An Introduction to Nonassociative Physics

Transcription

1 An Introduction to Nonassociative Physics Richard Szabo Action MP 1405 Quantum Structure of Spacetime Dualities and Generalized Geometries Corfu Summer Institute September 13, 2018

2 Outline Introduction: A brief history of nonassociativity in physics Magnetic Poisson brackets Classical & quantum dynamics in fields of magnetic charge Closed strings in locally non-geometric flux backgrounds M2-branes in locally non-geometric flux backgrounds M-waves in locally non-geometric KK-monopole backgrounds

3 Jordanian Quantum Mechanics If A, B are Hermitian operators, then so is A B = 1 2 (A B + B A) (Jordan 32): A B = B A, (A 2 B) A = A 2 (B A)

4 Jordanian Quantum Mechanics If A, B are Hermitian operators, then so is A B = 1 2 (A B + B A) (Jordan 32): A B = B A, (A 2 B) A = A 2 (B A) Commutative, nonassociative special Jordan algebra Sufficient to demand alternative : (A B) A = A (B A) use to define noncommutative Jordan algebras (Albert 46; Shafer 55)

5 Jordanian Quantum Mechanics If A, B are Hermitian operators, then so is A B = 1 2 (A B + B A) (Jordan 32): A B = B A, (A 2 B) A = A 2 (B A) Commutative, nonassociative special Jordan algebra Sufficient to demand alternative : (A B) A = A (B A) use to define noncommutative Jordan algebras (Albert 46; Shafer 55) Only 3 3 Hermitian matrices over octonions O are non-special (Jordan, von Neumann & Wigner 34; Zelmanov 84;... ) Octonionic quantum mechanics satisfies von Neumann axioms, no Hilbert space formulation (Günaydin, Piron & Ruegg 78)

6 Octonions Only 4 normed division algebras over R: R, C, H, O

7 Octonions Only 4 normed division algebras over R: R, C, H, O Octonions O: a 0 + a 1 e 1 + a 2 e a 7 e 7 noncommutative Jordan algebra with multiplication given by Fano plane: e 1 e 5 = e 7 = e 5 e 1 etc. e 2 A = 1 e A e B = δ AB + η ABC e C Symmetry: G 2 SO(7)

8 Octonions Only 4 normed division algebras over R: R, C, H, O Octonions O: a 0 + a 1 e 1 + a 2 e a 7 e 7 noncommutative Jordan algebra with multiplication given by Fano plane: e 1 e 5 = e 7 = e 5 e 1 etc. e 2 A = 1 e A e B = δ AB + η ABC e C Symmetry: G 2 SO(7) Rewrite e 4, e 5, e 6 = f 1, f 2, f 3 : [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 ) )

9 Nambu Dynamics Bi-Hamiltonian dynamics with Nambu Poisson 3-bracket on R 3 : {f, g, h} := ε ijk i f j g k h obeying Leibniz rule and fundamental identity (Nambu 73)

10 Nambu Dynamics Bi-Hamiltonian dynamics with Nambu Poisson 3-bracket on R 3 : {f, g, h} := ε ijk i f j g k h obeying Leibniz rule and fundamental identity (Nambu 73) Euler equations L + ω L = 0 for rotating rigid body in R 3 equivalent to bi-hamiltonian equations: L i = { L i, L 2, T } where T = 1 2 L ω and i = / L i

11 Nambu Dynamics Bi-Hamiltonian dynamics with Nambu Poisson 3-bracket on R 3 : {f, g, h} := ε ijk i f j g k h obeying Leibniz rule and fundamental identity (Nambu 73) Euler equations L + ω L = 0 for rotating rigid body in R 3 equivalent to bi-hamiltonian equations: L i = { L i, L 2, T } where T = 1 2 L ω and i = / L i Quantization? Nambu Heisenberg bracket := half-jacobiator : [A, B, C] NH = [A, B] C + [C, A] B + [B, C] A

12 Nambu Dynamics Bi-Hamiltonian dynamics with Nambu Poisson 3-bracket on R 3 : {f, g, h} := ε ijk i f j g k h obeying Leibniz rule and fundamental identity (Nambu 73) Euler equations L + ω L = 0 for rotating rigid body in R 3 equivalent to bi-hamiltonian equations: L i = { L i, L 2, T } where T = 1 2 L ω and i = / L i Quantization? Nambu Heisenberg bracket := half-jacobiator : [A, B, C] NH = [A, B] C + [C, A] B + [B, C] A Nambu later suggests to use nonassociative algebras to quantize 3-bracket as a Jacobiator: [A, B, C] := [[A, B], C] + [[C, A], B] + [[B, C], A] Related to formulating nonassociative quantum mechanics

13 Nonassociativity in String/M-Theory Closed string field theory: L -algebras (Strominger 87; Zwiebach 93) D-branes in curved backgrounds: H = db controls Jacobiator (Cornalba & Schiappa 01; Herbst, Kling & Kreuzer 01) Topological T-duality of principal torus bundles (Mathai & Rosenberg 04; Bouwknegt, Hannabuss & Mathai 06; Brodzki, Mathai, Rosenberg & Sz 08)

14 Nonassociativity in String/M-Theory Closed string field theory: L -algebras (Strominger 87; Zwiebach 93) D-branes in curved backgrounds: H = db controls Jacobiator (Cornalba & Schiappa 01; Herbst, Kling & Kreuzer 01) Topological T-duality of principal torus bundles (Mathai & Rosenberg 04; Bouwknegt, Hannabuss & Mathai 06; Brodzki, Mathai, Rosenberg & Sz 08) Multiple M2-branes & 3-algebras (Basu & Harvey 05; Bagger & Lambert 07) Open M2-branes in C-field backgrounds: Quantization of Nambu Poisson 3-brackets (Bergshoeff, Berman, van der Schaar & Sundell 00; Kawamoto & Sasakura 00; Chu & Smith 09; Sämann & Sz 12) G 2 - and Spin(7)-backgrounds of M-theory

15 Nonassociativity in String/M-Theory Closed string field theory: L -algebras (Strominger 87; Zwiebach 93) D-branes in curved backgrounds: H = db controls Jacobiator (Cornalba & Schiappa 01; Herbst, Kling & Kreuzer 01) Topological T-duality of principal torus bundles (Mathai & Rosenberg 04; Bouwknegt, Hannabuss & Mathai 06; Brodzki, Mathai, Rosenberg & Sz 08) Multiple M2-branes & 3-algebras (Basu & Harvey 05; Bagger & Lambert 07) Open M2-branes in C-field backgrounds: Quantization of Nambu Poisson 3-brackets (Bergshoeff, Berman, van der Schaar & Sundell 00; Kawamoto & Sasakura 00; Chu & Smith 09; Sämann & Sz 12) G 2 - and Spin(7)-backgrounds of M-theory In these lectures, we focus on two related occurences: Magnetic monopoles (Jackiw 85; Günaydin & Zumino 85) Locally non-geometric string & M-theory backgrounds (Blumenhagen & Plauschinn 10; Lüst 10; Mylonas, Schupp & Sz 12; Günaydin, Lüst & Malek 16; Kupriyanov & Sz 17; Lüst, Malek & Sz 17; Freidel, Leigh & Minic 17;... )

16 Magnetic Poisson Brackets M = R d configuration space x i, M momentum space p i, M = T M = M M phase space X I = (x i, p i ), with canonical symplectic 2-form ω 0 = dp i dx i

17 Magnetic Poisson Brackets M = R d configuration space x i, M momentum space p i, M = T M = M M phase space X I = (x i, p i ), with canonical symplectic 2-form ω 0 = dp i dx i B Ω 2 (M) magnetic field deforms ω 0 to almost symplectic form: ω B = ω 0 B

18 Magnetic Poisson Brackets M = R d configuration space x i, M momentum space p i, M = T M = M M phase space X I = (x i, p i ), with canonical symplectic 2-form ω 0 = dp i dx i B Ω 2 (M) magnetic field deforms ω 0 to almost symplectic form: ω B = ω 0 B θ B = σ 1 B defines magnetic Poisson brackets {f, g} B = θb IJ I f J g of f, g C (M): {x i, x j } B = 0, {x i, p j } B = δ i j, {p i, p j } B = B ij (x)

19 Magnetic Poisson Brackets M = R d configuration space x i, M momentum space p i, M = T M = M M phase space X I = (x i, p i ), with canonical symplectic 2-form ω 0 = dp i dx i B Ω 2 (M) magnetic field deforms ω 0 to almost symplectic form: ω B = ω 0 B θ B = σ 1 B defines magnetic Poisson brackets {f, g} B = θb IJ I f J g of f, g C (M): {x i, x j } B = 0, {x i, p j } B = δ i j, {p i, p j } B = B ij (x) H-twisted Poisson structure on M with H = db magnetic charge [θ B, θ B ] S = 3 θ B (dω B) gives nonassociative algebra with Jacobiators {f, g, h} B = [θ B, θ B ] IJK S I f J g K h: {p i, p j, p k } B = H ijk (x)

20 Magnetic Monopoles d = 3 and B ij = e ε ijk B k governs motion of electric charge e in a static magnetic field B on R 3

21 Magnetic Monopoles d = 3 and B ij = e ε ijk B k governs motion of electric charge e in a static magnetic field B on R 3 db = 0 gives classical Maxwell theory B = 0, B = A

22 Magnetic Monopoles d = 3 and B ij = e ε ijk B k governs motion of electric charge e in a static magnetic field B on R 3 db = 0 gives classical Maxwell theory B = 0, B = A Dirac monopole field on R 3 \ {0}: BD = 4π g δ (3) ( x) B D = g x x 3 = A D, AD = g x x n x x n

23 Magnetic Monopoles d = 3 and B ij = e ε ijk B k governs motion of electric charge e in a static magnetic field B on R 3 db = 0 gives classical Maxwell theory B = 0, B = A Dirac monopole field on R 3 \ {0}: BD = 4π g δ (3) ( x) In the lab: B D = g x x 3 = A D, AD = g x x n x x n Neutron scattering off spin ice pyrochlore lattices (Castelnovo, Moessner & Sondhi 08; Morris et al. 09;... )

24 Magnetic Monopoles d = 3 and B ij = e ε ijk B k governs motion of electric charge e in a static magnetic field B on R 3 db = 0 gives classical Maxwell theory B = 0, B = A Dirac monopole field on R 3 \ {0}: BD = 4π g δ (3) ( x) In the lab: B D = g x x 3 = A D, AD = g x x n x x n Neutron scattering off spin ice pyrochlore lattices (Castelnovo, Moessner & Sondhi 08; Morris et al. 09;... ) Smooth H = db 0 gives smooth distributions B 0 of magnetic charge

25 Locally Non-Geometric Fluxes Born reciprocity (x, p) (p, x) preserves ω 0, maps B Ω 2 (M) β Ω 2 (M ) with twisted Poisson brackets: {x i, x j } β = β ij (p), {x i, p j } β = δ i j, {p i, p j } β = 0

26 Locally Non-Geometric Fluxes Born reciprocity (x, p) (p, x) preserves ω 0, maps B Ω 2 (M) β Ω 2 (M ) with twisted Poisson brackets: {x i, x j } β = β ij (p), {x i, p j } β = δ i j, {p i, p j } β = 0 Twisting by R-flux R = dβ Ω 3 (M ) gives nonassociative configuration space: {x i, x j, x k } β = R ijk (p)

27 Locally Non-Geometric Fluxes Born reciprocity (x, p) (p, x) preserves ω 0, maps B Ω 2 (M) β Ω 2 (M ) with twisted Poisson brackets: {x i, x j } β = β ij (p), {x i, p j } β = δ i j, {p i, p j } β = 0 Twisting by R-flux R = dβ Ω 3 (M ) gives nonassociative configuration space: {x i, x j, x k } β = R ijk (p) R-flux model: Phase space of closed strings propagating in locally non-geometric R-flux backgrounds

28 Locally Non-Geometric Fluxes Born reciprocity (x, p) (p, x) preserves ω 0, maps B Ω 2 (M) β Ω 2 (M ) with twisted Poisson brackets: {x i, x j } β = β ij (p), {x i, p j } β = δ i j, {p i, p j } β = 0 Twisting by R-flux R = dβ Ω 3 (M ) gives nonassociative configuration space: {x i, x j, x k } β = R ijk (p) R-flux model: Phase space of closed strings propagating in locally non-geometric R-flux backgrounds M = T 3 with H-flux gives geometric and non-geometric fluxes via T-duality (Shelton, Taylor & Wecht 05) H ijk T i f i jk T j Q ij k T k R ijk

29 Locally Non-Geometric Fluxes Born reciprocity (x, p) (p, x) preserves ω 0, maps B Ω 2 (M) β Ω 2 (M ) with twisted Poisson brackets: {x i, x j } β = β ij (p), {x i, p j } β = δ i j, {p i, p j } β = 0 Twisting by R-flux R = dβ Ω 3 (M ) gives nonassociative configuration space: {x i, x j, x k } β = R ijk (p) R-flux model: Phase space of closed strings propagating in locally non-geometric R-flux backgrounds M = T 3 with H-flux gives geometric and non-geometric fluxes via T-duality (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)

30 Classical Motion in Fields of Magnetic Charge For d = 3, motion in magnetic field B (with or without sources) governed by Lorentz force p = e m p B, p = m x Hamiltonian equations Ẋ I = {X I, H} B for H = 1 2m p 2

31 Classical Motion in Fields of Magnetic Charge For d = 3, motion in magnetic field B (with or without sources) governed by Lorentz force p = e m p B, p = m x Hamiltonian equations Ẋ I = {X I, H} B for H = 1 2m p 2 B = constant: Motion follows helical trajectory with uniform velocity along B-direction

32 Classical Motion in Fields of Magnetic Charge For d = 3, motion in magnetic field B (with or without sources) governed by Lorentz force p = e m p B, p = m x Hamiltonian equations Ẋ I = {X I, H} B for H = 1 2m p 2 B = constant: Motion follows helical trajectory with uniform velocity along B-direction Dirac monopole field B = B D : (Bakas & Lüst 13) Conservation of Poincaré vector K confines motion to surface of cone, electric charge never reaches magnetic monopole and nonassociativity plays no role

33 Classical Motion in Fields of Magnetic Charge B = (0, 0, ρ z), constant magnetic charge ρ: (Kupriyanov & Sz 18) Motion follows Euler spiral with uniform velocity along z-direction

34 Classical Motion in Fields of Magnetic Charge B = (0, 0, ρ z), constant magnetic charge ρ: (Kupriyanov & Sz 18) Motion follows Euler spiral with uniform velocity along z-direction B = 1 3 ρ x, constant magnetic charge ρ: Motion is no longer integrable or confined, equivalent to motion in Dirac monopole field B D with additional frictional forces (Bakas & Lüst 13)

35 Classical Motion in Fields of Magnetic Charge B = (0, 0, ρ z), constant magnetic charge ρ: (Kupriyanov & Sz 18) Motion follows Euler spiral with uniform velocity along z-direction B = 1 3 ρ x, constant magnetic charge ρ: Motion is no longer integrable or confined, equivalent to motion in Dirac monopole field B D with additional frictional forces (Bakas & Lüst 13) Questions: What substitutes for canonical quantization of locally non-geometric closed strings? Is there a sensible nonassociative quantum mechanics?

36 Quantization of Magnetic Poisson Brackets Quantization: Linear map f O f on f C (M): [O f, O g ] = i O {f,g}b + O( 2 ) [O x i, O x j ] = 0, [O x i, O pj ] = i δ i j, [O pi, O pj ] = i B ij (O x )

37 Quantization of Magnetic Poisson Brackets Quantization: Linear map f O f on f C (M): [O f, O g ] = i O {f,g}b + O( 2 ) [O x i, O x j ] = 0, [O x i, O pj ] = i δ i j, [O pi, O pj ] = i B ij (O x ) Magnetic translation operators P v = exp ( i O p v ) : Pv 1 O x i P v = O x i +v i

38 Quantization of Magnetic Poisson Brackets Quantization: Linear map f O f on f C (M): [O f, O g ] = i O {f,g}b + O( 2 ) [O x i, O x j ] = 0, [O x i, O pj ] = i δ i j, [O pi, O pj ] = i B ij (O x ) Magnetic translation operators P v = exp ( i O p v ) : Pv 1 O x i P v = O x i +v i Representation of translation group R d? (Jackiw 85) P w P v = e i Φ2(x;v,w) P v+w, P w (P v P u ) = e i Φ3(x;u,v,w) (P w P v ) P u x ρ v w x u H v w

39 Quantization with db = 0 B = da is field strength of a (trivial) line bundle L M = R d

40 Quantization with db = 0 B = da is field strength of a (trivial) line bundle L M = R d O x i = x i, O pi = i i + A i (x) Represented on quantum Hilbert space H = L 2 (M, L) = L 2 (M)

41 Quantization with db = 0 B = da is field strength of a (trivial) line bundle L M = R d O x i = x i, O pi = i i + A i (x) Represented on quantum Hilbert space H = L 2 (M, L) = L 2 (M) Magnetic translations given by Wilson lines (parallel transport in L): x v v A x ( (P v ψ)(x) = exp i 1 (x;v) ) A ψ(x v)

42 Quantization with db = 0 B = da is field strength of a (trivial) line bundle L M = R d O x i = x i, O pi = i i + A i (x) Represented on quantum Hilbert space H = L 2 (M, L) = L 2 (M) Magnetic translations given by Wilson lines (parallel transport in L): x v v A x ( (P v ψ)(x) = exp i 1 (x;v) ) A ψ(x v) Defines weak projective representation of translation group R d on H: ( ω v,w (x) = exp i (P w P v ψ)(x) = ω v,w (x) (P v+w ψ)(x) 2 (x;w,v) 2-cocycle on R d with values in C (M, U(1)) ) ( B = e i 2 B(v,w) for B constant )

43 Quantization with db = 0 Magnetic Weyl correspondence f C (M) O f End(H): W (x, p) : H H, ( O f = M M ( W (x, p)ψ ) (y) = e i 2 p x e i p y (P xψ)(y) e i ω 0(X,Y ) f (Y ) ) dy W (X ) (2π) d dx (2π) d

44 Quantization with db = 0 Magnetic Weyl correspondence f C (M) O f End(H): W (x, p) : H H, ( O f = M M ( W (x, p)ψ ) (y) = e i 2 p x e i p y (P xψ)(y) e i ω 0(X,Y ) f (Y ) ) dy W (X ) (2π) d dx (2π) d Magnetic Moyal Weyl star product O f B g = O f O g ; e.g. for B constant: (f B g)(x ) = 1 (π ) d e 2 i ω B (Y,Z) f (X Y ) g(x Z) dy dz M M

45 Quantization with db = 0 Magnetic Weyl correspondence f C (M) O f End(H): W (x, p) : H H, ( O f = M M ( W (x, p)ψ ) (y) = e i 2 p x e i p y (P xψ)(y) e i ω 0(X,Y ) f (Y ) ) dy W (X ) (2π) d dx (2π) d Magnetic Moyal Weyl star product O f B g = O f O g ; e.g. for B constant: (f B g)(x ) = 1 (π ) d e 2 i ω B (Y,Z) f (X Y ) g(x Z) dy dz M M Canonical quantum mechanics = phase space quantum mechanics: Observables/states: (real) functions on phase space Operator product: star product, Traces: integration State function (density matrix): S 0, M S = 1 Expectation values: O = M O B S...

46 Quantization with H = db 0 Operator/state formulation of canonical quantization cannot handle nonassociative magnetic Poisson brackets

47 Quantization with H = db 0 Operator/state formulation of canonical quantization cannot handle nonassociative magnetic Poisson brackets For Dirac monopole B D = g x/ x 3 : Magnetic Poisson brackets are associative on M = R 3 \ {0}, = da D locally B D Quantum Hilbert space is H = L 2 (M, L) for a non-trivial line bundle L M 2 e g iff Dirac charge quantization: Z (Wu & Yang 76) Magnetic Weyl correspondence on M induces associative phase space star product (Soloviev 17)

48 Quantization with H = db 0 Operator/state formulation of canonical quantization cannot handle nonassociative magnetic Poisson brackets For Dirac monopole B D = g x/ x 3 : Magnetic Poisson brackets are associative on M = R 3 \ {0}, = da D locally B D Quantum Hilbert space is H = L 2 (M, L) for a non-trivial line bundle L M 2 e g iff Dirac charge quantization: Z (Wu & Yang 76) Magnetic Weyl correspondence on M induces associative phase space star product (Soloviev 17) For generic smooth distributions H Ω 3 (M), standard canonical quantization breaks down

49 Quantization with H = db 0 Operator/state formulation of canonical quantization cannot handle nonassociative magnetic Poisson brackets For Dirac monopole B D = g x/ x 3 : Magnetic Poisson brackets are associative on M = R 3 \ {0}, = da D locally B D Quantum Hilbert space is H = L 2 (M, L) for a non-trivial line bundle L M 2 e g iff Dirac charge quantization: Z (Wu & Yang 76) Magnetic Weyl correspondence on M induces associative phase space star product (Soloviev 17) For generic smooth distributions H Ω 3 (M), standard canonical quantization breaks down Can be studied perturbatively in H using noncommutative Jordan algebra of quantum moments (Bojowald, Brahma, Büyükçam & Strobl 14)

50 Deformation Quantization For any H = db Ω 3 (M), Kontsevich formality provides noncommutative and nonassociative star product on C (M)[[ ]]: f H g = f g + i 2 {f, g} B + n 2 ( i ) n n! b n (f, g) f... θ ρ g [f, g, h] H = 2 {f, g, h} B + n 3 ( i ) n n! t n (f, g, h)... [θ ρ,θ ρ ] S f g h where b n = U n (θ B,..., θ B ), t n = U n+1 ([θ B, θ B ] S, θ B,..., θ B ) are bi-/tri-differential operators (Mylonas, Schupp & Sz 12)

51 Deformation Quantization For any H = db Ω 3 (M), Kontsevich formality provides noncommutative and nonassociative star product on C (M)[[ ]]: f H g = f g + i 2 {f, g} B + n 2 ( i ) n n! b n (f, g) f... θ ρ g [f, g, h] H = 2 {f, g, h} B + n 3 ( i ) n n! t n (f, g, h)... [θ ρ,θ ρ ] S f g h where b n = U n (θ B,..., θ B ), t n = U n+1 ([θ B, θ B ] S, θ B,..., θ B ) are bi-/tri-differential operators (Mylonas, Schupp & Sz 12) For H constant, B ij (x) = 1 3 H ijk x k : (f H g)(x ) = 1 (π ) d M M e 2 i ω B (Y,Z) f (X Y ) g(x Z) dy dz

52 Deformation Quantization Nonassociative magnetic translations P v := e i p v give 3-cocycle: P v H P w = Π v,w (x) P v+w ( ) Pu H P v H P w ( ) = ω u,v,w (x) P u H Pv H P w where Π v,w (x) = e i 6 H(x,v,w) and ω u,v,w (x) = e i 6 H(u,v,w)

53 Deformation Quantization Nonassociative magnetic translations P v := e i p v give 3-cocycle: P v H P w = Π v,w (x) P v+w ( ) Pu H P v H P w ( ) = ω u,v,w (x) P u H Pv H P w where Π v,w (x) = e i 6 H(x,v,w) and ω u,v,w (x) = e i 6 H(u,v,w) Phase space formulation of nonassociative quantum mechanics is physically sensible and gives novel quantitative predictions (Mylonas, Schupp & Sz 13)

54 Deformation Quantization Nonassociative magnetic translations P v := e i p v give 3-cocycle: P v H P w = Π v,w (x) P v+w ( ) Pu H P v H P w ( ) = ω u,v,w (x) P u H Pv H P w where Π v,w (x) = e i 6 H(x,v,w) and ω u,v,w (x) = e i 6 H(u,v,w) Phase space formulation of nonassociative quantum mechanics is physically sensible and gives novel quantitative predictions (Mylonas, Schupp & Sz 13) R-flux model: Expectation values of oriented volume uncertainty operators V ijk = 1 2 [ x i, x j, x k ] R give quantum of volume V ijk = 1 2 l3 s R ijk For d = 3, no D0-branes in locally non-geometric string backgrounds (T-dual to Freed Witten anomaly for D3-branes on T 3 with H-flux) (Wecht 07)

55 Hilbert Space Formulation?

56 Hilbert Space Formulation? Symplectic realization: Double M to extended phase space (x i, x i, p i, p i ) with symplectic brackets: (Kupriyanov & Sz 18) {x i, p j } = { x i, p j } = {x i, p j } = δ i j {p i, p j } = B ij (x) x k ( k B ij (x) H ijk (x) ) {p i, p j } = { p i, p j } = 1 2 B ij(x)

57 Hilbert Space Formulation? Symplectic realization: Double M to extended phase space (x i, x i, p i, p i ) with symplectic brackets: (Kupriyanov & Sz 18) {x i, p j } = { x i, p j } = {x i, p j } = δ i j {p i, p j } = B ij (x) x k ( k B ij (x) H ijk (x) ) {p i, p j } = { p i, p j } = 1 2 B ij(x) O(3, 3) O(3, 3)-invariant Hamiltonian H = 1 m p I η IJ p J, p I = (p i, p i ), η = reproduces Lorentz force p = e m p B, p = m x ( )

58 Hilbert Space Formulation? Symplectic realization: Double M to extended phase space (x i, x i, p i, p i ) with symplectic brackets: (Kupriyanov & Sz 18) {x i, p j } = { x i, p j } = {x i, p j } = δ i j {p i, p j } = B ij (x) x k ( k B ij (x) H ijk (x) ) {p i, p j } = { p i, p j } = 1 2 B ij(x) O(3, 3) O(3, 3)-invariant Hamiltonian H = 1 m p I η IJ p J, p I = (p i, p i ), η = reproduces Lorentz force p = e m p B, p = m x ( ) Consistent Hamiltonian reduction eliminates auxiliary coordinates iff H = 0: No polarisation of extended symplectic algebra consistent with Lorentz force and nonassociative magnetic Poisson algebra

59 Hilbert Space Formulation? Symplectic realization: Double M to extended phase space (x i, x i, p i, p i ) with symplectic brackets: (Kupriyanov & Sz 18) {x i, p j } = { x i, p j } = {x i, p j } = δ i j {p i, p j } = B ij (x) x k ( k B ij (x) H ijk (x) ) {p i, p j } = { p i, p j } = 1 2 B ij(x) O(3, 3) O(3, 3)-invariant Hamiltonian H = 1 m p I η IJ p J, p I = (p i, p i ), η = reproduces Lorentz force p = e m p B, p = m x ( ) Consistent Hamiltonian reduction eliminates auxiliary coordinates iff H = 0: No polarisation of extended symplectic algebra consistent with Lorentz force and nonassociative magnetic Poisson algebra Higher structures: Use 2-Hilbert space of sections of a gerbe on M with field strength H = db 0 (Bunk, Müller & Sz 18)

60 Hilbert Space Formulation? Symplectic realization: Double M to extended phase space (x i, x i, p i, p i ) with symplectic brackets: (Kupriyanov & Sz 18) {x i, p j } = { x i, p j } = {x i, p j } = δ i j {p i, p j } = B ij (x) x k ( k B ij (x) H ijk (x) ) {p i, p j } = { p i, p j } = 1 2 B ij(x) O(3, 3) O(3, 3)-invariant Hamiltonian H = 1 m p I η IJ p J, p I = (p i, p i ), η = reproduces Lorentz force p = e m p B, p = m x ( ) Consistent Hamiltonian reduction eliminates auxiliary coordinates iff H = 0: No polarisation of extended symplectic algebra consistent with Lorentz force and nonassociative magnetic Poisson algebra Higher structures: Use 2-Hilbert space of sections of a gerbe on M with field strength H = db 0 (Bunk, Müller & Sz 18) Transgression: Send field strength of gerbe on M to field strength of line bundle on loop space C (S 1, M) (Sämann & Sz 12)

61 How do Closed Strings see Nonassociativity? Configuration space triproducts in R-flux model (Aschieri & Sz 15) (f g h)(x) = ( f (x) R g(x) ) R h(x) p=0 = k,k,k f (k) g(k ) h(k ) e i l 3 s 12 R(k,k,k ) e i (k+k +k ) x

62 How do Closed Strings see Nonassociativity? Configuration space triproducts in R-flux model (Aschieri & Sz 15) (f g h)(x) = ( f (x) R g(x) ) R h(x) p=0 = k,k,k f (k) g(k ) h(k ) e i l 3 s 12 R(k,k,k ) e i (k+k +k ) x Quantizes 3-bracket [f, g, h] = Asym(f g h) (Takhtajan 94) [x i, x j, x k ] = l 3 s R ijk

63 How do Closed Strings see Nonassociativity? Configuration space triproducts in R-flux model (Aschieri & Sz 15) (f g h)(x) = ( f (x) R g(x) ) R h(x) p=0 = k,k,k f (k) g(k ) h(k ) e i l 3 s 12 R(k,k,k ) e i (k+k +k ) 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 multiplication of tachyon vertex operators V k (z, z) = : e i k X (z, z) : in CFT scattering of momentum states in R-flux background: Vk V k V k R exp ( i l3 s 12 R(k, k, k ) ) (Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11)

64 How do Closed Strings see Nonassociativity? Configuration space triproducts in R-flux model (Aschieri & Sz 15) (f g h)(x) = ( f (x) R g(x) ) R h(x) p=0 = k,k,k f (k) g(k ) h(k ) e i l 3 s 12 R(k,k,k ) e i (k+k +k ) 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 multiplication of tachyon vertex operators V k (z, z) = : e i k X (z, z) : in CFT scattering of momentum states in R-flux background: Vk V k V k R exp ( i l3 s 12 R(k, k, k ) ) (Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11) Violates strong constraint of Double Field Theory (Blumenhagen, Fuchs, Hassler, Lüst & Sun 13)

65 How do Closed Strings see Nonassociativity? Configuration space triproducts in R-flux model (Aschieri & Sz 15) (f g h)(x) = ( f (x) R g(x) ) R h(x) p=0 = k,k,k f (k) g(k ) h(k ) e i l 3 s 12 R(k,k,k ) e i (k+k +k ) 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 multiplication of tachyon vertex operators V k (z, z) = : e i k X (z, z) : in CFT scattering of momentum states in R-flux background: Vk V k V k R exp ( i l3 s 12 R(k, k, k ) ) (Blumenhagen, Deser, Lüst, Plauschinn & Rennecke 11) Violates strong constraint of Double Field Theory (Blumenhagen, Fuchs, Hassler, Lüst & Sun 13) On-shell associativity of CFT amplitudes: f g h = f g h

66 Nonassociative Gravity? Nonassociative (Riemannian) differential geometry can be developed using quasi-hopf algebra 2-cochain (coboundary is a 3-cocyle) twist deformation techniques (Mylonas, Schupp & Sz 13; Aschieri & Sz 15; Barnes, Schenkel & Sz 15; Blumenhagen & Fuchs 16; Aschieri, Dimitrijević-Ćirić & Sz 17)

67 Nonassociative Gravity? Nonassociative (Riemannian) differential geometry can be developed using quasi-hopf algebra 2-cochain (coboundary is a 3-cocyle) twist deformation techniques (Mylonas, Schupp & Sz 13; Aschieri & Sz 15; Barnes, Schenkel & Sz 15; Blumenhagen & Fuchs 16; Aschieri, Dimitrijević-Ćirić & Sz 17) Metric formulation of nonassociative gravity on phase space: Ricci tensor, unique metric-compatible torsion-free connection, non-trivial real deformation of spacetime Ricci tensor: Ric ij = Ric ij + l3 s 12 Rabc ( k ( ag kl ( b g lm ) c Γ m ij ) j ( ag kl ( b g lm ) c Γ m ik ) ( + c g mn a(g lm Γ k lj ) bγ n ik a(g lm Γ k lk ) bγ n ij + (Γ l ik ag km aγ l ik g km ) b Γ n lj (Γl ij ag km aγ l ij g km ) b Γ n ) ) lk

68 Nonassociative Gravity? Nonassociative (Riemannian) differential geometry can be developed using quasi-hopf algebra 2-cochain (coboundary is a 3-cocyle) twist deformation techniques (Mylonas, Schupp & Sz 13; Aschieri & Sz 15; Barnes, Schenkel & Sz 15; Blumenhagen & Fuchs 16; Aschieri, Dimitrijević-Ćirić & Sz 17) Metric formulation of nonassociative gravity on phase space: Ricci tensor, unique metric-compatible torsion-free connection, non-trivial real deformation of spacetime Ricci tensor: Ric ij = Ric ij + l3 s 12 Rabc ( k ( ag kl ( b g lm ) c Γ m ij ) j ( ag kl ( b g lm ) c Γ m ik ) ( + c g mn a(g lm Γ k lj ) bγ n ik a(g lm Γ k lk ) bγ n ij + (Γ l ik ag km aγ l ik g km ) b Γ n lj (Γl ij ag km aγ l ij g km ) b Γ n ) ) lk Question: Is there an equivalent O(d, d)-invariant (off-shell) nonassociative version of the closed string effective action? S = 1 ( g Ric 1 16π G 12 e φ/3 H ijk H ijk 1 ) 6 iφ i φ + M [cf. Invariance of noncommutative Yang Mills theory of D-branes on T d under open string SO(d, d; Z) T-duality]

69 M-Theory Lift of the R-Flux Model (Günaydin, Lüst & Malek 16; Lüst, Malek & Syväri 17) S 1 M M S 1 radius λ string coupling g s

70 M-Theory Lift of the R-Flux Model (Günaydin, Lüst & Malek 16; Lüst, Malek & Syväri 17) 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 = R ε ijk f i jk Lift to M-theory on M = M S 1 x 4 : T-duality = U-duality

71 M-Theory Lift of the R-Flux Model (Günaydin, Lüst & Malek 16; Lüst, Malek & Syväri 17) 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 = R ε ijk f i jk Lift to M-theory on M = M S 1 x 4 : T-duality = U-duality Sends membrane wrapping (w ij ) H 2 ( M, Z) to momenta (p i ) 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)

72 M2-Brane Phase Space No D0-branes on M = p 4 = 0 along M-theory direction R µ,νραβ p µ = 0 = membrane has 7D phase space M: {x i, x j } = l 3 s 3 2 R4,ijk4 p k, {x 4, x i } = λ l3 s 3 2 R4,1234 p i {x i, p j } = δ i j x 4 + λ ε i jk x k, {x 4, p i } = λ 2 x i {p i, p j } = λ ε ijk p k {x i, x j, x k } = l 3 s 3 2 R4,ijk4 x 4, {x i, x j, x 4 } = λ2 l 3 s 3 2 R 4,ijk4 x k {p i, x j, x k } = λ l3 ( s 3 2 R4,1234 δ j i p k δ k i p j ), {p i, x j, x 4 } = λ2 l 3 s 3 2 R 4,ijk4 p k {p i, p j, x k } = λ 2 ε ij k x 4 λ ( δ j k x i δ i k x j ), {pi, p j, x 3 } = λ 3 ε ijk x k {p i, p j, p k } = 0

73 M2-Brane Phase Space No D0-branes on M = p 4 = 0 along M-theory direction R µ,νραβ p µ = 0 = membrane has 7D phase space M: {x i, x j } = l 3 s 3 2 R4,ijk4 p k, {x 4, x i } = λ l3 s 3 2 R4,1234 p i {x i, p j } = δ i j x 4 + λ ε i jk x k, {x 4, p i } = λ 2 x i {p i, p j } = λ ε ijk p k {x i, x j, x k } = l 3 s 3 2 R4,ijk4 x 4, {x i, x j, x 4 } = λ2 l 3 s 3 2 R 4,ijk4 x k {p i, x j, x k } = λ l3 ( s 3 2 R4,1234 δ j i p k δ k i p j ), {p i, x j, x 4 } = λ2 l 3 s 3 2 R 4,ijk4 p k {p i, p j, x k } = λ 2 ε ij k x 4 λ ( δ j k x i δ i k x j ), {pi, p j, x 3 } = λ 3 ε ijk x k {p i, p j, p k } = 0 Originates from nonassociative, alternative, octonion algebra O: (x A ) = (x i, x 4, p i ) = Λ (e A ) = 1 ( ) λ l 3 s R/3 f i, λ 2 3 l 3s R/3 e 7, λ e i Reduces to magnetic Poisson brackets for closed strings at λ = 0 (with x 4 = 1 central)

74 Quantization of M2-Brane Phase Space (Kupriyanov & Sz 17) G 2 -structure: R 7 carries cross product ( k η p ) A = η ABC k B p C, invariant under G 2 SO(7) Represented on O through X k = k A e A : X k η k = 1 2 [ X k, X k ]

75 Quantization of M2-Brane Phase Space (Kupriyanov & Sz 17) G 2 -structure: R 7 carries cross product ( k η p ) A = η ABC k B p C, invariant under G 2 SO(7) Represented on O through X k = k A e A : X k η k = 1 2 [ X k, X k ] Alternativity (X X ) X = X (X X ) defines octonion exponential: e X k = cos k + sin k k X k

76 Quantization of M2-Brane Phase Space (Kupriyanov & Sz 17) G 2 -structure: R 7 carries cross product ( k η p ) A = η ABC k B p C, invariant under G 2 SO(7) Represented on O through X k = k A e A : X k η k [ = 1 2 X k, X k ] Alternativity (X X ) X = X (X X ) defines octonion exponential: e X k = cos k + sin k k X k BCH formula e X k e X k = e X Bη ( k, k ) can be computed explicitly giving octonionic Weyl correspondence and star product: (f λ g)( x ) = f ( k ) g( k ) e i B η(λ k,λ k ) Λ 1 x k, k such that (f λ g)( x ) λ 0 (f R g)(x)

77 Quantization of M2-Brane Phase Space (Kupriyanov & Sz 17) G 2 -structure: R 7 carries cross product ( k η p ) A = η ABC k B p C, invariant under G 2 SO(7) Represented on O through X k = k A e A : X k η k [ = 1 2 X k, X k ] Alternativity (X X ) X = X (X X ) defines octonion exponential: e X k = cos k + sin k k X k BCH formula e X k e X k = e X Bη ( k, k ) can be computed explicitly giving octonionic Weyl correspondence and star product: (f λ g)( x ) = f ( k ) g( k ) e i B η(λ k,λ k ) Λ 1 x k, k such that (f λ g)( x ) λ 0 (f R g)(x) Triproduct ( f λ g λ h ) ( x ) = ( (f λ g) λ h ) (x µ, p i ) p=0 quantizes the 3-Lie algebra A 4 : [x µ, x ν, x α ] 1 = l 3 s R ε µναβ x β

78 Magnetic Monopoles & Quantum Gravity Apply canonical transformation (x, p) (p, x), l 3 s R 2 e ρ: [x i, x j ] = i λ ε ijk x k [p i, p j ] = i e ρ ε ijk x k, [p 4, p i ] = i λ e ρ x i [x i, p j ] = i δ ij p 4 + i λ ε ijk p k, [x i, p 4] = i λ 2 x i Reduces to magnetic brackets for electric charges at λ = 0 7-dimensional phase space with extra momentum mode p 4

79 Magnetic Monopoles & Quantum Gravity Apply canonical transformation (x, p) (p, x), l 3 s R 2 e ρ: [x i, x j ] = i λ ε ijk x k [p i, p j ] = i e ρ ε ijk x k, [p 4, p i ] = i λ e ρ x i [x i, p j ] = i δ ij p 4 + i λ ε ijk p k, [x i, p 4] = i λ 2 x i Reduces to magnetic brackets for electric charges at λ = 0 7-dimensional phase space with extra momentum mode p 4 Quaternion subalgebra: Setting ρ = 0 reveals noncommutative associative deformation of spacetime with [p i, p j ] = 0; restrict to λ 2 p 2 + p4 2 = 1: (Freidel & Livine 05) [x i, x j ] = i λ ε ijk x k, [x i, p j ] = i 1 λ 2 p 2 δ ij + i λ ε ijk p k Ponzano Regge spin foam model of 3D quantum gravity Uncontracted octonion algebra is related to monopoles in the spacetime of 3D quantum gravity, with λ = l P /!

80 M-Wave Phase Space (Lüst, Malek & Sz 17) Embed magnetic monopole into IIA string theory as a D6-brane, electric probes are D0-branes

81 M-Wave Phase Space (Lüst, Malek & Sz 17) Embed magnetic monopole into IIA string theory as a D6-brane, electric probes are D0-branes Lift to M-theory as KK-monopole, electric probes are M-waves along x 4 S 1 (graviton momentum modes p 4 = e/r 11 ): ds 2 11 = ds U d x d x + U 1 ( dx 4 + A d x ) 2 A = U, 2 U = ρ

82 M-Wave Phase Space (Lüst, Malek & Sz 17) Embed magnetic monopole into IIA string theory as a D6-brane, electric probes are D0-branes Lift to M-theory as KK-monopole, electric probes are M-waves along x 4 S 1 (graviton momentum modes p 4 = e/r 11 ): ds 2 11 = ds U d x d x + U 1 ( dx 4 + A d x ) 2 A = U, 2 U = ρ Parameters: l 2 s = l 3 P/R 11, g s = ( R 11 /l P ) 3/2 (Witten 95) M-theory string theory: g s, R 11 0 with l s finite λ l P R 1/3 11 0

83 M-Wave Phase Space (Lüst, Malek & Sz 17) Embed magnetic monopole into IIA string theory as a D6-brane, electric probes are D0-branes Lift to M-theory as KK-monopole, electric probes are M-waves along x 4 S 1 (graviton momentum modes p 4 = e/r 11 ): ds 2 11 = ds U d x d x + U 1 ( dx 4 + A d x ) 2 A = U, 2 U = ρ Parameters: l 2 s = l 3 P/R 11, g s = ( R 11 /l P ) 3/2 (Witten 95) M-theory string theory: g s, R 11 0 with l s finite λ l P R 1/ Non-geometric KK-monopole: ρ = smearing of Dirac monopoles, no local expression for A and metric Taub-NUT S1 R 3 = S 1 -gerbe over R 3

84 M-Theory Phase Space 3-Algebra Spin(7)-structure: [ξâ, ξ ˆB, ξĉ ] φ = φâ ˆBĈ ˆD ξ ˆD for ξ = (ξ 0, ξ ) = (1, e A ), where φ 0ABC = η ABC and φ ABCD = η ABCD ; Symmetry: Spin(7) SO(8)

85 M-Theory Phase Space 3-Algebra Spin(7)-structure: [ξâ, ξ ˆB, ξĉ ] φ = φâ ˆBĈ ˆD ξ ˆD for ξ = (ξ 0, ξ ) = (1, e A ), where φ 0ABC = η ABC and φ ABCD = η ABCD ; Symmetry: Spin(7) SO(8) 8D phase space coordinates X = (x µ, p µ) = (Λ ξ, λ ξ0) have 2 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

86 M-Theory Phase Space 3-Algebra Spin(7)-structure: [ξâ, ξ ˆB, ξĉ ] φ = φâ ˆBĈ ˆD ξ ˆD for ξ = (ξ 0, ξ ) = (1, e A ), where φ 0ABC = η ABC and φ ABCD = η ABCD ; Symmetry: Spin(7) SO(8) 8D phase space coordinates X = (x µ, p µ) = (Λ ξ, λ ξ0) have 2 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; breaks Spin(7) G 2, 8 = 7 1 G(X ) = p 4 gives M2-brane phase space, G(X ) = x 4 gives M-wave phase space; Born reciprocity is a Spin(7)-transformation