Higher Structures in Non-Geometric M-Theory

Transcription

1 Higher Structures in Non-Geometric M-Theory Richard Szabo Action MP 1405 Quantum Structure of Spacetime Noncommutativity and Physics: Quantum Spacetime Structures Bayrischzell April 24, 2017

2 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; Andriot, Larfors, Lüst & Patalong 12; Blair 14; Bakas & Lüst 15) 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 (Andriot, Hohm, Larfors, Lüst & Patalong 12) = ˆ [i β jk]

3 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)

4 Nonassociative quantum mechanics (Mylonas, Schupp & RS 13; Bojowald, Brahma, Büyükçam & Strobl 14; Bojowald, Brahma & Büyükçam 15) 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

5 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

6 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

7 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)

8 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)

9 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 Imaginary unit octonions: 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,...

10 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

11 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

12 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)

13 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,xj,x 4 = λ2 l 3 s 2 R 4,ijk4 p k λ ε k ij x 4 + δ k j x i δ k i x j, V p i,p j,x4 = λ3 2 2 ε ijk x k

14 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 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

15 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 Restrict to (x µ ) S 3 R 4 of radius 1 λ : f (l) g(l ) e 2 i λ B( λ 2 l, λ 2 l ) p [p i, p j ] = i λ ε ijk p k, [x i, p j ] = i λ Familiar from 3D quantum gravity (Freidel & Livine 06) 1 λ 2 x 2 δ i j + i λ ε i jk x k

16 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 XK ) X K (X K XK ) X K Spin(7) G 2 : k η k = k φ (1, 0 ) φ k Trivector: [ξâ, ξ ˆB, ξĉ ] φ = φâ ˆBĈ ˆD ξ ˆD for ξ = (ξ 0, ξ ) = (1, e A )

17 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

18 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 )

19 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