Emergent 4D gravity on covariant quantum spaces in the IKKT model

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1 Emergent 4D gravity on covariant quantum spaces in the IKKT model Harold Steinacker Department of Physics, University of Vienna Wroclaw, july 2016 H.S. : arxiv:

2 Motivation expect quantum structure of space-time at short distances (learning process...) Yang-Mills matrix models: NC gauge theory dynamical quantum geometry well-behavedness under quantization (UV/IR mixing!) SUSY IKKT model (quite unique!) describes branes M R 10, background independent closely related to IIB string theory (nonperturb.!) conjecture: 4D gravity emerges automatically on suitable (!) backgrounds (alternative to string theory compactification)

3 many years of trials: HS, Yang, Rivelles,... basic NC branes such as (deformed) R 4 θ don t quite work (?) universal dynamical metric propagating graviton covariant Einstein equations typically derivative coupling of graviton to matter, short-range how to proceed?

4 how to reconcile Lorentz invariance with quantum spacetime? in 2D: fuzzy sphere SN 2, covariant (invariant Poisson structure on SN 2 ) in 4D: any Poisson structure would break local SO(4) nevertheless: fully covariant fuzzy four-sphere SN 4 Grosse Klimcik Presnajder 1996; Castelino Lee Taylor 1997; Ramgoolam;... price to pay: internal structure, higher spin modes Snyder-type space Snyder 1947 [X µ, X ν ] im µν here: work out lowest spin modes on SN 4 in IKKT model (linearized) Einstein equations HS arxiv:

5 outline: fuzzy SN 4 fluctuation modes geometry: metric, vielbein, linearization realization in (IKKT) matrix model: equations of motion curvature, (lineariz.) Einstein equations one-loop corrections

6 The fuzzy four-sphere S 4 N Grosse Klimcik Presnajder 1996; Castelino, Lee, Taylor hermitian matrices X a, a = 1,..., 5 acting on H N Xa 2 = R 2 1l a covariance: X a A = End(H) transform as vectors of SO(5) explicitly: H N = (0, N) SO(5), [M ab, X c ] = i(δ ac X b δ bc X a ), M ab... so(5) generators acting on H N X a = r c α(γ a ) α βc β, [c β, c α] = δ β α, γ a... SO(5) Gamma matrices satisfy X a X a = R 2, R 2 = r 2 N(N + 4)/4 [X a, X b ] = ir 2 M ab, ɛ abcde X a X b X c X d = (N + 2)r 3 X e fully covariant under SO(5) (cf. Snyder space)

7 geometry recovered from optimally localized states: coherent states p = g Λ, g SO(5): p ( X a ) 2 p 2 min {p a = p X a p } = S 4 closer inspection: internal space C N+1 of optimally localized states at p S 4 internal squashed fuzzy SN+1 2 (radius 0) equivalently: N + 1 -sheeted cover of S 4

8 geometry recovered from optimally localized states: coherent states p = g Λ, g SO(5): p ( X a ) 2 p 2 min {p a = p X a p } = S 4 closer inspection: internal space C N+1 of optimally localized states at p S 4 internal squashed fuzzy SN+1 2 (radius 0) equivalently: N + 1 -sheeted cover of S 4

9 full geometry CP 3 ψ S 4 ψγ i ψ = x i... squashed S 2 N bundle over S4 (Hopf map) Medina-O Connor, Ramgoolam, Ho, Abe,... CP 3 quantized fuzzy CP 3 N, algebra of functions End(H) fuzzy S 4 N is really squashed CP3 N (hidden extra dimensions S2!) commutation relations: Θ ij := i[x i, X j ] θ ij (x, ξ)...poisson structure on CP 3 varies along fiber ξ S 2, selfdual average [θ ij (x, ξ)] S 2 = 0 local SO(4) invariance preserved! (rare) example 4D covariant quantum space Λ µµ Λ νν Θ µ ν = U Λ Θ µν U 1 Λ

10 fields and kinematics on S 4 N most general functions on SN 4 : φ End(H N ) = (n m, 2m) m n N.. much bigger than functions on S 4 : C N (S 4 ) := (n, 0) P N (X) n N (n, 2) modes: F bc (X)M bc = F a1...a n;bcx a 1...X an M bc etc. tower of higher spin modes origin: local SO(4) acts non-trivially on S 2 fiber important map: φ [φ] S 2 C N (S 4 ),... proj. on scalar fields, averaging over fiber

11 fields and kinematics on S 4 N most general functions on SN 4 : φ End(H N ) = (n m, 2m) m n N.. much bigger than functions on S 4 : C N (S 4 ) := (n, 0) P N (X) n N (n, 2) modes: F bc (X)M bc = F a1...a n;bcx a 1...X an M bc etc. tower of higher spin modes origin: local SO(4) acts non-trivially on S 2 fiber important map: φ [φ] S 2 C N (S 4 ),... proj. on scalar fields, averaging over fiber

12 local description: pick north pole p S 4 separate generators ( ) X a X µ =, x µ X µ, µ = 1,..., 4...normal coords at p similarly X 5 rescale X rx, ( ) M ab M µν P = µ P µ 0 where P µ = M µ5 P µ = 1 R g µνp ν, R = R N r algebra [P µ, X ν ] iδ ν µ, quantized volume element [P µ, P ν ] = i R 2 M µν 0 [X µ, X ν ] =: iθ µν = ir 2 M µν 0 ɛ abcd5 X a X b X c X d = (N + 2)r 3 R

13 Fluctuating S 4 N in the IKKT model IKKT or IIB model Ishibashi, Kawai, Kitazawa, Tsuchiya 1996 ( S[Y, Ψ] = Tr [Y a, Y b ][Y a, Y b ]δ aa δ bb + Ψγ ) a [Y a, Ψ] Y a = Y a Mat(N, C), a = 0,..., 9 (N ) gauge symmetry Y a UY a U 1, SO(10), SUSY...proposed as non-perturbative definition of IIB string theory nonperturbative results: Kim, Nishimura, Tsuchiya arxiv: ff Big Bang, 3+1-dim. brane emerges

14 perturbative approach: choose background solution = set of matrices {Y a, a = 1,..., 10} Y a y a : M R embedding of brane (=quantum space) (e.g. R 4 θ, S4 N ) fluctuations ( emergent ) gauge fields & dynamical geometry ( gravity ) (review: H.S. arxiv: ) quantum effects: lead to IIB supergravity in bulk (IKKT, Chepelev-Roiban, Kabat-Taylor, H.S,...) maximal SUSY essential to avoid pathological UV/IR mixing (non-local physics!)

15 add mass term : S[Y ] = 1 g 2 Tr ( [Y a, Y b ][Y a, Y b ] + µ 2 Y a Y a ) (scale, SUSY breaking, iε in Minkowski case) eom: ( µ2 )Y a = 0, = [Y a, [Y a,.]] add fluctuations S 4 N is solution for µ2 < 0 Y a = X a + A a expand action to second oder in A a S[Y ] = S[X] + 2 g 2 TrA a ( ( + 1 ) 2 µ2 )δb a + 2[[X a, X b ],. ] [X a, [X b,.]] A b key result: H.S. arxiv: Quantum fluctuations stabilize S 4 N even for µ2 > 0

16 Fluctuation modes on S 4 N organize fluctuations at p S 4 as where A µ = ξ µ + x µ R κ + θµν A ν, A ν = A ν + A νρ P ρ + A νρσ M ρσ +... ξ µ = ξ µ + ξ µν P ν + ξ µνρ M νρ +... κ = κ + κ µ P µ + κ µν M µν +... A 5 = κ rank 2 tensor field A νρ (x) = 1 2 (h νρ + a νρ ) h νρ = h ρν... metric fluctuation rank 3 tensor field A νρσ (x)m ρσ... spin connection field rank 1 field A ν (x)... gauge field (stack of SN 4 )

17 metric and vielbein consider transversal scalar field φ Y a = A a, a = 6,..., 9: lowest spin mode φ = φ(x) kinetic term [X α, φ][x α, φ] D α φd α φ = γ µν µ φ ν φ +..., D α := i[x α,.] effective metric vielbein γ µν [X α, X µ ][X α, X ν ] = g αβ θ αµ θ βν = g µν = g αβ e αµ e βν... metric rotates along internal S 2, average e αµ = θ αµ, e α = e αµ µ [θ µν ] S 2 = 0, [ γ µν ] S 2 = γ µν = 4 4 gµν... SO(5) invariant!

18 metric and vielbein consider transversal scalar field φ Y a = A a, a = 6,..., 9: lowest spin mode φ = φ(x) kinetic term [X α, φ][x α, φ] D α φd α φ = γ µν µ φ ν φ +..., D α := i[x α,.] effective metric vielbein γ µν [X α, X µ ][X α, X ν ] = g αβ θ αµ θ βν = g µν = g αβ e αµ e βν... metric rotates along internal S 2, average e αµ = θ αµ, e α = e αµ µ [θ µν ] S 2 = 0, [ γ µν ] S 2 = γ µν = 4 4 gµν... SO(5) invariant!

19 deformed background: Y a = X a + A a [Y a, φ][y a, φ] D a φd a φ = γ µν µ φ ν φ +..., D a := i[y a,.] e aµ [A] µ... vielbein effective metric γ µν [Y a, X µ ][Y a, X ν ] D a x µ D a x ν. where e αν [A] = i[x α + θ αν A ν, X µ ] = θ αµ θ αν [A νρ P ρ, X µ ] + O( A) = θ αβ (δβ ν + A βρg ρν ) +... using expansion A ν = A ν + A νρ P ρ + A νρσ M ρσ + O( A) and i[p ρ, X µ ] g ρµ

20 metric fluctuation: where γ µν = γ µν + δγ µν δγ µν = 4 4 hµν + θ µα θ νβ ( β ξ α + α ξ β ) note: non-derivative metric d.o.f. h µν due to presence of momentum generators P µ in SN 4 algebra (unlike previous attempts) full SO(5) covariance maintained on space with quantum structure, minimal length metric is universal, kinetic term for all fluctuations [Y a, A][Y a, A]

21 more on the effective metric kinetic term for (transversal) scalar fields: 2 Vol(M 4 ) S[φ] = 2 g 2 tr[y a, φ][y a, φ] dim H dim H 4 Vol(M 4 ) for effective metric 2g 2 M d 4 x G µν G µν µ φ ν φ G µν = α 4 4 γµν, α = rescaled vielbein ẽ αµ = α 2 2 e αµ metric fluctuation g 2 M d 4 x γ µν µ φ ν φ 4 4 γ 1 µν G µν = g µν + H µν, H µν = h µν 1 2 gµν h Lorentz-gauge µ h µν = 0 de Donder gauge µ H µν 1 2 νh = 0

22 more on the effective metric kinetic term for (transversal) scalar fields: 2 Vol(M 4 ) S[φ] = 2 g 2 tr[y a, φ][y a, φ] dim H dim H 4 Vol(M 4 ) for effective metric 2g 2 M d 4 x G µν G µν µ φ ν φ G µν = α 4 4 γµν, α = rescaled vielbein ẽ αµ = α 2 2 e αµ metric fluctuation g 2 M d 4 x γ µν µ φ ν φ 4 4 γ 1 µν G µν = g µν + H µν, H µν = h µν 1 2 gµν h Lorentz-gauge µ h µν = 0 de Donder gauge µ H µν 1 2 νh = 0

23 gauge transformations: Y a UY a U 1 leads to δa a = i[λ, X a ] + i[λ, A a ] expand Λ = Λ Λ abm ab U(1) SO(5)... - valued gauge trafos diffeos from δ v := i[v ρ P ρ,.] δh µν = ( µ v ρ + ρ v µ ) v ρ ρ h µν + (Λ h) µν δa µρσ = 1 2 µλ σρ (x) v ρ ρ A µρσ + (Λ A) µρσ etc.

24 gauge fixing:... simple in Yang-Mills matrix model!! gives 0 = i[x a, A a ] = i[x 5, A 5 ] i[x µ, A µ ]. 0 = µ A µ 0 = µ (h µν + a µν ) 0 = θ µν( 1 2 a νµ + ν ξ µ + R 2 ρ A ρµν ) (neglecting radial deformations κ)

25 classical action for gravitational modes: geometric fluctuations A µ (x) = θ µν( A νσ (x)p σ + A νσρ (x)m σρ) quadratic action ) S 2 [A] = 1 TrA g 2 a (( µ2 )δb a + 2i[Θab,. ] A b M 2( ) h µν ( + 8r 2 + µ 2 )h µν + a µν ( + 8r 2 + µ 2 )a µν 1 g 2 dim H Vol(M 4 ) coupling to matter δs[φ, A,...] = dim H 8 Vol(M 4 ) 8g 2 M d 4 x h µν T µν ( issue for conformal factor for scalar fields...)

26 equations of motion ( + 8r µ2 )h µν = 6 T µν ( + 8r µ2 )a µν = 0 ( + 4r µ2 )κ ( + 8r µ2 )A νσρ + r 2( A ρσν A σρν A ρνσ + A σνρ 2r 2 g νσ (A αβρ g αβ ) + 2r 2 g νρ (A αβσ g αβ ) ) can neglect radial fluctuations κ, set a µν = 0 particular solution for A αβρ : = 8 8R T = ρ A νσ σ A νρ A νσρ = 1 4 ( ρh νσ σ H νρ )... spin connection for H µν = h µν 1 2 g µνh, µ H µν 1 2 νh = 0...physical metric fluctuation (de Donder gauge)

27 equations of motion ( + 8r µ2 )h µν = 6 T µν ( + 8r µ2 )a µν = 0 ( + 4r µ2 )κ ( + 8r µ2 )A νσρ + r 2( A ρσν A σρν A ρνσ + A σνρ 2r 2 g νσ (A αβρ g αβ ) + 2r 2 g νρ (A αβσ g αβ ) ) can neglect radial fluctuations κ, set a µν = 0 particular solution for A αβρ : = 8 8R T = ρ A νσ σ A νρ A νσρ = 1 4 ( ρh νσ σ H νρ )... spin connection for H µν = h µν 1 2 g µνh, µ H µν 1 2 νh = 0...physical metric fluctuation (de Donder gauge)

28 curvature and Einstein equations linearized gravity: G µν = g µν + H µν = δ µν + δg µν + H µν near p S 4 linearized Ricci tensor R µν [g + H] R µν [g] µ ν H α α H µν (µ ρ H ν)ρ = 3 g R 2 µν α α H µν dedonder gauge lin. Einstein tensor G µν [g + H] 3 R 2 g µν α α h µν combine with eom for h µν (dropping insignificant mass term 1 ) R 2 G µν + 3 R 2 g µν 8πG N T µν Newton constant G N = 2 4π = R2 Nπ ( = Planck length!).

29 curvature and Einstein equations linearized gravity: G µν = g µν + H µν = δ µν + δg µν + H µν near p S 4 linearized Ricci tensor R µν [g + H] R µν [g] µ ν H α α H µν (µ ρ H ν)ρ = 3 g R 2 µν α α H µν dedonder gauge lin. Einstein tensor G µν [g + H] 3 R 2 g µν α α h µν combine with eom for h µν (dropping insignificant mass term 1 ) R 2 G µν + 3 R 2 g µν 8πG N T µν Newton constant G N = 2 4π = R2 Nπ ( = Planck length!).

30 summary so far: 4-D Einstein equations arise from fluctuations on fuzzy SN 4 in the Yang-Mills matrix model no Einstein-Hilbert term, no quantum effects needed spin connection A µρσ is solution, but dynamical torsion modes & tower of additional higher-spin modes non-linear model; expect (extension of) full Einstein theory IKKT model well-suited for quantization!! new view of string theory (no compactifications!) lots of open issues (fermions, Minkowski, non-lin. regime, torsion, conf. factor,...)

31 Quantization 1-loop effective action Γ eff [X] defined by e Γeff[X] = 1 loop well-behaved due to max. SUSY dadψe S[X+A,Ψ] here: one-loop for fluctuations around X a = X a + A a X a... S 4 N background simple gauge fixing: [X a, A a ] = 0 (cf. Lorentz gauge, Yang-Mills!)

32 1-loop effective action max. SUSY ( Γ 1loop [X] = 1 2 Tr log( + µ2 ( = 1 2 Tr where n= µ2 Tr 1 + O(µ 4 ) ) [Θab,.]) 2 log( ) ) 2 M(A) ab [Θab,.]) 1 log( M(ψ) 2 ab ( 1 ( 1( M (A) n ab [Θab,.] 1 2 µ2 ) ) ) n 1 2 ( 1 M (ψ) ab [Θab,.]) n M (ψ) ab M (A) ab iθ ab = [X a, X b ]...spinorial generators of so(5)...vector generators...background flux leading 4th order term ( ) Γ 1loop;4 [X] = 1 8 Tr ( 1 (M (A) ab [Θab,.]) ( 1 M (ψ) ab [Θab,.]) 4 ) = ( 1 4 Tr 1 [Θ a 1b 1,... 1 [Θ a 4b 4,.]]]] ( 4gb1 a 2 g b2 a 3 g b3 a 4 g b4 a 1 4g b1 a 2 g b2 a 4 g b4 a 3 g b3 a 1 4g b1 a 3 g b3 a 2 g b2 a 4 ) g b4 a 1 +g b1 a 2 g b2 a 1 g b3 a 4 g b4 a 3 + g b1 a 3 g b3 a 1 g b2 a 4 g b4 a 2 + g b1 a 4 g b4 a 1 g b2 a 3 g b3 a 2

33 to evaluate trace, use string state formalism Tr End(H) O = (dim H)2 (VolM) 2 M M H.S. arxiv: dxdy( x y )O( y x ) y x End(H)... string states, x... coherent state on M = CP 3 satisfy 1 1 ( y x ) y x x y [Θ ab,.]( y x ) 1 x y δθ ab (y, x) y x δθ ab (y, x) plug in leading n = 4 term = Θ ab (y) Θ ab (x)

34 where Γ 1loop;4 [X] = 1 4 (dim H) 2 (Vol(M)) 2 M M dxdy 3S 4[δΘ(x,y)] ( x y ) 4 S 4 [δθ] = 4tr(δΘgδΘgδΘgδΘg) (trδθgδθg) 2 = 4(δΘ ab + δθ + ab ) (δθcd δθ cd ), 0 δθ ± = δθ ± g δθ on 4-dimensional branes (IIB sugra! cf. H.S. arxiv: )

35 Results: vacuum energy: bare mass µ 2 > 0 stabilizes S 4 N (SUSY breaking, cf. H.S. arxiv: ) transversal fluctuations: no quadratic 1-loop corrections tangential fluctuations: h µν modes: Γ 1 loop;4 = 8π2 9 (dim H) 2 4 (VolM 4 ) 2 d 4 x ( hµν hµν 1 4 (gµν hµν ) 2) where h ρµ = h ρµ + ρ ξ µ + µ ξ ρ ( graviton mass mh 2 4 µ 2 2 ) 8 λ R 4 0 for R λ = 6g 2 dim H Euclidean (!) 1-loop equilibrium: m 2 h large & negative (or fine-tuning) in Minkowski case (?), R 2 expected much larger maybe ok (?)

36 comments: purely classical mechanism (not induced gravity or similar!) robust, protected from quantum corrections by max. SUSY quantum corrections negligible for R (maximal SUSY) Euclidean case: brane tension important, keeping R quite small large mass corrections (not good) Minkowski case: expect much larger R small quantum corrections some proposals for Minkowski signature, not clear cf. Gazeau & Toppan; Heckman & Verlinde; Buric & Madore

37 summary & outlook S 4 N S 4 N... fully SO(5) covariant quantum space as background in Yang-Mills matrix models Tr[X a, X b ][X a, X b ] η aa η bb + fermions fluctuations tower of higher spin modes 4-dimensional (lin.) Einstein equations spin connection, dynamical torsion waves IKKT matrix model suitable for quantization new view on string theory: no 10D compactification! issues, open questions: generalization to Lorentzian case conformal sector nonlinear case, higher spin modes, fermions,... (max. SUSY)

38 Fermions Dirac operator in the IIB model (SUSY!) /DΨ = Γ a [X a, Ψ] i( γ µ µ +...)Ψ comoving Clifford generators γ µ = Γ α ẽ αµ encode effective metric (up to conformal factor...) { γ µ, γ ν } = 2G µν cf. H.S. : arxiv: [hep-th] need expansion into higher spin modes (coupling to spin connection?)