Non-commutative & non-associative closed string geometry from flux compactifications

Transcription

1 Non-commutative & non-associative closed string geometry from flux compactifications Dieter Lüst, LMU (Arnold Sommerfeld Center) and MPI München Istanbul Conference, 4. August 2011

2 I) Introduction Closed string flux compactifications: Moduli stabilization string landscape AdS/CFT correspondence Generalized geometries Here: how does a closed string see space? What is the proper geometrical description of closed string flux background? Non-commutative & non-associative geometry! D.L., arxiv: ; R. Blumenhagen, E. Plauschinn, arxiv: ; R. Blumenhagen, A. Deser, D.L., E. Plauschinn, F. Rennecke, arxiv: ; D. Andriot, M. Larfors, D.L., P. Patalong: work in progress (See also: L. Cornalba, R. Schiappa (2001); P. Bouwknegt, K. Hannabuss, V. Mathai (2006))

3 Point particle in a (constant) magnetic field: Configuration space: M = T Q, B = rot A Langrange function: Canonical momenta: L = 1 2 (p i) 2 = 1 2 (ẋi A i ) 2 p i = L ẋ i =ẋi A i π ij = {x i,x j } =0, π ij = {p i,p j } =0, {x i,p j } = δ j i Mechanical momenta: p i =ẋ i = p i + A i π ij = {x i,x j } =0, π ij = {p i, p j } = ɛ ijk B k, {x i,p j } = δ j i Non-commutative (Poisson) algebra

4 Point particle in the field of a magnetic monopole: B H 2 (Q), H = db = ρ magn (B is non-closed) ρ magn... charge density of a magnetic monopole. π ij = {x i,x j } =0, π ij = {p i, p j } = H ijk x k, {x i,p j } = δ j i This leads to: π ijk = {{p i, p j }, p k } + perm. = H ijk Twisted Poisson structure. (C. Klimcik, T. Strobl, (2002); A. Alekseev, T. Strobl, (2005); C. Saemann, R. Szabo, arxiv: ) As we will see, we will get a twisted Poisson structure for closed strings, however for the position operators instead of the momentum operators.

5 Non-commutative geometry and string theory (a): Open strings: 2-dimensional D-branes with 2-form F-flux coordinates of open string end points become non-commutative: [X i (τ),x j (τ)] = ɛ ij Θ, Θ = 2πiα F 1+F 2 (A. Abouelsaood, C. Callan, C. Nappi, S. Yost (1987); J. Fröhlich, K. Gawedzki (1993); F. Lizzi, ER. Szabo (1997); A.Connes, M. Douglas, A. Schwarz (1997), V. Schomeru (1999);..

6 Non-commutative geometry and string theory (a): Open strings: 2-dimensional D-branes with 2-form F-flux coordinates of open string end points become non-commutative: [X i (τ),x j (τ)] = ɛ ij Θ, Θ = 2πiα F 1+F 2 constant (A. Abouelsaood, C. Callan, C. Nappi, S. Yost (1987); J. Fröhlich, K. Gawedzki (1993); F. Lizzi, ER. Szabo (1997); A.Connes, M. Douglas, A. Schwarz (1997), V. Schomeru (1999);..

7 Non-commutative geometry and string theory (b): Closed strings:

8 Non-commutative geometry and string theory (b): Closed strings: 3-dimensional backgrounds with 3-form flux we will show that coordinates of closed strings become non-commutative: [X i (τ, σ),x j (τ, σ)] F (3) ijk pk (D.L., arxiv: )

9 Non-commutative geometry and string theory (b): Closed strings: 3-dimensional backgrounds with 3-form flux we will show that coordinates of closed strings become non-commutative: [X i (τ, σ),x j (τ, σ)] F (3) ijk pk operator (D.L., arxiv: )

10 Non-commutative geometry and string theory (b): Closed strings: 3-dimensional backgrounds with 3-form flux we will show that coordinates of closed strings become non-commutative: [X i (τ, σ),x j (τ, σ)] F (3) ijk pk (D.L., arxiv: ) and even non-associative: operator (R. Blumenhagen, E. Plauschinn, arxiv: ) [[X i (τ, σ),x j (τ, σ)],x k (τ, σ)] + perm. F (3) ijk Non-commutative/non-associative gravity?

11 Non-commutative geometry and string theory (b): Closed strings: 3-dimensional backgrounds with 3-form flux we will show that coordinates of closed strings become non-commutative: [X i (τ, σ),x j (τ, σ)] F (3) ijk pk (D.L., arxiv: ) and even non-associative: operator (R. Blumenhagen, E. Plauschinn, arxiv: ) [[X i (τ, σ),x j (τ, σ)],x k (τ, σ)] + perm. F (3) ijk Non-commutative/non-associative gravity? constant

12 Outline: II) T-duality & twisted tori III) NC-NA Closed string geometry IV) Non-associative closed string CFT

13 II) T-duality & twisted tori How does a closed string see geometry? Consider compactification on a circle with radius R: X(τ, σ) =X L (τ + σ)+x R (τ σ) X L (τ + σ) = x 2 + p L(τ + σ)+i X R (τ σ) = x 2 + p R(τ σ)+i p L = 1 ( ) M 2 R +(α ) 1 NR, p R = 1 ( ) M 2 R (α ) 1 NR α 2 α 2 n 0 n 0 1 n α ne in(τ+σ), 1 n α ne in(τ σ) (KK momenta ) p = p L + p R = M R p = p L p R =(α ) 1 NR (dual momenta - winding modes)

14 T-duality: T : R α R, M N T : p p, p L p L, p R p R. Dual space coordinates: X(τ, σ) =X L X R (X, X) : Doubled geometry: (O. Hohm, C. Hull, B. Zwiebach (2009/10)) T-duality is part of diffeomorphism group. T : X X, X L X L, X R X R

15 Compactification on a 2-dimensional torus: Background: 2 complex background parameters: T-duality transformations: SL(2, Z) τ aτ + b τ : cτ + d SL(2, Z) ρ aρ + b ρ : cρ + d T-duality in x 1 R 1,R 2,e iα,b τ = e 2 e 1 = R 2 R 1 e iα, ρ = B + ir 1 R 2 sin α. They act as shifts/rotations on doubled coordinates. Mirror symmetry: τ ρ B Rτ

16 Three-dimensional backgrounds twisted 3-tori: (A. Dabholkar, C. Hull (2003) ; S. Hellerman, J. McGreevy, B. Williams (2004); J. Derendinger, C. Kounnas, P. Petropoulos, F. Zwirner (2004); J. Shelton, W. Taylor, B. Wecht (2005); G. Dall Agata, S. Ferrara (2005)...) Fibrations: 2-dim. torus that varies over a circle: T 2 x 1,x 2 M 3 S 1 x 3 The fibration is specified by its monodromy properties.

17 Three-dimensional backgrounds twisted 3-tori: (A. Dabholkar, C. Hull (2003) ; S. Hellerman, J. McGreevy, B. Williams (2004); J. Derendinger, C. Kounnas, P. Petropoulos, F. Zwirner (2004); J. Shelton, W. Taylor, B. Wecht (2005); G. Dall Agata, S. Ferrara (2005)...) Fibrations: 2-dim. torus that varies over a circle: T 2 x 1,x 2 M 3 S 1 x 3 The fibration is specified by its monodromy properties. Two (T-dual) cases: (i) Geometric spaces (manifolds) x 3 x 3 +2π τ(x 3 +2π) = aτ(x3 )+b cτ(x 3 )+d

18 Three-dimensional backgrounds twisted 3-tori: (A. Dabholkar, C. Hull (2003) ; S. Hellerman, J. McGreevy, B. Williams (2004); J. Derendinger, C. Kounnas, P. Petropoulos, F. Zwirner (2004); J. Shelton, W. Taylor, B. Wecht (2005); G. Dall Agata, S. Ferrara (2005)...) Fibrations: 2-dim. torus that varies over a circle: T 2 x 1,x 2 M 3 S 1 x 3 The fibration is specified by its monodromy properties. Two (T-dual) cases: (i) Geometric spaces (manifolds) x 3 x 3 +2π τ(x 3 +2π) = aτ(x3 )+b cτ(x 3 )+d

19 Three-dimensional backgrounds twisted 3-tori: (A. Dabholkar, C. Hull (2003) ; S. Hellerman, J. McGreevy, B. Williams (2004); J. Derendinger, C. Kounnas, P. Petropoulos, F. Zwirner (2004); J. Shelton, W. Taylor, B. Wecht (2005); G. Dall Agata, S. Ferrara (2005)...) Fibrations: 2-dim. torus that varies over a circle: T 2 x 1,x 2 M 3 S 1 x 3 The fibration is specified by its monodromy properties. Two (T-dual) cases: (i) Geometric spaces (manifolds) x 3 x 3 +2π τ(x 3 +2π) = aτ(x3 )+b cτ(x 3 )+d

20 Three-dimensional backgrounds twisted 3-tori: (A. Dabholkar, C. Hull (2003) ; S. Hellerman, J. McGreevy, B. Williams (2004); J. Derendinger, C. Kounnas, P. Petropoulos, F. Zwirner (2004); J. Shelton, W. Taylor, B. Wecht (2005); G. Dall Agata, S. Ferrara (2005)...) Fibrations: 2-dim. torus that varies over a circle: Two (T-dual) cases: (i) Geometric spaces (manifolds) x 3 x 3 +2π τ(x 3 +2π) = aτ(x3 )+b (ii) Non-geometric spaces (T-folds) ρ(x 3 +2π) = aρ(x3 )+b cρ(x 3 )+d x 3 x 3 +2π T 2 x 1,x 2 M 3 S 1 x 3 The fibration is specified by its monodromy properties. cτ(x 3 )+d

21 Three-dimensional backgrounds twisted 3-tori: (A. Dabholkar, C. Hull (2003) ; S. Hellerman, J. McGreevy, B. Williams (2004); J. Derendinger, C. Kounnas, P. Petropoulos, F. Zwirner (2004); J. Shelton, W. Taylor, B. Wecht (2005); G. Dall Agata, S. Ferrara (2005)...) Fibrations: 2-dim. torus that varies over a circle: Two (T-dual) cases: (i) Geometric spaces (manifolds) x 3 x 3 +2π τ(x 3 +2π) = aτ(x3 )+b (ii) Non-geometric spaces (T-folds) ρ(x 3 +2π) = aρ(x3 )+b cρ(x 3 )+d x 3 x 3 +2π T 2 x 1,x 2 M 3 S 1 x 3 The fibration is specified by its monodromy properties. cτ(x 3 )+d

22 Three-dimensional backgrounds twisted 3-tori: (A. Dabholkar, C. Hull (2003) ; S. Hellerman, J. McGreevy, B. Williams (2004); J. Derendinger, C. Kounnas, P. Petropoulos, F. Zwirner (2004); J. Shelton, W. Taylor, B. Wecht (2005); G. Dall Agata, S. Ferrara (2005)...) Fibrations: A ten-dimensional 2-dim. torus action that for varies non-geometrical over a circle: fluxes: Two (T-dual) cases: (i) Geometric spaces (manifolds) x 3 x 3 +2π τ(x 3 +2π) = aτ(x3 )+b (ii) Non-geometric spaces (T-folds) ρ(x 3 +2π) = aρ(x3 )+b cρ(x 3 )+d x 3 x 3 +2π T 2 x 1,x 2 M 3 S 1 x 3 The fibration is specified by its monodromy properties. (D. Andriot, M. Larfors, D.L. P. Patalong, arxiv: ) cτ(x 3 )+d

23 Two different kind of monodromies for the fibrations: (i) elliptic monodromies: finite order ( ) ( ) SL(2, Z) τ, SL(2, Z) ρ : or order 4 order 6 (ii) parabolic monodromies: infinite order SL(2, Z) τ, SL(2, Z) ρ : ( ) 1 n 0 1 or ( ) 1 0 n 1 Both types in general contain geometric spaces as well as non-geometric backgrounds.

24 III) NC-NA Closed string geometry 3.1) Open strings on D2-branes: (i) D2-branes with gauge F-flux Mixed D/N boundary conditions: [X 1 (τ, 0),X 2 (τ, 0)] = 2πiα F 12 1+(F 12 ) 2 T-duality in X 1 : (ii) D1-branes at angles Boundary conditions: (F 12 = da(x)) σ X 1 + F 12 τ X 2 = 0, σ X 2 F 12 τ X 1 = 0 T-duality (Seiberg-Witten map) N : σ X 1 + F 12 σ X 2 = 0, D : τ X 2 F 12 τ X 1 = 0. [X 1 (τ, 0),X 2 (τ, 0)] = 0 Geom. angle: ν = arccot F 12 π

25 Open string CFT with F-flux is exactly solvable Origin of open string non-commutativity: a) shifted oscillator frequencies due to boundary conditions: X 1 = x 1 α n Z α m Z X 2 = x 2 + i α n Z α n+ν n + ν e i(n+ν)τ sin[(n + ν)σ + θ 1 ] α m ν m ν e i(m ν)τ sin[(m ν)σ θ 1 ], α n+ν n + ν e i(n+ν)τ sin[(n + ν)σ + θ 1 ] i α m Z α m ν m ν e i(m ν)τ sin[(m ν)σ θ 1 ]. ν = arccot F 12 π (A. Abouelsaood, C. Callan, C. Nappi, S. Yost (1987); C. Chu, P. Ho (1999))

26 b) Open string CFT: X 1 (z 1 ),X 2 (z 2 ) = 2πiα F 12 1+(F 12 ) 2 log ( z 1 z 2 z 1 z 2 σ=σ =0 = 2πiα F 12 1+(F 12 ) 2 ɛ(τ 1 τ 2 ) This function has a jump when changing the order of z 1 and z 2 on the real line. N-point tachyon amplitude: T1... T N = exp ( i 1 n<m N ) p n,a θ ab p m,b ɛ(τ n τ n ) T 1... T N ) T V (p, τ) = :exp ( ip X(τ) ) : θ=0

27 b) Open string CFT: X 1 (z 1 ),X 2 (z 2 ) = 2πiα F 12 1+(F 12 ) 2 log Non-commutative gauge theories - Moyal-Weyl f 1 (x) f 2 (x)... f N (x) := exp [ i m<n Θab x m a x ] n b f1 (x 1 ) f 2 (x 2 )... f N (x N ) S d n x Tr ˆF ab ab ˆF ( z 1 z 2 z 1 z 2 σ=σ =0 = 2πiα F 12 1+(F 12 ) 2 ɛ(τ 1 τ 2 ) This function has a jump when changing the order of z 1 and z 2 on the real line. N-point tachyon amplitude: T1... T N = exp ( i 1 n<m N ) T V (p, τ) = :exp ( ip X(τ) ) : ) p n,a θ ab p m,b ɛ(τ n τ n ) T 1... T N θ=0 - product: x1 =...=x N =x

28 3.2) Closed strings on a 3-dim. space: Can the closed string also see a non-commutative space? What deformation is needed? Yes: one needs 3-form flux: F (3) = H/ ω/q/r (F (3) = F (2) )

29 3.2) Closed strings on a 3-dim. space: Can the closed string also see a non-commutative space? NS H-flux What deformation is needed? Yes: one needs 3-form flux: F (3) = H/ ω/q/r (F (3) = F (2) )

30 3.2) Closed strings on a 3-dim. space: Can the closed string also see a non-commutative metric space? What deformation is needed? flux Yes: one needs 3-form flux: F (3) = H/ ω/q/r (F (3) = F (2) )

31 3.2) Closed strings on a 3-dim. space: Can the closed string also see a non-commutative geom. space? What deformation is needed? flux Yes: one needs 3-form flux: F (3) = H/ ω/q/r (F (3) = F (2) )

32 3.2) Closed strings on a 3-dim. space: Can the closed string also see a non-commutative space? R-flux What deformation is needed? Yes: one needs 3-form flux: F (3) = H/ ω/q/r (F (3) = F (2) )

33 3.2) Closed strings on a 3-dim. space: Can the closed string also see a non-commutative space? R-flux What deformation is needed? Yes: one needs 3-form flux: (i) Geometric spaces (manifolds) [X 1 (τ, σ),x 2 (τ, σ)] = 0 F (3) = H/ ω/q/r (F (3) = F (2) )

34 3.2) Closed strings on a 3-dim. space: Can the closed string also see a non-commutative space? R-flux What deformation is needed? Yes: one needs 3-form flux: (i) Geometric spaces (manifolds) [X 1 (τ, σ),x 2 (τ, σ)] = 0 F (3) = H/ ω/q/r (F (3) = F (2) ) (ii) Non-geometric spaces (T-folds) [X 1 (τ, σ),x 2 (τ, σ)] 0 T-duality

35 3.2) Closed strings on a 3-dim. space: Can the closed string also see a non-commutative space? R-flux What deformation is needed? Yes: one needs 3-form flux: (i) Geometric spaces (manifolds) F (3) = H/ ω/q/r (F (3) = F (2) ) [X 1 (τ, σ),x 2 (τ, σ)] = 0 ([X 1 (τ, σ), X 2 (τ, σ)] 0) (ii) Non-geometric spaces (T-folds) More general: [X 1 (τ, σ),x 2 (τ, σ)] 0 T-duality Doubled geometry: Closed string non-commutativity in ( X, X )-space

36 Problem: Background is non-constant. CFT is in general not exactly solvable Ways to handle: Study SU(2) WZW model with H-flux (R. Blumenhagen, E. Plauschinn, arxiv: ) Consider monodromy properties and the corresponding closed string boundary conditions Shifted closed string mode expansion (D.L., arxiv: ) Consider sigma model perturbation theory for small fluxes (R. Blumenhagen, A. Deser, D.L., E. Plauschinn, F. Rennecke, arxiv: )

37 Specific example: elliptic monodromy C. Hull, R. Reid-Edwards (2009)) (i) Geometric space ( ω -flux ) (ω 123 x 3g x 1 x 2 x 3Rτ(x3 )) τ(x 3 )= (1 + i) cos(hx3 ) + sin(hx 3 ) cos(hx 3 ) (1 + i) sin(hx 3 ) Monodromy: τ(x 3 +2π) = 1/τ(x 3 ) (H Z)

38 Specific example: elliptic monodromy C. Hull, R. Reid-Edwards (2009)) (i) Geometric space ( ω -flux ) (ω 123 x 3g x 1 x 2 x 3Rτ(x3 )) τ(x 3 )= (1 + i) cos(hx3 ) + sin(hx 3 ) cos(hx 3 ) (1 + i) sin(hx 3 ) Monodromy: τ(x 3 +2π) = 1/τ(x 3 ) (H Z)

39 Specific example: elliptic monodromy C. Hull, R. Reid-Edwards (2009)) (i) Geometric space ( ω -flux ) (ω 123 x 3g x 1 x 2 x 3Rτ(x3 )) τ(x 3 )= (1 + i) cos(hx3 ) + sin(hx 3 ) cos(hx 3 ) (1 + i) sin(hx 3 ) Monodromy: τ(x 3 +2π) = 1/τ(x 3 ) (H Z)

40 Specific example: elliptic monodromy C. Hull, R. Reid-Edwards (2009)) (i) Geometric space ( ω -flux ) (ω 123 x 3g x 1 x 2 x 3Rτ(x3 )) τ(x 3 )= (1 + i) cos(hx3 ) + sin(hx 3 ) cos(hx 3 ) (1 + i) sin(hx 3 ) Monodromy: τ(x 3 +2π) = 1/τ(x 3 ) This induces the following Z 4 symmetric closed string boundary condition: X 3 (τ, σ +2π) =X 3 (τ, σ)+2πn 3 (H Z) X L (τ, σ +2π) = e iθ X L (τ, σ), θ = 2πN 3 H, X R (τ, σ +2π) = e iθ X R (τ, σ). (Complex coordinates: ) X L,R = X 1 L,R + ix 2 L,R winding number L-R symmetric order 4 rotation

41 Corresponding closed string mode expansion X L (τ + σ) = i X R (τ σ) = i α 2 α 2 n Z n Z Then one obtains: 1 n ν α n νe i(n ν)(τ+σ), 1 n + ν α n+νe i(n+ν)(τ σ) ν = θ 2π = N 3H, (shifted oscillators!) [X L (τ, σ), X L (τ, σ)] = [X R (τ, σ), X R (τ, σ)] = Θ Θ = α n Z 1 n ν = α π cot(πn 3 H) [X 1 (τ, σ),x 2 (τ, σ)] = [X 1 L + X 1 R,X 2 L + X 2 R]=0

42 T-dual geometry (mirror symmetry): (ii) Non-geometric space (Q-flux) ρ(x 3 )= (1 + i) cos(hx3 ) + sin(hx 3 ) cos(hx 3 ) (1 + i) sin(hx 3 ) τ(x 3 ) ρ(x 3 ) (H Z) H-field: H(x 3 )=H sin(2hx3 ) 6 cos(2hx 3 ) (2 sin(2hx 3 ) + cos(2hx 3 ) 3) 2 Monodromy: ρ(x 3 +2π) = 1/ρ(x 3 )

43 T-dual geometry (mirror symmetry): (ii) Non-geometric space (Q-flux) ρ(x 3 )= (1 + i) cos(hx3 ) + sin(hx 3 ) cos(hx 3 ) (1 + i) sin(hx 3 ) τ(x 3 ) ρ(x 3 ) (H Z) H-field: H(x 3 )=H sin(2hx3 ) 6 cos(2hx 3 ) (2 sin(2hx 3 ) + cos(2hx 3 ) 3) 2 Monodromy: ρ(x 3 +2π) = 1/ρ(x 3 )

44 T-dual geometry (mirror symmetry): (ii) Non-geometric space (Q-flux) ρ(x 3 )= (1 + i) cos(hx3 ) + sin(hx 3 ) cos(hx 3 ) (1 + i) sin(hx 3 ) τ(x 3 ) ρ(x 3 ) (H Z) H-field: H(x 3 )=H sin(2hx3 ) 6 cos(2hx 3 ) (2 sin(2hx 3 ) + cos(2hx 3 ) 3) 2 Monodromy: ρ(x 3 +2π) = 1/ρ(x 3 )

45 T-dual geometry (mirror symmetry): (ii) Non-geometric space (Q-flux) ρ(x 3 )= (1 + i) cos(hx3 ) + sin(hx 3 ) cos(hx 3 ) (1 + i) sin(hx 3 ) H-field: τ(x 3 ) ρ(x 3 ) (H Z) H(x 3 )=H sin(2hx3 ) 6 cos(2hx 3 ) (2 sin(2hx 3 ) + cos(2hx 3 ) 3) 2 Monodromy: ρ(x 3 +2π) = 1/ρ(x 3 ) This induces the following Z 4 asymmetric closed string boundary condition: X 3 (τ, σ +2π) =X 3 (τ, σ)+2πn 3 X L (τ, σ +2π) = e iθ X L (τ, σ), θ = 2πN 3 H, X R (τ, σ +2π) = e iθ X R (τ, σ). L-R a-symmetric order 4 rotation

46 Corresponding closed string mode expansion X L (τ + σ) = i X R (τ σ) = i α 2 α 2 n Z n Z 1 n ν α n νe i(n ν)(τ+σ), 1 n + ν α n νe i(n ν)(τ σ) ν = θ 2π = N 3H, Then one finally obtains: [X L (τ, σ), X L (τ, σ)] = [X R (τ, σ), X R (τ, σ)] = Θ [X 1 (τ, σ),x 2 (τ, σ)] = [X 1 L + X 1 R,X 2 L + X 2 R]=i Θ

47 T-duality in x 3 - direction R-flux Winding no. N 3 Momentum no. M 3 [X 1 (τ, σ),x 2 (τ, σ)] = iθ Θ = α n Z 1 n ν = α π cot(πm 3 H) Act on wave functions replace momentum number by momentum operator: M 3 α p 3, N 3 α p 3

48 Then one obtains the following non-commutative algebra: [X 1,X 2 ] il 3 sf (3) p 3 ([X i,x j ] iɛ ijk F (3) p k ) Use [p 3,X 3 ]= i = [[X 1,X 2 ],X 3 ] + perm. F (3) l 3 s Non-associative algebra (twisted Poisson structure)! This nicely agrees with the non-associative closed string structure found by Blumenhagen, Plauschinn in the SU(2) WZW model: arxiv:

49 Parabolic monodromies: D. Andriot, M. Larfors, D.L., P. Patalong: work in progress Start with flat torus and constant H-field. Chain of three T-dualities: Flux Commutators Three-brackets H-flux [ X 1, X 2 ] w 3 [ X 1, X 2, X 3 ] ω-flux [X 1, X 2 ] w 3 [X 1, X 2, X 3 ] Q-flux [X 1,X 2 ] w 3 [X 1,X 2, X 3 ] R-flux [X 1,X 2 ] p 3 [X 1,X 2,X 3 ]

50 IV) Non-associative closed string CFT R. Blumenhagen, A. Deser, D.L., E. Plauschinn, F. Rennecke, arxiv: General idea: Consider a flat background plus linear B-field: B ab = 1 3 H abc X c CFT-perturbation theory linear in H: S = S 0 + S 1, S 1 = 1 Correlation function: O1... O N = 1 Z (Theory is still conformal at linear order in H.) H abc 2πα 3 = O 1... O N Σ [dx] O 1... O N e S[X] d 2 zx a X b X c 0 O 1... O N S

51 Solution of classical equations of motion: X a = 1 2 Ha bc X b X c X0 a (z,z) =X a L(z)+X a R(z) X a (z,z) =X a 0(z,z)+ 1 2 Ha bc X b L(z) X c R(z) Tachyon vertex operator: V(z, z) = :exp ( ik L X L + ik R X R ) k a L = p a + wa α, kr a = p a wa α

52 T-duality: X a L (z) X a R (z) T-duality +X a L (z), X a R (z). Allowed momenta and winding: H-flux ω-flux Q-flux R-flux p 1,p 2,p 3 p 1,p 2,w 3 p 1,w 2,w 3 w 1,w 2,w 3 w 1,w 2,w 3 + w 1,w 2,p 3 + w 1,p 2,p 3 + p 1,p 2,p 3 +

53 T-duality: X a L (z) X a R (z) Allowed momenta and winding: T-duality +X a L (z), X a R (z). H-flux ω-flux Q-flux R-flux p 1,p 2,p 3 p 1,p 2,w 3 p 1,w 2,w 3 w 1,w 2,w 3 w 1,w 2,w 3 + w 1,w 2,p 3 + w 1,p 2,p 3 + p 1,p 2,p 3 + Remark: At higher order in H, the theory will flow to some new CFT: H-flux: R-flux: µ gab µ = α 4 Ha pq H bpq Flow to the SU(2) WZW model. Flow to an L-R asymmetric SU(2) WZW model.

54 Basic three-point function: X a (z 1,z 1 ) X b (z 2,z 2 ) X c (z 3,z 3 ) ( ) ( ) ( ) ] = α 2 12 [L Habc z 12 z z 13 + L 23 z z 21 + L 13 z 23 c.c. (z ij = z i z j ) (agrees with WZW-model computation: R. Blumenhagen, E. Plauschinn, arxiv: ) Rogers dilogarithm: L(x) = Li 2 (x)+ 1 2 log(x) log(1 x)

55 Basic three-point function: X a (z 1,z 1 ) X b (z 2,z 2 ) X c (z 3,z 3 ) ( ) ( ) ( ) ] = α 2 12 [L Habc z 12 z z 13 + L 23 z z 21 + L 13 z 23 c.c. (z ij = z i z j ) (agrees with WZW-model computation: R. Blumenhagen, E. Plauschinn, arxiv: ) Rogers dilogarithm: L(x) = Li 2 (x)+ 1 2 log(x) log(1 x) This function is discontinuous when z 1 z 2 =1,z 1 z 3 =0 It develops a jump when all three points approach each other, i.e. z 1 z 3 =0,z 2 z 3 =0

56 3-tachyon amplitudes: (i) 3 momentum states in H-background: 3 T1 T 2 T 3 = i=1 d2 z i δ (2) (z i zi 0) δ(p 1 + p 2 + p 3 ) [ [ ( exp iθ abc p 1,a p 2,b p 3,c L z12 ) ( z 13 L z12 )] ] z 13 θ. (ii) 3 momentum states in R-background: (corresponds to 3 winding states in H-background) + 3 T1 T 2 T 3 = i=1 d2 z i δ (2) (z i zi 0) δ(p 1 + p 2 + p 3 ) [ [ ( exp iθ abc p 1,a p 2,b p 3,c L z12 ) ( z 13 + L z12 )] ] z 13 θ. Behavior under permutations of the vertex operators: Vσ(1) V σ(2) V σ(3) ɛ = exp [ i ( 1+ɛ 2 (ɛ = 1 for H or R) ) ησ π 2 θ abc p 1,a p 2,b p 3,c ] V1 V 2 V 3 ɛ

57 N-tachyon amplitudes: V1 V 2... V N = V1 V 2... V N 0 exp [ iθ abc [ 1 i<j<k N p i,a p j,b p k,c L ( z ij ) ( zij ) ]] z ik L z ik θ E.g. the fluxed Virasoro-Shapiro amplitude with N=4: V1 V 2 V 3 V 4 = V1 V 2 V 3 V 4 0 [ ] exp [ iθ ] abc p 1,a p 2,b p 3,c L( z 12 z 13 ) L( z 12 z 14 )+L( z 13 z 14 ) L( z 23 z 24 ) c.c. θ

58 The phase factor appearing when commuting two vertex operators can be decoded in a deformed N-product: V p1 (x) N... N V pn (x) def = exp Non-associative exp [ m<n<r F abc y m a ( - product for functions: f 1 (y) f 2 (y)... f N (y) := ] f1 (y 1 ) f 2 (y 2 )... f N (y N ) y n b i π2 y r c 2 θabc 1 i<j<k N p i,a p j,b p k,c ) V P p i (x) (see also: K. Savvidy (2002)) y1 =...=y N =y

59 The phase factor appearing when commuting two vertex operators can be decoded in a deformed N-product: V p1 (x) N... N V pn (x) def = exp Non-associative exp [ m<n<r F abc y m a ( - product for functions: f 1 (y) f 2 (y)... f N (y) := ] f1 (y 1 ) f 2 (y 2 )... f N (y N ) y n b i π2 y r c 2 θabc 1 i<j<k N p i,a p j,b p k,c ) V P p i (x) (see also: K. Savvidy (2002)) y1 =...=y N =y Is there are non-commutative (non-associative) theory of gravity? Is there a map to commutative gravity (like SW-map for gauge theories)? (Non-commutative geometry & gravity: P. Aschieri, M. Dimitrijevic, F. Meyer, J. Wess (2005))

60 The phase factor appearing when commuting two vertex operators can be decoded in a deformed N-product: V p1 (x) N... N V pn (x) def = exp Non-associative exp [ m<n<r F abc y m a ( - product for functions: f 1 (y) f 2 (y)... f N (y) := ] f1 (y 1 ) f 2 (y 2 )... f N (y N ) y n b i π2 y r c 2 θabc 1 i<j<k N p i,a p j,b p k,c ) V P p i (x) (see also: K. Savvidy (2002)) y1 =...=y N =y Is there are non-commutative (non-associative) theory of gravity? Is there a map to commutative gravity (like SW-map for gauge theories)? (Non-commutative geometry & gravity: P. Aschieri, M. Dimitrijevic, F. Meyer, J. Wess (2005)) What is the generalization of quantum mechanics for this non-associative geometry? How to represent this algebra (octonians?)?

61 Murat Günaydin, Feza Gürsey: Quark structure and octonians (1973)

62 Murat Günaydin, Feza Gürsey: Quark structure and octonians (1973) Thank you for your attention!