Quantized cosmological spacetimes and higher spin in the IKKT model Harold Steinacker Department of Physics, University of Vienna ESI Vienna, july 2018 H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n Motivation Matrix Models ... natural framework for fundamental theory pre-geometric, constructive dynamical “quantum” (NC) spaces, gauge theory stringy features max. SUSY → inherit good behavior of critical string (UV) avoid string compactifications → need different mechanism for gravity & chirality IKKT: allows to describe “beginning of time”! H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n outline: matrix models & matrix geometry 4D covariant quantum spaces : fuzzy S 4 N , H 4 n cosmological space-times: M 3 , 1 & BB! fluctuations → higher spin gauge theory metric, vielbein; gravity? HS, arXiv:1606.00769 M. Sperling, HS arXiv:1707.00885 HS, arXiv:1709.10480, arXiv:1710.11495 M. Sperling, HS arXiv:1806.05907 H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n The IKKT model IKKT or IIB model Ishibashi, Kawai, Kitazawa, Tsuchiya 1996 � � [ X a , X b ][ X a ′ , X b ′ ] η aa ′ η bb ′ + ¯ Ψ γ a [ X a , Ψ] S [ X , Ψ] = − Tr X a = X a † ∈ Mat ( N , C ) , a = 0 , ..., 9 , N large gauge symmetry X a → UX a U − 1 , SO ( 9 , 1 ) , SUSY proposed as non-perturbative definition of IIB string theory quantized Schild action for IIB superstring reduction of 10 D SYM to point, N large add m 2 X a X a to set scale, IR regularization � dXd Ψ e iS [ X ] Z = Kim, Nishimura, Tsuchiya arXiv:1108.1540 ff H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n different points of view: classical solutions = “branes” justified by max. SUSY (cf. critical string thy) generically NC geometry, “matrix geometry” fluctuations → field theory, 3+1D physics, dynamical geometry UV/IR mixing → IKKT model → unique 4D NC gauge theory hypothesis space-time = (near-) classical solution of IIB model 10 bulk physics: sugra arises in M.M. from quantum effects (loops) Kabat-Taylor, IKKT,... “holographic” cf. HS arXiv:1606.00646 H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n “matrix geometry” ( ≈ NC geometry): S E ∼ Tr [ X a , X b ] 2 ⇒ config’s with small [ X a , X b ] � = 0 dominate i.e. “almost-commutative” configurations minimize � a � x | ∆ X 2 ∃ quasi-coherent states | x � , a | x � X a spectrum =: M ⊂ R 10 ≈ diag., � x | X a | x ′ � ≈ δ ( x − x ′ ) x a , x ∈ M hypothesis: classical solutions dominate “condensation” of matrices, geometry NC branes embedded in target space R 10 X a ∼ x a : → R 10 M ֒ cf. Q.M: replace functions x a � matrices / observables X a H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n typical examples: quantized Poisson manifolds Moyal-Weyl quantum plane R 4 θ : [ X a , X b ] = i θ ab 1 l quantized symplectic space ( R 4 , ω ) admits translations, no rotation invariance fuzzy 2-sphere S 2 N X 2 1 + X 2 2 + X 2 3 = R 2 [ X i , X j ] = i ǫ ijk X k N , fully covariant under SO ( 3 ) (Hoppe; Madore) generically: fluctuations → NC gauge theory, & dynamical geometry H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n issues for NC spaces / field theory: quantization → UV / IR mixing ֒ → max. SUSY model: IKKT, BFSS, BMN Lorentz / SO ( 4 ) covariance in 4D ? NC spaces: [ X µ , X ν ] =: i θ µν � = 0 obstacle: breaks Lorentz invariance ∃ fully covariant fuzzy four-sphere S 4 N Grosse-Klimcik-Presnajder 1996; Castelino-Lee-Taylor; Ramgoolam; Kimura; Abe Hasebe; Medina-O’Connor; Karabali-Nair; Zhang-Hu 2001 (QHE!) ... price to pay: “internal structure” → higher spin theory H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n covariant fuzzy four-sphere S 4 N 5 hermitian matrices X a , a = 1 , ..., 5 acting on H N � X 2 a = R 2 a covariance: X a ∈ End ( H N ) transform as vectors of SO ( 5 ) [ M ab , X c ] = i ( δ ac X b − δ bc X a ) , [ M ab , M cd ] = i ( δ ac M bd − δ ad M bc − δ bc M ad + δ bd M ac ) . M ab ... so ( 5 ) generators acting on H N H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n covariant fuzzy four-sphere S 4 N 5 hermitian matrices X a , a = 1 , ..., 5 acting on H N � X 2 a = R 2 a covariance: X a ∈ End ( H N ) transform as vectors of SO ( 5 ) [ M ab , X c ] = i ( δ ac X b − δ bc X a ) , [ M ab , M cd ] = i ( δ ac M bd − δ ad M bc − δ bc M ad + δ bd M ac ) . M ab ... so ( 5 ) generators acting on H N oscillator construction: Grosse-Klimcik-Presnajder 1996; ... = ψ † γ a ψ, [ ψ β , ψ † α ] = δ β X a α M ab = ψ † Σ ab ψ α N | 0 � ∼ = ( C 4 ) ⊗ S N ∼ H N = ψ † α 1 ...ψ † acting on = ( 0 , N ) so ( 5 ) H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n relations: = R 2 ∼ 1 4 r 2 N 2 X a X a = ir 2 M ab =: i Θ ab [ X a , X b ] ǫ abcde X a X b X c X d X e = ( N + 2 ) R 2 r 3 (volume quantiz.) geometry from coherent states | p � : { p a = � p | X a | p �} = S 4 closer inspection: degeneracy of coherent states, “internal” S 2 fiber cf. Karczmarek, Yeh, arXiv:1506.07188 H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n semi-classical picture: hidden bundle structure C P 3 ∋ ψ ↓ ↓ x a = ψ + Γ a ψ S 4 ∋ Ho-Ramgoolam, Medina-O’Connor, Abe, ... fuzzy case: oscillator construction [Ψ , Ψ † ] = δ → functions on fuzzy C P 3 N hidden extra dimensions S 2 ! fuzzy S 4 N is really fuzzy C P 3 N , Poisson tensor θ µν ( x , ξ ) ∼ − i [ X µ , X ν ] local SO ( 4 ) x rotates fiber ξ ∈ S 2 averaging over fiber → [ θ µν ( x , ξ )] 0 = 0 , local SO ( 4 ) preserved! ... 4D “covariant” quantum space H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n fields and harmonics on S 4 N ”functions“ on S 4 N : N N � � C s = End ( H N ) ∼ C s = ( n , 2 s ) ∋ s = 0 n = 0 ( n , 0 ) modes = scalar functions on S 4 : φ ( X ) = φ a 1 ... a n X a 1 ... X a n = ( n , 2 ) modes = selfdual 2-forms on S 4 φ bc ( X ) θ bc = φ a 1 ... a n b ; c X a 1 ... X a n θ bc = = fields on S 4 taking values in hs = ⊕ End ( H ) ∼ higher spin modes = would-be KK modes on S 2 (local SO ( 4 ) acts on S 2 fiber) H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n relation with spin s fields: one-to-one map ∼ C s = T ∗⊗ S s S 4 φ ( s ) = φ ( s ) b 1 ... b s ; c 1 ... c s ( x ) θ b 1 c 1 . . . θ b s c s �→ φ ( s ) c 1 ... c s ( x ) = φ ( s ) b 1 ... b s ; c 1 ... c s x b 1 . . . x b s { x c 1 , ..., { x c s , φ ( s ) ← φ ( s ) c 1 ... c s ( x ) } ... } c 1 ... c s ( x ) ... ”symbol“ of φ ∈ C s M. Sperling & HS, arXiv:1707.00885 C s ∼ = symm., traceless, tang., div.-free rank s tensor field on S 4 φ c 1 ... c s ( x ) x c i = 0 , φ c 1 ... c s ( x ) g c 1 c 2 = 0 , ∂ c i φ c 1 ... c s ( x ) = 0 . H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Fuzzy S 4 fuzzy H 4 Motivation Matrix geometry fields & kinematics Cosmological space-times towards gravity N n Poisson calculus: (semi-classical limit) M. Sperling & HS, 1806.05907 C P 3 = symplectic manifold, { x a , x b } = θ ab 1 ð a φ := − r 2 R 2 θ ab { x b , φ } , { x a , ·} = θ ab ð b satisfy T = g ac − ð a x c = P ac R 2 x a x c 1 matrix Laplacian: � = [ x a , [ x a , . ]] ∼ −{ x a , { x a , . }} = − r 2 R 2 ð a ð a covariant derivative: ∇ θ ab = 0 ∇ = P T ◦ ð , curvature R ab := R [ ð a , ð b ] = [ ∇ a , ∇ b ] − ∇ [ ð a , ð b ] ... Levi-Civita connection on S 4 H. Steinacker Quantized cosmological spacetimes and higher spin in the IKKT model
Recommend
More recommend
