Emerging from Matrix Models

Transcription

1 fom Matix Models Talk pesented by Daniel N. Blaschke Faculty of Physics, Mathematical Physics Goup Collaboato: H. Steinacke May 15, / 28

2 1 2 Cuvatue and Actions fo Matix Models 3 4 Reissne-Nodstöm 5 and Outlook 2 / 28

3 Matix models of Yang-Mills type S Y M = T[X a, X b ][X c, X d ]η ac η bd e.o.m: [X a, [X b, X c ]]η ab = 0 X a ae Hemitian matices acting on a Hilbet space H, and η ab is D dimensional at backgound metic xes signatue simplest solution of e.o.m.: [X a, X b ] = iθ ab = constant at Goenewold-Moyal space R θ X a = ( X µ, Φ i), µ = 1,..., 2n, i = 1,..., D 2n, so that Φ i (X) φ i (x) dene embedding M 2n R D (in semi-classical limit) 3 / 28

4 Matix models of Yang-Mills type S Y M = T[X a, X b ][X c, X d ]η ac η bd e.o.m: [X a, [X b, X c ]]η ab = 0 X a ae Hemitian matices acting on a Hilbet space H, and η ab is D dimensional at backgound metic xes signatue simplest solution of e.o.m.: [X a, X b ] = iθ ab = constant at Goenewold-Moyal space R θ X a = ( X µ, Φ i), µ = 1,..., 2n, i = 1,..., D 2n, so that Φ i (X) φ i (x) dene embedding M 2n R D (in semi-classical limit) 3 / 28

5 Matix models of Yang-Mills type S Y M = T[X a, X b ][X c, X d ]η ac η bd e.o.m: [X a, [X b, X c ]]η ab = 0 X a ae Hemitian matices acting on a Hilbet space H, and η ab is D dimensional at backgound metic xes signatue simplest solution of e.o.m.: [X a, X b ] = iθ ab = constant at Goenewold-Moyal space R θ X a = ( X µ, Φ i), µ = 1,..., 2n, i = 1,..., D 2n, so that Φ i (X) φ i (x) dene embedding M 2n R D (in semi-classical limit) 3 / 28

6 Yang-Mills Matix models II g µν (x) M 2n induced metic of 2n dimensional submanifold M 2n R D g µν (x) = µ x a ν x b η ab = η µν + µ φ i ν φ j η ij 4 / 28

7 Yang-Mills Matix models III M 2n endowed with a Poisson stuctue i[x µ, X ν ] {x µ, x ν } P B = θ µν (x) eective metic G µν = e σ θ µρ θ νσ g ρσ = (J 2 ) µ ρg ρν, e σ det θµν 1 det Gρσ special case: 2n = 4 det G µν = det g µν opens possibility fo special class of geometies whee G µν = g µν J 2 = 1 coesponds to a self-dual symplectic fom θ 1 i.e. Θ = 1 2 θ 1 µν dx µ dx ν, Θ = ±iθ e.o.m. µ θ 1 µν = 0 µν, 5 / 28

8 Yang-Mills Matix models III M 2n endowed with a Poisson stuctue i[x µ, X ν ] {x µ, x ν } P B = θ µν (x) eective metic G µν = e σ θ µρ θ νσ g ρσ = (J 2 ) µ ρg ρν, e σ det θµν 1 det Gρσ special case: 2n = 4 det G µν = det g µν opens possibility fo special class of geometies whee G µν = g µν J 2 = 1 coesponds to a self-dual symplectic fom θ 1 i.e. Θ = 1 2 θ 1 µν dx µ dx ν, Θ = ±iθ e.o.m. µ θ 1 µν = 0 µν, 5 / 28

9 Yang-Mills Matix models III M 2n endowed with a Poisson stuctue i[x µ, X ν ] {x µ, x ν } P B = θ µν (x) eective metic G µν = e σ θ µρ θ νσ g ρσ = (J 2 ) µ ρg ρν, e σ det θµν 1 det Gρσ special case: 2n = 4 det G µν = det g µν opens possibility fo special class of geometies whee G µν = g µν J 2 = 1 coesponds to a self-dual symplectic fom θ 1 i.e. Θ = 1 2 θ 1 µν dx µ dx ν, Θ = ±iθ e.o.m. µ θ 1 µν = 0 µν, 5 / 28

10 Yang-Mills Matix models III M 2n endowed with a Poisson stuctue i[x µ, X ν ] {x µ, x ν } P B = θ µν (x) eective metic G µν = e σ θ µρ θ νσ g ρσ = (J 2 ) µ ρg ρν, e σ det θµν 1 det Gρσ special case: 2n = 4 det G µν = det g µν opens possibility fo special class of geometies whee G µν = g µν J 2 = 1 coesponds to a self-dual symplectic fom θ 1 i.e. Θ = 1 2 θ 1 µν dx µ dx ν, Θ = ±iθ e.o.m. µ θ 1 µν = 0 µν, 5 / 28

11 Yang-Mills Matix models IV example: scala eld φ on M 4 in the semi-classical limit whee X a x a ae mee coodinates S[φ] = T[X a, φ][x c, φ]η ac d 4 x det θµν 1 {X a, φ} P B {X c, φ} P B η ac = d 4 x det G µν e σ θ µν µ x a ν φ θ ρσ ρ x c σ φ η ac = d 4 x det G µν G νσ ν φ σ φ, natual vecto elds: e a (f) := i[x a, f] θ µν µ x a ν f 6 / 28

12 Yang-Mills Matix models IV example: scala eld φ on M 4 in the semi-classical limit whee X a x a ae mee coodinates S[φ] = T[X a, φ][x c, φ]η ac d 4 x det θµν 1 {X a, φ} P B {X c, φ} P B η ac = d 4 x det G µν e σ θ µν µ x a ν φ θ ρσ ρ x c σ φ η ac = d 4 x det G µν G νσ ν φ σ φ, natual vecto elds: e a (f) := i[x a, f] θ µν µ x a ν f 6 / 28

13 Yang-Mills Matix models V Also possible to add U(N) gauge elds A to the matix model (fo simplicity, conside only 4 dimensions): Y µ = X µ θ µν A ν covaiant coodinates whee the A µ ae some U(N) valued elds. Field stength tenso appeas in semiclassical limit of commutato: [Y µ, Y ν ] i (1 θ ρσ A σ ρ ) θ µν iθ µρ θ νσ F ρσ S Y M Yang-Mills action Tuns out, that this descibes SU(N) gauge elds coupled to gavity whee U(1) gauge eld becomes geometical d.o.f. (see e.g. eview of H. Steinacke, axiv: ) 7 / 28

14 Yang-Mills Matix models V Also possible to add U(N) gauge elds A to the matix model (fo simplicity, conside only 4 dimensions): Y µ = X µ θ µν A ν covaiant coodinates whee the A µ ae some U(N) valued elds. Field stength tenso appeas in semiclassical limit of commutato: [Y µ, Y ν ] i (1 θ ρσ A σ ρ ) θ µν iθ µρ θ νσ F ρσ S Y M Yang-Mills action Tuns out, that this descibes SU(N) gauge elds coupled to gavity whee U(1) gauge eld becomes geometical d.o.f. (see e.g. eview of H. Steinacke, axiv: ) 7 / 28

15 Yang-Mills Matix models V Also possible to add U(N) gauge elds A to the matix model (fo simplicity, conside only 4 dimensions): Y µ = X µ θ µν A ν covaiant coodinates whee the A µ ae some U(N) valued elds. Field stength tenso appeas in semiclassical limit of commutato: [Y µ, Y ν ] i (1 θ ρσ A σ ρ ) θ µν iθ µρ θ νσ F ρσ S Y M Yang-Mills action Tuns out, that this descibes SU(N) gauge elds coupled to gavity whee U(1) gauge eld becomes geometical d.o.f. (see e.g. eview of H. Steinacke, axiv: ) 7 / 28

16 Enegy-momentum tenso when G µν = g µν S Y M T[X a, X b ] 2 : T ab = H ab H 4 ηab, [ ] H ab = 1 2 [X a, X c ], [X b, X c ] η cc, + H = H ab η ab, matix wad-identity: [X a, T a b ]η aa = 0 semiclassical limit: T ab e σ P ab N, H ab e σ P ab T, PT ab = g µν µ x a ν x b, PN ab = η ab PT ab whee P N,T ae the pojectos on the nomal esp. tangential space at p M 4. This means that P ab T µ x b = µ x a, P 2 T = P T, P ab N µ x b = 0, P 2 N = P N 8 / 28

17 Enegy-momentum tenso when G µν = g µν S Y M T[X a, X b ] 2 : T ab = H ab H 4 ηab, [ ] H ab = 1 2 [X a, X c ], [X b, X c ] η cc, + H = H ab η ab, matix wad-identity: [X a, T a b ]η aa = 0 semiclassical limit: T ab e σ P ab N, H ab e σ P ab T, PT ab = g µν µ x a ν x b, PN ab = η ab PT ab whee P N,T ae the pojectos on the nomal esp. tangential space at p M 4. This means that P ab T µ x b = µ x a, P 2 T = P T, P ab N µ x b = 0, P 2 N = P N 8 / 28

18 Enegy-momentum tenso when G µν = g µν S Y M T[X a, X b ] 2 : T ab = H ab H 4 ηab, [ ] H ab = 1 2 [X a, X c ], [X b, X c ] η cc, + H = H ab η ab, matix wad-identity: [X a, T a b ]η aa = 0 semiclassical limit: T ab e σ P ab N, H ab e σ P ab T, PT ab = g µν µ x a ν x b, PN ab = η ab PT ab whee P N,T ae the pojectos on the nomal esp. tangential space at p M 4. This means that P ab T µ x b = µ x a, P 2 T = P T, P ab N µ x b = 0, P 2 N = P N 8 / 28

19 Enegy-momentum tenso when G µν = g µν S Y M T[X a, X b ] 2 : T ab = H ab H 4 ηab, [ ] H ab = 1 2 [X a, X c ], [X b, X c ] η cc, + H = H ab η ab, matix wad-identity: [X a, T a b ]η aa = 0 semiclassical limit: T ab e σ P ab N, H ab e σ P ab T, PT ab = g µν µ x a ν x b, PN ab = η ab PT ab whee P N,T ae the pojectos on the nomal esp. tangential space at p M 4. This means that P ab T µ x b = µ x a, P 2 T = P T, P ab N µ x b = 0, P 2 N = P N 8 / 28

20 Cuvatue Can use pojecto P N to wite down covaiant deivatives g, i.e. P ab N µ ν x b = (η ab P ab T ) µ ν x b = ( µ ν Γ ρ µν ρ )x a = µ ν x a fom which follows µ x a ν ρ x a = 0 and P ab N µ ν x b = µ ν x a. Riemann tenso: R ρσνµ = R ρσν τ τ x a µ x a = [ ρ, σ ] ν x a µ x a = σ µ x a ρ ν x a σ ν x a µ ρ x a ( G.-C. theo.) = P ab N ( σ µ x a ρ ν x b σ ν x a µ ρ x b ) R = g x a g x a µ ν x a µ ν x a 9 / 28

21 Cuvatue Can use pojecto P N to wite down covaiant deivatives g, i.e. P ab N µ ν x b = (η ab P ab T ) µ ν x b = ( µ ν Γ ρ µν ρ )x a = µ ν x a fom which follows µ x a ν ρ x a = 0 and P ab N µ ν x b = µ ν x a. Riemann tenso: R ρσνµ = R ρσν τ τ x a µ x a = [ ρ, σ ] ν x a µ x a = σ µ x a ρ ν x a σ ν x a µ ρ x a ( G.-C. theo.) = P ab N ( σ µ x a ρ ν x b σ ν x a µ ρ x b ) R = g x a g x a µ ν x a µ ν x a 9 / 28

22 Cuvatue Can use pojecto P N to wite down covaiant deivatives g, i.e. P ab N µ ν x b = (η ab P ab T ) µ ν x b = ( µ ν Γ ρ µν ρ )x a = µ ν x a fom which follows µ x a ν ρ x a = 0 and P ab N µ ν x b = µ ν x a. Riemann tenso: R ρσνµ = R ρσν τ τ x a µ x a = [ ρ, σ ] ν x a µ x a = σ µ x a ρ ν x a σ ν x a µ ρ x a ( G.-C. theo.) = P ab N ( σ µ x a ρ ν x b σ ν x a µ ρ x b ) R = g x a g x a µ ν x a µ ν x a 9 / 28

23 Cuvatue Can use pojecto P N to wite down covaiant deivatives g, i.e. P ab N µ ν x b = (η ab P ab T ) µ ν x b = ( µ ν Γ ρ µν ρ )x a = µ ν x a fom which follows µ x a ν ρ x a = 0 and P ab N µ ν x b = µ ν x a. Riemann tenso: R ρσνµ = R ρσν τ τ x a µ x a = [ ρ, σ ] ν x a µ x a = σ µ x a ρ ν x a σ ν x a µ ρ x a ( G.-C. theo.) = P ab N ( σ µ x a ρ ν x b σ ν x a µ ρ x b ) R = g x a g x a µ ν x a µ ν x a 9 / 28

24 One-loop eective action D. Klamme and H. Steinacke, JHEP 02 (2010) 074: ( 1 S Ψ = T 4 [Xa, X b ][X a, X b ] + 1 ) 2 ψγ a [X a, Ψ] Γ Ψ = k [ g 16π 2 4Λ 4 + Λ 2( 1 3 R µ σ µ σ e σ θ µν θ ρσ R µνρσ + 1 ) 4 gx a g x a ] + O(log Λ) 10 / 28

25 One-loop eective action D. Klamme and H. Steinacke, JHEP 02 (2010) 074: ( 1 S Ψ = T 4 [Xa, X b ][X a, X b ] + 1 ) 2 ψγ a [X a, Ψ] Γ Ψ = k [ g 16π 2 4Λ 4 + Λ 2( 1 3 R µ σ µ σ e σ θ µν θ ρσ R µνρσ + 1 ) 4 gx a g x a ] + O(log Λ) 10 / 28

26 Extensions to the matix model action compae with: ( S 6 = T α X a X a + β ) 2 [Xc, [X a, X b ]][X c, [X a, X b ]] α + β g e σ (2π) 2 g x a g x a + β (2π) 2 with X a [X b, [X b, X a ]]. ( ) g 1 2 θµρ θ ηα R µρηα 2R + e σ µ σ µ σ extensions of ode 10 ) S E-H = T (2T ab X a X b T ab H ab 2 g e 2σ (2π) 2 R 11 / 28

27 Extensions to the matix model action compae with: ( S 6 = T α X a X a + β ) 2 [Xc, [X a, X b ]][X c, [X a, X b ]] α + β g e σ (2π) 2 g x a g x a + β (2π) 2 with X a [X b, [X b, X a ]]. ( ) g 1 2 θµρ θ ηα R µρηα 2R + e σ µ σ µ σ extensions of ode 10 ) S E-H = T (2T ab X a X b T ab H ab 2 g e 2σ (2π) 2 R 11 / 28

28 Deviations fom G = g ode 10-tems lead to E-H action only fo G = g, but vaiation equies G µν = g µν + h µν d.o.f: φ i and A µ, i.e. vaiations g µν = η µν + µ φ i (x) ν φ j (x)η ij, θ 1 µν = θ 1 µν + F µν, F µν = µ A ν ν A µ δ φ g µν = δ φ G µν =: h (φ) µν, δ A θ 1 µν = µ δa ν ν δa µ, G µν = g µν + δ A G µν Seveal matix tems ae semiclassically equivalent fo G = g, but not fo G g. futhe tems possible (wok in pogess) 12 / 28

29 Deviations fom G = g ode 10-tems lead to E-H action only fo G = g, but vaiation equies G µν = g µν + h µν d.o.f: φ i and A µ, i.e. vaiations g µν = η µν + µ φ i (x) ν φ j (x)η ij, θ 1 µν = θ 1 µν + F µν, F µν = µ A ν ν A µ δ φ g µν = δ φ G µν =: h (φ) µν, δ A θ 1 µν = µ δa ν ν δa µ, G µν = g µν + δ A G µν Seveal matix tems ae semiclassically equivalent fo G = g, but not fo G g. futhe tems possible (wok in pogess) 12 / 28

30 Deviations fom G = g ode 10-tems lead to E-H action only fo G = g, but vaiation equies G µν = g µν + h µν d.o.f: φ i and A µ, i.e. vaiations g µν = η µν + µ φ i (x) ν φ j (x)η ij, θ 1 µν = θ 1 µν + F µν, F µν = µ A ν ν A µ δ φ g µν = δ φ G µν =: h (φ) µν, δ A θ 1 µν = µ δa ν ν δa µ, G µν = g µν + δ A G µν Seveal matix tems ae semiclassically equivalent fo G = g, but not fo G g. futhe tems possible (wok in pogess) 12 / 28

31 Deviations fom G = g ode 10-tems lead to E-H action only fo G = g, but vaiation equies G µν = g µν + h µν d.o.f: φ i and A µ, i.e. vaiations g µν = η µν + µ φ i (x) ν φ j (x)η ij, θ 1 µν = θ 1 µν + F µν, F µν = µ A ν ν A µ δ φ g µν = δ φ G µν =: h (φ) µν, δ A θ 1 µν = µ δa ν ν δa µ, G µν = g µν + δ A G µν Seveal matix tems ae semiclassically equivalent fo G = g, but not fo G g. futhe tems possible (wok in pogess) 12 / 28

32 Embedding of metic ( ds 2 = 1 c ) ( dt 2 S + 1 c ) 1 d dω 2 Conside Eddington-Finkelstein coodinates and dene: t = t S + ( ), = + c ln 1 c, ( ds 2 = 1 c need 3 exta dimensions: ) dt c dtd + ( 1 + c ) d dω 2 φ 1 + iφ 2 = φ 3 e iω(t+), φ 3 = 1 ω c, whee φ 3 is time-like 13 / 28

33 Embedding of metic ( ds 2 = 1 c ) ( dt 2 S + 1 c ) 1 d dω 2 Conside Eddington-Finkelstein coodinates and dene: t = t S + ( ), = + c ln 1 c, ( ds 2 = 1 c need 3 exta dimensions: ) dt c dtd + ( 1 + c ) d dω 2 φ 1 + iφ 2 = φ 3 e iω(t+), φ 3 = 1 ω c, whee φ 3 is time-like 13 / 28

34 Embedding of metic ( ds 2 = 1 c ) ( dt 2 S + 1 c ) 1 d dω 2 Conside Eddington-Finkelstein coodinates and dene: t = t S + ( ), = + c ln 1 c, ( ds 2 = 1 c need 3 exta dimensions: ) dt c dtd + ( 1 + c ) d dω 2 φ 1 + iφ 2 = φ 3 e iω(t+), φ 3 = 1 ω c, whee φ 3 is time-like 13 / 28

35 Embedding of metic II 7-dim. embedding given by x a = 1 c ω 1 c ω t cos ϕ sin ϑ sin ϕ sin ϑ cos ϑ cos (ω(t + )) sin (ω(t + )) 1 ω c with backgound metic η ab = diag(, +, +, +, +, +, ). 14 / 28

36 Embedded black hole 15 / 28

37 Symplectic fom Requie Θ = iθ, so that G µν = e σ θ µρ θ νσ g ρσ = g µν and lim e σ = const. 0. Solution: Θ = ie dt S + B dϕ, E = c 1 (cos ϑd γ sin ϑdϑ) = d(f() cos ϑ), ( B = c 1 2 sin ϑ cos ϑdϑ + sin 2 ϑd ) = c 1 2 d(2 sin 2 ϑ), ( γ = 1 ) c, f() = c 1 γ, f = c 1 = const., fom which follows e σ = c 2 1 ( 1 ) c sin2 ϑ c 2 1e σ. 16 / 28

38 Symplectic fom Requie Θ = iθ, so that G µν = e σ θ µρ θ νσ g ρσ = g µν and lim e σ = const. 0. Solution: Θ = ie dt S + B dϕ, E = c 1 (cos ϑd γ sin ϑdϑ) = d(f() cos ϑ), ( B = c 1 2 sin ϑ cos ϑdϑ + sin 2 ϑd ) = c 1 2 d(2 sin 2 ϑ), ( γ = 1 ) c, f() = c 1 γ, f = c 1 = const., fom which follows e σ = c 2 1 ( 1 ) c sin2 ϑ c 2 1e σ. 16 / 28

39 Symplectic fom Requie Θ = iθ, so that G µν = e σ θ µρ θ νσ g ρσ = g µν and lim e σ = const. 0. Solution: Θ = ie dt S + B dϕ, E = c 1 (cos ϑd γ sin ϑdϑ) = d(f() cos ϑ), ( B = c 1 2 sin ϑ cos ϑdϑ + sin 2 ϑd ) = c 1 2 d(2 sin 2 ϑ), ( γ = 1 ) c, f() = c 1 γ, f = c 1 = const., fom which follows e σ = c 2 1 ( 1 ) c sin2 ϑ c 2 1e σ. 16 / 28

40 Daboux coodinates x µ D = {H ts, t S, H ϕ, ϕ} coesponding to Killing vecto elds V ts = ts, V ϕ = ϕ whee the symplectic fom Θ is constant: Θ = ic 1 dh ts dt S + c 1 dh ϕ dϕ, = c 1 d (ih ts dt S + H ϕ dϕ), H ts = γ cos ϑ, Relations to the Killing vecto elds: E = c 1 dh ts = c 1 E µ dx µ = i Vts Θ, B = c 1 dh ϕ = c 1 B µ dx µ = i Vϕ Θ, H ϕ = sin 2 ϑ E µ = Vtsθ ν νµ 1, B µ = Vϕ ν θνµ 1, ds 2 D = γdt 2 S + e σ γ dh2 ts + 2 sin 2 ϑdϕ 2 + e σ 2 sin 2 ϑ dh2 ϕ 17 / 28

41 Daboux coodinates x µ D = {H ts, t S, H ϕ, ϕ} coesponding to Killing vecto elds V ts = ts, V ϕ = ϕ whee the symplectic fom Θ is constant: Θ = ic 1 dh ts dt S + c 1 dh ϕ dϕ, = c 1 d (ih ts dt S + H ϕ dϕ), H ts = γ cos ϑ, Relations to the Killing vecto elds: E = c 1 dh ts = c 1 E µ dx µ = i Vts Θ, B = c 1 dh ϕ = c 1 B µ dx µ = i Vϕ Θ, H ϕ = sin 2 ϑ E µ = Vtsθ ν νµ 1, B µ = Vϕ ν θνµ 1, ds 2 D = γdt 2 S + e σ γ dh2 ts + 2 sin 2 ϑdϕ 2 + e σ 2 sin 2 ϑ dh2 ϕ 17 / 28

42 Daboux coodinates x µ D = {H ts, t S, H ϕ, ϕ} coesponding to Killing vecto elds V ts = ts, V ϕ = ϕ whee the symplectic fom Θ is constant: Θ = ic 1 dh ts dt S + c 1 dh ϕ dϕ, = c 1 d (ih ts dt S + H ϕ dϕ), H ts = γ cos ϑ, Relations to the Killing vecto elds: E = c 1 dh ts = c 1 E µ dx µ = i Vts Θ, B = c 1 dh ϕ = c 1 B µ dx µ = i Vϕ Θ, H ϕ = sin 2 ϑ E µ = Vtsθ ν νµ 1, B µ = Vϕ ν θνµ 1, ds 2 D = γdt 2 S + e σ γ dh2 ts + 2 sin 2 ϑdϕ 2 + e σ 2 sin 2 ϑ dh2 ϕ 17 / 28

43 Daboux coodinates x µ D = {H ts, t S, H ϕ, ϕ} coesponding to Killing vecto elds V ts = ts, V ϕ = ϕ whee the symplectic fom Θ is constant: Θ = ic 1 dh ts dt S + c 1 dh ϕ dϕ, = c 1 d (ih ts dt S + H ϕ dϕ), H ts = γ cos ϑ, Relations to the Killing vecto elds: E = c 1 dh ts = c 1 E µ dx µ = i Vts Θ, B = c 1 dh ϕ = c 1 B µ dx µ = i Vϕ Θ, H ϕ = sin 2 ϑ E µ = Vtsθ ν νµ 1, B µ = Vϕ ν θνµ 1, ds 2 D = γdt 2 S + e σ γ dh2 ts + 2 sin 2 ϑdϕ 2 + e σ 2 sin 2 ϑ dh2 ϕ 17 / 28

44 Sta poduct A Moyal type sta poduct can easily be dened as with (g h)(x D ) = g(x D )e i 2 θ µν D = ɛ µθ µν D ν h(x D ), 0 i 0 0 i , whee ɛ = 1/c 1 1 denotes the expansion paamete. 18 / 28

45 Sta poduct II... o in embedding coodinates: [ ( ( iɛ i ze σ c (g h)(x) = g(x) exp t 2 2 γ + ) ie σ z t +(( t ) z ce σ z +( 2 x x + ) ( y y 1 ) x x 2 +y 2 y y ) )] x h(x) whee cae must be taken with the sequence of opeatos and the side they act on. Highe odes in this sta poduct lead to non-commutative coections to the embedding geomety, e.g.: φ 1 φ 1 + φ 2 φ 2 φ 3 φ 3 19 / 28

46 Sta poduct II... o in embedding coodinates: [ ( ( iɛ i ze σ c (g h)(x) = g(x) exp t 2 2 γ + ) ie σ z t +(( t ) z ce σ z +( 2 x x + ) ( y y 1 ) x x 2 +y 2 y y ) )] x h(x) whee cae must be taken with the sequence of opeatos and the side they act on. Highe odes in this sta poduct lead to non-commutative coections to the embedding geomety, e.g.: φ 1 φ 1 + φ 2 φ 2 φ 3 φ 3 19 / 28

47 Sta commutatos fo geomety i [ x a, x b] = ɛe σ 0 cy cx izf 2 i (1) izf 21 (1) cy 0 e σ cyz yf (γ) yf 21 (γ) cx e σ 0 cxz xf (γ) xf 21 (γ) i cyz cxz 3 0 iωφ 3 2 iωφ 1 0 izf + 12 (1) yf + 12 (γ) xf + 12 (γ) iωφ 2 0 iωzφ2 3 izf 21 (1) xf 21 (γ) yf 21 (γ) izφ3 yγφ O(ɛ 3 ), with izφ yγφ3 2 2 xγφ iωzφ 3φ iωzφ iωφ 2 3 iωzφ φ xγφ3 iωzφ 2 0 3φ 2 iωzφ 3φ ( ) Y f ij ± (Y ) = 2 φ i ± ωφ j. 20 / 28

48 Embedding of Reissne-Nodstöm metic metic in spheical coodinates x µ = {t,, ϑ, ϕ}: ) ) 1 ds 2 2m q2 = (1 + 2 d t 2 2m q2 + (1 + 2 d dω which has two concentic hoizons at h = (m ± ) m 2 q 2 Shift the time-coodinate accoding to ( ) 1 t = t + ( ), with d 1 2m + q2 d, 2 and aive at ds 2 2m q2 = (1 + + ( m q2 2 ) ( 2m dt ) d dω. ) q2 2 dtd 21 / 28

49 Embedding of Reissne-Nodstöm metic metic in spheical coodinates x µ = {t,, ϑ, ϕ}: ) ) 1 ds 2 2m q2 = (1 + 2 d t 2 2m q2 + (1 + 2 d dω which has two concentic hoizons at h = (m ± ) m 2 q 2 Shift the time-coodinate accoding to ( ) 1 t = t + ( ), with d 1 2m + q2 d, 2 and aive at ds 2 2m q2 = (1 + + ( m q2 2 ) ( 2m dt ) d dω. ) q2 2 dtd 21 / 28

50 Embedding of metic II 10-dimensional embedding M 1,3 R 4,6 with additional coodinates φ i given by φ 1 + iφ 2 = φ 3 e iω(t+), φ 3 = 1 2m ω, φ 4 + iφ 5 = φ 6 e iω(t+), φ 6 = q ω φ 3, φ 4 and φ 5 ae time-like coodinates. 22 / 28

51 Symplectic fom and Daboux coodinates Θ = 1 ɛ H t = γ cos ϑ, γ = (1 ( idh t d t + dh ϕ dϕ ), ) 2m q2 + 2, e σ = γ sin 2 ϑ + H ϕ = 2 2 ) (1 q2 2 sin 2 ϑ, ) 2 (1 q2 2 cos 2 ϑ ds 2 D = γd t 2 + e σ γ dh2 t + 2 sin 2 ϑdϕ 2 + e σ 2 sin 2 ϑ dh2 ϕ 23 / 28

52 Symplectic fom and Daboux coodinates Θ = 1 ɛ H t = γ cos ϑ, γ = (1 ( idh t d t + dh ϕ dϕ ), ) 2m q2 + 2, e σ = γ sin 2 ϑ + H ϕ = 2 2 ) (1 q2 2 sin 2 ϑ, ) 2 (1 q2 2 cos 2 ϑ ds 2 D = γd t 2 + e σ γ dh2 t + 2 sin 2 ϑdϕ 2 + e σ 2 sin 2 ϑ dh2 ϕ 23 / 28

53 Symplectic fom and Daboux coodinates Θ = 1 ɛ H t = γ cos ϑ, γ = (1 ( idh t d t + dh ϕ dϕ ), ) 2m q2 + 2, e σ = γ sin 2 ϑ + H ϕ = 2 2 ) (1 q2 2 sin 2 ϑ, ) 2 (1 q2 2 cos 2 ϑ ds 2 D = γd t 2 + e σ γ dh2 t + 2 sin 2 ϑdϕ 2 + e σ 2 sin 2 ϑ dh2 ϕ 23 / 28

54 Sta poduct fo geomety A Moyal type sta poduct can again be dened as (g h)(x D ) = g(x D )e i 2 with the same block-diagonal θ µν as befoe. µθ µν D ν h(x D ),... and once moe, highe odes in the sta poduct lead to non-commutative coections to the embedding geomety, e.g.: φ 1 φ 1 + φ 2 φ 2 φ 3 φ 3, φ 4 φ 4 + φ 5 φ 5 φ 6 φ / 28

55 Sta poduct fo geomety A Moyal type sta poduct can again be dened as (g h)(x D ) = g(x D )e i 2 with the same block-diagonal θ µν as befoe. µθ µν D ν h(x D ),... and once moe, highe odes in the sta poduct lead to non-commutative coections to the embedding geomety, e.g.: φ 1 φ 1 + φ 2 φ 2 φ 3 φ 3, φ 4 φ 4 + φ 5 φ 5 φ 6 φ / 28

56 Sta commutatos fo geomety i [x µ, x ν ] θ µν = ɛe σ (1 γ)y 0 + iq2 xz (1 γ)x 4 (1 γ)y 0 e (1 γ)x e ς 0 iβ i [φ i, x µ ] ɛe σ yzη 2 + iq2 yz iβ 4 ς yzη 2 xzη 2 xzη 0 2 izαf 12( + 1 2) yf 12( + γ 2 ) iq2 xzωφ 2 xf 12( + γ 2 ) 4 iq2 yzωφ 2 4 izαf21( 1 2) yf21( γ 2 ) + iq2 xzωφ 1 xf21( γ 2 ) 4 + iq2 yzωφ 1 4 izφ 3α yγφ 3 xγφ izαf + 45 (1) yf + 45 (γ) iq2 xzωφ 5 xf (γ) iq2 yzωφ 5 4 izαf 54 (1) yf 54 (γ) + iq2 xzωφ 4 xf 4 54 (γ) + iq2 yzωφ 4 4 izφ 6α yγφ 6 xγφ iωφ 2 β iωφ 1 β iωφ 5 β iωφ 4 β 25 / 28

57 Sta commutatos fo geomety II i [φ i, φ j ] ɛe σ 0 iωzφ 2 3 α iωzφ 3φ 2α 2 2 iωzφ 1φ 5α 2 2 iωzαg φ 2 2 iωzφ 3φ 5α 2 with iωzφ 2 3 α iωzφ 3φ 2α iωzφ 1φ 5α iωzαg φ iωzφ 3φ 5α iωzφ 3φ 1α iωzαg φ iωzφ 2φ 4α iωzφ 3φ 4α iωzφ 3φ 1α iωzφ 2 0 3φ 5α 2 2 iωzφ3φ4α iωzαg φ iωzφ 3φ 5α iωzφ α iωzφ 5φ 6α 2 2 iωzφ 2φ 4α iωzφ 3φ 4α iωzφ α iωzφ 0 4φ 6α 2 2 iωzφ 3φ 4α iωzφ 0 5φ 6α iωzφ 4φ 6α ( ) ) Y f ± ij (Y ) = φ i ± ωφ j, α = (1 q2 2, ( )) e ς = (γ + 2 z2 m 2 q2 2, β = (1 q2 z 2 ) 4, g φ = (φ 3 φ 6 + φ 1 φ 5 ) = (φ 3 φ 6 + φ 2 φ 4 ). 26 / 28

58 and Outlook Have shown, that E-H action can emege in the famewok of matix models. Discussed explicit embeddings of and geometies including self-dual symplectic foms. Open questions: deviations fom G = g, highe ode quantum eects, etc. (wok in pogess). 27 / 28

59 and Outlook Have shown, that E-H action can emege in the famewok of matix models. Discussed explicit embeddings of and geometies including self-dual symplectic foms. Open questions: deviations fom G = g, highe ode quantum eects, etc. (wok in pogess). 27 / 28

60 and Outlook Have shown, that E-H action can emege in the famewok of matix models. Discussed explicit embeddings of and geometies including self-dual symplectic foms. Open questions: deviations fom G = g, highe ode quantum eects, etc. (wok in pogess). 27 / 28

61 Refeences D. N. Blaschke, H. Steinacke, Cuvatue and Actions fo Matix Models, submitted to Class. Quant. Gav., [axiv: ]. D. N. Blaschke, H. Steinacke, Cuvatue and Actions fo Matix Models II, wok in pogess D. N. Blaschke, H. Steinacke, fom Matix Models, [axiv: ]. Thank you fo you attention! 28 / 28