EMERGENT GEOMETRY FROM QUANTISED SPACETIME

Transcription

1 EMERGENT GEOMETRY FROM QUANTISED SPACETIME M.SIVAKUMAR UNIVERSITY OF HYDERABAD February 24, 2011 H.S.Yang and MS - (Phys Rev D ) Quantum Field Theory -IISER-Pune

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3 Organisation Quantised space time

4 Organisation Quantised space time Matrix model-emergent geometry

5 Organisation Quantised space time Matrix model-emergent geometry Mass deformed IKKT model

6 Organisation Quantised space time Matrix model-emergent geometry Mass deformed IKKT model Conclusion

7 Quantised Spacetime-Motivation Gravity -fundamental interaction-geometry.

8 Quantised Spacetime-Motivation Gravity -fundamental interaction-geometry. Gen Rel + quantum = quantum structure of spacetime

9 Quantised Spacetime-Motivation Gravity -fundamental interaction-geometry. Gen Rel + quantum = quantum structure of spacetime quantised spacetime : Moyal spacetime

10 Quantised Spacetime-Motivation Gravity -fundamental interaction-geometry. Gen Rel + quantum = quantum structure of spacetime quantised spacetime : Moyal spacetime Pre-geometric notions:matrix models to symplectic geometry to Riemann geometry?

11 Emergent gravity-motivations Sakharaov paradigm:induced gravity-

12 Emergent gravity-motivations Sakharaov paradigm:induced gravity- GTR -Thermodynamics nexus(jacobson-padmanabhan..)

13 Emergent gravity-motivations Sakharaov paradigm:induced gravity- GTR -Thermodynamics nexus(jacobson-padmanabhan..) Thermodynamics > stat mech of atoms.

14 Emergent gravity-motivations Sakharaov paradigm:induced gravity- GTR -Thermodynamics nexus(jacobson-padmanabhan..) Thermodynamics > stat mech of atoms. GTR dynamics of spacetime > emergent spacetime

15 Emergent gravity-motivations Sakharaov paradigm:induced gravity- GTR -Thermodynamics nexus(jacobson-padmanabhan..) Thermodynamics > stat mech of atoms. GTR dynamics of spacetime > emergent spacetime Dark-energy, quantum gravity.. Gravity not fundamental?

16 Emergent gravity-motivations Sakharaov paradigm:induced gravity- GTR -Thermodynamics nexus(jacobson-padmanabhan..) Thermodynamics > stat mech of atoms. GTR dynamics of spacetime > emergent spacetime Dark-energy, quantum gravity.. Gravity not fundamental? emergent gravity from quantised spacetime?rivelles(2002),hs Yang(2004) (Steinacker-2008)

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18 quantised spacetime-moyal Plane Endow spacetime with a symplectic structure B ab

19 quantised spacetime-moyal Plane Endow spacetime with a symplectic structure B ab quantize the spacetime y a with its Poisson structure θ ab (B 1 ) ab, described by {y a, y b } = iθ ab

20 quantised spacetime-moyal Plane Endow spacetime with a symplectic structure B ab quantize the spacetime y a with its Poisson structure θ ab (B 1 ) ab, described by {y a, y b } = iθ ab Deformation :(f.g)(x) (f g)(x)

21 quantised spacetime-moyal Plane Endow spacetime with a symplectic structure B ab quantize the spacetime y a with its Poisson structure θ ab (B 1 ) ab, described by {y a, y b } = iθ ab Deformation :(f.g)(x) (f g)(x) [f, g] > {f, g} θ +..

22 NC EM as General Relativity? U(1) gauge transformation is translation: exp ik.y f (x) exp iky = f (x + θk). Infinitesmally ib ab [y b, f ] = a f

23 NC EM as General Relativity? U(1) gauge transformation is translation: exp ik.y f (x) exp iky = f (x + θk). Infinitesmally ib ab [y b, f ] = a f NC U(1): No local gauge invariant observables,non-linearity :similarity to GTR.

24 NC EM as General Relativity? U(1) gauge transformation is translation: exp ik.y f (x) exp iky = f (x + θk). Infinitesmally ib ab [y b, f ] = a f NC U(1): No local gauge invariant observables,non-linearity :similarity to GTR. NC scalar-u(1)coupling scalar- linear gravity coupling. (Rivelles):UV-IR mixing =effect of emergent gravity (Grosse-Steinacker)

25 NC EM as General Relativity? U(1) gauge transformation is translation: exp ik.y f (x) exp iky = f (x + θk). Infinitesmally ib ab [y b, f ] = a f NC U(1): No local gauge invariant observables,non-linearity :similarity to GTR. NC scalar-u(1)coupling scalar- linear gravity coupling. (Rivelles):UV-IR mixing =effect of emergent gravity (Grosse-Steinacker) the gauge symmetry acting on U(1) gauge fields as A A + dφ is a diffeomorphism symmetry generated by a vector field X φ satisfying L Xφ B = 0,

26 NC EM as General Relativity? U(1) gauge transformation is translation: exp ik.y f (x) exp iky = f (x + θk). Infinitesmally ib ab [y b, f ] = a f NC U(1): No local gauge invariant observables,non-linearity :similarity to GTR. NC scalar-u(1)coupling scalar- linear gravity coupling. (Rivelles):UV-IR mixing =effect of emergent gravity (Grosse-Steinacker) the gauge symmetry acting on U(1) gauge fields as A A + dφ is a diffeomorphism symmetry generated by a vector field X φ satisfying L Xφ B = 0, Darboux theorem of symplectic geometry as equivalence principle; can always locally eliminate EM by coordinate transformation.(h.s.yang) Can U(1) gauge theory on symplectic lead to gravity?

27 NC EM as General Relativity? U(1) gauge transformation is translation: exp ik.y f (x) exp iky = f (x + θk). Infinitesmally ib ab [y b, f ] = a f NC U(1): No local gauge invariant observables,non-linearity :similarity to GTR. NC scalar-u(1)coupling scalar- linear gravity coupling. (Rivelles):UV-IR mixing =effect of emergent gravity (Grosse-Steinacker) the gauge symmetry acting on U(1) gauge fields as A A + dφ is a diffeomorphism symmetry generated by a vector field X φ satisfying L Xφ B = 0, Darboux theorem of symplectic geometry as equivalence principle; can always locally eliminate EM by coordinate transformation.(h.s.yang) Can U(1) gauge theory on symplectic lead to gravity? Can quantised spacetime be emergent?

28 IKKT model (1996) Matrix models spacetime structure. S IKKT = 1 4 Tr[X a, X b ][X a, X b ]

29 IKKT model (1996) Matrix models spacetime structure. S IKKT = 1 4 Tr[X a, X b ][X a, X b ] Eqn of motion: [X a, [X a, X b ]] = 0,

30 IKKT model (1996) Matrix models spacetime structure. S IKKT = 1 4 Tr[X a, X b ][X a, X b ] Eqn of motion: [X a, [X a, X b ]] = 0, A classical solution is given by X a cl = y a, with [y a, y b ] = iθ ab. (NC) spacetime is a solution and not given apriori.

31 IKKT model-fluctuation expand X a θ ab Db around the Moyal vacuum.

32 IKKT model-fluctuation expand X a θ ab Db around the Moyal vacuum. Da (y) = B ab y b + Âa(y)

33 IKKT model-fluctuation expand X a θ ab Db around the Moyal vacuum. Da (y) = B ab y b + Âa(y) UXU symmetry leads to Gauge transform of A i[ D a (y), D b (y)] = a  b (y) b  a (y) i[âa(y), Âb(y)] B ab = F ab (y) B ab.

34 IKKT model-fluctuaion Then the IKKT matrix model becomes NC U(1) gauge theory S NC = 1 4g d 2n yg ac G bd( F B )ab ( F ) B YM 2 cd where G ab = θ ac θ bc

35 IKKT model-fluctuaion Then the IKKT matrix model becomes NC U(1) gauge theory S NC = 1 4g d 2n yg ac G bd( F B )ab ( F ) B YM 2 cd where G ab = θ ac θ bc equations of motion for NC gauge fields : D[a Fbc] = 0

36 IKKT model-fluctuaion Then the IKKT matrix model becomes NC U(1) gauge theory S NC = 1 4g d 2n yg ac G bd( F B )ab ( F ) B YM 2 cd where G ab = θ ac θ bc equations of motion for NC gauge fields : D[a Fbc] = 0 How metric is related to Gauge field?

37 metric from gauge field-hs Yang Poisson algebra provides f X f -Hamiltonian vector field X f (g) = {g, f } θ

38 metric from gauge field-hs Yang Poisson algebra provides f X f -Hamiltonian vector field X f (g) = {g, f } θ f and the corresponding Hamiltonian vector field X f is the Lie algebra homomophism in the sense X {f,g}θ = [X f, X g ]

39 metric from gauge field-hs Yang Poisson algebra provides f X f -Hamiltonian vector field X f (g) = {g, f } θ f and the corresponding Hamiltonian vector field X f is the Lie algebra homomophism in the sense X {f,g}θ = [X f, X g ] Given D a i[ D a (y), f (y)] = θ µν D a(y) y ν f (y) y µ +

40 metric from gauge field-hs Yang Poisson algebra provides f X f -Hamiltonian vector field X f (g) = {g, f } θ f and the corresponding Hamiltonian vector field X f is the Lie algebra homomophism in the sense X {f,g}θ = [X f, X g ] Given D a i[ D a (y), f (y)] = θ µν D a(y) y ν f (y) y µ +

41 metric from gauge field-hs Yang Poisson algebra provides f X f -Hamiltonian vector field X f (g) = {g, f } θ f and the corresponding Hamiltonian vector field X f is the Lie algebra homomophism in the sense X {f,g}θ = [X f, X g ] Given D a i[ D a (y), f (y)] = θ µν D a(y) y ν f (y) y µ + Vector fields V a [](y) to order θ is θ µν D a(y) y ν yµ which form a set of vector fields. But V a are not orthonormal. Orthonormal E a = (λ) 1 V a where λ 2 = det V µ a ds 2 = g ab E a E b

42 Emergent gravity and dark energy- HS Yang (JHEP) eqn of motion for gauge field Einstein eqn for the metric.

43 Emergent gravity and dark energy- HS Yang (JHEP) eqn of motion for gauge field Einstein eqn for the metric. Flat spacetime comes from A (0) a = B ab y b whose field strength is constant = B ab Flat spacetime condensation of gauge field

44 Emergent gravity and dark energy- HS Yang (JHEP) eqn of motion for gauge field Einstein eqn for the metric. Flat spacetime comes from A (0) a = B ab y b whose field strength is constant = B ab Flat spacetime condensation of gauge field Energy associated with vacuum is M 4 P

45 Emergent gravity and dark energy- HS Yang (JHEP) eqn of motion for gauge field Einstein eqn for the metric. Flat spacetime comes from A (0) a = B ab y b whose field strength is constant = B ab Flat spacetime condensation of gauge field Energy associated with vacuum is M 4 P Vacuum energy is used to make flat spacetime, hence does not gravitate! Soln to CC problem!

46 Emergent gravity and dark energy- HS Yang (JHEP) eqn of motion for gauge field Einstein eqn for the metric. Flat spacetime comes from A (0) a = B ab y b whose field strength is constant = B ab Flat spacetime condensation of gauge field Energy associated with vacuum is M 4 P Vacuum energy is used to make flat spacetime, hence does not gravitate! Soln to CC problem! Generalization to constant curvature space?

47 2d Constant curvature manifolds from Matrix models-hs Yang &MSK S mcs = κtr ( i 3! ε ABC X A [X B, X C ] λ 2 X AX A ).

48 2d Constant curvature manifolds from Matrix models-hs Yang &MSK S mcs = κtr ( i 3! ε ABC X A [X B, X C ] λ 2 X AX A ). Equation of motion : [X A, X B ] = iλε AB C X C

49 2d Constant curvature manifolds from Matrix models-hs Yang &MSK S mcs = κtr ( i 3! ε ABC X A [X B, X C ] λ 2 X AX A ). Equation of motion : [X A, X B ] = iλε AB C X C Invariant ( ) R 2 g AB X A X B. They describe a two-dimensional manifold M 2.

50 2d Constant curvature manifolds from Matrix models-hs Yang &MSK S mcs = κtr ( i 3! ε ABC X A [X B, X C ] λ 2 X AX A ). Equation of motion : [X A, X B ] = iλε AB C X C Invariant ( ) R 2 g AB X A X B. They describe a two-dimensional manifold M 2. M 2 is symplectic Local coordinates: 1 2 B abdy a dy b = dy 1 dy 2.

51 2d Constant curvature manifolds from Matrix models-hs Yang &MSK S mcs = κtr ( i 3! ε ABC X A [X B, X C ] λ 2 X AX A ). Equation of motion : [X A, X B ] = iλε AB C X C Invariant ( ) R 2 g AB X A X B. They describe a two-dimensional manifold M 2. M 2 is symplectic Local coordinates: 1 2 B abdy a dy b = dy 1 dy 2. M 2 as a hypersurface embedded in IR 3 or IR 2,1 and described by L A = L A (y a ), X A = L A is classical solution.

52 2d Constant curvature manifolds from Matrix models-hs Yang &MSK S mcs = κtr ( i 3! ε ABC X A [X B, X C ] λ 2 X AX A ). Equation of motion : [X A, X B ] = iλε AB C X C Invariant ( ) R 2 g AB X A X B. They describe a two-dimensional manifold M 2. M 2 is symplectic Local coordinates: 1 2 B abdy a dy b = dy 1 dy 2. M 2 as a hypersurface embedded in IR 3 or IR 2,1 and described by L A = L A (y a ), X A = L A is classical solution. For eg:ds 2 y a = (sinh t, ϕ) L 1 = y, L 2 = 1 + y 2 sin ϕ, L 3 = 1 + y 2 cos ϕ,where y = sinh t. V (0) A = θab L A y b y. a

53 d=2 Constant curvature manifolds from Matrix models

54 d=2 Constant curvature manifolds from Matrix models

55 d=2 Constant curvature manifolds from Matrix models fluctuation of the surface M 2 around the vacuum geometry X A (y) = L A (y) + A A (y).

56 d=2 Constant curvature manifolds from Matrix models fluctuation of the surface M 2 around the vacuum geometry X A (y) = L A (y) + A A (y). The fluctuating coordinate system satisfies the following Poisson bracket relation {X A, X B } θ = ε AB C X C + F AB where F AB = {L A, A B } θ {L B, A A } θ + {A A, A B } θ + ɛ AB C A C.

57 d=2 Constant curvature manifolds from Matrix models fluctuation of the surface M 2 around the vacuum geometry X A (y) = L A (y) + A A (y). The fluctuating coordinate system satisfies the following Poisson bracket relation {X A, X B } θ = ε AB C X C + F AB where F AB = {L A, A B } θ {L B, A A } θ + {A A, A B } θ + ɛ AB C A C. Note F AB should satisfy Jacobi identity.this gives ɛ ABC {X A, {X B, X C } θ } θ = 0. This constraint can be solved by taking F AB (X ) = ɛ ABC F (X ) X C

58 d=2 Constant curvature manifolds from Matrix models {X A, X B } θ = ε ABC G(X ) where the polynomial G(X ) is X C defined in M = IR 3 p,p and given by G(X ) = F (X ) 1 2 g ABX A X B + ρ.

59 d=2 Constant curvature manifolds from Matrix models {X A, X B } θ = ε ABC G(X ) where the polynomial G(X ) is X C defined in M = IR 3 p,p and given by G(X ) = F (X ) 1 2 g ABX A X B + ρ. function F (X ) is an arbitrary polynomial in 3 variables in M defines a two-dimensional surface M 2

60 d=2 Constant curvature manifolds from Matrix models {X A, X B } θ = ε ABC G(X ) where the polynomial G(X ) is X C defined in M = IR 3 p,p and given by G(X ) = F (X ) 1 2 g ABX A X B + ρ. function F (X ) is an arbitrary polynomial in 3 variables in M defines a two-dimensional surface M 2 determine the embedding coordinate by solving the polynomial equation G(X ) = 0.The metric of the two-dimensional surface given by the vector fields V A = θ ab X A (y) y b y +... a

61 d=2 Constant curvature manifolds from Matrix models {X A, X B } θ = ε ABC G(X ) where the polynomial G(X ) is X C defined in M = IR 3 p,p and given by G(X ) = F (X ) 1 2 g ABX A X B + ρ. function F (X ) is an arbitrary polynomial in 3 variables in M defines a two-dimensional surface M 2 determine the embedding coordinate by solving the polynomial equation G(X ) = 0.The metric of the two-dimensional surface given by the vector fields V A = θ ab X A (y) y b y +... a g AB X A y a X B y b is the induced metric for arbitrary potential

62 Constant curvature Matrix model for d=2n Mass deformed IKKT model: S m = S IKKT + m 2 TrX a X a Linearized form :S κ = Tr ( 1 4 M abm ab 1 2κ M ab[x a, X b ] + d 1 2κ X ax a)

63 Constant curvature Matrix model for d=2n Mass deformed IKKT model: S m = S IKKT + m 2 TrX a X a Linearized form :S κ = Tr ( 1 4 M abm ab 1 2κ M ab[x a, X b ] + d 1 2κ X ax a) Symmetry: (X a, M ab ) U(X a, M ab )U

64 Constant curvature Matrix model for d=2n Mass deformed IKKT model: S m = S IKKT + m 2 TrX a X a Linearized form :S κ = Tr ( 1 4 M abm ab 1 2κ M ab[x a, X b ] + d 1 2κ X ax a) Symmetry: (X a, M ab ) U(X a, M ab )U Equation of motion: [X a, X b ] = κm ab [M ab, X b ] + (d 1)X a = 0.

65 Constant curvature Matrix model for d=2n Mass deformed IKKT model: S m = S IKKT + m 2 TrX a X a Linearized form :S κ = Tr ( 1 4 M abm ab 1 2κ M ab[x a, X b ] + d 1 2κ X ax a) Symmetry: (X a, M ab ) U(X a, M ab )U Equation of motion: [X a, X b ] = κm ab [M ab, X b ] + (d 1)X a = 0. The above eqn is has Snyder algebra as solution: [X a, X b ] = κm ab, [X a, M bc ] = g ac X b g ab X c, [M ab, M cd ] = g ac M bd g ad M bc g bc M ad + g bd M ac

66 Constant curvature Matrix model for d=2n Mass deformed IKKT model: S m = S IKKT + m 2 TrX a X a Linearized form :S κ = Tr ( 1 4 M abm ab 1 2κ M ab[x a, X b ] + d 1 2κ X ax a) Symmetry: (X a, M ab ) U(X a, M ab )U Equation of motion: [X a, X b ] = κm ab [M ab, X b ] + (d 1)X a = 0. The above eqn is has Snyder algebra as solution: [X a, X b ] = κm ab, [X a, M bc ] = g ac X b g ab X c, [M ab, M cd ] = g ac M bd g ad M bc g bc M ad + g bd M ac Comments: Snyder algebra is Lorentz algebra in d + 1 dim {M ab &X a } = M AB For d=3 M AB = i λ ɛab C X C forms Snyder algebra.

67 Vaccuum geometry in mass-deformed IKKT To map Matrix algebra to NC * algebra: Snyder algebra from deformation quantisation of a Poisson manifold whose Poisson tensor :Π = 1 2 Lab (x) x a x b

68 Vaccuum geometry in mass-deformed IKKT To map Matrix algebra to NC * algebra: Snyder algebra from deformation quantisation of a Poisson manifold whose Poisson tensor :Π = 1 2 Lab (x) x a x b

69 Vaccuum geometry in mass-deformed IKKT To map Matrix algebra to NC * algebra: Snyder algebra from deformation quantisation of a Poisson manifold whose Poisson tensor :Π = 1 2 Lab (x) x a x b the vector fields for the background gauge fields D a (0) (x) are given by ad bd (0) a [ f ](x) = i[ D a (0) (x), f (x)] = V a (0)µ (x) f (x) x µ +

70 Vaccuum geometry in mass-deformed IKKT To map Matrix algebra to NC * algebra: Snyder algebra from deformation quantisation of a Poisson manifold whose Poisson tensor :Π = 1 2 Lab (x) x a x b the vector fields for the background gauge fields D a (0) (x) are given by ad bd (0) a [ f ](x) = i[ D a (0) (x), f (x)] = V a (0)µ (x) f (x) x µ + The vector fields (V a (0), S (0) ab ) Γ(TM back) as ( differential Lorentz generators of SO(5 p, p) S (0) AB = κ x B x x A A ) x B

71 Vaccuum geometry in mass-deformed IKKT To map Matrix algebra to NC * algebra: Snyder algebra from deformation quantisation of a Poisson manifold whose Poisson tensor :Π = 1 2 Lab (x) x a x b the vector fields for the background gauge fields D a (0) (x) are given by ad bd (0) a [ f ](x) = i[ D a (0) (x), f (x)] = V a (0)µ (x) f (x) x µ + The vector fields (V a (0), S (0) ab ) Γ(TM back) as ( differential Lorentz generators of SO(5 p, p) S (0) AB = κ x B x x A A consider a invariant of the 5d Lorentz algebra g AB x A x B = ( 1) R 2. ) x B

72 Vaccuum geometry in mass-deformed IKKT To map Matrix algebra to NC * algebra: Snyder algebra from deformation quantisation of a Poisson manifold whose Poisson tensor :Π = 1 2 Lab (x) x a x b the vector fields for the background gauge fields D a (0) (x) are given by ad bd (0) a [ f ](x) = i[ D a (0) (x), f (x)] = V a (0)µ (x) f (x) x µ + The vector fields (V a (0), S (0) ab ) Γ(TM back) as ( differential Lorentz generators of SO(5 p, p) S (0) AB = κ x B x x A A consider a invariant of the 5d Lorentz algebra g AB x A x B = ( 1) R 2. ) x B

73 Vaccuum geometry in mass-deformed IKKT To map Matrix algebra to NC * algebra: Snyder algebra from deformation quantisation of a Poisson manifold whose Poisson tensor :Π = 1 2 Lab (x) x a x b the vector fields for the background gauge fields D a (0) (x) are given by ad bd (0) a [ f ](x) = i[ D a (0) (x), f (x)] = V a (0)µ (x) f (x) x µ + The vector fields (V a (0), S (0) ab ) Γ(TM back) as ( differential Lorentz generators of SO(5 p, p) S (0) AB = κ x B x x A A consider a invariant of the 5d Lorentz algebra g AB x A x B = ( 1) R 2. ds 2 = G (0) ab dx a dx b = (det G (0) ab )g ABV a (0)A ) x B V (0)B b dx a dx b = (det G (0) ab )(g abx g 55x a x b )dx a dx b = g AB x A x a xb x b dx a dx b

74 fluctuation Emergent Vacuum geometry is Constant curvature space in d-dimensions such as S d, ds d and AdS d, depending on embedded flat Euclidean or Lorentzian spacetime in (d + 1)-dimensions.

75 fluctuation Emergent Vacuum geometry is Constant curvature space in d-dimensions such as S d, ds d and AdS d, depending on embedded flat Euclidean or Lorentzian spacetime in (d + 1)-dimensions. Consider fluctuations of X a κg ab Db (x) around the vacuum (0) solution as: D a (x) = D a (x) + Âa(x) (0) where D a (x) = g ab κ x b

76 fluctuation Emergent Vacuum geometry is Constant curvature space in d-dimensions such as S d, ds d and AdS d, depending on embedded flat Euclidean or Lorentzian spacetime in (d + 1)-dimensions. Consider fluctuations of X a κg ab Db (x) around the vacuum (0) solution as: D a (x) = D a (x) + Âa(x) (0) where D a (x) = g ab κ x b

77 fluctuation Emergent Vacuum geometry is Constant curvature space in d-dimensions such as S d, ds d and AdS d, depending on embedded flat Euclidean or Lorentzian spacetime in (d + 1)-dimensions. Consider fluctuations of X a κg ab Db (x) around the vacuum (0) solution as: D a (x) = D a (x) + Âa(x) (0) where D a (x) = g ab κ x b [ D a, D b ] = iκ 1 Lab (x) + [ D (0) a, Âb] [ D (0) b, Âa] + [Âa, Âb] i F ab.

78 fluctuation Emergent Vacuum geometry is Constant curvature space in d-dimensions such as S d, ds d and AdS d, depending on embedded flat Euclidean or Lorentzian spacetime in (d + 1)-dimensions. Consider fluctuations of X a κg ab Db (x) around the vacuum (0) solution as: D a (x) = D a (x) + Âa(x) (0) where D a (x) = g ab κ x b [ D a, D b ] = iκ 1 Lab (x) + [ D a (0), Âb] [ D (0) b, Âa] + [Âa, Âb] i F ab. the action for the fluctuations in Ŝ κ = κ2 4 Tr Hg ac g bd Fab F cd + (d 1)κ 2 Tr H g ab Da D b

79 fluctuation Emergent Vacuum geometry is Constant curvature space in d-dimensions such as S d, ds d and AdS d, depending on embedded flat Euclidean or Lorentzian spacetime in (d + 1)-dimensions. Consider fluctuations of X a κg ab Db (x) around the vacuum (0) solution as: D a (x) = D a (x) + Âa(x) (0) where D a (x) = g ab κ x b [ D a, D b ] = iκ 1 Lab (x) + [ D a (0), Âb] [ D (0) b, Âa] + [Âa, Âb] i F ab. the action for the fluctuations in Ŝ κ = κ2 4 Tr Hg ac g bd Fab F cd + (d 1)κ 2 Tr H g ab Da D b Respects NC U(1) symmetry

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81 Conclusion and open problems

82 Conclusion and open problems New Matrix model constructed and emergent fluctuating geometry studied. Spacetime satisfying Snyder algebra is classical solution. Spacetime again is emergent concept-background independence.

83 Conclusion and open problems New Matrix model constructed and emergent fluctuating geometry studied. Spacetime satisfying Snyder algebra is classical solution. Spacetime again is emergent concept-background independence. The correspondence between NC -algebra and,generalized vector fields extended to mass deformed IKKT model to get constant curvature space. Fluctuation about vacuum geometry is NC massive gauge theory.

84 Conclusion and open problems New Matrix model constructed and emergent fluctuating geometry studied. Spacetime satisfying Snyder algebra is classical solution. Spacetime again is emergent concept-background independence. The correspondence between NC -algebra and,generalized vector fields extended to mass deformed IKKT model to get constant curvature space. Fluctuation about vacuum geometry is NC massive gauge theory. Einstein gravity with CC from massive gauge theory?

85 Conclusion and open problems New Matrix model constructed and emergent fluctuating geometry studied. Spacetime satisfying Snyder algebra is classical solution. Spacetime again is emergent concept-background independence. The correspondence between NC -algebra and,generalized vector fields extended to mass deformed IKKT model to get constant curvature space. Fluctuation about vacuum geometry is NC massive gauge theory. Einstein gravity with CC from massive gauge theory? Nonlinear Snyder algebra? S G = Tr ( 1 4 M abm ab 1 2κ M ab[x a, X b ] + 1 2κ G(X )).

86 Conclusion and open problems New Matrix model constructed and emergent fluctuating geometry studied. Spacetime satisfying Snyder algebra is classical solution. Spacetime again is emergent concept-background independence. The correspondence between NC -algebra and,generalized vector fields extended to mass deformed IKKT model to get constant curvature space. Fluctuation about vacuum geometry is NC massive gauge theory. Einstein gravity with CC from massive gauge theory? Nonlinear Snyder algebra? S G = Tr ( 1 4 M abm ab 1 2κ M ab[x a, X b ] + 1 2κ G(X )). G(X ) is quadratic > usual Snyder algebra > Constant curvature.if G(X ) cubic and above >what geometry?

87 Broad picture and more to think Matrix model NC spacetime NC algebra fluctuating coordinates- NC U(1) gauge field fluctuating geometry.

88 Broad picture and more to think Matrix model NC spacetime NC algebra fluctuating coordinates- NC U(1) gauge field fluctuating geometry. Pure gauge fluctuation is Killing symmetry of vacuum geometry Dynamics of NCU(1) gauge field dynamics of gravity Opposite to KK!

89 Broad picture and more to think Matrix model NC spacetime NC algebra fluctuating coordinates- NC U(1) gauge field fluctuating geometry. Pure gauge fluctuation is Killing symmetry of vacuum geometry Dynamics of NCU(1) gauge field dynamics of gravity Opposite to KK! A link between symplective geometry and Riemann geometry! Symmetries of Matrix model symmetries of NC spacetime symmetries of general relativity

90 Broad picture and more to think Matrix model NC spacetime NC algebra fluctuating coordinates- NC U(1) gauge field fluctuating geometry. Pure gauge fluctuation is Killing symmetry of vacuum geometry Dynamics of NCU(1) gauge field dynamics of gravity Opposite to KK! A link between symplective geometry and Riemann geometry! Symmetries of Matrix model symmetries of NC spacetime symmetries of general relativity

91 Broad picture and more to think Matrix model NC spacetime NC algebra fluctuating coordinates- NC U(1) gauge field fluctuating geometry. Pure gauge fluctuation is Killing symmetry of vacuum geometry Dynamics of NCU(1) gauge field dynamics of gravity Opposite to KK! A link between symplective geometry and Riemann geometry! Symmetries of Matrix model symmetries of NC spacetime symmetries of general relativity Emergent time?emergent matter?

92 Broad picture and more to think Matrix model NC spacetime NC algebra fluctuating coordinates- NC U(1) gauge field fluctuating geometry. Pure gauge fluctuation is Killing symmetry of vacuum geometry Dynamics of NCU(1) gauge field dynamics of gravity Opposite to KK! A link between symplective geometry and Riemann geometry! Symmetries of Matrix model symmetries of NC spacetime symmetries of general relativity Emergent time?emergent matter? THANKS!