Emergent space-time and gravity in the IIB matrix model

Transcription

1 Emergent space-time and gravity in the IIB matrix model Harold Steinacker Department of physics Veli Losinj, may 2013

2 Geometry and physics without space-time continuum aim: (toy-?) model for quantum theory of all fund. interactions (gravity!) pre-geometric geometry, gravity emerge at low energies quantum structure of space-time at L P candidates: string theory: vast landscape of possible vacua... (??) here: Matrix Models related to string theory, more predictive dynamical NC space-time accessible, novel tools!

3 Geometry and physics without space-time continuum aim: (toy-?) model for quantum theory of all fund. interactions (gravity!) pre-geometric geometry, gravity emerge at low energies quantum structure of space-time at L P candidates: string theory: vast landscape of possible vacua... (??) here: Matrix Models related to string theory, more predictive dynamical NC space-time accessible, novel tools!

4 Analogy: quantized (phase) space in QM class. mechanics phase space R 2 quantum mechanics quantized phase space R 2 functions f (q, p), Heisenberg algebra f (Q, P) {p, q} = 1 [P, Q] = i 1l NC (matrix) geometry: same math, interpreted as physical space(-time) need metric structure, dynamical (from M.M.!)

5 for practical purposes (& this talk): replace quantum space by Poisson manifold: (M, {.,.})... 2n-dimensional manifold with Poisson structure {f, g} = iθ µν (x) µ f ν g θ µν... dim. L 2 quantization map: such that Q : C(M) L(H) Q(f ) Q(g) = Q(fg) + O(θ) [Q(f ), Q(g)] = Q(i{f, g}) + O(θ 2 ) ( nice ) Φ L(H) = Mat(, C) quantized function on M furthermore: (2π) n Tr Q(φ) d 2n x θ 1 µν φ(x)

6 IIB model = IKKT matrix model S[X] = Tr ( [X a, X b ][X a, X b ]η aa η bb + Ψγ ) a [X a, Ψ] X a = X a Mat(N, C), a = 0,..., 9 (N ) gauge symmetry X a UX a U 1, SO(9, 1), SUSY Ishibashi, Kawai, Kitazawa and Tsuchiya 1996 { 1) nonpert. def. of IIB string theory (on R 10 ) (IKKT ) 2) N = 4 SUSY Yang-Mills gauge thy. on noncommutative R 4 θ dynamical NC branes M R 10 ( 4D gravity!? H.S ff) semi-classical version: S[x] = ( ) θ 1 {x a, x b }{x a, x b }η aa η bb + Ψiγ a {x a, Ψ} M x a : M R 9,1, a = 0,..., 9

7 IIB model = IKKT matrix model S[X] = Tr ( [X a, X b ][X a, X b ]η aa η bb + Ψγ ) a [X a, Ψ] X a = X a Mat(N, C), a = 0,..., 9 (N ) gauge symmetry X a UX a U 1, SO(9, 1), SUSY Ishibashi, Kawai, Kitazawa and Tsuchiya 1996 { 1) nonpert. def. of IIB string theory (on R 10 ) (IKKT ) 2) N = 4 SUSY Yang-Mills gauge thy. on noncommutative R 4 θ dynamical NC branes M R 10 ( 4D gravity!? H.S ff) semi-classical version: S[x] = ( ) θ 1 {x a, x b }{x a, x b }η aa η bb + Ψiγ a {x a, Ψ} M x a : M R 9,1, a = 0,..., 9

8 configuration space: 10 matrices X a Mat(N, C) generate algebra of functions on noncommutative brane(s) semi-classical: embedding X a x a : M R 9,1 geometry! commutation relations [X a, X b ] i{x a, x b } = iθ µν (x) µ x a ν x b encode Poisson-tensor field θ µν (x) on M R 9,1... B - field; geometry, not elmag!

9 dynamical H. Steinacker NC branes M R 10 Emergent space-time and gravity( in the4d IIB matrix gravity model? ) action S[x] = θ 1 {x a, x b }{x a, x b } = θ 1 θ µν θ µ ν g µµ g νν M M g µν = µ x a ν x b η ab embedding metric (cf. closed string m.) equations of motion: vary embedding of M R 9,1 : δs δx a = 0 0 = {x b, {x c, x a }} δ bc = e σ G x a G µν (x) = e σ θ µµ (x)θ νν (x) g µ ν (x) effective metric (cf. open string e 2σ = θ 1 µν g µν vary Poisson structure: δs δf = 0, θµν = θ µν + θ µµ F µ ν θ ν ν µ (e σ θ 1 µν ) = e σ G ρν θ ρµ µ (Gg)

10 relation with string theory D-brane solutions in IIB string theory reproduced flat 2n- dim. quantum planes R 2n θ, [X µ, X ν ] = iθ µν 1l compact extra dim M 4 T 2, etc. addition of solutions: assume X(1) a, X (2) a are solutions ( ) X a new solution X a = (1) 0 0 X(2) a intersecting( branes, stacks ) (cf. string theory) X a (11) φ (12) X a = φ (21) X a (22) interaction in SUGRA reproduced at one loop (IKKT,...)

11 fluctuations on brane nonabelian gauge theory consider stack of coincident branes ( ) ( Y a Y µ X = Y i = µ ) 1l n φ i 1l n tangential fluctuations su(n) gauge fields ( Y a = (1 + A ρ X ρ ) µ ) 1l n φ i 1l n effective action: S YM = d 4 x G e σ tr F, F G + 2 η(x) tr F F (H.S., JHEP 0712:049 (2007), JHEP 0902:044,(2009) ) IKKT model on stack of branes SU(n) N = 4 SYM coupled to metric G µν (x)

12 results: fluctuations of matrices X A around stack of branes SU(n) NC Yang-Mills gauge theory coupled to G ab (x) universal effective metric G ab (x) on such branes, dynamical prospects: well suited for quantization, predictive can be put on computer (Lorentzian!) Kim, Nishimura, Tsuchiya dims at late times intersecting branes chiral fermions A. Chatzistavrakidis, H.S., G. Zoupanos (2011) all ingredients for physics

13 Quantization Quantization of matrix model: Z = dx a dψ e S[X] S[Ψ] = e S eff path integral 2 interpretations: 1 as NC N = 4 SYM on R 4 θ : X µ = X µ + θ µν A ν UV finite 2 on M 4 R 10 : U(1) absorbed in θ µν (x), g µν induced gravity action (Sakharov 1967) S eff d 4 x G ( Λ 4 + cλ 2 4 R[G] +... ) IKKT model quantum theory for 4D geometry gravity

14 assume g = G (Euclidean): assume eff. action S = d 4 x g ( 2Λ 4 + Λ 2 4 R) + S matter leads to δs = d 4 x g δg µν ( Λ 4 g µν + 8πT µν Λ 2 4 Gµν ) = 2 δφ i µ ( g ( Λ 4 g µν + 8πT µν Λ 2 4 Gµν )) ν φ i since g µν = η µν + µ φ i ν φ i locally 1 Einstein branch Λ 4 g µν + Λ 2 4 Gµν = 8πT µν 2 harmonic branch Λ 4 g φ = (8πT µν Λ 2 4 Gµν ) µ ν φ prototype: flat space R 4 θ R10, even for Λ 0!

15 Einstein branch embedding theorems realize 4D geometries on M R 9,1 illustration: Schwarzschild geometry M R 7 (Blaschke, H.S. arxiv:1005:0499) complicated due to θ µν etc. need full 1-loop effective action (at least)

16 harmonic branch Λ 4 g φ = (8πT µν Λ 2 4 Gµν ) µ ν φ protected from cosm. const. problem Einstein equations not established: tangential fluctuations Ricci-flat metric perturbations Rivelles [hep-th/ ] similar for zero modes of compactifications A. Polychronakos, H.S., J. Zahn 2013 embedding perturbations of R 4 non-linear treatment (?) extrinsic curvature µ ν φ 0 of M R 9,1 coupling to T µν at linearized level, Newtonian gravity H.S., JHEP 1301 (2013), JHEP 0912 (2009) on fuzzy (A)dS 2 D. Jurman, H.S. in progress

17 example: emergent gravity on fuzzy (A)dS 2 D. Jurman, H.S. in progress... solution of matrix model S[X] = 1 gym 2 Tr ([X a, X b ][X a, X b ] + ig f abc [X a, X b ]X c) solutions X a... fuzzy (A)dS 2 study perturbations X a = X a + A a due to T µν δ T Ric[G] = 8πG N ( T Ric 1 µ ν T µν) where 8πG N = L2 NC 2R 2, T µν = θ µµ T νν T µ ν

18 summary, conclusion matrix-models Tr[X a, X b ][X a, X b ] η aa η bb + fermions dynamical NC branes emergent gravity tangential fluctuations gauge theory on brane complicated dynamics of geometry B-field i.e. θ µν part of geometry, dynamics metric not fundamental UV finite for 4D geometries!... more to be discovered

19 references N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A Large N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/ ]. H. Steinacker, Emergent Geometry and Gravity from Matrix Models: an Introduction, arxiv: [hep-th]. topical review, Class. Quantum Grav. 27 (2010) H. Steinacker, Emergent Gravity from Noncommutative Gauge Theory. JHEP 12 (2007) 049. [arxiv: v1 (hep-th)] H. Steinacker, Emergent Gravity and Noncommutative Branes from Yang-Mills Matrix Models, Nucl. Phys. B 810:1-39,2009. arxiv: [hep-th]. A. Polychronakos, H. Steinacker, J. Zahn, compactified branes and 4-dimensional geometry in the IKKT model arxiv: D. N. Blaschke and H. Steinacker, Curvature and Gravity Actions for Matrix Models, arxiv: [hep-th].class. Quant. Grav. 27:165010,2010. H. Steinacker, Non-commutative geometry and matrix models, arxiv: [hep-th].