Nonassociative Geometry and Non-Geometric Backgrounds

Transcription

1 Nonassociative Geometry and Non-Geometric Backgrounds Richard Szabo Action MP 1405 Quantum Structure of Spacetime Division of Theoretical Physics Seminar Rudjer Bos kovic Institute, Zagreb April 18, 2018

2 D-branes and the Lie algebra su(2)

3 D-branes and the Lie algebra su(2) D1 D3

4 D-branes and the Lie algebra su(2) D1 D3 (BPS) monopoles D1-branes ending on D3-branes

5 D-branes and the Lie algebra su(2) D1 D3 (BPS) monopoles D1-branes ending on D3-branes From perspective of D1-branes described by Nahm equations: (Nahm 80; Diaconescu 97; Tsimpis 98) dt i ds + εijk [T j, T k ] = 0 (s = x 6 )

6 D-branes and the Lie algebra su(2) D1 D3 (BPS) monopoles D1-branes ending on D3-branes From perspective of D1-branes described by Nahm equations: (Nahm 80; Diaconescu 97; Tsimpis 98) dt i ds + εijk [T j, T k ] = 0 (s = x 6 ) Solution: T i (s) = 1 s X i, [X i, X j ] = ε ijk X k fuzzy S 2 (Myers 99)

7 M-branes and the 3-Lie algebra A4

8 M-branes and the 3-Lie algebra A M2 M5

9 M-branes and the 3-Lie algebra A M2 M5 Lift monopoles to M-theory self-dual strings

10 M-branes and the 3-Lie algebra A M2 M5 Lift monopoles to M-theory self-dual strings M2-branes ending on M5-branes described by Basu Harvey equations: (Basu & Harvey 05) dt i ds + εijkl [T j, T k, T l ] = 0 (s = x 6 )

11 M-branes and the 3-Lie algebra A M2 M5 Lift monopoles to M-theory self-dual strings M2-branes ending on M5-branes described by Basu Harvey equations: (Basu & Harvey 05) dt i ds + εijkl [T j, T k, T l ] = 0 (s = x 6 ) Solution: T i (s) = 1 2s X i, [X i, X j, X k ] = ε ijkl X l fuzzy S 3?

12 Open strings and B-fields

13 Open strings and B-fields Nahm equations boundary condition for open strings (Chu & Smith 09)

14 Open strings and B-fields Nahm equations boundary condition for open strings (Chu & Smith 09) Turning on constant 2-form B-field on D3-brane induces shift in Nahm equations, accounted for by Heisenberg algebra: [X i, X j ] = i θ ij, θ B

15 Open strings and B-fields Nahm equations boundary condition for open strings (Chu & Smith 09) Turning on constant 2-form B-field on D3-brane induces shift in Nahm equations, accounted for by Heisenberg algebra: [X i, X j ] = i θ ij, θ B Quantization of Poisson bracket on R 2 (Chu & Ho 98; Schomerus 99; Seiberg & Witten 99;... )

16 Open strings and B-fields Nahm equations boundary condition for open strings (Chu & Smith 09) Turning on constant 2-form B-field on D3-brane induces shift in Nahm equations, accounted for by Heisenberg algebra: [X i, X j ] = i θ ij, θ B Quantization of Poisson bracket on R 2 (Chu & Ho 98; Schomerus 99; Seiberg & Witten 99;... ) Quantization captured by associative Moyal Weyl star-product of fields encoded in scattering amplitudes: (f g)(x) = f (k) g(k ) e i 2 k θk e i (k+k ) x k,k

17 Open strings and B-fields Nahm equations boundary condition for open strings (Chu & Smith 09) Turning on constant 2-form B-field on D3-brane induces shift in Nahm equations, accounted for by Heisenberg algebra: [X i, X j ] = i θ ij, θ B Quantization of Poisson bracket on R 2 (Chu & Ho 98; Schomerus 99; Seiberg & Witten 99;... ) Quantization captured by associative Moyal Weyl star-product of fields encoded in scattering amplitudes: (f g)(x) = f (k) g(k ) e i 2 k θk e i (k+k ) x k,k Low-energy dynamics described by noncommutative gauge theory on D3-brane

18 Open membranes and C-fields

19 Open membranes and C-fields Basu Harvey equations boundary condition of open membranes

20 Open membranes and C-fields Basu Harvey equations boundary condition of open membranes Modification from constant 3-form C-field on M5-brane accounted for by Nambu Heisenberg algebra: (Nambu 73) [X i, X j, X k ] = i Θ ijk, Θ C Quantization of Nambu Poisson 3-bracket on R 3

21 Open membranes and C-fields Basu Harvey equations boundary condition of open membranes Modification from constant 3-form C-field on M5-brane accounted for by Nambu Heisenberg algebra: (Nambu 73) [X i, X j, X k ] = i Θ ijk, Θ C Quantization of Nambu Poisson 3-bracket on R 3 Agrees via transgression with noncommutative loop space from quantization of open membranes: (Bergshoeff, Berman, van der Schaar & Sundell 00; Kawamoto & Sasakura 00; Sämann & RS 12) [ X i (τ), X j (τ ) ] = i Θ ijk Ẋ k (τ) Ẋ (τ) 2 δ(τ τ ) +

22 Open membranes and C-fields Basu Harvey equations boundary condition of open membranes Modification from constant 3-form C-field on M5-brane accounted for by Nambu Heisenberg algebra: (Nambu 73) [X i, X j, X k ] = i Θ ijk, Θ C Quantization of Nambu Poisson 3-bracket on R 3 Agrees via transgression with noncommutative loop space from quantization of open membranes: (Bergshoeff, Berman, van der Schaar & Sundell 00; Kawamoto & Sasakura 00; Sämann & RS 12) [ X i (τ), X j (τ ) ] = i Θ ijk Ẋ k (τ) Ẋ (τ) 2 δ(τ τ ) + Quantization unknown

23 Non-geometric fluxes

24 Non-geometric fluxes Quantum geometry of closed strings: Closed strings winding and propagating in non-geometric flux compactifications probe noncommutative/nonassociative deformations of geometry (Blumenhagen & Plauschinn 10; Lüst 10; Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11; Condeescu, Florakis & Lüst 12; Mylonas, Schupp & RS 12; Andriot, Larfors, Lüst & Patalong 12; Davidović, Nikolić & Sazdović 13; Blair 14; Bakas & Lüst 15; Nikolić & Obrić 18; Chatzistavrakidis, Jonke, Khoo & RS 18;... )

25 Non-geometric fluxes Quantum geometry of closed strings: Closed strings winding and propagating in non-geometric flux compactifications probe noncommutative/nonassociative deformations of geometry (Blumenhagen & Plauschinn 10; Lüst 10; Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11; Condeescu, Florakis & Lüst 12; Mylonas, Schupp & RS 12; Andriot, Larfors, Lüst & Patalong 12; Davidović, Nikolić & Sazdović 13; Blair 14; Bakas & Lüst 15; Nikolić & Obrić 18; Chatzistavrakidis, Jonke, Khoo & RS 18;... ) M = T 3 with H-flux gives geometric and non-geometric fluxes via T-duality (Hull 05; Shelton, Taylor & Wecht 05) H ijk T i f i jk T j Q ij k T k R ijk

26 Non-geometric fluxes Quantum geometry of closed strings: Closed strings winding and propagating in non-geometric flux compactifications probe noncommutative/nonassociative deformations of geometry (Blumenhagen & Plauschinn 10; Lüst 10; Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11; Condeescu, Florakis & Lüst 12; Mylonas, Schupp & RS 12; Andriot, Larfors, Lüst & Patalong 12; Davidović, Nikolić & Sazdović 13; Blair 14; Bakas & Lüst 15; Nikolić & Obrić 18; Chatzistavrakidis, Jonke, Khoo & RS 18;... ) M = T 3 with H-flux gives geometric and non-geometric fluxes via T-duality (Hull 05; Shelton, Taylor & Wecht 05) H ijk T i f i jk T j Q ij k T k R ijk Sends string winding (w i ) H 1 (T 3, Z) = Z 3 to momenta (p i )

27 Non-geometric fluxes Quantum geometry of closed strings: Closed strings winding and propagating in non-geometric flux compactifications probe noncommutative/nonassociative deformations of geometry (Blumenhagen & Plauschinn 10; Lüst 10; Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11; Condeescu, Florakis & Lüst 12; Mylonas, Schupp & RS 12; Andriot, Larfors, Lüst & Patalong 12; Davidović, Nikolić & Sazdović 13; Blair 14; Bakas & Lüst 15; Nikolić & Obrić 18; Chatzistavrakidis, Jonke, Khoo & RS 18;... ) M = T 3 with H-flux gives geometric and non-geometric fluxes via T-duality (Hull 05; Shelton, Taylor & Wecht 05) H ijk T i f i jk T j Q ij k T k R ijk Sends string winding (w i ) H 1 (T 3, Z) = Z 3 to momenta (p i ) T ijk In Double Field Theory: H ijk = [i B jk] R ijk = ˆ [i β jk] (Andriot, Hohm, Larfors, Lüst & Patalong 12)

28 Parabolic R-flux model

29 Parabolic R-flux model Twisted Poisson structure on phase space M = T M (Lüst 10) [x i, x j ] = i l3 s 3 Rijk p k, [x i, p j ] = i δ i j, [p i, p j ] = 0 Closed string nonassociativity (Jacobiator) [x i, x j, x k ] = l 3 s R ijk

30 Parabolic R-flux model Twisted Poisson structure on phase space M = T M (Lüst 10) [x i, x j ] = i l3 s 3 Rijk p k, [x i, p j ] = i δ i j, [p i, p j ] = 0 Closed string nonassociativity (Jacobiator) [x i, x j, x k ] = l 3 s R ijk Quantization captured by nonassociative phase space star product (Mylonas, Schupp & RS 12; Bakas & Lüst 13; Kupriyanov & Vassilevich 15) (f g)(x) = k,k f (k) g(k ) e i 2 (k p k x kx k p ) e i l3 s 6 R (kx k x ) p e i (k+k ) x

31 Parabolic R-flux model Twisted Poisson structure on phase space M = T M (Lüst 10) [x i, x j ] = i l3 s 3 Rijk p k, [x i, p j ] = i δ i j, [p i, p j ] = 0 Closed string nonassociativity (Jacobiator) [x i, x j, x k ] = l 3 s R ijk Quantization captured by nonassociative phase space star product (Mylonas, Schupp & RS 12; Bakas & Lüst 13; Kupriyanov & Vassilevich 15) (f g)(x) = k,k f (k) g(k ) e i 2 (k p k x kx k p ) e i l3 s 6 R (kx k x ) p e i (k+k ) x Nonassociative BCH formula captured by 3-cocycle: ( e i k x e i k x ) e i k x = e i l3 s 6 R k x (k x k x ) e i k x ( e i k x e i k x )

32 Parabolic R-flux model Twisted Poisson structure on phase space M = T M (Lüst 10) [x i, x j ] = i l3 s 3 Rijk p k, [x i, p j ] = i δ i j, [p i, p j ] = 0 Closed string nonassociativity (Jacobiator) [x i, x j, x k ] = l 3 s R ijk Quantization captured by nonassociative phase space star product (Mylonas, Schupp & RS 12; Bakas & Lüst 13; Kupriyanov & Vassilevich 15) (f g)(x) = k,k f (k) g(k ) e i 2 (k p k x kx k p ) e i l3 s 6 R (kx k x ) p e i (k+k ) x Nonassociative BCH formula captured by 3-cocycle: ( e i k x e i k x ) e i k x = e i l3 s 6 R k x (k x k x ) e i k x ( e i k x e i k x ) Closure, 3-cyclicity: f g = f g, (f g) h = f (g h)

33 Nonassociative quantum mechanics (Mylonas, Schupp & RS 13; Bojowald, Brahma, Büyükçam & Strobl 14; Bojowald, Brahma & Büyükçam 15; Kupriyanov & RS 18; Bunk, Müller & RS 18)

34 Nonassociative quantum mechanics (Mylonas, Schupp & RS 13; Bojowald, Brahma, Büyükçam & Strobl 14; Bojowald, Brahma & Büyükçam 15; Kupriyanov & RS 18; Bunk, Müller & RS 18) Phase space quantum mechanics:

35 Nonassociative quantum mechanics (Mylonas, Schupp & RS 13; Bojowald, Brahma, Büyükçam & Strobl 14; Bojowald, Brahma & Büyükçam 15; Kupriyanov & RS 18; Bunk, Müller & RS 18) Phase space quantum mechanics: Observables: Real-valued functions f (x, p)

36 Nonassociative quantum mechanics (Mylonas, Schupp & RS 13; Bojowald, Brahma, Büyükçam & Strobl 14; Bojowald, Brahma & Büyükçam 15; Kupriyanov & RS 18; Bunk, Müller & RS 18) Phase space quantum mechanics: Observables: Real-valued functions f (x, p) Dynamics: f t = i [H, f ]

37 Nonassociative quantum mechanics (Mylonas, Schupp & RS 13; Bojowald, Brahma, Büyükçam & Strobl 14; Bojowald, Brahma & Büyükçam 15; Kupriyanov & RS 18; Bunk, Müller & RS 18) Phase space quantum mechanics: Observables: Real-valued functions f (x, p) f Dynamics: = i t [H, f ] States: Normalized phase space wavefunctions ψ a Statistical probabilities µ a [0, 1]

38 Nonassociative quantum mechanics (Mylonas, Schupp & RS 13; Bojowald, Brahma, Büyükçam & Strobl 14; Bojowald, Brahma & Büyükçam 15; Kupriyanov & RS 18; Bunk, Müller & RS 18) Phase space quantum mechanics: Observables: Real-valued functions f (x, p) f Dynamics: = i t [H, f ] States: Normalized phase space wavefunctions ψ a Statistical probabilities µ a [0, 1] Expectation values: f = µ a ψa (f ψ a ) = a f S S = a µ a ψ a ψ a

39 Nonassociative quantum mechanics (Mylonas, Schupp & RS 13; Bojowald, Brahma, Büyükçam & Strobl 14; Bojowald, Brahma & Büyükçam 15; Kupriyanov & RS 18; Bunk, Müller & RS 18) Phase space quantum mechanics: Observables: Real-valued functions f (x, p) f Dynamics: = i t [H, f ] States: Normalized phase space wavefunctions ψ a Statistical probabilities µ a [0, 1] Expectation values: f = µ a ψa (f ψ a ) = a f S S = a µ a ψ a ψ a Positivity, reality,... using closure, 3-cyclicity, Hermiticity (f g) = g f, and unitality f 1 = f = 1 f

40 Spacetime quantization

41 Spacetime quantization Coarse-graining of spacetime with R-flux: Nonassociating observables cannot have common eigen-states x I S = λ I S

42 Spacetime quantization Coarse-graining of spacetime with R-flux: Nonassociating observables cannot have common eigen-states x I S = λ I S Oriented area and volume (uncertainty) operators: A IJ = Im ( [ x I, x J ] ) = i ( x I x J x J x I ) V IJK = 1 3 Re( x I [ x J, x K ] + x K [ x I, x J ] + x J [ x K, x I ] ) Shifted coordinates x I := x I x I

43 Spacetime quantization Coarse-graining of spacetime with R-flux: Nonassociating observables cannot have common eigen-states x I S = λ I S Oriented area and volume (uncertainty) operators: A IJ = Im ( [ x I, x J ] ) = i ( x I x J x J x I ) V IJK = 1 3 Re( x I [ x J, x K ] + x K [ x I, x J ] + x J [ x K, x I ] ) Shifted coordinates x I := x I x I Minimal area and volume: A x i,p j = δ i j, A ij = l3 s 3 Rijk p k, Quantized spacetime with cells of minimal volume V ijk = l3 s 2 R ijk l3 s 2 R ijk

44 Spacetime quantization Coarse-graining of spacetime with R-flux: Nonassociating observables cannot have common eigen-states x I S = λ I S Oriented area and volume (uncertainty) operators: A IJ = Im ( [ x I, x J ] ) = i ( x I x J x J x I ) V IJK = 1 3 Re( x I [ x J, x K ] + x K [ x I, x J ] + x J [ x K, x I ] ) Shifted coordinates x I := x I x I Minimal area and volume: A x i,p j = δ i j, A ij = l3 s 3 Rijk p k, Quantized spacetime with cells of minimal volume V ijk = l3 s 2 R ijk l3 s 2 R ijk Freed Witten anomaly: No D3-branes on T 3 with H-flux No D0-branes in R-flux background (Wecht 07) T ijk

45 Nonassociative geometry

46 Nonassociative geometry Configuration space triproducts (Aschieri & RS 15) (f g h)(x) = ( f (x) g(x) ) h(x) p=0

47 Nonassociative geometry Configuration space triproducts (Aschieri & RS 15) (f g h)(x) = ( f (x) g(x) ) h(x) p=0 = f (k x ) g(k x) h(k x ) e i l 3 s k x,k x,k x 12 R kx (k x k x ) e i (kx +k x +k x ) x

48 Nonassociative geometry Configuration space triproducts (Aschieri & RS 15) (f g h)(x) = ( f (x) g(x) ) h(x) p=0 = f (k x ) g(k x) h(k x ) e i l 3 s k x,k x,k x 12 R kx (k x k x ) e i (kx +k x +k x ) x Quantizes 3-bracket [f, g, h] = Asym(f g h) (Takhtajan 94) [x i, x j, x k ] = l 3 s R ijk

49 Nonassociative geometry Configuration space triproducts (Aschieri & RS 15) (f g h)(x) = ( f (x) g(x) ) h(x) p=0 = f (k x ) g(k x) h(k x ) e i l 3 s k x,k x,k x 12 R kx (k x k x ) e i (kx +k x +k x ) x Quantizes 3-bracket [f, g, h] = Asym(f g h) (Takhtajan 94) [x i, x j, x k ] = l 3 s R ijk Agrees with triproducts of tachyon vertex operators after T-duality in CFT perturbation theory around flat space with constant H-flux (Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11)

50 Nonassociative geometry Configuration space triproducts (Aschieri & RS 15) (f g h)(x) = ( f (x) g(x) ) h(x) p=0 = f (k x ) g(k x) h(k x ) e i l 3 s k x,k x,k x 12 R kx (k x k x ) e i (kx +k x +k x ) x Quantizes 3-bracket [f, g, h] = Asym(f g h) (Takhtajan 94) [x i, x j, x k ] = l 3 s R ijk Agrees with triproducts of tachyon vertex operators after T-duality in CFT perturbation theory around flat space with constant H-flux (Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11) Violates strong constraint of Double Field Theory (Blumenhagen, Fuchs, Hassler, Lüst & Sun 14)

51 Nonassociative geometry Configuration space triproducts (Aschieri & RS 15) (f g h)(x) = ( f (x) g(x) ) h(x) p=0 = f (k x ) g(k x) h(k x ) e i l 3 s k x,k x,k x 12 R kx (k x k x ) e i (kx +k x +k x ) x Quantizes 3-bracket [f, g, h] = Asym(f g h) (Takhtajan 94) [x i, x j, x k ] = l 3 s R ijk Agrees with triproducts of tachyon vertex operators after T-duality in CFT perturbation theory around flat space with constant H-flux (Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11) Violates strong constraint of Double Field Theory (Blumenhagen, Fuchs, Hassler, Lüst & Sun 14) On-shell associativity: f g h = f g h

52 (Günaydin, Lüst & Malek 16) M-theory lift of R-flux model

53 M-theory lift of R-flux model (Günaydin, Lüst & Malek 16) S 1 M M S 1 radius λ string coupling g s

54 M-theory lift of R-flux model (Günaydin, Lüst & Malek 16) S 1 M M S 1 radius λ string coupling g s Generate string R-flux starting from twisted torus M = T 3 : T jk R ijk f i jk

55 M-theory lift of R-flux model (Günaydin, Lüst & Malek 16) S 1 M M S 1 radius λ string coupling g s Generate string R-flux starting from twisted torus M = T 3 : T jk R ijk f i jk Lift to M-theory on M = M S 1 x 4 : T-duality = U-duality

56 M-theory lift of R-flux model (Günaydin, Lüst & Malek 16) S 1 M M S 1 radius λ string coupling g s Generate string R-flux starting from twisted torus M = T 3 : T jk R ijk f i jk Lift to M-theory on M = M S 1 x 4 : T-duality = U-duality U µνρ In SL(5) Exceptional Field Theory: C µνρ Ω µνρ with R µ,νραβ = ˆ µ[ν Ω ραβ] (Not a 5-vector!) (Blair & Malek 14)

57 M-theory lift of R-flux model (Günaydin, Lüst & Malek 16) S 1 M M S 1 radius λ string coupling g s Generate string R-flux starting from twisted torus M = T 3 : T jk R ijk f i jk Lift to M-theory on M = M S 1 x 4 : T-duality = U-duality U µνρ In SL(5) Exceptional Field Theory: C µνρ Ω µνρ with R µ,νραβ = ˆ µ[ν Ω ραβ] (Not a 5-vector!) (Blair & Malek 14) Choice R 4,µναβ = R ε µναβ breaks SL(5) SO(4)

58 M-theory phase space

59 M-theory phase space Sends membrane wrapping (w ij ) H 2( M, Z) to momenta (p i )

60 M-theory phase space Sends membrane wrapping (w ij ) H 2( M, Z) to momenta (p i ) No D0-branes on M = p 4 = 0 along M-theory direction: H 2( M, Z) = H 1( T 3, Z) H 2( T 3, Z) = Z 2 Z [H] Z 2 Restored for R = 0: H 2(T 3, Z) = Z 3

61 M-theory phase space Sends membrane wrapping (w ij ) H 2( M, Z) to momenta (p i ) No D0-branes on M = p 4 = 0 along M-theory direction: H 2( M, Z) = H 1( T 3, Z) H 2( T 3, Z) = Z 2 Z [H] Z 2 Restored for R = 0: H 2(T 3, Z) = Z 3 R µ,νραβ p µ = 0 = 7D phase space M: [x i, x j ] = i l 3 s 3 R4,ijk4 p k, [x 4, x i ] = i λ l3 s 3 R4,1234 p i [x i, p j ] = i δ i j x 4 + i λ ε i jk x k, [x 4, p i ] = i λ 2 x i [p i, p j ] = i λ ε ijk p k [x i, x j, x k ] = i l 3 s 3 R4,ijk4 x 4, [x i, x j, x 4 ] = i λ2 l 3 s 3 R 4,ijk4 x k [p i, x j, x k ] = i λ l 3 ( s 3 R4,1234 δ j i p k δ k i p j ), [p i, x j, x 4 ] = i λ2 l 3 s 3 R 4,ijk4 p k [p i, p j, x k ] = i λ 2 ε ij k x 4 i λ ( δ k j x i δ k i x j ), [pi, p j, x 4 ] = i λ 3 ε ijk x k [p i, p j, p k ] = 0

62 M-theory phase space Sends membrane wrapping (w ij ) H 2( M, Z) to momenta (p i ) No D0-branes on M = p 4 = 0 along M-theory direction: H 2( M, Z) = H 1( T 3, Z) H 2( T 3, Z) = Z 2 Z [H] Z 2 Restored for R = 0: H 2(T 3, Z) = Z 3 R µ,νραβ p µ = 0 = 7D phase space M: [x i, x j ] = i l 3 s 3 R4,ijk4 p k, [x 4, x i ] = i λ l3 s 3 R4,1234 p i [x i, p j ] = i δ i j x 4 + i λ ε i jk x k, [x 4, p i ] = i λ 2 x i [p i, p j ] = i λ ε ijk p k [x i, x j, x k ] = i l 3 s 3 R4,ijk4 x 4, [x i, x j, x 4 ] = i λ2 l 3 s 3 R 4,ijk4 x k [p i, x j, x k ] = i λ l 3 ( s 3 R4,1234 δ j i p k δ k i p j ), [p i, x j, x 4 ] = i λ2 l 3 s 3 R 4,ijk4 p k [p i, p j, x k ] = i λ 2 ε ij k x 4 i λ ( δ k j x i δ k i x j ), [pi, p j, x 4 ] = i λ 3 ε ijk x k [p i, p j, p k ] = 0 Reduces to string R-flux algebra at λ = 0 (with x 4 = 1 central)

63 Octonionic phase space

64 Octonionic phase space Originates from nonassociative, alternative, octonion algebra O: (x A ) = (x i, x 4 (, p i ) = Λ (e A ) = 1 λ l 4 2 s R/3 f i, λ 3 l 4 s R/3 e 7, λ e i )

65 Octonionic phase space Originates from nonassociative, alternative, octonion algebra O: (x A ) = (x i, x 4 (, p i ) = Λ (e A ) = 1 λ l 4 2 s R/3 f i, Imaginary unit octonions: λ 3 l 4 s R/3 e 7, λ e i ) e A e B = δ AB 1+η ABC e C, η ABC = +1 for ABC = 123, 435, 471,...

66 Octonionic phase space Originates from nonassociative, alternative, octonion algebra O: (x A ) = (x i, x 4 (, p i ) = Λ (e A ) = 1 λ l 4 2 s R/3 f i, Imaginary unit octonions: λ 3 l 4 s R/3 e 7, λ e i ) e A e B = δ AB 1+η ABC e C, η ABC = +1 for ABC = 123, 435, 471,... Octonionic commutator [e A, e B ] = 2 η ABC e C (f i = e i+3, i = 1, 2, 3): has components [e i, e j ] = 2 ε ijk e k, [e 7, e i ] = 2 f i [f i, f j ] = 2 ε ijk e k, [e 7, f i ] = 2 e i [e i, f j ] = 2 (δ ij e 7 ε ijk f k )

67 Octonionic phase space Originates from nonassociative, alternative, octonion algebra O: (x A ) = (x i, x 4 (, p i ) = Λ (e A ) = 1 λ l 4 2 s R/3 f i, Imaginary unit octonions: λ 3 l 4 s R/3 e 7, λ e i ) e A e B = δ AB 1+η ABC e C, η ABC = +1 for ABC = 123, 435, 471,... Octonionic commutator [e A, e B ] = 2 η ABC e C (f i = e i+3, i = 1, 2, 3): has components [e i, e j ] = 2 ε ijk e k, [e 7, e i ] = 2 f i [f i, f j ] = 2 ε ijk e k, [e 7, f i ] = 2 e i [e i, f j ] = 2 (δ ij e 7 ε ijk f k ) Jacobiator: [e A, e B, e C ] = 12 η ABCD e D = 6 ( (e A e B ) e C e A (e B e C ) ) η ABCD = +1 for ABCD = 1267, 1425, 1346,...

68 (Kupriyanov & RS 17) G 2 -structures

69 G 2 -structures (Kupriyanov & RS 17) 4 normed algebras over R: R, C, H, O

70 G 2 -structures (Kupriyanov & RS 17) 4 normed algebras over R: R, C, H, O 4 real inner product spaces with cross product: (R 0, 0) (R 1, 0) (R 3, ) (R 7, η) 0 0 (k p) i = ε ijk k j p k ( k η p ) A = η ABC k B p C SO(3) G 2 SO(7)

71 G 2 -structures (Kupriyanov & RS 17) 4 normed algebras over R: R, C, H, O 4 real inner product spaces with cross product: (R 0, 0) (R 1, 0) (R 3, ) (R 7, η) 0 0 (k p) i = ε ijk k j p k ( k η p ) A = η ABC k B p C SO(3) G 2 SO(7) Jacobiator: J η( k, k, k ) = ( k η k ) η k + ( k η k ) η k + ( k η k ) η k 0

72 G 2 -structures (Kupriyanov & RS 17) 4 normed algebras over R: R, C, H, O 4 real inner product spaces with cross product: (R 0, 0) (R 1, 0) (R 3, ) (R 7, η) 0 0 (k p) i = ε ijk k j p k ( k η p ) A = η ABC k B p C SO(3) G 2 SO(7) Jacobiator: J η( k, k, k ) = ( k η k ) η k + ( k η k ) η k + ( k η k ) η k 0 Represented on O through X k = k A e A : [ ] [ X k η k = 1 X k, X 2 k, X Jη( k, k, k ) = 1 X k, X 4 k, X k ]

73 G 2 -structures (Kupriyanov & RS 17) 4 normed algebras over R: R, C, H, O 4 real inner product spaces with cross product: (R 0, 0) (R 1, 0) (R 3, ) (R 7, η) 0 0 (k p) i = ε ijk k j p k ( k η p ) A = η ABC k B p C SO(3) G 2 SO(7) Jacobiator: J η( k, k, k ) = ( k η k ) η k + ( k η k ) η k + ( k η k ) η k 0 Represented on O through X k = k A e A : [ ] [ X k η k = 1 X k, X 2 k, X Jη( k, k, k ) = 1 X k, X 4 k, X k ] Alternativity (X X ) X = X (X X ) defines octonion exponential: e X k = cos k 1 + sin k k X k

74 Octonionic BCH formula

75 Octonionic BCH formula BCH formula e X k e X k = e X Bη ( k, k ) can be computed explicitly in terms of vector star sums of p, p B 7 R 7 : p η p ( = ɛ p, p 1 p 2 p + 1 p 2 p p η p )

76 Octonionic BCH formula BCH formula e X k e X k = e X Bη ( k, k ) can be computed explicitly in terms of vector star sums of p, p B 7 R 7 : p η p ( = ɛ p, p 1 p 2 p + 1 p 2 p p η p ) Noncommutativity/nonassociativity: p η p p η p = 2 p η p A η( p, p, p ) = ( p η p ) η p p η ( p η p ) = 2 3 J η( p, p, p )

77 Octonionic BCH formula BCH formula e X k e X k = e X Bη ( k, k ) can be computed explicitly in terms of vector star sums of p, p B 7 R 7 : p η p ( = ɛ p, p 1 p 2 p + 1 p 2 p p η p ) Noncommutativity/nonassociativity: p η p p η p = 2 p η p A η( p, p, p ) = ( p η p ) η p p η ( p η p ) = 2 3 J η( p, p, p ) Extend to all k R 7 : B η ( k, k ) = sin 1 p η p p η p p η p p= k sin( k )/ k

78 Octonionic BCH formula BCH formula e X k e X k = e X Bη ( k, k ) can be computed explicitly in terms of vector star sums of p, p B 7 R 7 : p η p ( = ɛ p, p 1 p 2 p + 1 p 2 p p η p ) Noncommutativity/nonassociativity: p η p p η p = 2 p η p A η( p, p, p ) = ( p η p ) η p p η ( p η p ) = 2 3 J η( p, p, p ) Extend to all k R 7 : B η ( k, k ) = sin 1 p η p p η p p η p p= k sin( k )/ k = k + k 2 k η k + O( 2 )

79 Quantization of M-theory phase space

80 Quantization of M-theory phase space Nonassociative phase space star product: (f λ g)( x ) = f ( k ) g( k ) e i B η(λ k,λ k ) Λ 1 x k, k

81 Quantization of M-theory phase space Nonassociative phase space star product: (f λ g)( x ) = f ( k ) g( k ) e i B η(λ k,λ k ) Λ 1 x k, k Nonassociative BCH formula captured by 2-group addition : ( e i k x λ e i k x ) λ e i k x = e i A η(λ k,λ k,λ k ) e i k x λ ( e i k x λ e i k x ) A η( k, k, k ) = sin 1 ( p η p ) η p ( p η p ) η p A η( p, p, p ) p= k sin( k )/ k

82 Quantization of M-theory phase space Nonassociative phase space star product: (f λ g)( x ) = f ( k ) g( k ) e i B η(λ k,λ k ) Λ 1 x k, k Nonassociative BCH formula captured by 2-group addition : ( e i k x λ e i k x ) λ e i k x = e i A η(λ k,λ k,λ k ) e i k x λ ( e i k x λ e i k x ) A η( k, k, k ) = sin 1 ( p η p ) η p ( p η p ) η p A η( p, p, p ) p= k sin( k )/ k Using x 4 λ f = x 4 f + O(λ) reduces to string star product: lim (f λ g)( x ) = (f g)(x) λ 0

83 Quantization of M-theory phase space Nonassociative phase space star product: (f λ g)( x ) = f ( k ) g( k ) e i B η(λ k,λ k ) Λ 1 x k, k Nonassociative BCH formula captured by 2-group addition : ( e i k x λ e i k x ) λ e i k x = e i A η(λ k,λ k,λ k ) e i k x λ ( e i k x λ e i k x ) A η( k, k, k ) = sin 1 ( p η p ) η p ( p η p ) η p A η( p, p, p ) p= k sin( k )/ k Using x 4 λ f = x 4 f + O(λ) reduces to string star product: lim (f λ g)( x ) = (f g)(x) λ 0 Gauge-equivalent closed, 3-cyclic star product: f λ g = D 1( Df λ Dg ), D = 1 + O(λ)

84 Spacetime quantization in M-theory

85 Spacetime quantization in M-theory Using λ we can calculate:

86 Spacetime quantization in M-theory Using λ we can calculate: Minimal areas: A ij = l3 s 3 R 4,ijk4 p k, A 4i = λ l3 s 3 R 4,1234 p i A x i,p j = δ i j x 4 + λ ε i jk x k, A x 4,p i = λ 2 x i A p i,p j = λ εijk p k

87 Spacetime quantization in M-theory Using λ we can calculate: Minimal areas: A ij = l3 s 3 R 4,ijk4 p k, A 4i = λ l3 s 3 R 4,1234 p i A x i,p j = δ i j x 4 + λ ε i jk x k, A x 4,p i = λ 2 x i A p i,p j = λ εijk p k Minimal volumes: V ijk = l3 s 2 R 4,ijk4 x 4, V ij4 = λ2 l 3 s 2 V p i,xj,x k = λ l3 s 2 V p i,p j,xk = λ 2 2 R 4,ijk4 x k R 4,1234 ( δ j i p k δ k i p j ), V p i,x j,x 4 = λ2 l 3 s 2 R 4,ijk4 p k λ k εij x 4 + δ k j x i δ k i x j p, V i,p j,x 4 = λ3 2 2 εijk x k

88 Nonassociative geometry in M-theory

89 Nonassociative geometry in M-theory Configuration space triproducts: ( f λ g λ h ) ( x ) = ( (f λ g) λ h ) (x µ, p i ) p=0 = f ( k ) g( k ) h( k ) e i TΛ ( k, k, k ) Λ 1 x k, k, k

90 Nonassociative geometry in M-theory Configuration space triproducts: ( f λ g λ h ) ( x ) = ( (f λ g) λ h ) (x µ, p i ) p=0 = f ( k ) g( k ) h( k ) e i TΛ ( k, k, k ) Λ 1 x k, k, k T Λ ( k, k, k ) = sin 1 ( p η p ) η p ( p η p ) η p + ɛ p, p 1 p η p 2 p + ɛ p, p ( Aη ( p, p, p ) + ɛ p, p 1 p η p 2 p 1 p η p 2 p ) p=λ k sin( Λ k )/ Λ k

91 Nonassociative geometry in M-theory Configuration space triproducts: ( f λ g λ h ) ( x ) = ( (f λ g) λ h ) (x µ, p i ) p=0 = f ( k ) g( k ) h( k ) e i TΛ ( k, k, k ) Λ 1 x k, k, k T Λ ( k, k, k ) = sin 1 ( p η p ) η p ( p η p ) η p + ɛ p, p 1 p η p 2 p + ɛ p, p ( Aη ( p, p, p ) + ɛ p, p 1 p η p 2 p 1 p η p 2 p ) p=λ k sin( Λ k )/ Λ k ( = Λ k + k + k + 2 ( 1 2 2Λ Aη(Λ k, Λ k, Λ k ) + Λ k + Λ k 2 k + Λ k + Λ k 2 k + Λ k + Λ k 2 ) ) k + O(λ)

92 Nonassociative geometry in M-theory Configuration space triproducts: ( f λ g λ h ) ( x ) = ( (f λ g) λ h ) (x µ, p i ) p=0 = f ( k ) g( k ) h( k ) e i TΛ ( k, k, k ) Λ 1 x k, k, k T Λ ( k, k, k ) = sin 1 ( p η p ) η p ( p η p ) η p + ɛ p, p 1 p η p 2 p + ɛ p, p ( Aη ( p, p, p ) + ɛ p, p 1 p η p 2 p 1 p η p 2 p ) p=λ k sin( Λ k )/ Λ k ( = Λ k + k + k + 2 ( 1 2 2Λ Aη(Λ k, Λ k, Λ k ) + Λ k + Λ k 2 k + Λ k + Λ k 2 k + Λ k + Λ k 2 ) ) k + O(λ) Quantizes 3-bracket [f, g, h] λ = Asym(f λ g λ h) for A 4 : [x µ, x ν, x α ] 1 = l 3 s R ε µναβ x β

93 Noncommutative M-theory momentum space

94 Noncommutative M-theory momentum space Setting x µ = 0 reveals noncommutative associative deformation of momentum space with η, independent of R-flux: (Guedes, Oriti & Raasakka 13; Kupriyanov & Vitale 15) (f λ g)(p) = l,l f (l) g(l ) e 2 i λ B( λ 2 l, λ 2 l ) p

95 Noncommutative M-theory momentum space Setting x µ = 0 reveals noncommutative associative deformation of momentum space with η, independent of R-flux: (Guedes, Oriti & Raasakka 13; Kupriyanov & Vitale 15) (f λ g)(p) = l,l f (l) g(l ) e 2 i λ B( λ 2 l, λ 2 l ) p Restrict to (x µ ) S 3 R 4 of radius 1 λ : [p i, p j ] = i λ ε ijk p k, [x i, p j ] = i λ 1 λ 2 x 2 δ i j + i λ ε i jk x k

96 Noncommutative M-theory momentum space Setting x µ = 0 reveals noncommutative associative deformation of momentum space with η, independent of R-flux: (Guedes, Oriti & Raasakka 13; Kupriyanov & Vitale 15) (f λ g)(p) = l,l f (l) g(l ) e 2 i λ B( λ 2 l, λ 2 l ) p Restrict to (x µ ) S 3 R 4 of radius 1 λ : [p i, p j ] = i λ ε ijk p k, [x i, p j ] = i λ 1 λ 2 x 2 δ i j + i λ ε i jk x k Familiar from 3D quantum gravity (Freidel & Livine 06)

97 Spin(7)-structures

98 Spin(7)-structures Triple cross product of K = (KÂ) = (k 0, k ) R 8 = R R 7 : (K φ K φ K )Â := φâ ˆBĈ ˆD K ˆB K Ĉ K ˆD φ 0ABC = η ABC, φ ABCD = η ABCD

99 Spin(7)-structures Triple cross product of K = (KÂ) = (k 0, k ) R 8 = R R 7 : (K φ K φ K )Â := φâ ˆBĈ ˆD K ˆB K Ĉ K ˆD φ 0ABC = η ABC, φ ABCD = η ABCD K φ K φ K = ( k ( k η k ), 1 12 J η( k, k, k ) k 0 ( k η k ) k 0 ( k η k ) k 0 ( k η k ) )

100 Spin(7)-structures Triple cross product of K = (KÂ) = (k 0, k ) R 8 = R R 7 : (K φ K φ K )Â := φâ ˆBĈ ˆD K ˆB K Ĉ K ˆD φ 0ABC = η ABC, φ ABCD = η ABCD K φ K φ K = ( k ( k η k ), 1 12 J η( k, k, k ) k 0 ( k η k ) k 0 ( k η k ) k 0 ( k η k ) ) Preserved by Spin(7) SO(8), extends representation [ ] X k η k = 1 2 XK, X K on octonions XK = k k A e A O: X K φ K φ K = ( ) 1 2 (XK X K ) X K (X K X K ) X K

101 Spin(7)-structures Triple cross product of K = (KÂ) = (k 0, k ) R 8 = R R 7 : (K φ K φ K )Â := φâ ˆBĈ ˆD K ˆB K Ĉ K ˆD φ 0ABC = η ABC, φ ABCD = η ABCD K φ K φ K = ( k ( k η k ), 1 12 J η( k, k, k ) k 0 ( k η k ) k 0 ( k η k ) k 0 ( k η k ) ) Preserved by Spin(7) SO(8), extends representation [ ] X k η k = 1 2 XK, X K on octonions XK = k k A e A O: X K φ K φ K = ( ) 1 2 (XK X K ) X K (X K X K ) X K Spin(7) G 2 : k η k = k φ (1, 0 ) φ k

102 Spin(7)-structures Triple cross product of K = (KÂ) = (k 0, k ) R 8 = R R 7 : (K φ K φ K )Â := φâ ˆBĈ ˆD K ˆB K Ĉ K ˆD φ 0ABC = η ABC, φ ABCD = η ABCD K φ K φ K = ( k ( k η k ), 1 12 J η( k, k, k ) k 0 ( k η k ) k 0 ( k η k ) k 0 ( k η k ) ) Preserved by Spin(7) SO(8), extends representation [ ] X k η k = 1 2 XK, X K on octonions XK = k k A e A O: X K φ K φ K = ( ) 1 2 (XK X K ) X K (X K X K ) X K Spin(7) G 2 : k η k = k φ (1, 0 ) φ k Trivector: [ξâ, ξ ˆB, ξ Ĉ ] φ = φâ ˆBĈ ˆD ξ ˆD for ξ = (ξ 0, ξ ) = (1, e A )

103 Covariant M-theory phase space 3-algebra

104 Covariant M-theory phase space 3-algebra 8D phase space coordinates X = (x µ, p µ ) = (Λ ξ, λ 2 ξ 0) have SO(4) SO(4)-symmetric 3-brackets: [x i, x j, x k ] φ = l3 s 2 R 4,ijk4 x 4, [x i, x j, x 4 ] φ = λ2 l 3 s 2 R 4,ijk4 x k [p i, x j, x k ] φ = λ2 l 3 s 2 R 4,ijk4 p 4 λ l3 s 2 R 4,ijk4 p k [p i, x j, x 4 ] φ = λ2 l 3 s 2 R 4,1234 δ j i p 4 λ2 l 3 s [p i, p j, x k ] φ = λ2 2 ε ij k x λ 2 2 R 4,ijk4 p k ( δ k j x i δ k i x j ) [p i, p j, x 4 ] φ = 2 λ 3 2 ε ijk x k, [p i, p j, p k ] φ = 2 2 λ ε ijk p 4 [p 4, x i, x j ] φ = λ l3 s 2 R 4,ijk4 p k, [p 4, x i, x 4 ] φ = λ2 l 3 s 2 R 4,1234 p i, [p 4, p i, x j ] φ = 2 λ 2 δ j i x 4 2 λ 2 2 ε i jk x k [p 4, p i, x 4 ] φ = 2 λ 3 2 x i, [p 4, p i, p j ] φ = 2 λ 2 2 ε ijk p k

105 Covariant M-theory phase space 3-algebra 8D phase space coordinates X = (x µ, p µ ) = (Λ ξ, λ 2 ξ 0) have SO(4) SO(4)-symmetric 3-brackets: [x i, x j, x k ] φ = l3 s 2 R 4,ijk4 x 4, [x i, x j, x 4 ] φ = λ2 l 3 s 2 R 4,ijk4 x k [p i, x j, x k ] φ = λ2 l 3 s 2 R 4,ijk4 p 4 λ l3 s 2 R 4,ijk4 p k [p i, x j, x 4 ] φ = λ2 l 3 s 2 R 4,1234 δ j i p 4 λ2 l 3 s [p i, p j, x k ] φ = λ2 2 ε ij k x λ 2 2 R 4,ijk4 p k ( δ k j x i δ k i x j ) [p i, p j, x 4 ] φ = 2 λ 3 2 ε ijk x k, [p i, p j, p k ] φ = 2 2 λ ε ijk p 4 [p 4, x i, x j ] φ = λ l3 s 2 R 4,ijk4 p k, [p 4, x i, x 4 ] φ = λ2 l 3 s 2 R 4,1234 p i, [p 4, p i, x j ] φ = 2 λ 2 δ j i x 4 2 λ 2 2 ε i jk x k [p 4, p i, x 4 ] φ = 2 λ 3 2 x i, [p 4, p i, p j ] φ = 2 λ 2 2 ε ijk p k [f, g] G := [f, g, G] φ for any constraint G(X ) = 0;

106 Covariant M-theory phase space 3-algebra 8D phase space coordinates X = (x µ, p µ ) = (Λ ξ, λ 2 ξ 0) have SO(4) SO(4)-symmetric 3-brackets: [x i, x j, x k ] φ = l3 s 2 R 4,ijk4 x 4, [x i, x j, x 4 ] φ = λ2 l 3 s 2 R 4,ijk4 x k [p i, x j, x k ] φ = λ2 l 3 s 2 R 4,ijk4 p 4 λ l3 s 2 R 4,ijk4 p k [p i, x j, x 4 ] φ = λ2 l 3 s 2 R 4,1234 δ j i p 4 λ2 l 3 s [p i, p j, x k ] φ = λ2 2 ε ij k x λ 2 2 R 4,ijk4 p k ( δ k j x i δ k i x j ) [p i, p j, x 4 ] φ = 2 λ 3 2 ε ijk x k, [p i, p j, p k ] φ = 2 2 λ ε ijk p 4 [p 4, x i, x j ] φ = λ l3 s 2 R 4,ijk4 p k, [p 4, x i, x 4 ] φ = λ2 l 3 s 2 R 4,1234 p i, [p 4, p i, x j ] φ = 2 λ 2 δ j i x 4 2 λ 2 2 ε i jk x k [p 4, p i, x 4 ] φ = 2 λ 3 2 x i, [p 4, p i, p j ] φ = 2 λ 2 2 ε ijk p k [f, g] G := [f, g, G] φ for any constraint G(X ) = 0; e.g. G(X ) = 2 λ p 2 4 or G(X ) = R µ,νραβ p µ

107 Covariant M-theory phase space 3-algebra 8D phase space coordinates X = (x µ, p µ ) = (Λ ξ, λ 2 ξ 0) have SO(4) SO(4)-symmetric 3-brackets: [x i, x j, x k ] φ = l3 s 2 R 4,ijk4 x 4, [x i, x j, x 4 ] φ = λ2 l 3 s 2 R 4,ijk4 x k [p i, x j, x k ] φ = λ2 l 3 s 2 R 4,ijk4 p 4 λ l3 s 2 R 4,ijk4 p k [p i, x j, x 4 ] φ = λ2 l 3 s 2 R 4,1234 δ j i p 4 λ2 l 3 s [p i, p j, x k ] φ = λ2 2 ε ij k x λ 2 2 R 4,ijk4 p k ( δ k j x i δ k i x j ) [p i, p j, x 4 ] φ = 2 λ 3 2 ε ijk x k, [p i, p j, p k ] φ = 2 2 λ ε ijk p 4 [p 4, x i, x j ] φ = λ l3 s 2 R 4,ijk4 p k, [p 4, x i, x 4 ] φ = λ2 l 3 s 2 R 4,1234 p i, [p 4, p i, x j ] φ = 2 λ 2 δ j i x 4 2 λ 2 2 ε i jk x k [p 4, p i, x 4 ] φ = 2 λ 3 2 x i, [p 4, p i, p j ] φ = 2 λ 2 2 ε ijk p k [f, g] G := [f, g, G] φ for any constraint G(X ) = 0; e.g. G(X ) = 2 λ p 2 4 or G(X ) = R µ,νραβ p µ SO(4)-invariance: S (R 4 ) (Günaydin, Lüst & Malek 16) Trivector modelled on negative chirality spinors

108 Vector trisums

109 Vector trisums Restrict X P X P = ( p 0 p 0 p p ) 1 + p 0 X p + p 0 X p + X p η p to P, P S 7 = Spin(7)/G 2 R 8 : X p η p = Im( X P X P ), ɛ p, p = sgn Re ( X P X P )

110 Vector trisums Restrict X P X P = ( p 0 p 0 p p ) 1 + p 0 X p + p 0 X p + X p η p to P, P S 7 = Spin(7)/G 2 R 8 : X p η p = Im( X P X P ), ɛ p, p = sgn Re ( X P X P ) Extend to vector trisum X p φ p φ p = Im( (X P X P ) X P ) : p φ p φ p 1 p η p 2 p + ɛ p, p 1 p η ( p ) 2 p ( = ɛ p, p, p ɛ p, p + ɛ p, p 1 p η p 2 p + A η( p, p, p ) + 1 p 2 ( p η p ) + 1 p 2 ( p η p ) + 1 p 2 ( p η p ))

111 Vector trisums Restrict X P X P = ( p 0 p 0 p p ) 1 + p 0 X p + p 0 X p + X p η p to P, P S 7 = Spin(7)/G 2 R 8 : X p η p = Im( X P X P ), ɛ p, p = sgn Re ( X P X P ) Extend to vector trisum X p φ p φ p = Im( (X P X P ) X P ) : p φ p φ p 1 p η p 2 p + ɛ p, p 1 p η ( p ) 2 p ( = ɛ p, p, p ɛ p, p + ɛ p, p 1 p η p 2 p + A η( p, p, p ) + 1 p 2 ( p η p ) + 1 p 2 ( p η p ) + 1 p 2 ( p η p )) Spin(7) G 2 : p η p = p φ 0 φ p

112 Vector trisums Restrict X P X P = ( p 0 p 0 p p ) 1 + p 0 X p + p 0 X p + X p η p to P, P S 7 = Spin(7)/G 2 R 8 : X p η p = Im( X P X P ), ɛ p, p = sgn Re ( X P X P ) Extend to vector trisum X p φ p φ p = Im( (X P X P ) X P ) : p φ p φ p 1 p η p 2 p + ɛ p, p 1 p η ( p ) 2 p ( = ɛ p, p, p ɛ p, p + ɛ p, p 1 p η p 2 p + A η( p, p, p ) + 1 p 2 ( p η p ) + 1 p 2 ( p η p ) + 1 p 2 ( p η p )) Spin(7) G 2 : p η p = p φ 0 φ p Nonassociativity: Asym( p φ p φ p ) = Im(X P φ P φ P )

113 Phase space nonassociative geometry

114 Phase space nonassociative geometry Extend vector trisum to all k R 7 R 8 : B φ ( k, k, k ) = sin 1 p φ p φ p p p φ p φ p φ p φ p p= k sin( k )/ k

115 Phase space nonassociative geometry Extend vector trisum to all k R 7 R 8 : B φ ( k, k, k ) = sin 1 p φ p φ p p p φ p φ p φ p φ p p= k sin( k )/ k = k + k + k + ( k η k + k η k + k η ) k + 2 ( 2 2 Aη( k, k, k ) k + k 2 k k + k 2 k k + k 2 ) k + O ( 3)

116 Phase space nonassociative geometry Extend vector trisum to all k R 7 R 8 : B φ ( k, k, k ) = sin 1 p φ p φ p p p φ p φ p φ p φ p p= k sin( k )/ k = k + k + k + ( k η k + k η k + k η ) k + 2 ( 2 2 Aη( k, k, k ) k + k 2 k k + k 2 k k + k 2 ) k + O ( 3) Phase space triproducts: (f λ g λ h)( x ) = k, k, k f ( k ) g( k ) h( k ) e i B φ (Λ k,λ k,λ k ) Λ 1 x

117 Phase space nonassociative geometry Extend vector trisum to all k R 7 R 8 : B φ ( k, k, k ) = sin 1 p φ p φ p p p φ p φ p φ p φ p p= k sin( k )/ k = k + k + k + ( k η k + k η k + k η ) k + 2 ( 2 2 Aη( k, k, k ) k + k 2 k k + k 2 k k + k 2 ) k + O ( 3) Phase space triproducts: (f λ g λ h)( x ) = k, k, k f ( k ) g( k ) h( k ) e i B φ (Λ k,λ k,λ k ) Λ 1 x f λ g = f λ 1 λ g, f λ g λ h = (f λ g λ h) p=0

118 Phase space nonassociative geometry Extend vector trisum to all k R 7 R 8 : B φ ( k, k, k ) = sin 1 p φ p φ p p p φ p φ p φ p φ p p= k sin( k )/ k = k + k + k + ( k η k + k η k + k η ) k + 2 ( 2 2 Aη( k, k, k ) k + k 2 k k + k 2 k k + k 2 ) k + O ( 3) Phase space triproducts: (f λ g λ h)( x ) = k, k, k f ( k ) g( k ) h( k ) e i B φ (Λ k,λ k,λ k ) Λ 1 x f λ g = f λ 1 λ g, f λ g λ h = (f λ g λ h) p=0 3-bracket [f, g, h] λ = Asym(f λ g λ h) obeys: [f, g, 1] λ = 3 [f, g] λ, lim λ 0 [x i, x j, x k ] λ p=0 = l 3 s R ijk