Non-Associative Geometry and Double Field Theory

Transcription

1 Non-Associative Geometry and Double Field Theory DIETER LÜST (LMU, MPI) Corfu, September 15,

2 Outline: I) Introduction II) Non-geometric Backgrounds & Non-Commutativity/Non-Associativity (world sheet point of view) III) Double field theory (target space point of view) IV) Dimensional Reduction of DFT 2

3 I) Introduction Jacobi identity for three operators in QM: Jac(F, G, H) = [F, G, H] = [F, [G, H]] + [H, [F, G]] + [G, [H, F]] = F (GH) (FG)H F (HG) (FH)G +... Algebraically zero for associative operators. 3

4 I) Introduction Jacobi identity for three operators in QM: Jac(F, G, H) = [F, G, H] = [F, [G, H]] + [H, [F, G]] + [G, [H, F]] = F (GH) (FG)H F (HG) (FH)G +... Algebraically zero for associative operators. Poisson bracket: {f,g} =! I f@ J g! IJ... Poisson structure The Jacobi identity of the Poisson bracket is zero if d! =0 3

5 { { Why is d! =0 in standard Hamiltonian mechanics? Choose local coordinates q i,p i Hamiltonian dynamics is defined on the cotangent bundle, which defines a 2D-dim manifold: T? M Q I =(q 1,...,q D ; p 1,...,p D ) 2 M Poisson structure = Symplectic structure:! ij = The symplectic structure of =) d! =0 4 2 T? M! =! IJ dq I ^ dq J = dv, v = q i dp i p i dq i 0 1D 1 D 0. T? M is exact.

6 Deformation quantization - Star product: f? g = e i I f@ J g = fg + i I f@ J g +... e.g.! IJ = 0 1D 1 D 0, f = q i,g= p j Then the commutator is obtained: [q i,p j ]? = i j. 5

7 Deformation quantization - Star product: f? g = e i I f@ J g = fg + i I f@ J g +... e.g.! IJ = 0 1D 1 D 0, f = q i,g= p j Then the commutator is obtained: [q i,p j ]? = i j. Quasi Poisson tensor: d! 6= 0 (f? g)? h f? (g? h)! L! I f@ J g@ K h 6= 0 Non-associativity is possible if d! 6= 0! 5

8 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) Non-geometric backgrounds: 6 P. Bouwknegt, K. Hannabuss, Mathai (2003) L. Cornalba, R. Schiappa (2001)

9 Non-geometric backgrounds: 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; Ibanez, Nilles, Quevedo, 1987;..., Faraggi, Rizos, Sonmez, 2014) Are related to non-commutative/non-associative geometry (Blumenhagen, Plauschinn; Lüst, 2010; Blumenhagen, Deser, Lüst, Rennecke, Pluaschin, 2011; Condeescu, Florakis, Lüst; 2012, Andriot, Larfors, Lüst, Patalong, 2012))

10 Non-geometric spaces Double Field Theory CY SUGRA Flux comp. Non-Geometric spaces DFT

11 II) Non-geometric backgrounds & non-commutativity/non-associativity (word-sheet) 9

12 II) Non-geometric backgrounds & non-commutativity/non-associativity (word-sheet) Consider D-dimensional toroidal string backgrounds: Doubling of closed string coordinates and momenta: - Coordinates: O(D,D) vector X M =( X i,x i ) (X i = X i L( + )+X R ( ) Xi = X i L( + ) X R ( )) - Momenta: O(D,D) vector p M =( p i,p i ) 9

13 II) Non-geometric backgrounds & non-commutativity/non-associativity (word-sheet) Consider D-dimensional toroidal string backgrounds: Doubling of closed string coordinates and momenta: - Coordinates: O(D,D) vector X M =( X i,x i ) (X i = X i L( + )+X R ( ) Xi = X i L( + ) X R ( )) - Momenta: O(D,D) vector p M =( p i,p i ) winding T-duality momentum 9

14 II) Non-geometric backgrounds & non-commutativity/non-associativity (word-sheet) Consider D-dimensional toroidal string backgrounds: Doubling of closed string coordinates and momenta: - Coordinates: O(D,D) vector - O(D,D) transformations: Mix in general with Xi.! Act asymmetrically on and. X i X i! A C B D X i X i, = X M =( X i,x i ) (X i = X i L( + )+X R ( ) Xi = X i L( + ) X R ( )) - Momenta: O(D,D) vector p M =( p i,p i ) A C B D X i X L X R 2 O(D, D) 9

15 II) Non-geometric backgrounds & non-commutativity/non-associativity (word-sheet) Consider D-dimensional toroidal string backgrounds: Doubling of closed string coordinates and momenta: - Coordinates: O(D,D) vector - O(D,D) transformations: Mix in general with Xi.! Act asymmetrically on and. X i X i! A C B D X i X i Generalized metric: H MN = X M =( X i,x i ) (X i = X i L( + )+X R ( ) Xi = X i L( + ) X R ( )) - Momenta: O(D,D) vector p M =( p i,p i ) X i X L X R A B, = 2 O(D, D) C D G ij G ik B kj B ik G kj G ij B ik G kl B lj H MN! P 9M H PQ Q N

16 Non-geometric backgrounds & non-geometric fluxes: - 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) Q-space will become non-commutative: C. Hull (2004) 10

17 Non-geometric backgrounds & non-geometric fluxes: - 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) Q-space will become non-commutative: - Non-geometric R-fluxes: spaces that are even locally not anymore manifolds. R-space will become non-associative: C. Hull (2004) [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 10

18 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 X 3 Metric, B-field of T 2 : depends on S 1 ) H MN (X 3 ) (M,N =1,...,4) 11

19 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 Metric, B-field of T 2 : depends on X 3 S 1 ) H MN (X 3 ) (M,N =1,...,4) The fibration is specified by its monodromy properties. O(2,2) monodromy: H MN (X 3 +2 ) = O(2,2) H PQ (X 3 ) 1 O(2,2) 11

20 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 Metric, B-field of T 2 : depends on X 3 S 1 ) H MN (X 3 ) (M,N =1,...,4) The fibration is specified by its monodromy properties. O(2,2) monodromy: H MN (X 3 +2 ) = O(2,2) H PQ (X 3 ) 1 Non-geometric spaces: Monodromy mixes O(2,2) X i $ X i Acts asymmetrically on XL,X i R i (i =1, 2) 11

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

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

23 (Non-)geometric backgrounds with parabolic monodromy and single 3-form fluxes: 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

24 (Non-)geometric backgrounds with parabolic monodromy and single 3-form fluxes: 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

25 (Non-)geometric backgrounds with parabolic monodromy and single 3-form fluxes: 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

26 (Non-)geometric backgrounds with parabolic monodromy and single 3-form fluxes: 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

27 (Non-)geometric backgrounds with parabolic monodromy and single 3-form fluxes: 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 - CFT & canonical T-duality D. Andriot, M. Larfors, D. L., P. Patalong, arxiv: C. Blair, arxiv: I. Bakas, D.L. to appear soon 17

28 Q-flux: O(2,2) monodromy mixed closed string boundary (DN) conditions: O(2, 2){ X 3 Q(, +2 ) = X 3 Q(, )+2 p 3 =) XQ(, 1 +2 ) = XQ(, 1 ) 2 p 3 Q X Q2 (, ), XQ(, 2 +2 ) = XQ(, 2 )+2 p 3 Q X Q1 (, ), X Q1 (, +2 ) = X Q1 (, ), X Q2 (, +2 ) = X Q2 (, ). 15

29 Q-flux: O(2,2) monodromy mixed closed string boundary (DN) conditions: O(2, 2){ X 3 Q(, +2 ) = X 3 Q(, )+2 p 3 =) XQ(, 1 +2 ) = XQ(, 1 ) 2 p 3 Q X Q2 (, ), XQ(, 2 +2 ) = XQ(, 2 )+2 p 3 Q X Q1 (, ), X Q1 (, +2 ) = X Q1 (, ), X Q2 (, +2 ) = X Q2 (, ). winding number along base direction 15

30 Q-flux: O(2,2) monodromy mixed closed string boundary (DN) conditions: O(2, 2){ X 3 Q(, +2 ) = X 3 Q(, )+2 p 3 =) XQ(, 1 +2 ) = XQ(, 1 ) 2 p 3 Q X Q2 (, ), XQ(, 2 +2 ) = XQ(, 2 )+2 p 3 Q X Q1 (, ), X Q1 (, +2 ) = X Q1 (, ), X Q2 (, +2 ) = X Q2 (, ). winding number along base direction [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 15

31 Q-flux: O(2,2) monodromy mixed closed string boundary (DN) conditions: O(2, 2){ X 3 Q(, +2 ) = X 3 Q(, )+2 p 3 =) XQ(, 1 +2 ) = XQ(, 1 ) 2 p 3 Q X Q2 (, ), XQ(, 2 +2 ) = XQ(, 2 )+2 p 3 Q X Q1 (, ), X Q1 (, +2 ) = X Q1 (, ), X Q2 (, +2 ) = X Q2 (, ). winding number along base direction [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 : [X 1 Q(, ),X 2 Q(, )] = i 2 0 ) + i 2 ( 0 ) 2 6 Q p3 15

32 Q-flux: O(2,2) monodromy mixed closed string boundary (DN) conditions: O(2, 2){ X 3 Q(, +2 ) = X 3 Q(, )+2 p 3 =) XQ(, 1 +2 ) = XQ(, 1 ) 2 p 3 Q X Q2 (, ), XQ(, 2 +2 ) = XQ(, 2 )+2 p 3 Q X Q1 (, ), X Q1 (, +2 ) = X Q1 (, ), X Q2 (, +2 ) = X Q2 (, ). winding number along base direction [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 : [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. 0 ) + i 2 ( 0 ) 2 15

33 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 16

34 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 17

35 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: ) 17

36 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 18

37 Star products, 3-products and CFT 19

38 Star products, 3-products and CFT Open string non-commutativity: Constant Poisson structure: [x i,x j ]= ij 19

39 Star products, 3-products and CFT Open string non-commutativity: Constant Poisson structure: [x i,x j ]= ij Vertex operators at boundary of disc: Worldsheet has SL(2, R)/Z 2 Non-commutativity is possible: symmetry. 12 = 21, 123 = 231, 123 6=

40 Star products, 3-products and CFT Open string non-commutativity: Constant Poisson structure: [x i,x j ]= ij Vertex operators at boundary of disc: Worldsheet has SL(2, R)/Z 2 Non-commutativity is possible: 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, J. Wess (2000);... ) 19 Z symmetry. 12 = 21, 123 = 231, 123 6= 132 d n x (g? f) S d n x Tr ˆF ab ˆF ab

41 Closed strings: Vertex operators on sphere: Worldsheet has SL(2, C)/Z 2 symmetry. Non-commutativity appears impossible: 12 = 21, 123 =

42 Closed strings: Vertex operators on sphere: Worldsheet has SL(2, C)/Z 2 symmetry. Non-commutativity appears impossible: Non-associative algebra: 12 = 21, 123 = 231 [x i,x j ]= ijk p k, [x i,p j ]=i~ ij, [p i,p j ]=0 [x i,x j,x k ]=[[x i,x j ],x k ]+cycl. perm. = R ijk 20

43 Non-commutativity? p 2-product: (f 1? p f 2 )(~x, ~p) =e i 2 IJ J (f 1 f 2 ) ~x; ~p 6-dimensional Poisson tensor: IJ (p) = 0 R ijk p D. Mylonas, P. Schupp, R.Szabo, arxiv: , arxiv: , arxiv: I. Bakas, D.Lüst, arxiv: i j j i 0 1 A ; R ijk = 2 R 6 ijk 21

44 Non-commutativity? p 2-product: (f 1? p f 2 )(~x, ~p) =e i 2 IJ J (f 1 f 2 ) ~x; ~p 6-dimensional Poisson tensor: IJ (p) = 0 R ijk p D. Mylonas, P. Schupp, R.Szabo, arxiv: , arxiv: , arxiv: I. Bakas, D.Lüst, arxiv: i j j i 0 1 A ; R ijk = 2 R 6 ijk Mathematical framework for non-associative algebras: Group cohomology, 3-Cocycles, 2-cochains and 3-products 21 (talk of 2013)

45 3-product: R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: D. Mylonas, P. Schupp, R.Szabo, arxiv: , arxiv: , arxiv: I. Bakas, D.Lüst, arxiv: (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 3-product is non-associative. It is consistent with the 3-bracket among the coordinates: It obeys the 3-cyclicity property: f 1 = X i,f 2 = X j,f 3 = X k : f f f 3 =[X i,x j,x k ]=R ijk Z d n x (f f 2 ) 4 3 f 3 = 22 Z d n xf (f f 3 )

46 Three point function in CFT: R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: X i (z 1, z 1 ) X j (z 2, z 2 ) X c (z 3, z 3 ) = R ijkh L z 12 z 13 + L z 12 z 13 ) X i,x j,x k h := lim X i (z 1, z 1 ), X b (z 2, z 2 ),X c (z 3, z 3 ) i +cycl. = R ijk zi!z i 23

47 Three point function in CFT: R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: X i (z 1, z 1 ) X j (z 2, z 2 ) X c (z 3, z 3 ) = R ijkh L z 12 z 13 + L z 12 z 13 ) X i,x j,x k h := lim X i (z 1, z 1 ), X b (z 2, z 2 ),X c (z 3, z 3 ) i +cycl. = R ijk zi!z 4 3 : Scattering of 3 momentum states in R-background: (corresponds to 3 winding states in H-background) 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 i R ijk p 1,i p 2,j p 3,k. ( =0, 1) 23

48 Three point function in CFT: R. Blumenhagen, A. Deser, D.Lüst, E. Plauschinn, F. Rennecke, arxiv: X i (z 1, z 1 ) X j (z 2, z 2 ) X c (z 3, z 3 ) = R ijkh L z 12 z 13 + L z 12 z 13 ) X i,x j,x k h := lim X i (z 1, z 1 ), X b (z 2, z 2 ),X c (z 3, z 3 ) i +cycl. = R ijk zi!z 4 3 : Scattering of 3 momentum states in R-background: (corresponds to 3 winding states in H-background) 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 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! 23 i i R ijk p 1,i p 2,j p 3,k. ( =0, 1)

49 III) Double field theory (target space point of view) W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...) 24

50 III) Double field theory (target space point of view) W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...) O(D,D) invariant effective string action containing momentum and winding coordinates at the same time: Z S X M =( x m,x m DFT = d 2D Xe 2 0 R ) R =4H M N N H MN 4H M N 0 +4@ M H N M H N H KL 1 2 N H L H MK 24

51 III) Double field theory (target space point of view) W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...) O(D,D) invariant effective string action containing momentum and winding coordinates at the same time: Z S X M =( x m,x m DFT = d 2D Xe 2 0 R ) R =4H M N N H MN 4H M N 0 +4@ M H N M H N H KL 1 2 N H L H MK Covariant fluxes of DFT: (Geissbuhler, Marques, Nunez, Penas; Aldazabal, Marques, Nunez) F ABC = D [A E B M E C]M, D A = E A M. Comprise all fluxes (Q,f,Q,R) into one covariant expression: F abc = H abc, F a bc = F a bc, F c ab = Q c ab, F abc = R abc 24

52 DFT action in flux formulation: S DFT = Z apple 1 dx e 2d F A F A 0S AA0 + F ABC F A 0 B 0 C 0 4 SAA0 BB0 1 0S CC0 12 SAA BB0 S CC0 1 6 F ABC F ABC F A F A (Looks similar to scalar potential in gauged SUGRA.) 25

53 DFT action in flux formulation: S DFT = Z apple 1 dx e 2d F A F A 0S AA0 + F ABC F A 0 B 0 C 0 4 SAA0 BB0 1 0S CC0 12 SAA BB0 S CC0 1 6 F ABC F ABC F A F A (Looks similar to scalar potential in gauged SUGRA.) 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 depend only on one kind of coordinates. The strong constraint defines a D-dim. hypersurface (brane) in 2D-dim. double geometry. 25

54 Non-associative deformations in double field theory: DFT generalization of the 3-product: f 4 3 g 4 3 h (X) =fgh+ `4s (For general functions f(x, P) (R. Blumenhagen, M. Fuchs, F. Hassler, D.Lüst, R. Sun, arxiv: ) 6 F ABC D A f D B g D C h + O(`8s) the phase space is 4D-dimensional.) I. Bakas, D. L., arxiv:

55 Non-associative deformations in double field theory: DFT generalization of the 3-product: f 4 3 g 4 3 h (X) =fgh+ `4s (For general functions f(x, P) Non-vanishing R-flux: (R. Blumenhagen, M. Fuchs, F. Hassler, D.Lüst, R. Sun, arxiv: ) f = x i,g= x j,h= x k : f 4 3 g 4 3 h =[x i,x j,x k ]=`4s R ijk f = x i,g= x j,h= x k : 6 F ABC D A f D B g D C h + O(`8s) the phase space is 4D-dimensional.) I. Bakas, D. L., arxiv: f 4 3 g 4 3 h =[ x i, x j, x k ]=0 26

56 Non-associative deformations in double field theory: DFT generalization of the 3-product: f 4 3 g 4 3 h (X) =fgh+ `4s (For general functions f(x, P) Non-vanishing R-flux: (R. Blumenhagen, M. Fuchs, F. Hassler, D.Lüst, R. Sun, arxiv: ) f = x i,g= x j,h= x k : f 4 3 g 4 3 h =[x i,x j,x k ]=`4s R ijk f = x i,g= x j,h= x k : Non-vanishing H-flux: 6 F ABC D A f D B g D C h + O(`8s) the phase space is 4D-dimensional.) 26 I. Bakas, D. L., arxiv: f 4 3 g 4 3 h =[ x i, x j, x k ]=0 f = x i,g= x j,h= x k : f 4 3 g 4 3 h =[ x i, x j, x ]=`4s k H ijk f = x i,g= x j,h= x k : f 4 3 g 4 3 h =[x i,x j,x k ]=0

57 General functions f, g and h (conformal fields in CFT): Consider the additional term in the DFT tri-product: F ABC D A f D B g D C h 27

58 General functions f, g and h (conformal fields in CFT): Consider the additional term in the DFT tri-product: F ABC D A f D B g D C h Imposing the strong constraint on f, g and h the additional term vanishes and the tri-product becomes the normal product. 27

59 IV) Dimensional Reduction of DFT Consistent DFT solutions: 2(D-d) linear independent Killing vectors: L K J I H MN =0 28 R MN =0 DFT and generalized Scherk-Schwarz ansatz (O(D,D) twists) gives rise to effective theory in D-d dimensions: S e = Z O. Hohm, D. Lüst, B. Zwiebach, arxiv: ; F. Hassler, D. Lüst, arxiv: see also: A. Dabholkar, C. Hull, 2002, 2005; C. Hull, R. Reid-Edwards, 2005, 2006, 2007 dx (D d)p ge 2 R +4@ µ 1 12 H µ H µ 1 4 H MNF Mµ Fµ N D µh MN D µ H MN V

60 The corresponding backgrounds are in general nongeometric and go beyond dimensional reduction of SUGRA. (i) DFT on spaces satisfying the strong constraint (SC) (ii) Mild violation of SC: Rewriting of SUGRA, geometric spaces Killing vectors violate the SC. Patching of coordinate charts correspond to generalized coordinate transformations that violate the SC. (iii) Strong violation of SC: Background fields violate the SC. However the fluxes have to obey the closure constraint - consistent gauge algebra in the effective theory. 29 M. Grana, D. Marques, arxiv:

61 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes with fluxes (only subset) D d SUGRA embedding tensor gauged SUGRA 30

62 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes i = 0 SC with fluxes (only subset) Double Field Theory D d SUGRA embedding tensor gauged SUGRA 31

63 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes i = 0 SC with fluxes (only subset) Double Field Theory Violation of SC Non-geometric space D d SUGRA embedding tensor gauged SUGRA 32

64 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes i = 0 SC Double Field Theory D with fluxes (only subset) Violation of SC Non-geometric space d SUGRA embedding tensor gauged SUGRA e.o.m. Vacuum 33

65 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes i = 0 SC Double Field Theory Violation of SC Non-geometric space uplift Asymmetric orbifold CFT D with fluxes (only subset) d SUGRA embedding tensor gauged SUGRA e.o.m. Vacuum 34

66 Effective scalar potential: V = 1 4 F K I L F JKL H IJ F IKMF JLN H IJ H KL H MN R MN =0 Minkowski vacua: V = 0 and K MN = V =0 H MN This leads to additional conditions on the fluxes. F IKM 35

67 Simplest non-trivial solutions: d=3 dim. backgrounds: T 2 X 1 X 2 S 1 S 1 X 3

68 Simplest non-trivial solutions: d=3 dim. backgrounds: T 2 X 1 X 2 S 1 S 1 X 3 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. - CFT: beta-functions are non-vanishing at quadratic order in fluxes. - Effective scalar potential: no Minkowski minima ( AdS)

69 Elliptic background spaces: Multiple fluxes: These backgrounds do satisfy R MN =0. 37

70 Elliptic background spaces: Multiple fluxes: These backgrounds do satisfy R MN =0. Single elliptic geometric space (Solvmanifold): f 2 13 = f 1 23 = f Symmetric orbifold. Z L 4 Z R 4 37

71 Elliptic background spaces: Multiple fluxes: These backgrounds do satisfy R MN =0. Single elliptic geometric space (Solvmanifold): f 2 13 = f 1 23 = f Symmetric orbifold. Z L 4 Z R 4 Single elliptic T-dual, non-geometric space: H 123 = Q 12 3 = H Asymmetric Z L 4 Z R 4 orbifold. 37

72 Elliptic background spaces: Multiple fluxes: These backgrounds do satisfy R MN =0. Single elliptic geometric space (Solvmanifold): f 2 13 = f 1 23 = f Symmetric orbifold. Z L 4 Z R 4 Single elliptic T-dual, non-geometric space: H 123 = Q 12 3 = H Asymmetric Double elliptic, genuinely non-geometric space: H 123 = Q 12 3 = H, f 2 13 = f 1 23 = f Asymmetric Z L 4 Z R 4 Z L 4 37 orbifold. orbifold.

73 Monodromy of double elliptic background: c.s.: Kahler: (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) 38

74 Monodromy of double elliptic background: Background (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) satisfies cos(fx 3 ) 0 sin(fx 3 ), f 2 1 strong c.s.: 4 constraint + Z, Kahler: (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) 38

75 Monodromy of double elliptic background: Background (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) satisfies cos(fx 3 ) 0 sin(fx 3 ), f 2 1 strong c.s.: 4 constraint + Z, Kahler: (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) 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

76 Monodromy of double elliptic background: Background (x 3 ) = 0 cos(fx 3 )+sin(fx 3 ) satisfies cos(fx 3 ) 0 sin(fx 3 ), f 2 1 strong c.s.: 4 constraint + Z, Kahler: (x 3 ) = 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) 0 sin(hx 3 ), H Z. =) (2 ) = 1 (0), (2 ) = 1 (0) 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 Killing vectors do not satisfy strong constraint. However their algebra closes! 38

77 There situations, where the strong constraint even for the background can be violated. - This seems to be the case for certain very asymmetric orbifolds. C. Condeescu, I. Florakis, C. Kounnas, D.Lüst, arxiv: (x 3, x 3 )= 0 cos(f 4 x 3 + f 2 x 3 )+sin(f 4 x 3 + f 2 x 3 ) cos(f 4 x 3 + f 2 x 3 ) 0 sin(f 4 x 3 + f 2 x 3 ), f 4,g Z (x 3, x 3 )= 0 cos(g 4 x 3 + g 2 x 3 )+sin(g 4 x 3 + g 2 x 3 ) cos(g 4 x 3 + g 2 x 3 ) 0 sin(g 4 x 3 + g 2 x 3 ), f 2,g Z Fluxes: Parameter f 4 f 2 g 4 g 2 Fluxes f, f Q, Q H, Q f,r Asymmetric Z L 4 Z R 2 orbifold with H, f,q,r-fluxes. This (partially?) solves a so far existing puzzle between effective SUGRA and uplift/string compactification. 39

78 V) Outlook & open questions 40

79 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

80 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/ for functions that satisfy strong constraint in DFT. 40

81 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/ for functions that satisfy strong constraint in DFT. Non-associativity is an off-shell phenomenon! 40

82 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/ for functions that satisfy strong constraint in DFT. Non-associativity is an off-shell phenomenon! 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

83 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/ for functions that satisfy strong constraint in DFT. Non-associativity is an off-shell phenomenon! 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)) DFT allows for consistent reduction on non-geometric backgrounds that go beyond Sugra and also beyond generalized geometry interesting applications for cosmology. 40