Non-Associativity, Double Field Theory and Applications

Transcription

1 Non-Associativity, Double Field Theory and Applications DIETER LÜST (LMU, MPI) Recent Developments in String Theory, Monte Verita, Ascona, July 25,

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 V) Application: De Sitter and Inflation (if time) 2

3 I) Introduction 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))

4 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) 4 P. Bouwknegt, K. Hannabuss, Mathai (2003) L. Cornalba, R. Schiappa (2001)

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

6 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 T-duality with Xi. winding! Act asymmetrically on X L momentum and X R. X i A B X i A B, Λ = O(D, D) X i C D X i C D Generalized metric: H MN = X M =( X i,x i ) (X i = X i L(τ + σ)+x R (τ σ) Xi = X i L(τ + σ) X R (τ σ)) - Momenta: O(D,D) vector p M =( p i,p i ) G ij G ik B kj B ik G kj X i G ij B ik G kl B lj H MN Λ P 6M H PQ Λ Q N

7 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: Diff(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 )+ = 0 7

8 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 +2π) = Λ O(2,2) H PQ (X 3 ) Λ 1 Non-geometric spaces: Monodromy mixes O(2,2) X i X i Acts asymmetrically on XL,X i R i (i =1, 2) 8

9 Non geometric torus, metric is patched together by a T-duality transformation: G ij G ij 9

10 Non geometric torus, metric is patched together by a T-duality transformation: G ij G ij 10

11 (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

12 Q-flux: O(2,2) monodromy mixed closed string boundary (DN) conditions: O(2, 2){ X 3 Q(τ, σ +2π) = X 3 Q(τ, σ)+2π p 3 = XQ(τ, 1 σ +2π) = XQ(τ, 1 σ) 2π p 3 Q X Q2 (τ, σ), XQ(τ, 2 σ +2π) = XQ(τ, 2 σ)+2π p 3 Q X Q1 (τ, σ), X Q1 (τ, σ +2π) = X Q1 (τ, σ), X Q2 (τ, σ +2π) = X Q2 (τ, σ). winding number along base direction [X 1 Q(τ, σ),x 2 Q(τ, σ )] = i 2 Q p3 n=0 1 n 2 e in(σ σ) (σ σ) n=0 1 n e in(σ σ) + i 2 (σ σ) 2 σ σ : [X 1 Q(τ, σ),x 2 Q(τ, σ)] = i π2 6 Q p3 The non-commutativity of the torus (fibre) coordinates is determined by the winding in the circle (base) direction. 12

13 Corresponding uncertainty relation: ( X 1 Q) 2 ( X 2 Q) 2 L 6 s Q 2 p 3 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 13

14 R-flux background: p 3 p 3, T-duality in x 3 -direction R-flux XQ,3 X 3 R For the case of non-geometric R-fluxes one gets: Use [XR,X 1 R]= i 2 π2 6 Rp 3 [X 3 R,p 3 ]=i = R Q Non-associative algebra: [[X 1 R(τ, σ),x 2 R(τ, σ)],x 3 R(τ, σ)] + perm. = π2 6 R Corresponding classical uncertainty relations : ( X 1 R) 2 ( X 2 R) 2 L 6 s R 2 p 3 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: ) 14

15 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 15

16 Mathematical framework to describe non-geometric string backgrounds and the non-associative algebras: 3-Cocycles, 2-cochains and 3-products Open string non-commutativity: Constant Poisson structure: [x i,x j ]=θ ij Moyal-Weyl star-product: (f 1 f 2 )(x) =e iθij x 1 i x 2 j 2-cyclicity: f 1 (x 1 ) f 2 (x 2 ) x Non-commutative gauge theories: d n x (f g) = S (N. Seiberg, E. Witten (1999); J. Madore, S. Schraml, P. Schupp, J. Wess (2000);... ) d n x (g f) d n x Tr ˆF ab ˆF ab 16

17 Closed strings: Non-associative algebra: [x i,x j ]= ijk p k [x i,p j ]=iδ ij, [p i,p j ]=0 [x i,x j,x k ]=[[x i,x j ],x k ]+cycl. perm. = R ijk Non-commutativity p 2-product: (f 1 p f 2 )(x, p) =e i 2 θij (p) I J (f 1 f 2 ) x; p 6-dimensional Poisson tensor: θ IJ (p) = R ijk p k D. Mylonas, P. Schupp, R.Szabo, arxiv: , arxiv: , arxiv: I. Bakas, D.Lüst, arxiv: δ i j δ j i 0 ; R ijk = π2 R 6 ijk

18 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 1 3 f 2 3 f 3 )(x) =((f 1 p f 2 ) p f 3 )(x) (f 1 3 f 2 3 f 3 )(x) =e irijk x 1 i x 2 j 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 1 3 f 2 3 f 3 =[X i,x j,x k ]=R ijk d n x (f 1 3 f 2 ) 3 f 3 = 18 d n xf 1 3 (f 2 3 f 3 )

19 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 ijk L z 12 z 13 + L z12 z 13 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) : Vσ(1) V σ(1) V σ(1) R = V 1 V 2 V 3 iη R exp σ R ijk p 1,i p 2,j p 3,k. (η σ =0, 1) However this non-associative phase is vanishing, when going on-shell in CFT and using momentum conservation: X i,x j,x k := lim X i (z 1, z 1 ), X b (z 2, z 2 ),X c (z 3, z 3 ) +cycl. = R ijk zi z p 1 = (p 2 + p 3 ) On-shell CFT amplitudes are associative! 19

20 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: S X M =( x m,x m DFT = d 2D Xe 2φ R ) R =4H MN M φ N φ M N H MN 4H MN M φ N φ +4 M H MN N φ HMN M H KL N H KL 1 2 HMN N H KL 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 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 20

21 DFT action in flux formulation: S DFT = 1 dx e 2d F A F A S AA + F ABC F A B C η 4 SAA BB η CC 1 S 12 SAA BB S CC 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, arxiv: ) M M =0, M f 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. 21

22 Non-associative deformations in double field theory: DFT generalization of the 3-product: Non-vanishing R-flux: (R. Blumenhagen, M. Fuchs, F. Hassler, D.Lüst, R. Sun, arxiv: ) f 3 g 3 h (X) =fgh+ 4 s 6 F ABC D A f D B g D C h + O( 8 s) (For general functions f(x, P) f = x i,g= x j,h= x k : f 3 g 3 h =[x i,x j,x k ]= 4 s R ijk f = x i,g= x j,h= x k : Non-vanishing H-flux: the phase space is 4D-dimensional.) 22 I. Bakas, D. L., arxiv: f 3 g 3 h =[ x i, x j, x k ]=0 f = x i,g= x j,h= x k : f 3 g 3 h =[ x i, x j, x k ]= 4 s H ijk f = x i,g= x j,h= x k : f 3 g 3 h =[x i,x j,x k ]=0

23 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. 23

24 IV) Dimensional Reduction of DFT Consistent DFT solutions: 2(D-d) linear independent Killing vectors: L K J I H MN =0 24 R MN =0 DFT and generalized Scherk-Schwarz ansatz (O(D,D) twists) gives rise to effective theory in D-d dimensions: S eff = 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) ge 2φ R +4 µ φ µ φ 1 12 H µνρh µνρ 1 4 H MNF Mµν F N µν D µh MN D µ H MN V

25 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. 25 M. Grana, D. Marques, arxiv:

26 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 26

27 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes SFT i = 0 SC with fluxes (only subset) Double Field Theory D d SUGRA embedding tensor gauged SUGRA 27

28 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes SFT i = 0 SC with fluxes (only subset) Double Field Theory Violation of SC Non-geometric space D d SUGRA embedding tensor gauged SUGRA 28

29 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes SFT i = 0 SC Double Field Theory D d SUGRA with fluxes (only subset) Violation of SC Non-geometric space embedding tensor gauged SUGRA e.o.m. Vacuum 29

30 Dimensional reduction of double field theory: Generalized Scherk-Schwarz compactifications string theory simplification (truncation) matching amplitudes SUGRA without fluxes SFT i = 0 SC Double Field Theory Violation of SC Non-geometric space uplift Asymmetric orbifold CFT D d SUGRA with fluxes (only subset) embedding tensor gauged SUGRA e.o.m. Vacuum 30

31 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 = δv δh MN =0 This leads to additional conditions on the fluxes. F IKM 31

32 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)

33 Elliptic background spaces: Multiple fluxes: These backgrounds do satisfy R MN =0. Single elliptic geometric space (Solvmanifold): Single elliptic T-dual, non-geometric space: H 123 = Q 12 3 = H Double elliptic, genuinely non-geometric space: f 2 13 = f 1 23 = f Symmetric orbifold. Z L 4 Z R 4 Asymmetric H 123 = Q 12 3 = H, f 2 13 = f 1 23 = f Asymmetric Z L 4 Z R 4 Z L 4 33 orbifold. orbifold.

34 Monodromy of double elliptic background: Background τ(x 3 ) = τ 0 cos(fx 3 )+sin(fx 3 ) satisfies cos(fx 3 ) τ 0 sin(fx 3 ), f 1 strong c.s.: 4 constraint + Z, Kahler: ρ(x 3 ) = ρ 0 cos(hx 3 )+sin(hx 3 ) cos(hx 3 ) ρ 0 sin(hx 3 ), H Z. = τ(2π) = 1 τ(0), ρ(2π) = 1 ρ(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 ) Killing vectors do not satisfy strong constraint. However their algebra closes! 34

35 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: τ(x 3, x 3 )= τ 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 ) τ 0 sin(f 4 x 3 + f 2 x 3 ), ρ(x 3, x 3 )= ρ 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 ) ρ 0 sin(g 4 x 3 + g 2 x 3 ), f 4,g Z 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. 35

36 V) De Sitter and Inflation F. Hassler, D. Lüst, S. Massai, arxiv: Effective scalar potential of double elliptic backgrounds: V (τ, ρ) = 1 R 2 f f 1 f 2 (τ 2 R τ 2 I )+f 2 2 τ 4 2τ 2 I 36 + H2 +2HQ(ρ 2 R ρ2 I )+Q2 ρ 4 2ρ 2 I The potential is positive semi-definite. No up-lift is needed! Vacuum structure: Minkowski vacua: HQ > 0 ρ R =0, ρ I = H Q, V min =0 e.g. H = Q =1/4 Asymmetric orbifold. Z L 4 0

37 de Sitter vacua: HQ < 0 (ρ R) 2 +(ρ I) 2 = H Q, V min = 4HQ > 0 However here the radius R is not stabilized. Another option: SO(2,2) gauging V (ρ, τ) = H2 2ρ 2 I 1 + 2(ρ 2 R ρ 2 I)+ ρ 4 + H2 ρ I τ I (1 + ρ 2 )(1 + τ 2 )+ H2 2τ 2 I 1 + 2(τ 2 R τ 2 I )+ τ 4 ρ = τ = i with V min =4H 2 All moduli τ and ρ have positive mass square. 37

38 Inflation from non-geometric backgrounds: There are some attractive features for inflation: The potentials are positive with quadratic and quartic couplings that depend on the (non)-geometric fluxes. The non-trivial monodromies allow for enlarged field range of the inflaton field. (McAllsiter, Silverstein, Westphal, 2008) No up-lift is needed! One needs to tune fluxes to obtain slow roll inflation. ( Orbifolds with high order of twist!) Realization of monodromy inflation in order to obtain a visible tensor to scalar ratio (gravitational waves). 38

39 Enlarged field range for parabolic monodromy ρ ρ +1 or τ τ +1 : τ, ρ i Infinite field range for τ R or ρ R. 39

40 Enlarged field range for elliptic Z 4 monodromy ρ 1 ρ or τ 1 τ : τ, ρ i 1 2 Infinite field range for combinations of τ R and τ I or combinations of ρ R and ρ I

41 Enlarged field range for elliptic Z 6 monodromy ρ 1 or : ρ +1 τ 1 τ +1 τ, ρ i Infinite field range for combinations of τ R and τ I or combinations of ρ R and ρ I. 41

42 Simple elliptic model for non-geometric inflation: Expect fluxes Kinetic energy: Inflaton field: Inflaton potential: H, Q 1 N L kin = 1 4ρ 2 I φ = ρ R 2ρ I ( ρ R ) 2 +( ρ I ) 2 V (φ, ρ I )=V 0 (ρ I )+m 2 (ρ I ) φ 2 + λ(ρ I ) φ 4 V 0 (ρ I )= H2 2HQρ 2 I + Q2 ρ 4 I 2ρ 2 I, m 2 (ρ I )=4HQ +4Q 2 ρ 2 I, λ(ρ I )=8Q 2 ρ 2 I 42

43 Minimization with respect to : ρ I Inflaton mass and self-coupling: 1 m 2 =4HQ + ρ I Ms 2, ρ I V 0 =0 λ =8HQρ I g 2 sm 2 P λ m 2 = 2(ρ I )3 1+(ρ I )2 (M 2 P = 1 g 2 s M 2 s ρ I) Small λ small value for ρ I. 43

44 Slow roll inflation with 60 e-foldings and n s 0.967, r (BICEP2) n s =1 6 +2η, = M 2 P 2 φ V V 2, η = M 2 P r = 16 2 φ V m M P, V 1/ M P φ 15M P V H Q 10 5, ρ I 10 2 Need very small fluxes (large monodromy N 10 5 ) and sub-stringy value for the volume of the fibre. 44

45 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. 45

46 Dear Organizers, many thanks for this excellent and stimulating meeting at this wonderful conference site. So let us give a long applause to Marcos, Matthias, Matthias & Niklas! 46