Non-Commutative/Non-Associative Geometry and Non-geometric String Backgrounds

Transcription

1 Non-Commutative/Non-Associative Geometry and Non-geometric String Backgrounds DIETER LÜST (LMU, MPI) Workshop on Non-Associativity in Physics and Related Mathematical Structures, PennState, 1st. May

2 Non-Commutative/Non-Associative Geometry and Non-geometric String Backgrounds DIETER LÜST (LMU, MPI) O. Hohm, D.L., B. Zwiebach, arxiv: ; I. Bakas, D.L., arxiv: ; R. Blumenhagen, M. Fuchs, F. Hassler, D.L., R. Sun, arxiv: ; F. Hassler, D.L., arxiv: ; A. Betz, R. Blumenhagen, D.L., F. Rennecke, arxiv: Workshop on Non-Associativity in Physics and Related Mathematical Structures, PennState, 1st. May

3 Outline: I) Introduction II) Non-commutative/non-associative closed string geometry and non-geometric string backgrounds III) Some Remarks on Double Field Theory Talk by Ralph Blumenhagen 2

4 I) Introduction Two complementary approaches to quantum gravity: 3

5 I) Introduction Two complementary approaches to quantum gravity: - Canonical quantum gravity (LQG,CDT) for point-like fields: Discrete (non-commutative) fuzzy space-time: 3

6 I) Introduction Two complementary approaches to quantum gravity: - Canonical quantum gravity (LQG,CDT) for point-like fields: Discrete (non-commutative) fuzzy space-time: - String theory (finitely extended objects): Smooth geometry (resolution of singularities) 3

7 I) Introduction Two complementary approaches to quantum gravity: - Canonical quantum gravity (LQG,CDT) for point-like fields: Discrete (non-commutative) fuzzy space-time: - String theory (finitely extended objects): Smooth geometry (resolution of singularities) What is the relation between these two approaches? 3

8 As we will see, non-geometric string backgrounds and T-duality will provide a very interesting and new classical relation between fuzzy space and finite extension of the string. 4

9 As we will see, non-geometric string backgrounds and T-duality will provide a very interesting and new classical relation between fuzzy space and finite extension of the string. Non-commutative and non-associative closed string geometry: R. Blumenhagen, E. Plauschinn, arxiv: ; D. Lüst, arxiv: ; R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arxiv: D. Andriot, M. Larfors, D.Lüst, P. Patalong, arxiv: A. Bakas, D.Lüst, arxiv: R. Blumenhagen, M. Fuchs, F. Hassler, D. Lüst, R. Sun, arxiv:

10 As we will see, non-geometric string backgrounds and T-duality will provide a very interesting and new classical relation between fuzzy space and finite extension of the string. Fundamental closed string non- Non-commutative and non-associative closed associativity in WZWmodel with H-flux string geometry: R. Blumenhagen, E. Plauschinn, arxiv: ; D. Lüst, arxiv: ; R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arxiv: D. Andriot, M. Larfors, D.Lüst, P. Patalong, arxiv: A. Bakas, D.Lüst, arxiv: R. Blumenhagen, M. Fuchs, F. Hassler, D. Lüst, R. Sun, arxiv:

11 As we will see, non-geometric string backgrounds and T-duality will provide a very interesting and new classical relation between fuzzy space and finite extension of the string. Fundamental closed string non- Non-commutative and non-associative closed associativity in WZWmodel with H-flux string geometry: R. Blumenhagen, E. Plauschinn, arxiv: ; D. Lüst, arxiv: ; R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arxiv: D. Andriot, M. Larfors, D.Lüst, P. Patalong, arxiv: A. Bakas, D.Lüst, arxiv: Closed string noncommutativity in tori with non-geometric fluxes and T-duality, derived non-associativity follows as violation of Jacobi identity R. Blumenhagen, M. Fuchs, F. Hassler, D. Lüst, R. Sun, arxiv:

12 As we will see, non-geometric string backgrounds and T-duality will provide a very interesting and new classical relation between fuzzy space and finite extension of the string. string geometry: closed string non- Non-commutative and Non-associativity non-associative in CFT s with closed Closed string noncommutativity in tori with Fundamental non-geometric fluxes and T-duality, derived non-associativity follows associativity in WZWmodel with H-flux geometric and T-dual nongeometric as violation of Jacobi identity fluxes R. Blumenhagen, E. Plauschinn, arxiv: ; D. Lüst, arxiv: ; R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arxiv: D. Andriot, M. Larfors, D.Lüst, P. Patalong, arxiv: A. Bakas, D.Lüst, arxiv: R. Blumenhagen, M. Fuchs, F. Hassler, D. Lüst, R. Sun, arxiv:

13 As we will see, non-geometric string backgrounds and T-duality will provide a very interesting and new classical relation between fuzzy space and finite extension of the string. string geometry: closed string non- Non-commutative and Non-associativity non-associative in CFT s with closed Closed string noncommutativity in tori with Fundamental non-geometric fluxes and T-duality, derived non-associativity follows associativity in WZWmodel with H-flux geometric and T-dual nongeometric as violation of Jacobi identity fluxes R. Blumenhagen, E. Plauschinn, arxiv: ; D. Lüst, arxiv: ; R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: C. Condeescu, I. Florakis, D. L., JHEP 1204 (2012), 121, arxiv: D. Andriot, M. Larfors, D.Lüst, P. Patalong, arxiv: A. Bakas, D.Lüst, arxiv: R. Blumenhagen, M. Fuchs, F. Hassler, D. Lüst, R. Sun, arxiv: New classical uncertainty relations - minimal volume due to finite string length! 4

14 Non-associativity in physics: Jordan & Malcev algebras, octonions M. Günaydin, F. Gürsey (1973); M. Günaydin, D. Minic, arxiv: Nambu dynamics Y. Nambu (1973); D. Minic, H. Tze (2002); M. Axenides, E. Floratos (2008) Magnetic monopoles R. Jackiw (1985); M. Günaydin, B. Zumino (1985) Closed string field theory A. Strominger (1987), B. Zwiebach (1993) T-duality and principle torus bundles D-branes in curved backgrounds Multiple M2-branes and 3-algebras J. Bagger, N. Lambert (2007) 5 P. Bouwknegt, K. Hannabuss, Mathai (2003) L. Cornalba, R. Schiappa (2001)

15 II) Non-geometric backgrounds and noncommutative & non-associative geometry 6

16 II) Non-geometric backgrounds and noncommutative & non-associative geometry Closed string background fields: G ij,b ij, 6

17 II) Non-geometric backgrounds and noncommutative & non-associative geometry Closed string background fields: G ij,b ij, Generalized metric: H MN = G ij G ik B kj B ik G kj G ij B ik G kl B lj 6

18 II) Non-geometric backgrounds and noncommutative & non-associative geometry Closed string background fields: G ij,b ij, Generalized metric: H MN = T-duality - O(D,D) transformations: H MN! P M H PQ Q, 2 O(D, D) N They contain: B ij! B ij +2 ij, G ij G ik B kj B ik G kj G ij B ik G kl B lj R! L 2 s/r 6

19 II) Non-geometric backgrounds and noncommutative & non-associative geometry Closed string background fields: G ij,b ij, Generalized metric: H MN = T-duality - O(D,D) transformations: H MN! P String length M H(not PQ hbar!) Q N They contain: B ij! B ij +2 ij, G ij G ik B kj B ik G kj G ij B ik G kl B lj, 2 O(D, D) R! L 2 s/r 6

20 II) Non-geometric backgrounds and noncommutative & non-associative geometry Closed string background fields: G ij,b ij, Generalized metric: H MN = T-duality - O(D,D) transformations: H MN! P String length M H(not PQ hbar!) Q N They contain: B ij! B ij +2 ij, G ij G ik B kj B ik G kj G ij B ik G kl B lj, 2 O(D, D) R! L 2 s/r Doubling of closed string coordinates and momenta: - Coordinates: O(D,D) vector X M =( X i,x i ) - Momenta: O(D,D) vector p M =( p i,p i ) (Here D=3) 6 winding T-duality momentum

21 Non-geometric backgrounds are generic within the landscape of string compactifications. Several potentially interesting applications in string phenomenology and cosmology. They are only consistent in string theory. Make use of string symmetries, T-duality T-folds, Left-right asymmetric spaces Asymmetric orbifolds (Kawai, Lewellen, Tye, 1986; Lerche, D.L. Schellekens, 1986, Antoniadis, Bachas, Kounnas, 1987; Narain, Sarmadi, Vafa, 1987) They can be potentially used for the construction of de Sitter vacua T-dualiy: classical bounce (pre-big bang) models (Brandenberger, Vafa, 1989; Meissner, Veneziano, 1991; Gasperini, Veneziano, 1993)

22 - Non-geometric Q-fluxes: spaces that are locally still Riemannian manifolds but not anymore globally. (Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht, 2005; Dabholkar, Hull, 2005) Transition functions between two coordinate patches are given in terms of O(D,D) T-duality transformations: Di (M D )! O(D, D) C. Hull (2004) Q-space will become non-commutative: [X i,x j ] 6= 0 8

23 - Non-geometric Q-fluxes: spaces that are locally still Riemannian manifolds but not anymore globally. (Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht, 2005; Dabholkar, Hull, 2005) Transition functions between two coordinate patches are given in terms of O(D,D) T-duality transformations: Di (M D )! O(D, D) C. Hull (2004) Q-space will become non-commutative: [X i,x j ] 6= 0 - Non-geometric R-fluxes: spaces that are even locally not anymore manifolds. R-space will become non-associative: [X i,x j,x k ] := [[X i,x j ],X k ]+cycl. perm. = = (X i X j ) X k X i (X j X k )+ 6= 0 8

24 Example: Three-dimensional flux backgrounds: 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. Metric, B-field of T 2 : H MN (X 3 ) S 1 O(2,2) monodromy: H MN (X 3 +2 ) = O(2,2) H PQ (X 3 ) 1 O(2,2) 9

25 Example: Three-dimensional flux backgrounds: 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. Metric, B-field of T 2 : H MN (X 3 ) S 1 O(2,2) monodromy: Complex structure of : T 2 Kähler parameter of : H MN (X 3 +2 ) = O(2,2) H PQ (X 3 ) 1 O(2,2) (X 3 +2 ) = a (X3 )+b c (X 3 )+d T 2 (X 3 +2 ) = a0 (X 3 )+b 0 c 0 (X 3 )+d 0 9

26 Torus 10

27 Torus with non-constant B-field (H-flux), B-field is patched together by a B-field (gauge) transformation: B! B +2 H 11

28 Non geometric torus, metric is patched together by a T-duality transformation: G ij! G ij 12

29 Non geometric torus, metric is patched together by a T-duality transformation: G ij! G ij 13

30 3-dimensional fibration: (X 3 +2 ) = 1 (X 3 ) T 2 X 1 X 2 S 1 S 1 X 3 Twisted torus with f-flux 14

31 3-dimensional fibration: (X 3 +2 ) = 1 (X 3 ) T 2 X 1 X 2 S 1 S 1 X 3 Non-geometric space with Q-flux 15

32 (i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes: Chain of four T-dual spaces: (a) Geometric space: 3-dimensional torus with H - flux 0 1 R B 12 = HXH ,H B 12 = H G ij A, 0 R R 2 3 (X 3 H)=iR 1 R 2 HX 3 H (Bianchi I) X 3 H! X 3 H +2 R 3 ) g O(2,2) : (X 3 H +2 R 3 )= (X 3 H)+2 HR 3 16

33 (i) (Non-)geometric backgrounds with parabolic monodromy and constant 3-form fluxes: Chain of four T-dual spaces: (a) Geometric space: 3-dimensional torus with H - flux 0 1 R B 12 = HXH ,H B 12 = H G ij A, 0 R R 2 3 (X 3 H)=iR 1 R 2 HX 3 H X 3 H! X 3 H +2 R 3 ) g O(2,2) : (X 3 H +2 R 3 )= (X 3 H)+2 HR 3 T-duality in X 1 : (b) Geometric spaces: twisted 3-torus with f - flux G ij = 0 1 R 2 1 fx 3 f R 2 1 R fx 3 f R1 2 0 fx 3 f R R C A, B ij =0 (X 3 f )=ir 1 R 2 (Bianchi I) (f H) (Bianchi II) fx 3 f X 3 f! X 3 f +2 R 3 ) g O(2,2) : (X 3 f +2 R 3 )= (X 3 f )+2 fr 3 16

34 T-duality in X 2 : (c) Non-geometric space: T-fold with Q-flux G ij = 0 (X 3 Q)= F R F R R C A,B ij = F 0 1 QX 0 3 Q 0 R1 2R2 2 C 0 0A,F= QX 3 Q R1 2R2 2 1 QX 3 Q ir 1 R 2 ) g O(2,2) : (X 3 Q +2 R 3 )= (Q f H) 1 2 QX3 Q A R 1 R 2 (XQ 3 ) 1+2 R 3 Q (XQ 3 ) 1 This does not correspond to a standard diffeomorphism but to a T-duality transformation. 17

35 T-duality in X 2 : (c) Non-geometric space: T-fold with Q-flux G ij = 0 (X 3 Q)= F R F R R C A,B ij = F 0 1 QX 0 3 Q 0 R1 2R2 2 C 0 0A,F= QX 3 Q R1 2R2 2 1 QX 3 Q ir 1 R 2 ) g O(2,2) : (X 3 Q +2 R 3 )= (Q f H) 1 2 QX3 Q A R 1 R 2 (XQ 3 ) 1+2 R 3 Q (XQ 3 ) 1 This does not correspond to a standard diffeomorphism but to a T-duality transformation. T-duality in X 3 : (d) Non-geometric space with R-flux Now the Buscher rules for T-duality cannot be applied. There exist no locally defined metric and B-field. 17

36 Summary: Flat torus with H-flux Twisted torus with f-flux Nongeometric space with Q-flux T x1 T x2 T x3 Nongeometric space with R-flux 17

37 Summary: Flat torus with H-flux Twisted torus with f-flux Nongeometric space with Q-flux T x1 T x2 T x3 Nongeometric space with R-flux [X i H,f,X j H,f ]=0 17

38 Summary: Flat torus with H-flux Twisted torus with f-flux Nongeometric space with Q-flux T x1 T x2 T x3 Nongeometric space with R-flux [X i H,f,X j H,f ]=0 [X 1 Q,X 2 Q] ' Q p 3 17

39 Summary: Flat torus with H-flux Twisted torus with f-flux Nongeometric space with Q-flux T x1 T x2 T x3 Nongeometric space with R-flux [XH,f,X i j H,f ]=0 [XQ,X 1 Q] 2 ' Q p 3 [[XR,X 1 R],X 2 R] 3 ' R 17

40 Summary: Flat torus with H-flux Twisted torus with f-flux Nongeometric space with Q-flux T x1 T x2 T x3 Nongeometric space with R-flux [XH,f,X i j H,f ]=0 [XQ,X 1 Q] 2 ' Q p 3 [[XR,X 1 R],X 2 R] 3 ' R They can be computed by - standard world-sheet quantization of the closed string D. Andriot, M. Larfors, D. L., P. Patalong, arxiv: CFT & canonical T-duality I. Bakas, D.L. to appear soon 17

41 Q-flux: [XQ(, 1 ),XQ(, 2 0 )] = i X 2 Q 1 p3 n 2 e in( n6=0 0 ) ( 0 ) X n6=0 1 n e in( 0 ) + i 2 ( 0 ) 2 19

42 Q-flux: [XQ(, 1 ),XQ(, 2 0 )] = i X 2 Q 1 p3 n 2 e in( n6=0 winding number 0 ) ( 0 ) X n6=0 1 n e in( 0 ) + i 2 ( 0 ) 2 19

43 Q-flux: [XQ(, 1 ),XQ(, 2 0 )] = i X 2 Q 1 p3 n 2 e in( n6=0 0 ) ( 0 ) X n6=0 1 n e in( 0 ) + i 2 ( 0 ) 2! 0 : [X 1 Q(, ),X 2 Q(, )] = i 2 6 Q p3 The non-commutativity of the torus (fibre) coordinates is determined by the winding in the circle (base) direction. 19

44 Corresponding uncertainty relation: ( X 1 Q) 2 ( X 2 Q) 2 L 6 s Q 2 h p 3 i 2 The spatial uncertainty in the - directions grows with the dual momentum in the third direction: non-local strings with winding in third direction. X 1,X 2 20

45 R-flux background: T-duality in x 3 -direction R-flux Use p 3! p 3, XQ,3 X 3 R For the case of non-geometric R-fluxes one gets: [XR,X 1 R]= 2 i 2 6 Rp 3 [X 3 R,p 3 ]=i =) R Q Non-associative algebra: [[X 1 R(, ),X 2 R(, )],X 3 R(, )] + perm. = 2 6 R 21

46 R-flux background: T-duality in x 3 -direction R-flux Use p 3! p 3, XQ,3 X 3 R For the case of non-geometric R-fluxes one gets: [XR,X 1 R]= 2 i 2 6 Rp 3 [X 3 R,p 3 ]=i =) R Q Non-associative algebra: [[X 1 R(, ),X 2 R(, )],X 3 R(, )] + perm. = 2 6 R Corresponding classical uncertainty relations : ( X 1 R) 2 ( X 2 R) 2 L 6 s R 2 hp 3 i 2 Volume: ( X 1 R) 2 ( X 2 R) 2 ( X 3 R) 2 L 6 s R 2 (see also: D. Mylonas, P. Schupp, R.Szabo, arxiv: ) 21

47 The algebra of commutation relation looks different in each of the four duality frames. Non-vanishing commutators and 3-brackets: T-dual frames Commutators Three-brackets H-flux [ x 1, x 2 ] H p 3 [ x 1, x 2, x 3 ] H f-flux [x 1, x 2 ] f p 3 [x 1, x 2, x 3 ] f Q-flux [x 1,x 2 ] Q p 3 [x 1,x 2, x 3 ] Q R-flux [x 1,x 2 ] Rp 3 [x 1,x 2,x 3 ] R However: R-flux & winding coordinates: [ x i, x j, x k ]=0 22

48 The algebra of commutation relation looks different in each of the four duality frames. Non-vanishing commutators and 3-brackets: T-dual frames Commutators Three-brackets H-flux [ x 1, x 2 ] H p 3 [ x 1, x 2, x 3 ] H f-flux [x 1, x 2 ] f p 3 [x 1, x 2, x 3 ] f Q-flux [x 1,x 2 ] Q p 3 [x 1,x 2, x 3 ] Q R-flux [x 1,x 2 ] Rp 3 [x 1,x 2,x 3 ] R However: R-flux & winding coordinates: [ x i, x j, x k ]=0 T-duality is mapping these commutators and 3-brackets onto each other: T-duality 22

49 (ii) (Non-)geometric backgrounds with elliptic monodromy and non-geometric fluxes. They can be described in terms of twisted tori and (a)symmetric freely acting orbifolds. A. Dabholkar, C. Hull (2002, 2005) C. Hull; R. Read-Edwards (2005, 2006, 2007, 2009) D. Lüst, JHEP 1012 (2011) 063, arxiv: , C. Condeescu, I. Florakis, D. Lüst, JHEP 1204 (2012), 121, arxiv: C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arxiv: In general not T-dual to a geometric space! (Only consistent in string theory (respectively in DFT).) The fibre torus depends on the third coordinate in a more complicate way. 17

50 The corresponding commutators can be explicitly derived in CFT. More complicate, non-linear commutation relations: R-frame: [x 1,x 2 ]=[ x 1,x 2 ]=[x 1, x 2 ]=[ x 1, x 2 ]=i (p 3 ) (p 3 )= 2 cot( p 3R) This algebra cannot be T-dualized to a commutative algebra! The string always moves on a noncommutative/non-associative fuzzy space: 24

51 Mathematical framework to describe non-geometric string backgrounds and the non-associative algebras: D. Mylonas, P. Schupp, R.Szabo, arxiv: , arxiv: , arxiv: I. Bakas, D. Lüst, arxiv: ; 3-Cocycles, 2-cochains and star-products Group theory cohomology - Hochschild; Stasheff; Cartan, Eilenberg,... 25

52 Mathematical framework to describe non-geometric string backgrounds and the non-associative algebras: D. Mylonas, P. Schupp, R.Szabo, arxiv: , arxiv: , arxiv: I. Bakas, D. Lüst, arxiv: ; 3-Cocycles, 2-cochains and star-products Group theory cohomology - Hochschild; Stasheff; Cartan, Eilenberg,... Open string non-commutativity: Constant Poisson structure: [x i,x j ]= ij 25

53 Mathematical framework to describe non-geometric string backgrounds and the non-associative algebras: D. Mylonas, P. Schupp, R.Szabo, arxiv: , arxiv: , arxiv: I. Bakas, D. Lüst, arxiv: ; 3-Cocycles, 2-cochains and star-products Group theory cohomology - Hochschild; Stasheff; Cartan, Eilenberg,... Open string non-commutativity: Constant Poisson structure: Moyal-Weyl star-product: (f 1? f 2 )(~x) =e i x 1 x 2 j f 1 (~x 1 ) f 2 (~x 2 ) ~x 2-cyclicity: Non-commutative gauge theories: Z d n x (f? g) = (N. Seiberg, E. Witten (1999); J. Madore, S. Schraml, P. Schupp, 25 J. Wess (2000);... ) Z [x i,x j ]= ij d n x (g? f) S d n x Tr ˆF ab ˆF ab

54 Are the similar structures for closed strings? Tri-product. Deformed theory of gravity? Possibly yes, but only off-shell. 26

55 Are the similar structures for closed strings? Tri-product. Deformed theory of gravity? Possibly yes, but only off-shell. Recall: closed string parabolic model in R-flux frame: Non-associative algebra: [x i,x j ] ls~ 3 1 R ijk p k [x i,p j ]=i~ ij, [p i,p j ]=0 [x 1,x 2,x 3 ]:=[[x 1,x 2 ],x 3 ]+cycl. perm. lsr 3 26

56 3-cocycles in Lie group cohomology Consider the following group elements (loops): U(~a, ~ b)=e i(~a ~x+~ b ~p) Now we want to consider the product of two or three group elements in order to derive the noncommutative/non-associative phases in the group products (BCH formula). 27

57 Group cohomology: U(~a 1, ~ b 1 )U(~a 2, ~ b 2 )=e i 2 R 12 '(~a 1,~a 2 ) U(~a 2, ~ b 2 )U(~a 1, ~ b 1 ). Non-commutativity is determined by the following 2-cochain: ' 2 (~a 1,~a 2 )=(~a 1 ~a 2 ) ~p 28

58 Product law of three group elements becomes non-associative: U(~a 1, ~ b 1 )U(~a 2, ~ b 2 ) U(~a 3, ~ b 3 )=e i R 2 (~a 1 ~a 2 ) ~a 3 U(~a 1, ~ b 1 ) U(~a 2, ~ b 2 )U(~a 3, ~ b 3 ). Non-associativity is determined by the 3-cocycle: ' 3 (~a 1,~a 2,~a 3 )=(~a 1 ~a 2 ) ~a 3 3-cocycle: volume of tetrahedon: Volume(~a 1,~a 2,~a 3 )= 1 6 (~a 1 ~a 2 ) ~a 3 29

59 Derivation of the star product The multiplication of the group elements and the use of Weyl s correspondence rule lead to star 2- and 3-products for the multiplication of functions. f(~x, ~p) (f 1? p f 2 )(~x, ~p) =e i 2 IJ J (f 1 f 2 ) ~x; ~p D. Mylonas, P. Schupp, R.Szabo, arxiv: dimensional Poisson tensor: IJ (p) = 0 R ijk p i j j i 0 1 A ; R ijk = 2 R 6 ijk (The full phase space including also dual coordinates and dual momenta is 12-dimensional!) 30 I. Bakas, D. Lüst, arxiv:

60 It leads to the following 3-product: (f f f 3 )(~x) =((f 1? p f 2 )? p f 3 )(~x) (f f f 3 )(~x) =e x 1 x 2 x 3 k f1 (~x 1 ) f 2 (~x 2 ) f 3 (~x 3 ) ~x This delta-product is non-associative. It is consistent with the 3-bracket among the coordinates: f 1 = x i,f 2 = x j,f 3 = x k : f f f 3 =[x i,x j,x k ]=`4s R ijk It obeys the 3-cyclicity property: Z d n x (f f 2 ) 4 3 f 3 = 31 Z d n xf (f f 3 )

61 was already derived in CFT from the multiplication of 3 tachyon vertex operators: R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: Scattering of 3 momentum states in R-background: (corresponds to 3 winding states in H-background) 4 3 V i (z, z) =:exp ip i X i (z, z) : V (1) V (1) V (1) R = V 1 V 2 V 3 R exp i R ijk p 1,i p 2,j p 3,k. However this non-associative phase is vanishing, when going on-shell in CFT and using momentum conservation: p 1 = (p 2 + p 3 ) On-shell CFT amplitudes are associative! 32 ( =0, 1)

62 IV) Double geometry - double field theory W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...) Effective field theory description of non-geometric spaces: 33

63 IV) Double geometry - double field theory W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...) Effective field theory description of non-geometric spaces: (i) Geometric (H,f)-spaces: Standard supergravity. 33

64 IV) Double geometry - double field theory W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...) Effective field theory description of non-geometric spaces: (i) Geometric (H,f)-spaces: Standard supergravity. (ii) Non-geometric (Q,R)-spaces: -supergravity, algebroids (still T-dual to geometric spaces) D. Andriot, M. Larfors, D. Lüst, P. Patalong, arxiv: , D. Andriot, O. Hohm, M. Larfors, D. Lüst, P. Patalong, arxiv: ,16, arxiv: D. Andriot, A. Betz, arxiv: , R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, arxiv: , arxiv: , R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid, arxiv:

65 IV) Double geometry - double field theory W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...) Effective field theory description of non-geometric spaces: (i) Geometric (H,f)-spaces: Standard supergravity. (ii) Non-geometric (Q,R)-spaces: -supergravity, algebroids (still T-dual to geometric spaces) D. Andriot, M. Larfors, D. Lüst, P. Patalong, arxiv: , D. Andriot, O. Hohm, M. Larfors, D. Lüst, P. Patalong, arxiv: ,16, arxiv: D. Andriot, A. Betz, arxiv: , R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, arxiv: , arxiv: , R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid, arxiv: (iii) Non-geometric (H,f,Q,R)-spaces: Double field theory O. Hohm, D. Lüst, B. Zwiebach, arxiv: , R. Blumenhagen, M. Fuchs, Hassler, D. Lüst, R. Sun, arxiv: , F. Hassler, D.Lüst arxiv:

66 Double field theory: O(D,D) invariant effective string action containing momentum and winding coordinates at the same time: X M =( x m,x m ) 34

67 Double field theory: O(D,D) invariant effective string action containing momentum and winding coordinates at the same time: X M =( x m,x m ) Covariant flux formulation of DFT. (Geissbuhler, Marques, Nunez, Penas; Aldazabal, Marques, Nunez) Comprises all fluxes (Q,f,Q,R) into one covariant expression: F ABC = D [A E B M E C]M, D A = E A M. 34

68 Double field theory: O(D,D) invariant effective string action containing momentum and winding coordinates at the same time: X M =( x m,x m ) Covariant flux formulation of DFT. (Geissbuhler, Marques, Nunez, Penas; Aldazabal, Marques, Nunez) Comprises all fluxes (Q,f,Q,R) into one covariant expression: F ABC = D [A E B M E C]M, D A = E A M. The DFT action is invariant under generalized, non-associative diffeomorphisms: M =( m, E A M = L E A M = P E A M +(@ M P [ 1, 2 ] M C = N N M 2 [L 1, L 2 ]=L [ 1, 2 ] C P M )E A P N@ M 2 N ( 1 $ 2 ) m )

69 The generalized diffeomorphisms contain simultaneous coordinate and B -, - gauge field transformations. Generalized diffeomorphisms act on the generalized coordinates in a non-associative way: The Courant bracket violates the Jacobi identity. C. Hull, B. Zwiebach, arxiv: ; O. Hohm, B. Zwiebach, arxiv: ; O. Hohm, D. Lüst, B. Zwiebach, arxiv:1309:

70 However for generalized functions f(x) (e.g. the background fields) one has to require the strong constraint (string level matching condition): (CFT origin of the strong constraint: A. Betz, R. Blumenhagen, D. Lüst, F. Rennecke, M M M g = D A f D A g =0 Functions must depend only on one kind of coordinates. The strong constraint defines a D-dim. hypersurface (brane) in 2D-dim. double geometry. Then the algebra of diffeomorphisms on functions closes and becomes associative. 36

71 Dimensional reduction of double field theory: O. Hohm, D. Lüst, B. Zwiebach, arxiv: ; F. Hassler, D. Lüst, arxiv:

72 Dimensional reduction of double field theory: O. Hohm, D. Lüst, B. Zwiebach, arxiv: ; F. Hassler, D. Lüst, arxiv: Consistent DFT solutions (generalized Scherk-Schwarz compactifications): R MN =0 37

73 Dimensional reduction of double field theory: O. Hohm, D. Lüst, B. Zwiebach, arxiv: ; F. Hassler, D. Lüst, arxiv: Consistent DFT solutions (generalized Scherk-Schwarz compactifications): R MN =0 The corresponding backgrounds are in general nongeometric and go beyond dimensional reduction of 10D SUGRA or generalized geometry. 37

74 Dimensional reduction of double field theory: O. Hohm, D. Lüst, B. Zwiebach, arxiv: ; F. Hassler, D. Lüst, arxiv: Consistent DFT solutions (generalized Scherk-Schwarz compactifications): R MN =0 The corresponding backgrounds are in general nongeometric and go beyond dimensional reduction of 10D SUGRA or generalized geometry. Killing vectors, L E A M =0, correspond to generalized diffeomorphisms that depend on and x m. x m 37

75 Dimensional reduction of double field theory: 37 O. Hohm, D. Lüst, B. Zwiebach, arxiv: ; F. Hassler, D. Lüst, arxiv: Consistent DFT solutions (generalized Scherk-Schwarz compactifications): R MN =0 The corresponding backgrounds are in general nongeometric and go beyond dimensional reduction of 10D SUGRA or generalized geometry. Killing vectors, L E A M =0, correspond to generalized diffeomorphisms that depend on and x m. x m Patching of coordinate charts correspond to generalized coordinate transformations of the form where the gauge functions and. x m X 0M = X M, in general depend on x m

76 Consider the following 3+3 dimensional backgrounds: (i) Parabolic background spaces: Single fluxes: or or H 123 f 1 23 Q 12 3 or R 123 These backgrounds do not satisfy R MN =0. 38

77 Consider the following 3+3 dimensional backgrounds: (i) Parabolic background spaces: Single fluxes: or or H 123 f 1 23 Q or R 123 These backgrounds do not satisfy R MN =0. (ii) Elliptic background spaces: Multiple fluxes: These backgrounds do satisfy R MN =0. Single elliptic geometric space: f13 2 = f23 1 = f Single elliptic T-dual, non-geometric space: H 123 = Q 12 3 = H Double elliptic, genuinely non-geometric space: H 123 = Q 12 3 = H, f13 2 = f23 1 = f

78 E.g. double elliptic background: (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) cos(fx 3 ) 0 sin(fx 3 ), f Z, (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) 39

79 E.g. double elliptic background: (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) cos(fx 3 ) 0 sin(fx 3 ), f Z, (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) Patching is generated by the coordinate transformation: (x 1,x 2, x 1, x 2 ) = 1 2 x1 x 2 + x 1 x 2 x 1 x 2 x 1 x x 1x 1 + x 2 x (x1 ) 2 +(x 2 ) 2 +( x 1 ) 2 +( x 2 ) 2. 39

80 E.g. double elliptic background: (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) cos(fx 3 ) 0 sin(fx 3 ), f Z, (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) Patching is generated by the coordinate transformation: (x 1,x 2, x 1, x 2 ) = 1 2 x1 x 2 + x 1 x 2 x 1 x 2 x 1 x x 1x 1 + x 2 x (x1 ) 2 +(x 2 ) 2 +( x 1 ) 2 +( x 2 ) 2. Corresponding Killing vectors of background: KĴÎ = (Hx3 + f x 3 1 ) 2 (Hx2 + f x 2 1 ) 2 (fx3 + H x 3 1 ) 2 (fx2 + H x 2 ) B A

81 E.g. double elliptic background: (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) cos(fx 3 ) 0 sin(fx 3 ), f Z, (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) Patching is generated by the coordinate transformation: (x 1,x 2, x 1, x 2 ) = 1 2 x1 x 2 + x 1 x 2 x 1 x 2 x 1 x x 1x 1 + x 2 x (x1 ) 2 +(x 2 ) 2 +( x 1 ) 2 +( x 2 ) 2. Corresponding Killing vectors of background: KĴÎ = (Hx3 + f x 3 1 ) 2 (Hx2 + f x 2 1 ) 2 (fx3 + H x 3 1 ) 2 (fx2 + H x 2 ) B A Background satisfies strong constraint

82 E.g. double elliptic background: (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) cos(fx 3 ) 0 sin(fx 3 ), f Z, (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) Patching is generated by the coordinate transformation: Patching does not satisfy strong (x 1,x 2, x 1, x 2 ) = 1 2 x1 x 2 + x 1 x 2 x 1 x 2 x 1 x x 1x 1 + x 2 x constraint (x1 ) 2 +(x 2 ) 2 +( x 1 ) 2 +( x 2 ) 2. Corresponding Killing vectors of background: KĴÎ = (Hx3 + f x 3 1 ) 2 (Hx2 + f x 2 1 ) 2 (fx3 + H x 3 1 ) 2 (fx2 + H x 2 ) B A Background satisfies strong constraint

83 E.g. double elliptic background: (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) cos(fx 3 ) 0 sin(fx 3 ), f Z, (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) Patching is generated by the coordinate transformation: Patching does not satisfy strong (x 1,x 2, x 1, x 2 ) = 1 2 x1 x 2 + x 1 x 2 x 1 x 2 x 1 x x 1x 1 + x 2 x constraint (x1 ) 2 +(x 2 ) 2 +( x 1 ) 2 +( x 2 ) 2. Corresponding Killing vectors of background: Killing 0 vectors 0do KĴÎ = (Hx3 + f x 3 1 ) 2 (Hx2 + f x 2 1 ) 2 (fx3 + H x 3 1 ) 2 (fx2 + H x 2 ) not satisfy 0 strong constraint. 0 B However C 1 their algebra 0 A closes! 1 39 Background satisfies strong constraint

84 V) Outlook & open questions 40

85 V) Outlook & open questions Non-commutative & non-associative closed string geometry arises in the presence of non-geometric fluxes (like open string non-commutativity on D- branes with gauge flux). This leads to a non-associative tri-product (like the star-product). 40

86 V) Outlook & open questions Non-commutative & non-associative closed string geometry arises in the presence of non-geometric fluxes (like open string non-commutativity on D- branes with gauge flux). This leads to a non-associative tri-product (like the star-product). However the non-associativity is not visible in on-shell CFT amplitudes. 40

87 V) Outlook & open questions Non-commutative & non-associative closed string geometry arises in the presence of non-geometric fluxes (like open string non-commutativity on D- branes with gauge flux). This leads to a non-associative tri-product (like the star-product). However the non-associativity is not visible in on-shell CFT amplitudes. Non-associativity is an off-shell phenomenon! 40

88 V) Outlook & open questions Non-commutative & non-associative closed string geometry arises in the presence of non-geometric fluxes (like open string non-commutativity on D- branes with gauge flux). This leads to a non-associative tri-product (like the star-product). However the non-associativity is not visible in on-shell CFT amplitudes. Non-associativity is an off-shell phenomenon! Are there situations, where the strong constraint for the background can be relaxed? - This seems to be the case for certain very asymmetric orbifolds. C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arxiv:

89 V) Outlook & open questions Non-commutative & non-associative closed string geometry arises in the presence of non-geometric fluxes (like open string non-commutativity on D- branes with gauge flux). This leads to a non-associative tri-product (like the star-product). However the non-associativity is not visible in on-shell CFT amplitudes. Non-associativity is an off-shell phenomenon! Are there situations, where the strong constraint for the background can be relaxed? - This seems to be the case for certain very asymmetric orbifolds. C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arxiv: Is there a non-commutative (non-associative) theory of gravity? (A. Chamseddine, G. Felder, J. Fröhlich (1992), J. Madore (1992); L. Castellani (1993) P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, J. Wess (2005), L. Alvarez-Gaume, F. Meyer, M. Vazquez-Mozo (2006)) 40