Non-commutative geometry and matrix models II

Transcription

1 Non-commutative geometry and matrix models II Harold Steinacker Fakultät für Physik Zakopane, march 2011

2 outline part II: embedded NC spaces, matrix models, and emergent gravity noncommutative gauge theory Yang-Mills matrix models general geometry in matrix models (embedded NC spaces, curvature) nonabelian gauge fields, fermions, SUSY quantization of M.M: heat kernel expansion, UV/IR mixing aspects of (emergent) gravity, outlook

3 Yang-Mills matrix models dynamical embedded NC spaces gravity well suited for quantization note: Z = dx a e S[X] S[X] = Tr[X a, X b ][X a, X b ]δ aa δ bb + (matter) matrix configuration X a... matrix geometry ( background ) integration over space of geometries emergent (dominant, effective) geometry very closely related to NC gauge theory D = 10, add Majorana-Weyl fermions IKKT model (=dim-red. of D = 10 SYM) nonperturb. def. of IIB string theory Ishibashi, Kawai, Kitazawa and Tsuchiya hep-th/ more generally: intersecting spaces, stacks, etc.

4 Gauge theory on R 4 θ Let [ X µ, X ν ] = i θ µν, X µ L(H) (Moyal-Weyl) consider fluctuations around R 4 θ : recall [ X µ, φ] = iθ µν ν φ X µ = X µ θ µν Aν [X µ, X ν ] = i θ µν + i θ µµ θ νν ( µ A ν ν A µ + i[a µ, A ν ]) = i θ µν + i θ µµ θνν F µ ν = i θ µµ θνν ( θ 1 µν + F µ ν ) F µν (x)... u(1) field strength gauge transformations: X µ UX µ U 1 = U( X µ θ µν A ν )U 1 = X µ + U[ X µ, U 1 ] + θ µν UA ν U 1 = X µ + θ µν (U ν U 1 + UA ν U 1 ) infinites: U = e iλ(x), δa µ = i µ Λ(X) + i[λ(x), A µ ]

5 Yang-Mills action: or S YM [X] = Tr[X µ, X ν ][X µ, X ν ]δ µµ δ νν = ρ d x(f µν + i θ µν )(F µ ν + i θ µ ν Ḡ )Ḡµµ νν Tr([X µ, X ν ] i θ µν )([X µ, X ν ] i θ µ ν )δ µµ δ νν = ρ d 4 xf µν F µ ν Ḡµµ Ḡ νν (same up to surface term Tr[X, X] = F 0 )... NC U(1) gauge theory on R 4 θ, effective metric Ḡ µν = θ µµ θνν δ µ ν, reduces to usual U(1) gauge theory on R 4 invariant under gauge trafo X µ UX µ U 1, 1 ρ = θ µν 1/2 (as classical F.T.!!) F µν UF µν U 1 symplectomorphism no local observables! (need trace)

6 Yang-Mills action: or S YM [X] = Tr[X µ, X ν ][X µ, X ν ]δ µµ δ νν = ρ d x(f µν + i θ µν )(F µ ν + i θ µ ν Ḡ )Ḡµµ νν Tr([X µ, X ν ] i θ µν )([X µ, X ν ] i θ µ ν )δ µµ δ νν = ρ d 4 xf µν F µ ν Ḡµµ Ḡ νν (same up to surface term Tr[X, X] = F 0 )... NC U(1) gauge theory on R 4 θ, effective metric Ḡ µν = θ µµ θνν δ µ ν, reduces to usual U(1) gauge theory on R 4 invariant under gauge trafo X µ UX µ U 1, 1 ρ = θ µν 1/2 (as classical F.T.!!) F µν UF µν U 1 symplectomorphism no local observables! (need trace)

7 coupling to scalar fields: consider ( ) S[X, φ i ] = Tr [X µ, X ν ][X µ, X ν ]δ µµ δ νν + [X µ, φ i ][X µ, φ i ]δ µµ = ρ ) d 4 x (F µν F µ ν Ḡ Ḡµµ νν + D µ φ i D ν φ i Ḡµν (dropping surface terms) gauge transformation [X µ, φ] = i θ µν ( ν + i[a µ,.])φ =: i θ µν D µ φ φ i Uφ i U 1 (adjoint) same form as S[X] = Tr[X a, X b ][X a, X b ]δ aa δ bb, a = 1,..., 4+k

8 note: extremely simple origin of gauge fields: arbitrary fluctuations X µ X µ + A µ (A µ = θ µν A ν ) configuration space = {4 hermitian matrices X a } works only on NC spaces! matrix models Tr[X, X][X, X] gauge-invariant YM action generalized easily to U(n) theories but U(1) sector does not decouple from SU(n) sector one-loop: UV/IR mixing not QED, problem except in N = 4 SUSY case: finite (!?)... nevertheless phys. wrong for U(1) sector: proper interpretation in terms of (emergent) geometry, gravity.

9 try something similar for fuzzy sphere: S[X] = 1 g 2 Tr ( [X a, X b ][X a, X b ] 4iε abc X a X b X c 2X a X a ) = 1 g 2 Tr ([X a, X b ] iε abc X c )([X a, X b ] iε abc X c ) = 1 g 2 Tr F ab F ab 0 where X a Mat(N, C), solutions (minima!): a = 1, 2, 3 and F ab := [X a, X b ] iε abc X c field strength F ab = 0 [X a, X b ] = iε abc X c X a = λ a, λ a... rep. of su(2) any rep. of su(2) is a solution! X a = concentric fuzzy spheres S 2 M i! geometry & topology dynamical! λ a (M 1 ) λ a (M 2 ) λ a (M k )

10 try something similar for fuzzy sphere: S[X] = 1 g 2 Tr ( [X a, X b ][X a, X b ] 4iε abc X a X b X c 2X a X a ) = 1 g 2 Tr ([X a, X b ] iε abc X c )([X a, X b ] iε abc X c ) = 1 g 2 Tr F ab F ab 0 where X a Mat(N, C), solutions (minima!): a = 1, 2, 3 and F ab := [X a, X b ] iε abc X c field strength F ab = 0 [X a, X b ] = iε abc X c X a = λ a, λ a... rep. of su(2) any rep. of su(2) is a solution! X a = concentric fuzzy spheres S 2 M i! geometry & topology dynamical! λ a (M 1 ) λ a (M 2 ) λ a (M k )

11 expand around solution: X a = λ a + A a Mat(N, C) F ab = [λ a, A b ] [λ b, A a ] iε abc A c + [A a, A b ] F = F ab ξ a ξ b = da + AA can be interpreted in terms of { U(1) gauge theory on S 2 N (tang. fluct. if) λ a A a = 0 coupled to scalar field D µ φd µ φ (radial fluctuations) X a = λ a (1 + φ) can fix geometry, suppress radial field by adding constraint S[X] = Tr (([X a, X b ] iε abc X c )([X a, X b ] iε abc X c ) +(X a X a C N ) 2) indeed deformed Maxwell theory on SN 2, as classical F.T.

12 however: recall fuzzy sphere: near north pole x a = (0, 0, 1) X 3 = 1 (X 1 ) 2 (X 2 ) 2 expect: radial deformation X 3 = λ 3 + A 3 = φ(x 1, X 2 )... deformation of embedding, geometry! geometry NC gauge theory??? consider deformed fuzzy spaces, effective geometry

13 Yang-Mills Matrix Models, reconsidered let g ab = δ ab... SO(D) (resp. g ab = g ab... SO(n, m)) ( ) S = Tr [X a, X b ][X a, X b ]g aa g bb + fermions X a = X a Mat(, C), a = 1,..., D gauge symmetry X a UX a U 1, or ( ) S = Tr ([X a, X b ] iθ ab 1l)([X a, X b ] iθ a b 1l)g aa g bb +... (up to boundary terms Tr[X, X]) { } NC space(-time) pre-geometric; solutions (emergent) metric (gravity) { } nonabelian gauge fields... fluctuations of NC space gravitons D = 10 : quantization well-defined (?!)

14 Space-time & geometry from matrix models: e.o.m.: δs = 0 [X a, [X a, X b ]]g aa = 0 solutions: ( NC spaces) 1) prototype (d=4): [X a, X b ] = iθ ab 1l, rank θ ab = 4 split X a = ( X µ, Φ i ), µ = 1,..., 4 [ X µ, X ν] = i θ µν 1l Φ i = 0 interpretation: }... R 4 θ X a : R 4 θ R10... embedded quantum plane fluctuations X a = X a + δx a propagating fields on R 4 θ

15 Noncommutative spaces and Poisson structure (M, θ µν (x))... 2n-dimensional manifold with Poisson structure Its quantization M θ is NC algebra such that I : C(M) A = Mat(, C) f (x) ˆf (X) x a X a, e ikx e ikx such that [ˆf (X), ĝ(x)] = I(i{f (x), g(x)}) + O(θ 2 ) ( nice ) Φ Mat(, C) quantized function on M furthermore: (2π) 2 Tr(φ(X)) d 4 x ρ(x) φ(x) ρ(x) = Pfaff (θ 1 µν )... symplectic volume (cf. Bohr-Sommerfeld quantization)

16 2) generic solutions (d=4): deformations of R 4 θ R10 X a = (X µ, Φ i (X µ )), µ = 1,..., 4; A = Mat(, C) generated by X µ [X µ, X ν ] i{x µ, x ν } = iθ µν (x), generic NC space R D interpretation: X a x a : M 4 R dim (or 3+1-dim.) space(time), brane quantized Poisson-MF (M, θ µν (x))

17 Effective geometry of NC brane: consider scalar field coupled to Matrix Model ( test particle ) use [f, ϕ] iθ µν (x) µ f ν ϕ S[ϕ] = Tr [X a, ϕ][x b, ϕ] g ab (U(H) gauge inv.!) d 4 x θµν 1 θ µ µ µ x a µ ϕ θ ν ν ν x b ν ϕ g ab = d 4 x G µν G µν (x) µ ϕ ν ϕ G µν (x) = e σ θ µµ (x)θ νν (x) g µ ν (x) effective metric g µν (x) = µ x a ν x b g ab induced metric on M 4 θ e 2σ = θ 1 µν g µν, G µν = g µν for dim(m) = 4 ϕ couples to metric G µν (x), determined by θ µν (x) & embedding φ i (x)... quantized Poisson manifold with metric (M, θ µν (x), G µν (x))

18 same metric G µν for gauge fields, fermions all matter couples to dynamical metric G µν effective gravity however: metric is not fundamental d.o.f. rather: matrices X a resp. (φ i, θ µν ) resp. (φ i, F µν ) dynamics of gravity NOT given by Einstein equations not GR (long distances!), may be close enough to observation (?) note: D = 10 just enough to describe most general g µν (x) in d = 4 (locally) A. Friedman (1961)

19 class of embedded NC spaces X a : M R D is stable under small deformations consider small deformation X a = X a + A a by assumption locally X a = (X µ, φ i (X µ )) (x µ, φ i (x µ )) X µ generate A = Mat(N, C) A a = A a (x µ ), smooth X a = (X µ + A µ, φ + A i ) ( x µ, φ i ( x µ )) : M R D [ X µ, X ν ] i{ x µ, x ν }...new Poisson bracket... deformed embedded NC space M

20 dynamics of geometry def. := [X a, [X b,.]]g ab... matrix Laplacian on A result: (M, ω) symplectic manifold, ω = 1 2 θ 1 µν dx µ dx ν x a : M R D... embedding in R D induced metric g µν and G µν as above. Then: {x a, {x b, ϕ}}g ab = e σ G ϕ µ G (eσ θ 1 µν ) = G νρ θ ρµ ( e σ µ η + µ x a G x b g ab ) for ϕ C (M), G... Levi-Civita, G... Laplace- Op. w.r.t. G µν, and η(x) := 1 4 eσ G µν g µν. cf. fuzzy sphere, torus etc! (H.S., 2008) Hence: φ e σ G φ(x)

21 proof: either Tr ϕ [X a, [X a, ϕ]] = Tr [X a, ϕ ][X a, ϕ] d 4 x θµν 1 ϕ {X a, {X a, ϕ}} = d 4 x G µν G µν (x) µ ϕ ν ϕ d 4 x G µν e σ ϕ {X a, {X a, ϕ}} = d 4 x G µν ϕ G ϕ or {X a, {X a, ϕ}} = θ µρ µ x a ρ (θ νη ν x a η ϕ) = θ µρ ρ ( µ x a θ νη ν x a η ϕ) = θ µρ ρ (θ νη g µν η ϕ) = θ µρ θ νη g µν ρ η ϕ + θ µρ ρ (θ νη g µν ) η ϕ = e σ (G ρη ρ η ϕ Γ η η ϕ) = e σ G ϕ,

22 proof: either Tr ϕ [X a, [X a, ϕ]] = Tr [X a, ϕ ][X a, ϕ] d 4 x θµν 1 ϕ {X a, {X a, ϕ}} = d 4 x G µν G µν (x) µ ϕ ν ϕ d 4 x G µν e σ ϕ {X a, {X a, ϕ}} = d 4 x G µν ϕ G ϕ or {X a, {X a, ϕ}} = θ µρ µ x a ρ (θ νη ν x a η ϕ) = θ µρ ρ ( µ x a θ νη ν x a η ϕ) = θ µρ ρ (θ νη g µν η ϕ) = θ µρ θ νη g µν ρ η ϕ + θ µρ ρ (θ νη g µν ) η ϕ = e σ (G ρη ρ η ϕ Γ η η ϕ) = e σ G ϕ,

23 in particular: matrix e.o.m: [X a, [X b, X a ]]g aa = 0 G Φ i = 0, G x µ = 0 µ (e σ θµν 1 ) = e σ G ρν θ ρµ µ η η = 1 4 eσ G µν g µν... covariant formulation in semi-classical limit in particular: M 4 R D is harmonic embedding (w.r.t. G µν ) minimal surface

24 dynamics of NC structure θ µν : S YM = Tr[X a, X b ][X a, X b ] d 4 x ge σ η Euclidean case: at p M, diagonalize g µν = diag(1, 1, 1, 1) using SO(4) standard form θ µν = θ 0 α 0 0 α ±α α 1 0 effective metric G µν = diag(α 2, α 2, α 2, α 2 ). Note. 1 4 Gµν g µν = e σ η = 1 2 (α2 + α 2 ) 1 ω = ±ω e σ η = 1 G µν = g µν S YM minimal minimum of S YM θ µν (A)SD G µν = g µν.

25 more structure: define Then hence J η γ = e σ/2 θ ηγ g γ γ = e σ/2 G ηγ θ 1 γ γ. G µν = J µ ρ J ν ρ gρρ = (J 2 ) µ ρ g ρν, G µν g νρ = (J 2 ) µ ρ, J 2 = δ g = G... almost-complex structure (M, J, e σ/2 g µν ) almost-kähler g = G note: g = G e.o.m. for θ µν reduces to follows from ω = ±ω µ θ 1 µν = 0

26 more structure: define Then hence J η γ = e σ/2 θ ηγ g γ γ = e σ/2 G ηγ θ 1 γ γ. G µν = J µ ρ J ν ρ gρρ = (J 2 ) µ ρ g ρν, G µν g νρ = (J 2 ) µ ρ, J 2 = δ g = G... almost-complex structure (M, J, e σ/2 g µν ) almost-kähler g = G note: g = G e.o.m. for θ µν reduces to follows from ω = ±ω µ θ 1 µν = 0

27 special class of solutions: g µν = G µν, G φ i = 0 µ θµν 1 = 0 holds for (anti)self-dual symplectic structure θ 1 µν, (θ 1 ) = ±θ 1 Euclidean (θ 1 ) = ±iθ 1 Minkowski (Wick rotation X 0 it ) then S MM Tr[X a, X b ][X a, X b ] = d 4 x g µν... same structure as vacuum energy / cosm. const.

28 semi-classical derivation of e.o.m.: S = Tr[X a, X b ][X a, X b ] 1 (2π) n geometrical d.o.f: δθ 1 µν = µδa ν νδa µ δφ i d 2n x G e σ η. δs = 1 ( ) 2 d 2n x θµν 1 g µνθ µµ δθ νν g µ ν + gµνθµµ θ νν δg µ ν + η(x) θµν δθνµ 1 = 1 ( ) 2 d 2n x θµν 1 e 2σ G ηµ θµν 1 G νρ δθρη 1 + e σ G µν δg µν + η(x) θ µν δθνµ 1 = d 2n x ( ) G G ηµ G νρ e σ θµν 1 ρδa η e σ η θ ρη ρδa η + G µν µφ i νδφ i = d 2n x ( ) ( ) GδA η G ηµ G νρ ρ(e σ θµν 1 ) ρ(e σ η θ ρη ) + δφ i ν G G µν µφ i = d 2n x ( G (δa η G ηµ G νρ ρ(e σ θµν 1 ) 1 ) ) G ρ( Ge σ ηθ ρη ) + δφ i G φ i = d 2n x ( ) ) G (δa η G ηµ G νρ ρ(e σ θµν 1 ) e σ θ ρη ρη + δφ i G φ i

29 (sufficiently) generic 4D geometry in M.M.: 1 take some nice (M 4, g µν ) (e.g. asympt. flat, glob. hyperbolic,...) 2 choose embedding x a : M R 10 (Friedman etal) 3 equip M with (anti)selfdual symplectic form ω = θµν 1 dx µ dx ν, g (ω) = ±ω (almost-kähler) construct quantization of (M, ω): I : C(M) A = Mat(, C) in particular: X a x a 4 effective metric G µν g µν, encoded in in M.M. (examples: fuzzy spaces = quantized coadjoint orbits, e.g. SN 2 R3 )

30 su(n) gauge fields: include fluctuations: where effective action: same model, new vacuum ( ) ( ) Y Y a µ X = Y i = µ 1l n φ i 1l n Y a = (1 + A ρ ρ ) ( X µ 1l n φ i 1l n + Φ i A µ = θ µν A ν,α λ α, λ α su(n) Φ i = Φ i α λ α (n coinciding branes) S YM = d 4 x G e σ G µµ G νν tr F µν F µ ν + 2 η(x) tr F F (H.S., JHEP 0712:049 (2007), JHEP 0902:044,(2009) )... su(n) Yang-Mills coupled to metric G µν (x) )

31 fermions S[Ψ] = Tr Ψ /DΨ = Tr ΨΓ a [X a, Ψ] d 4 x ρ(x) Ψiγ µ (x) µ Ψ, note γ µ (x) = Γ a θ νµ ν x a {γ µ, γ ν } = {Γ a, Γ b }θ µ µ µ x a θ ν ν ν x b = 2θ µ µ θ ν ν g µ ν = 2e σ G µν (x) naturally SUSY (IKKT model with D = 10) couple to G µν, but non-standard spin connection (submanifold!)

32 global SO(9, 1) symmetry: can use to fix φ i p = 0 = φ i p... analogous to Riemannian normal coordinates bottom line: U(1) sector is geometry scalar fields describe embedding M 4 R 10, θ µν & φ i completely absorbed in g µν, G µν (semi-classically) dynamics, propagators due to [X a,.][x a,.] fluctuations of branes dyn. geometry, nonabelian gauge fields couples naturally to matter

33 global SO(9, 1) symmetry: can use to fix φ i p = 0 = φ i p... analogous to Riemannian normal coordinates bottom line: U(1) sector is geometry scalar fields describe embedding M 4 R 10, θ µν & φ i completely absorbed in g µν, G µν (semi-classically) dynamics, propagators due to [X a,.][x a,.] fluctuations of branes dyn. geometry, nonabelian gauge fields couples naturally to matter expect good quantum theory (including gravity): action NC N = 4 U(1) SYM

34 global SO(9, 1) symmetry: can use to fix φ i p = 0 = φ i p... analogous to Riemannian normal coordinates bottom line: U(1) sector is geometry scalar fields describe embedding M 4 R 10, θ µν & φ i completely absorbed in g µν, G µν (semi-classically) dynamics, propagators due to [X a,.][x a,.] fluctuations of branes dyn. geometry, nonabelian gauge fields couples naturally to matter expect good quantum theory (including gravity): action NC N = 4 U(1) SYM

35 U(1) gauge fields as gravitons δθ 1 µν = F µν on R 4 θ therefore G µν (x) = η µν h µν (+O(F 2 )) F µν (x)... u(1) field strength h µν = η νν θν ρ F ρµ + η µµ θµ η F ην 1 2 η µν ( θ ρη F ρη )... linearized metric fluctuation e.o.m: [X µ, [X ν, X µ ]]η µµ = 0 µ F µν = 0 R µν [G] = 0 ( µ h µν = 0... harm. gauge) while R µνρη 0 on-shell d.o.f. of gravitons on Minkowski space i.e.: NC U(1) on R 4 θ as gravitons cf. Rivelles [hep-th/ ] cf. Kitazawa [hep-th/ ]

36 higher-order terms, curvature result: H ab := 1 2 [[X a, X c ], [X b, X c ]] + T ab := H ab 1 4 gab H, H := H ab g ab = [X c, X d ][X c, X d ], X := [X b, [X b, X]] for 4-dim. M R D with g µν = G µν : Tr ( ) 2T ab X a X b T ab H ab 2 (2π) d 4 x g e 2σ R 2 Tr([[X a, X c ], [X c, X b ]][X a, X b ] 2 X a X a ) 1 (2π) d 4 x g e ( σ e σ θ µη θ ρα R µηρα 2R + µ σ µ σ ) (Blaschke, H.S. arxiv: ) (cf. Arnlind, Hoppe, Huisken arxiv: ) Einstein-Hilbert- type action for gravity as matrix model pre-geometric version of (quantum?) gravity, background indep.!

37 derivation: (assume g = G) H ab = 1 2 [[X a, X c ], [X b, X c ]] + e σ G µν µ x a ν x b g=g = e σ P ab T, T ab = H ab 1 4 ηab H e σ P ab N P N, P T... projector on normal / tangential bundle of M R D. note R νµλκ = P ab N ( κ ν x a λ µ x b + κ µ x a ν λ x b ) = κ ν x a λ µ x a + κ µ x a ν λ x a (i.e. Gauss-Codazzi theorem) and T bc [X a, [X a, T bc ]] hence e 2σ P bc N µ µ (e σ g bc e σ ν x b ν x c )) = e ((D 2σ 4) e σ 2PN bc(eσ µ ν x b µ ν x c ) ( ) = e 2σ (D 4) e σ 2e σ µ ν x a µ ν x a 2T ab X a X b T bc T bc e 2σ( ) (D 4) e σ 2e σ R noting that H e σ = η, result follows. )

38 vacuum energy / cosm.const in matrix model: recall g = G for general G g can show where Tr L H2 H ab H ab (2π) 2 d 4 x Λ 4 (x) g. Λ(x) = LΛ 2 NC = Le σ/2 L... cutoff length in matrix model (recall Λ 4 NC = gµν θ 1 µν = eσ ) note: is different from action Tr[X a, X b ][X a, X b ] 1 (2π) 2 d 4 x g e σ η g=g = 1 (2π) n d 2n x g.

39 proof: H ab = 1 2 [[X a, X c ], [X b, X c ]] + e σ G µν µ x a ν x b = e σ (J 2 P T ) ab, in normal coords, J 2 = diag(α 2, α 2, α 2, α 2 ) 2 EV char. equation implies (note H = e σ trj 2 ) hence Tr L H2 H ab H ab J 4 = 1 2 (trj 2 ) J 2 δ H ab H ab 1 2 H2 e 2σ P ab T (P T ) ab = 4Λ 8 NC L (2π) 2 d 4 x ge σ e σ = 1 2(2π) 2 d 4 x Λ 4 (x) g.

40 Quantization of M.M. Z = dx a dψ e S[X] S[Ψ]...non-perturbative! includes integration over geometries!! probably ill-defined in general (UV/IR mixing = ind. gravity) ONE model with well-defined (finite!?) quantization: N = 4 NC SYM on R 4 θ (IKKT) model, D = 10 Ishibashi, Kawai, Kitazawa and Tsuchiya 1996, ff fully SO(9, 1) and U(H) invariant

41 2 interpretations: Z = dx a dψ e S[X] S[Ψ] = e S eff 1 on R 4 θ : X µ = X µ + θ µν A ν, X µ...moyal-weyl NC SYM on R 4 θ, UV/IR mixing in U(1) sector IKKT model, D = 10: N = 4 SYM, perturb. finite!(?) 2 on M 4 R 10 : U(1) absorbed in θ µν (x), g µν gravity, induced E-H. action S eff d 4 x G ( Λ 4 + cλ 2 4 R[G] +... ) (R[G] due to UV/IR mixing in NC gauge theory) explanation for UV/IR mixing & U(1) entanglement D = 10 good quantization!! (maximal SUSY)

42 2 interpretations: Z = dx a dψ e S[X] S[Ψ] = e S eff 1 on R 4 θ : X µ = X µ + θ µν A ν, X µ...moyal-weyl NC SYM on R 4 θ, UV/IR mixing in U(1) sector IKKT model, D = 10: N = 4 SYM, perturb. finite!(?) 2 on M 4 R 10 : U(1) absorbed in θ µν (x), g µν gravity, induced E-H. action S eff d 4 x G ( Λ 4 + cλ 2 4 R[G] +... ) (R[G] due to UV/IR mixing in NC gauge theory) explanation for UV/IR mixing & U(1) entanglement D = 10 good quantization!! (maximal SUSY)

43 induced action: fermionic loop S[Ψ] = Tr Ψγ a [X a, Ψ] induced effective action: Γ eff := 1 ( 2 Tr log /D 2) 1 2 Tr dα α L... cutoff length heat kernel expansion: 0 α /D2 Tre = n 2 /D e α e 1 αl 2 =: Γ L [X]. α n 4 2 Γ (n) commutative case: Γ (n)... Seeley-de Witt coeff., Γ eff = d 4 x (Λ 4 G + Λ 2 R[G] +...) NC case:... induced gravity (Sakharov 1967) coupling to gravity compute induced gravity

44 NC heat kernel expansion perturbation of flat background expect: 1 2 Tr 0 dα α effective M.M. Γ eff = TrL(X), /D 2 = /D V ( ) e α /D2 e α /D2 0 e 1 αl 2 = k>0 O(V k ) =: TrL(X) invar. under SO(D) and U( ) (cf. Grosse Wohlgenannt 2008) complication: UV/IR mixing, additional divergences Λ n, n Z, Λ = L Λ 2 NC, Λ NC = θ 1 µν 1/8... NC scale mild UV/IR mixing: finite Λ, such that p 2 Λ 2 Λ 4 NC then semi-class. approx. ok even in loops or: N = 4 model: finite (?!) 1,

45 for computation: use NC gauge theory point of view ( ) ( ) X perturbation of R 4 θ : X a µ θ = + µν A ν 0 φ i /D 2 = + V, Ḡ µν = Λ 4 NC θ µµ θνν δ µ ν ) V Ψ = (2[A iḡµν µ, νψ] + [ µa ν, Ψ] + i[a µ, [A ν, Ψ]] + δ ij [ϕ i, [ϕ j, Ψ]] +Λ 4 ( NC Σµν[F µν, Ψ] + 2Σ µi θ µν [ νφ i + i[a ν, φ i ], Ψ] iσ ij [[φ i, φ j ], Ψ] ) compute all dimension 6 operators in effective gauge theory

46 ... long computation (one-loop NC YM, generic external fields A µ, ϕ i )

47 effective gauge theory induced gauge theory up to dimension 6: Γ eff = tr1l 16 Λ 4 Λ 4 NC 1 2 Λ4 NC d 4 x (2π) g (g αβ D 2 α ϕ i D β ϕ i ( θ µν F νµ θ ρσ F σρ + ( θ σσ F σσ )(F θf θ) ) ) 2 θ νµ F µα g αβ ν ϕ i β ϕ i ( θ µν F µν )g αβ β ϕ i α ϕ i + h.o. ( + tr1l Λ 2 d 4 x 4 24 (2π) g Λ4 NC F ρη g F στ Ḡ ρσḡητ 12Λ 8 NC g ϕ i ϕ i ) Λ4 NC ( θ µν F µν ) G ( θ ρσ F ρσ ) Λ6 d 4 x Λ 8 (2π) g(...) NC all of this is due to UV/IR mixing! (D. Blaschke, H.S. M. Wohlgenannt arxiv: )

48 effective generalized matrix model: ( ) X re-assemble effective action: X a µ = + 0 ( ) θ µν A ν φ i Γ L [X] = Tr L 4 1 2H 2 H ab H ab + 1 L L 2 10,curv [X]+... d 4 x Λ 4 (x) g(x) L 10,curv [X] = c 1 [X c, H ab ][X c, H ab ] + c 2 H cd [X c, [X a, X b ]][X d, [X a, X b ]] +... H ab = [X a, X c ][X b, X c ] + (a b), H = H ab η ab (D. Blaschke, H.S. M. Wohlgenannt arxiv: ) SO(D) manifest, broken by background (e.g. R 4 θ ) highly non-trivial predictions for (NC) gauge theory expect generalization to nonabelian N = 4 SYM: full SO(9, 1)! effective generalized matrix model = powerful new tool for (NC) gauge theory and (emergent) gravity

49 effective generalized matrix model: ( ) X re-assemble effective action: X a µ = + 0 Γ L [X] = Tr L 4 1 2H 2 H ab H ab + 1 L L 2 10,curv [X]+... ( ) θ µν A ν φ i L 10,curv [X] = c 1 [X c, H ab ][X c, H ab ] + c 2 H cd [X c, [X a, X b ]][X d, [X a, X b ]] +... H ab = [X a, X c ][X b, X c ] + (a b), H = H ab η ab (D. Blaschke, H.S. M. Wohlgenannt arxiv: ) SO(D) manifest, broken by background (e.g. R 4 θ ) highly non-trivial predictions for (NC) gauge theory expect generalization to nonabelian N = 4 SYM: full SO(9, 1)! effective generalized matrix model = powerful new tool for (NC) gauge theory and (emergent) gravity

50 SO(9, 1) resp. SO(10) symmetry: ) ( θ e.g. [X a, X b ] = µν + θ µµ θ νν F µ ν θ µν D νφ i is θ µν D νφ j [φ i, φ j SO(9, 1) ] multiplet only possible due to NC! ( X SO(9, 1) acts on X a = µ θ ) µν A ν, φ i non-linearly realized (cf. SSB) on ( ) θ µν A ν φ i can use to fix φ i p = 0 = φ i p... analogous to Riemannian normal coordinates

51 full IKKT model around R 4 θ : ( N = 4 SYM on R4 θ!) background field method X a X a + Y a, fully SO(9, 1) covariant, e.g. ( ( ) Γ 1 loop = 1 2 Tr log(1l + Σ (Y ) ab 1 [Θ ab,.]) 1 2 log(1l + Σ (ψ) ab 1 [Θ ab,.]) = [X a, [X a,.]] Θ rs = [X r, X s ] Σ rs = SO(9, 1) generator effective generalized M.M. (work in progress, D. Blaschke, H.S.) SO(9, 1) invariant formalism, broken spontaneously through R 4 θ NC essential.

52 dynamics of emergent NC gravity assume effective action S d 4 x g ( 2Λ 4 + Λ 2 4R) + 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 = g µν + µ φ i ν φ i 1 Einstein branch Λ 4 g µν + Λ 2 4G µν = 8πT µν 2 harmonic branch Λ 4 g φ = (8πT µν Λ 2 4G µν ) µ ν φ prototype: flat space R 4 θ R10, even for Λ 0!

53 illustration of Einstein branch example: Schwarzschild geometry (Blaschke, H.S. arxiv:1005:0499) embedding M R 7, asymptotically flat (harmonic), e σ const t r cos ϕ sin ϑ r sin ϕ sin ϑ x a r cos ϑ = 1 r cr, ω cos (ω(t + r)) 1 r cr ω sin (ω(t + r)) 1 ω r cr with g ab = diag(, +, +, +, +, +, ). central singularity: embedding with complexified SD symplectic form θ 1 = iθ 1, θ 1 const for r

54 issues remain: θ µν degenerate on a circle on horizon gauge coupling depends on e σ θ µν, not good extrinsic term such as Tr X a X a G x a G x a may arise need to understand (quantum) effective action show: predominantly intrinsic geometry GR Lorentz violating effects due to θ µν must be very small (maybe average out θ µν?) probably need something like M 4 K, intersecting branes,... singularities? presumably resolved by fuzzyness...

55 Fuzzy extra dimensions in field theory e.g. SN 2 may arise in ordinary 4D gauge theory through Higgs effect: consider SU(N ) Yang-Mills theory on 4-D Minkowski space M 4 S YM = ( ) d 4 1 y Tr F 4g µνf 2 µν + (D µ φ a ) D µ φ a V (φ) A µ... su(n ) - valued gauge fields, D µ = µ + [A µ,.], and φ a = φ a, a = 1, 2, 3... scalar fields in adjoint of SU(N ) global SO(3) symmetry, gauge symmetry φ a U φ a U, U = U(y) U(N ) V (φ)... renormalizable potential respecting the symmetries

56 Fuzzy extra dimensions in field theory e.g. SN 2 may arise in ordinary 4D gauge theory through Higgs effect: consider SU(N ) Yang-Mills theory on 4-D Minkowski space M 4 S YM = ( ) d 4 1 y Tr F 4g µνf 2 µν + (D µ φ a ) D µ φ a V (φ) A µ... su(n ) - valued gauge fields, D µ = µ + [A µ,.], and φ a = φ a, a = 1, 2, 3... scalar fields in adjoint of SU(N ) global SO(3) symmetry, gauge symmetry φ a U φ a U, U = U(y) U(N ) V (φ)... renormalizable potential respecting the symmetries

57 (almost) most general potential respecting the symmetries: ( V (φ) = Tr a 2 (φ a φ a b 1l) 2 + c + 1 g ) 2 F ab F ab for suitable constants a, b, c, g, where F ab = [φ a, φ b ] iε abc φ c vacuum = minimum of V (φ), achieved if F ab = [φ a, φ b ] iε abc φ c = 0, a(φ a φ a b) = 0 φ a... representation of SU(2) with Casimir b = C 2 (N) for some N N J (N) a φ a = J (N) a 1l n... generator of the N-dimensional irrep of SU(2) (assume N = Nn)

58 note: Mat(N, C) = Mat(N, C) Mat(n, C) = C(S 2 N ) Mat(n, C) interpretation: φ a (y)... generate u(n)-valued functions on SN 2 M4 ( ) y µ therefore: φ a... functions on M 4 SN 2 R7 Higgs effect: U(N ) gauge symmetry broken to U(n) spontaneously generated extra dimensions (= commutant of φ a = J (N) a ) model describes 6-dimensional U(n) gauge theory on M 4 S 2 N finite tower of massive Kaluza-Klein modes due to Higgs effect (also true if add fermions to model) P. Aschieri, T. Grammatikopoulos, H.S., G. Zoupanos 2006; Madore, Manousselis; etc.... same mechanisms as in string theory, within renormalizable QFT! full matrix model same applies to space-time itself

59 Y-M. Matrix Models + fermions: contain all ingredients for theory of fund. interactions. a priori only SU(n) gauge groups symmetry breaking, contact with particle physics: possible mechanisms: extra-dimensional fuzzy spaces M 4 K R 10 add cubic terms to matrix model extra-dim. fuzzy S 2, interesting low-energy gauge groups, including SU(3) SU(2) U(1) ( U(1) anomalous ) (P. Aschieri, T. Grammatikopoulos, H.S., G. Zoupanos 2006; Madore, Manousselis; Aoki, Azuma, Iso,...; H. Grosse, F. Lizzi, H.S. arxiv: ) however non-chiral difference to string theory: R D bulk unphysical, nothing propagates in bulk predictive framework

60 Summary, outlook Matrix (fuzzy) geometry : quantized symplectic spaces M R D generic class, many examples matrix-model Tr[X a, X b ][X a, X b ] g aa g bb dynamical matrix geometries emergent gravity & gauge thy not same as G.R., but might be close enough (extrinsic geometry, physics of vacuum energy,...) can address curvature, etc. suitable for quantizing gravity! (IKKT model D = 10, maximal SUSY) new powerful techniques: effective generalized matrix models... to be developed... new, more work is needed

61 Cosmological solution assume: vacuum energy Λ 4 energy density ρ D. Klammer, H. S., PRL 102 (2009) look for harmonic embedding x a = 0 of FRW metric Ansatz ds 2 = dt 2 + a(t) 2 (dχ 2 + sinh 2 (χ)dω 2 ), x a a(t) (t, χ, θ, ϕ) = ( cos ψ(t) sin ψ(t) ) 0 x c(t) sinh(χ) sin θ cos ϕ sinh(χ) sin θ sin ϕ sinh(χ) cos θ cosh(χ) Evolution a(t), Ψ(t), x c (t) determined by x a = 0 solution of M.M + leading term d 4 x GΛ 4 R10 (cf. B. Nielsen, JGP 4, (1987) ) in Γ 1 loop

62 harmonic embedding g x a = 0 leads to analog of Friedmann equations H 2 = ȧ2 a 2 = b 2 a 10 + d 2 a 8 k a 2. ä a = 3d 2 a 8 + 4b 2 a 10. largely independent of detailed matter/energy content as long as Λ 4 ρ k = 1 (negative spatial curvature) most interesting

63 Implications: 1) early universe: big bounce: ȧ = 0 for a = a min b 1/4 ( bound for energy density ρ vs. vacuum energy Λ 4 ) inflation-like phase a(t) t 2 4 b, ends at a(t exit ) = geometric mechanism (no scalar field required), no fine-tuning 3 d

64 2) late evolution (now): ȧ 1 approaches Milne-like universe (k = 1, spatial curvature), in remarkably good agreement with observation (age yr, type Ia supernovae) different physics for early universe (recombination etc.) A. Benoit-Levy and G. Chardin, [arxiv: ] CMB acoustic peak argued to be at correct scale (?) no fine-tuning of cosm. const., no need for dark energy!