]ˆ Ψ = (ˆ Λ 1 ⋆ ˆ Λ 2 − ˆ Λ 2 ⋆ ˆ Λ 1 ) ⋆ ˆ [ δ ⋆ , δ ⋆ ⋆ Ψ 1 � � = 1 [ˆ Λ A , ˆ Λ B 2 ] { T A , T B } + { ˆ Λ A , ˆ Λ B ⋆ ˆ Ψ = i ˆ Λ 3 ⋆ ˆ Ψ = δ ⋆ 3 ˆ ⋆ ⋆ 2 } [ T A , T B ] Ψ . 1 1 2 NC closure rule [ δ ⋆ , δ ⋆ Λ 2 ] = δ ⋆ ⋆ Λ 2 ] . ˆ ˆ − i [ˆ Λ 1 ⋆ , ˆ Λ 1 consistently generalizes its commutative counterpart. 1 Universal enveloping algebra (UEA) approach; infinite number of dofs. 2 Seiberg-Witten (SW) map; induced NC transformations, Λ ˆ V µ = ˆ V µ ( V µ + δ ǫ V µ ) − ˆ δ ⋆ V µ ( V µ ) . SW map between NC and classical fields : Λ ǫ = ǫ − 1 ˆ 4 θ ρσ { V ρ , ∂ σ ǫ } + O ( θ 2 ) , V µ = V µ − 1 ˆ 4 θ ρσ { V ρ , ∂ σ V µ + F σµ } + O ( θ 2 ) . D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14

AdS algebra AdS algebra so ( 2 , 3 ) has ten generators M AB = − M BA ( A , B = 0 , 1 , 2 , 3 , 5) [ M AB , M CD ] = i ( η AD M BC + η BC M AD − η AC M BD − η BD M AC ) η AB = (+ , − , − , − , +) . Split the generators into six AdS rotations M ab and four AdS translations M a 5 ( a , b = 0 , 1 , 2 , 3) to obtain [ M a 5 , M b 5 ] = − iM ab [ M ab , M c 5 ] = i ( η bc M a 5 − η ac M b 5 ) [ M ab , M cd ] = i ( η ad M bc + η bc M ad − η ac M bd − η bd M ac ) Introduce M ab := M ab and P a := l − 1 M a 5 = α M a 5 , where l is AdS radius and α = l − 1 [ P a , P b ] = − i α 2 M ab [ M ab , P c ] = i ( η bc P a − η ac P b ) , [ M ab , M cd ] = i ( η ad M bc + η bc M ad − η ac M bd − η bd M ac ) In the limit α → 0 (or l → ∞ ), AdS algebra → Poincaré algebra (WI contraction). M AB = i 4 [Γ A , Γ B ] , { Γ A , Γ B } = 2 η AB , Γ A = ( i γ a γ 5 , γ 5 ) In this particular representation, M ab = i 4 [ γ a , γ b ] = 1 2 σ ab and M a 5 = − 1 2 γ a . D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14

AdS gauge theory of gravity AdS gauge field ω µ = 1 M AB = 1 σ ab − 1 2 ω AB 4 ω ab 2 ω a 5 γ a µ µ µ AdS field strength � � σ ab γ a ab − ( ω a 5 µ ω b 5 − ω b 5 µ ω a 5 a 5 F µν = ∂ µ ω ν − ∂ ν ω µ − i [ ω µ , ω ν ] = R ν ) − F µν ν µν 4 2 ab = ∂ µ ω ab − ∂ ν ω ab + ω a µ c ω cb − ω a ν c ω cb R µν ν µ ν µ a 5 = D L µ ω a 5 − D L ν ω a 5 F µν ν µ Gauge parameter ǫ = 1 2 ǫ AB M AB = ∂ µ ǫ ab − ǫ a δ ǫ ω ab c ω cb + ǫ b c ω ca − ǫ a 5 ω 5 b + ǫ b 5 ω 5 a µ µ µ µ µ = ∂ µ ǫ a 5 − ǫ a δ ǫ ω a 5 c ω c 5 + ǫ 5 c ω ca µ µ µ ab = − ǫ ac F µν c + ǫ bc F b µν c − ǫ a 5 F a µν 5 + ǫ b 5 F b a δ ǫ F µν µν 5 a 5 = − ǫ ac F µν c + ǫ 5 c F 5 a δ ǫ F µν µν c Set ǫ a 5 = 0 and identify ω ab with spin-connection and ω a 5 = e a µ / l . µ µ D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14

AdS gravity action Introduce an auxiliary field φ = φ A Γ A ; it is a space-time scalar and internal space 5-vector transforming in the adjoint representation of SO ( 2 , 3 ) , that is δ ǫ φ = i [ ǫ, φ ] . � il d 4 x ε µνρσ F µν F ρσ φ S AdS = 64 π G N Auxiliary field is constrained by φ 2 = η AB φ A φ B = l 2 . To break SO ( 2 , 3 ) to SO ( 1 , 3 ) set φ a = 0 and φ 5 = l (physical gauge) φ | g.f. = l γ 5 . � � � � � + l 2 1 R − 6 d 4 x mn rs 16 ǫ µνρσ R S AdS | g.f. = − e R ρσ ǫ mnrs µν l 2 16 π G N Cosmological constant Λ = − 3 / l 2 vanishes under WI contraction. D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14

SO ( 2 , 3 ) ⋆ model of pure NC gravity 1 NC deformation of GR (NC Einstein-Hilbert action ) : � il d 4 x ε µνρσ ˆ F µν ⋆ ˆ F ρσ ⋆ ˆ S NC = φ . 64 π G N After SW expansion and symmetry breaking : � �� � R + θ αβ θ γδ � 7 � 1 2 l 4 R αβγδ − 15 ρ d 4 x S NC | g.f. = − − g 16 l 4 T αβ T γδρ + ... . 16 π G N 2 NC field equations; deformation of Minkowski space ; interpretation of θ -constant noncommutativity; Fermi inertial coordinates g 00 = 1 − R 0 m 0 n x m x n , g 0 i = − 2 g ij = − δ ij − 1 3 R 0 min x m x n , 3 R imjn x m x n . c ´ M. Dimitrijevi´ Ciri´ c, B. Nikoli´ c and V. Radovanovi´ c, NC SO ( 2 , 3 ) ⋆ gravity : noncommutativity as a source of curvature and torsion , Phys. Rev. D 96 , 064029 (2017) D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14

OSp(4|1) SUGRA 1 Orthosymplectic supergroup OSp ( 4 | 1 ) has 14 generators - 10 AdS generators M AB and 4 fermionic generators ˆ ˆ Q α comprising a single Majorana spinor. Bosonic sector SO ( 2 , 3 ) ∼ Sp ( 4 ) . 2 Supermatrix for the OSp ( 4 | 1 ) gauge field Ω µ is given by � √ αψ µ � M AB + √ α ¯ Ω µ = 1 ω µ ˆ µ ˆ 2 ω AB ψ α Q α = √ α ¯ . µ ψ µ 0 3 Action with OSp ( 4 | 1 ) gauge symmetry : � il d 4 x ε µνρσ F µν ( I 5 × 5 − 2 l 2 Φ 2 ) F ρσ Φ . 1 S 41 = 32 π G N Auxiliary field � � � l γ 5 � 1 4 π + i φ a γ a γ 5 + φ 5 γ 5 λ 0 Φ = = | g.f. . − ¯ 0 0 λ π In the physical gauge, the action exactly reduces to N = 1 AdS 4 SUGRA action � � � � � S 41 | g.f. = − 1 � R ( e , ω ) − 6 α 2 � ρ + i α d 4 x + 2 ε µνρσ ( ¯ D L e ψ µ γ 5 γ ν 2 γ ρ ψ σ ) . 2 κ 2 4 The leading non-vanishing NC correction is quadratic in θ µν . D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14

OSp(4|2) SUGRA Orthosymplectic group OSp ( 4 | 2 ) has 19 generators - 10 AdS generators ˆ M AB , 8 fermionic generators ˆ Q I α ( α = 1 , 2 , 3 , 4 ; I = 1 , 2 ) comprising a pair of Majorana spinors, and an additional bosonic generator ˆ T . √ αψ 1 √ αψ 2   ω µ ω µ + √ α ¯ µ µ √ α ¯ µ ˆ Q α + α A µ ˆ ψ α  ψ 1  . Ω µ = ˆ T = 0 i α A µ µ √ α ¯ ψ 2 − i α A µ 0 µ c ´ M. Dimitrijevi´ Ciri´ c, D. Goˇ canin, N. Konjik and V. Radovanovi´ c, “Noncommutative Electrodynamics from SO ( 2 , 3 ) ⋆ Model of Noncommutative Gravity”, Eur. Phys. J. C 78 (2018) no.7, 548. Majorana spinors, ψ 1 µ and ψ 2 µ , can be combined into an SO ( 2 ) doublet, � ψ 1 � µ Ψ µ = . ψ 2 µ Charged Dirac vector-spinors ψ ± µ , related by C -conjugation, ψ − µ = C ¯ µ = ψ 1 µ ± i ψ 2 ψ + T µ . � � � il I 6 × 6 − 1 d 4 x ε µνρσ F µν 2 l 2 Φ 2 S 42 = F ρσ Φ . 32 π G N D. Goˇ canin & V. Radovanovi´ c, “Canonical Deformation of Dragoljub Goˇ canin Noncommutative OSp ( 4 | 2 ) SUGRA / 14