Non-Associative Flux Algebra in String and M-theory from Octonions

Transcription

1 Non-Associative Flux Algebra in String and M-theory from Octonions DIETER LÜST (LMU, MPI) Corfu, September 15th,

2 Non-Associative Flux Algebra in String and M-theory from Octonions DIETER LÜST (LMU, MPI) In collaboration with M. Günaydin & E. Malek, arxiv: Corfu, September 15th,

3 This talk is dedicated to my friend Ioannis Bakas 2

4 Outline: I) Introduction II) Non-associative R-flux algebra for closed strings III) R-flux algebra from octonions IV) M-theory up-lift of R-flux background V) Non-associative R-flux algebra in M-theory 3

5 I) Introduction Geometry in general depends on, with what kind of objects you test it. Point particles in classical Einstein gravity see continuous Riemannian manifolds. - [x i,x j ]=0 Strings may see space-time in a different way. We expect the emergence of a new kind of stringy geometry. 4

6 Closed strings in non-geometric R-flux backgrounds non-associative phase space algebra: x i,x j = i l3 s ~ Rijk p k x i,p j = i~ ij, p i,p j =0 D.L., arxiv: ; R. Blumenhagen, E. Plauschinn, arxiv: =) x i,x j,x k 1 3 x 1,x 2,x 3 +cycl. perm. = l 3 sr ijk 5

7 Closed strings in non-geometric R-flux backgrounds non-associative phase space algebra: x i,x j = i l3 s ~ Rijk p k x i,p j = i~ ij, p i,p j =0 D.L., arxiv: ; R. Blumenhagen, E. Plauschinn, arxiv: =) x i,x j,x k 1 x 1,x 2,x 3 +cycl. perm. = l 3 3 sr ijk This algebra can be derived from closed string CFT. R. Blumenhagen, A. Deser, D.L., E. Plauschinn, F. Rennecke, arxiv: C. Condeescu, I. Florakis, D. L., arxiv: D. Andriot, M. Larfors, D.L., P. Patalong:arXiv: C. Blair, arxiv: I. Bakas, D.L., arxiv:

8 Closed strings in non-geometric R-flux backgrounds non-associative phase space algebra: x i,x j = i l3 s ~ Rijk p k x i,p j = i~ ij, p i,p j =0 D.L., arxiv: ; R. Blumenhagen, E. Plauschinn, arxiv: =) x i,x j,x k 1 x 1,x 2,x 3 +cycl. perm. = l 3 3 sr ijk This algebra can be derived from closed string CFT. R. Blumenhagen, A. Deser, D.L., E. Plauschinn, F. Rennecke, arxiv: C. Condeescu, I. Florakis, D. L., arxiv: D. Andriot, M. Larfors, D.L., P. Patalong:arXiv: C. Blair, arxiv: I. Bakas, D.L., arxiv: This algebra is also closely related to double field theory. R. Blumenhagen, M. Fuchs, F. Hassler, D.L., R. Sun, arxiv:

9 Two questions: 6

10 Two questions: On the mathematical side: How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions? 6

11 Two questions: On the mathematical side: How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions? On the physics side: Can one lift the R-flux algebra of closed strings to M-theory? 6

12 Two questions: On the mathematical side: How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions? On the physics side: Our conjecture: the answers to these two questions are closely related Can one lift the R-flux algebra of closed strings to M-theory? 6

13 II) Non-geometric string flux backgrounds Three-dimensional string flux backgrounds: Chain of three T-duality transformations: H ijk T i! f i jk T j! Q ij k T k! R ijk, (i, j, k =1,...,3) (Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht (2005); Dabholkar, Hull, 2005) (i) T 3 with H-flux: ds 2 =(dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2, B 12 = Nx 3 H-flux: H 123 = N 7

14 (ii) Twisted torus tilde T 3 : T-duality along x 1 ds 2 = dx 1 Nx 3 dx 2 2 +(dx 2 ) 2 +(dx 3 ) 2, B 2 =0 T 3 is a U(1) bundle over T 2 : Globally defined 1-forms: 1 = dx 1 Nx 3 dx 2, 2 = dx 2, 3 = dx 3 d i = f i jk j ^ k Geometric flux: f 1 23 = N 8

15 (iii) Q-flux background: T-duality along x 2 ds 2 = (dx1 ) 2 +(dx 2 ) 2 1+N 2 (x 3 ) 2 +(dx 3 ) 2, B 23 = Nx 3 1+N 2 (x 3 ) 2 This background is globally not well defined, but it is patched together by a T-duality transformation. T - fold C. Hull (2004) 9

16 (iii) Q-flux background: T-duality along x 2 ds 2 = (dx1 ) 2 +(dx 2 ) 2 1+N 2 (x 3 ) 2 +(dx 3 ) 2, B 23 = Nx 3 1+N 2 (x 3 ) 2 This background is globally not well defined, but it is patched together by T-duality transformation. T - fold C. Hull (2004) To make it well defined use double field theory: W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...) SO(3,3) double field theory: Coordinates: (x 1,x 2,x 3 ; x 1, x 2, x 3 ) 10

17 The dual background can then by described by dual metric and a bi-vector: M. Grana, R. Minasian, M. Petrini, D. Waldram (2008); D. Andriot, O. Hohm, M. Larfors, D.L., P. Patalong (2011,2012); R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid (2013); D. Andriot, A. Betz (2013) T ij B ij (x)! ij (x) = 1 2 T ij g(x)! ĝ(x) = 1 2 (g B) 1 (g + B) 1, (g B) 1 +(g + B) 1 1. Q ij k k ij 11

18 The dual background can then by described by dual metric and a bi-vector: M. Grana, R. Minasian, M. Petrini, D. Waldram (2008); D. Andriot, O. Hohm, M. Larfors, D.L., P. Patalong (2011,2012); R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid (2013); D. Andriot, A. Betz (2013) T ij B ij (x)! ij (x) = 1 2 T ij g(x)! ĝ(x) = 1 2 (g B) 1 (g + B) 1, (g B) 1 +(g + B) 1 1. Q ij k k ij For the Q-flux background one obtains: ˆds 2 =(dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2, 12 = Nx 3 Q-flux: Q 12 3 = N 11

19 Then one obtains from the CFT of the Q-flux background the following commutation relation among the coordinates: I. Bakas, D.L., arxiv: Sigma-model for non-geometric backgrounds: A. Chatzistavrakidis, L. Jonke, O. Lechtenfeld, arxiv: x 1,x 2 = N p 3 In general: winding number = dual momentum x i,x j = i l2 s ~ I S 1 k Q ij k (x) dxk = i l3 s ~ Qij k pk 12

20 (iv) R-flux background: T-duality along x 3 Buscher rule fails and one would get a background that is even locally not well defined. 13

21 (iv) R-flux background: T-duality along x 3 Buscher rule fails and one would get a background that is even locally not well defined. R-flux can be defined in double field theory: x k T k! x k T k ij (x k )! ij ( x k ) R ijk =3ˆ@ [k ij] 13

22 In our case we get: ˆds 2 =(dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2, 12 = N x 3 R-flux: R 123 = N Strong constraint of DFT is violated by this background. But it is still a consistent CFT background. 14

23 In our case we get: ˆds 2 =(dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2, 12 = N x 3 R-flux: R 123 = N Strong constraint of DFT is violated by this background. But it is still a consistent CFT background. Now for the R-flux background we obtain: x i,x j = i l3 s ~ Rijk p k x i,p j = i~ ij, p i,p j =0 =) x i,x j,x k 1 3 x 1,x 2,x 3 +cycl. perm. = l 3 sr ijk 14

24 In our case we get: ˆds 2 =(dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2, 12 = N x 3 R-flux: R 123 = N Strong constraint of DFT is violated by this background. But it is still a consistent CFT background. Now for the R-flux background we obtain: x i,x j = i l3 s ~ Rijk p k x i,p j = i~ ij, p i,p j =0 momentum =) x i,x j,x k 1 3 x 1,x 2,x 3 +cycl. perm. = l 3 sr ijk 14

25 Two remarks: Mathematical framework to describe nongeometric string backgrounds: Group theory cohomology. 3-cycles, 2-cochains,? - products D. Mylonas, P. Schupp, R.Szabo, arxiv: , arxiv: , arxiv: I. Bakas, D.Lüst, arxiv: The same algebra appear in the context of the magnetic monopole. R. Jackiw (1985); M. Günaydin, B. Zumino (1985) I. Bakas, D.L., arxiv:

26 III) R-flux algebra from octonions There exist four division algebras: over R, C, Q, O Division algebra of real octonions O : non-commutative, non-associative Besides the identity, there are seven imaginary units e A e A e B = AB + ABC e C (A =1...,7) ABC =1() (ABC) = (123), (516), (624), (435), (471), (572), (673) 16

27 Fano plane mnemonic: e 6 e 1 e 2 e 7 e 5 e 3 e 4 Remark: Octonions generate a simple Malcev algebra M. Günaydin, F. Gürsey (1973); M. Günaydin, D. Minic, arxiv:

28 Split indices: e i, e (i+3) = f i, for i =1, 2, 3 and e 7 18

29 Split indices: e i, e (i+3) = f i, for i =1, 2, 3 and e 7 [e i,e j ] = 2 ijk e k, [e 7,e i ]=2f i, [f i,f j ] = 2 ijk e k, [e 7,f i ]= 2e i, [e i,f j ] = 2 ij e 7 2 ijk f k [e i,e j,f k ] = 4 ijk e 7 8 k[i f j], [e i,f j,f k ] = 8 i[j e k], [f i,f j,f k ] = 4 ijk e 7, [e i,e j,e 7 ] = 4 ijk f k, [e i,f j,e 7 ] = 4 ijk e k, [f i,f j,e 7 ] = 4 ijk f k Associator [X, Y, Z] (XY )Z X(YZ) 18

30 Contraction of octonionic Malcev algebra: p 1 N p i = i 2 e i, x i = i 1/2 2 f i, I = i 3/2 p N 2 e 7 19

31 Contraction of octonionic Malcev algebra: p 1 N p i = i 2 e i, x i = i 1/2 2 f i, I = i 3/2! 0 p N 2 e 7 [f i,f j ]= 2 ijk e k =) [x i,x j ]=in ijk p k [e i,e j ]=2 ijk e k =) [p i,p j ]=0 [f i,e j ]= je i 7 + i jkf k =) x i,p j = i i j I [x i,i]=0=[p i,i] [f i,f j,f k ]= 4 ijk e 7 =) x i,x j,x k = N ijk I Agrees with non-associative R-flux algebra! 19

32 What is the role of un-contracted algebra? 20

33 What is the role of un-contracted algebra? Lift of R-flux algebra to non-geometric M-theory background: 20

34 What is the role of un-contracted algebra? Lift of R-flux algebra to non-geometric M-theory background: - e 7 additional M-theory coordinate Four coordinates: f 1,f 2,f 3,e 7 20

35 What is the role of un-contracted algebra? Lift of R-flux algebra to non-geometric M-theory background: - e 7 additional M-theory coordinate Four coordinates: f 1,f 2,f 3,e 7 - but no additional momentum. Three momenta: e 1,e 2,e 3 20

36 What is the role of un-contracted algebra? Lift of R-flux algebra to non-geometric M-theory background: - e 7 additional M-theory coordinate Four coordinates: f 1,f 2,f 3,e 7 - but no additional momentum. Three momenta: e 1,e 2,e 3 Seven dimensional phase space! 20

37 What is the role of un-contracted algebra? Lift of R-flux algebra to non-geometric M-theory background: Will be closely related to SL(4) /SO(4) exceptional field theory. - e 7 additional M-theory coordinate Four coordinates: f 1,f 2,f 3,e 7 - but no additional momentum. Three momenta: e 1,e 2,e 3 Seven dimensional phase space! 20

38 IV) M-theory up-lift of R-flux background Consider IIA string: the duality chain splits into two possible T-dualities: H ijk T ij! Q ij k, fi jk T jk! R ijk 21

39 IV) M-theory up-lift of R-flux background Consider IIA string: the duality chain splits into two possible T-dualities: H ijk T ij! Q ij k, fi jk T jk! R ijk 21

40 IV) M-theory up-lift of R-flux background Consider IIA string: the duality chain splits into two possible T-dualities: H ijk T ij! Q ij k, fi jk T jk! R ijk Uplift to M-theory: add additional circle S 1 x 4 3-dim IIA flux background background two T-dualities 3 U-dualities (Need third duality along the M-theory circle to ensure right dilaton shift.) 21 4-dim M-theory flux

41 (i) twisted torus T 3 S 1 x 4 ds 2 4 = dx 1 Nx 3 dx 2 2 +(dx 2 ) 2 +(dx 3 ) 2 +(dx 4 ) 2, C 3 =0 22

42 (i) twisted torus T 3 S 1 x 4 ds 2 4 = dx 1 Nx 3 dx 2 2 +(dx 2 ) 2 +(dx 3 ) 2 +(dx 4 ) 2, C 3 =0 (ii) R-flux background: dualise along x 2,x 3 and x 4. This leads to a locally not well defined space. 22

43 (i) twisted torus T 3 S 1 x 4 ds 2 4 = dx 1 Nx 3 dx 2 2 +(dx 2 ) 2 +(dx 3 ) 2 +(dx 4 ) 2, C 3 =0 (ii) R-flux background: dualise along x 2,x 3 and x 4. This leads to a locally not well defined space. Use SL(5) exceptional field theory: 10 generalized coordinates: D. Berman, M. Perry, arxiv: x A $ x [ab] (A =1,...,10 ; a, b =1...,5) 4 coordinates of T 4 : x = x 5, ( =1,...,4) 6 dual coordinates: ( x 41, x 42, x 43 ; x 21, x 31, x 32 ) 22 wrapped F1 wrapped D2

44 (i) twisted torus T 3 S 1 x 4 ds 2 4 = dx 1 Nx 3 dx 2 2 +(dx 2 ) 2 +(dx 3 ) 2 +(dx 4 ) 2, C 3 =0 (ii) R-flux background: dualise along x 2,x 3 and x 4. This leads to a locally not well defined space. Use SL(5) exceptional field theory: 10 generalized coordinates: 10 of SL(5) D. Berman, M. Perry, arxiv: x A $ x [ab] (A =1,...,10 ; a, b =1...,5) 4 coordinates of T 4 : x = x 5, ( =1,...,4) 6 dual coordinates: ( x 41, x 42, x 43 ; x 21, x 31, x 32 ) 22 wrapped F1 wrapped D2

45 (i) twisted torus T 3 S 1 x 4 ds 2 4 = dx 1 Nx 3 dx 2 2 +(dx 2 ) 2 +(dx 3 ) 2 +(dx 4 ) 2, C 3 =0 (ii) R-flux background: dualise along x 2,x 3 and x 4. This leads to a locally not well defined space. Use SL(5) exceptional field theory: D. Berman, M. Perry, arxiv: generalized coordinates: 10 of SL(5) 5 of SL(5) x A $ x [ab] (A =1,...,10 ; a, b =1...,5) 4 coordinates of T 4 : x = x 5, ( =1,...,4) 6 dual coordinates: ( x 41, x 42, x 43 ; x 21, x 31, x 32 ) 22 wrapped F1 wrapped D2

46 (i) twisted torus T 3 S 1 x 4 ds 2 4 = dx 1 Nx 3 dx 2 2 +(dx 2 ) 2 +(dx 3 ) 2 +(dx 4 ) 2, C 3 =0 (ii) R-flux background: dualise along x 2,x 3 and x 4. This leads to a locally not well defined space. Use SL(5) exceptional field theory: 10 generalized coordinates: 10 of SL(5) D. Berman, M. Perry, arxiv: x A $ x [ab] (A =1,...,10 ; a, b =1...,5) 4 coordinates of T 4 : 4 of SO(4) 5 of SL(5) x = x 5, ( =1,...,4) 6 dual coordinates: ( x 41, x 42, x 43 ; x 21, x 31, x 32 ) 22 wrapped F1 wrapped D2

47 (i) twisted torus T 3 S 1 x 4 ds 2 4 = dx 1 Nx 3 dx 2 2 +(dx 2 ) 2 +(dx 3 ) 2 +(dx 4 ) 2, C 3 =0 (ii) R-flux background: dualise along x 2,x 3 and x 4. This leads to a locally not well defined space. Use SL(5) exceptional field theory: 10 generalized coordinates: 10 of SL(5) D. Berman, M. Perry, arxiv: x A $ x [ab] (A =1,...,10 ; a, b =1...,5) 4 coordinates of T 4 : 4 of SO(4) 5 of SL(5) x = x 5, ( =1,...,4) 6 dual coordinates: ( x 41, x 42, x 43 ; x 21, x 31, x 32 ) 6 of SO(4) wrapped F1 wrapped D2 22

48 SL(5) Flux-background: C. Blair, E. Malek, arxiv: dual metric: tri-vector: ĝ = 1+V 2 1/3 1+V 2 g V V, = 1+V 2 1 g g g C, R-flux: ˆds 2 7 = 1+V 2 1/3 ds 2 7. V = 1 3! e ˆ@ + C R, =4ˆ@ [ 23

49 SL(5) Flux-background: C. Blair, E. Malek, arxiv: dual metric: tri-vector: ĝ = 1+V 2 1/3 1+V 2 g V V, = 1+V 2 1 g g g C, R-flux: ˆds 2 7 = 1+V 2 1/3 ds 2 7. V = 1 3! e ˆ@ + 23, C R, =4ˆ@ [ A particular choice of R-flux breaks SL(5) to SO(4).

50 In this way we obtain a well defined R-flux background in M-theory, which is dual to twisted torus: ˆds 2 7 =(dx 1 ) 2 +(dx 2 ) 2 +(dx 3 ) 2 +(dx 4 ) 2, 134 = N x 24 R 4,1234 = N The R-flux breaks the section condition of exceptional field theory. But it should be still a consistent M-theory background. 24

51 Four coordinates: x 1,x 2,x 3,x 4 What are the possible conjugate momenta (or windings)? 25

52 Four coordinates: x 1,x 2,x 3,x 4 What are the possible conjugate momenta (or windings)? Consider cohomology of twisted torus: H 1 ( T 3 S 1, R) =R 3 25

53 Four coordinates: x 1,x 2,x 3,x 4 What are the possible conjugate momenta (or windings)? Consider cohomology of twisted torus: H 1 ( T 3 S 1, R) =R 3 Alternatively consider Freed-Witten anomaly: R-Flux with momentum p 4 along R-Flux with D0 branes. Dualize to IIB: H-flux with D3-branes This is forbidden by the Freed-Witten anomaly. 25 x 4 No momentum modes along the direction! x 4

54 So we see that the phase space space of R-flux background in M-theory is seven-dimensional: x 1,x 2,x 3,x 4 ; p 1,p 2,p 3 26

55 So we see that the phase space space of R-flux background in M-theory is seven-dimensional: x 1,x 2,x 3,x 4 ; p 1,p 2,p 3 4 of SO(4) 3 of SO(4) 26

56 So we see that the phase space space of R-flux background in M-theory is seven-dimensional: x 1,x 2,x 3,x 4 ; p 1,p 2,p 3 4 of SO(4) 3 of SO(4) Missing momentum condition in covariant terms: p R, =0 This condition is not the same as section condition. 26

57 V) Non-associative R-flux algebra in M-theory Identify X i = 1 2 ip Nl 3/2 s 1/2 f i, X 4 = 1 2 ip Nl 3/2 s 3/2 e 7, P i = 1 2 i~ e i 27

58 V) Non-associative R-flux algebra in M-theory Identify X i = 1 2 ip Nl 3/2 s 1/2 f i, X 4 = 1 2 ip Nl 3/2 s 3/2 e 7, P i = 1 2 i~ e i Octonionic algebra = conjectured M-theory algebra [P i,p j ] = i ~ ijk P k, X 4,P i = i 2 ~X i, X i,x j = il3 s ~ R 4,ijk4 P k, X 4,X i = i l3 s ~ R4,1234 P i, X i,p j = i~ j ix4 + i ~ i jkx k, X,X,X = lsr 3 4, X, Pi,X j,x k =2 l 3 sr 4,1234 [j i P k], P i,x j,x 4 = 2 l 3 sr 4,ijk4 P k, [P i,p j,x k ] = 2 ~ 2 ijk X 4 +2 ~ 2 k[ix j], [P i,p j,x 4 ] = 3 ~ 2 ijk X k, [P i,p j,p k ] = 0. 27

59 Remarks: 28

60 Remarks: Natural identification of contraction parameter : String coupling g s : / g s / R 4 28

61 Remarks: Natural identification of contraction parameter : String coupling g s : / g s / R 4 The non-associative M-theory algebra is not SL(5) invariant. However the algebra is SO(4) invariant: X : (2, 2) of SU(2, 2) P i : (3, 1) of SU(2, 2) Lie subalgebra [P i,p j ]= i ~ ijk P k 28

62 Remarks: Natural identification of contraction parameter : String coupling g s : / g s / R 4 The non-associative M-theory algebra is not SL(5) invariant. However the algebra is SO(4) invariant: X : (2, 2) of SU(2, 2) P i : (3, 1) of SU(2, 2) Lie subalgebra [P i,p j ]= i ~ ijk P k Further Modification compared to the string case: X i,p j = i~ i j X 4 + i ~ i jkx k 28

63 V) Outlook & open questions Non-associative algebras occur in M-theory at many places: - Multiple M2-brane theories and 3-algebras J. Bagger, N. Lambert (2007) - Spin(7),G 2 backgrounds I.Bakas, E. Floratos, A. Kehagias, hep-th/

64 V) Outlook & open questions Non-associative algebras occur in M-theory at many places: - Multiple M2-brane theories and 3-algebras J. Bagger, N. Lambert (2007) - Spin(7),G 2 backgrounds I.Bakas, E. Floratos, A. Kehagias, hep-th/ What is the meaning of the odd-dimensional phase space? Derivation from M-brane sigma model See also:d. Mylonas, P. Schupp, R.Szabo, arxiv:

65 V) Outlook & open questions Non-associative algebras occur in M-theory at many places: - Multiple M2-brane theories and 3-algebras J. Bagger, N. Lambert (2007) - Spin(7),G 2 backgrounds I.Bakas, E. Floratos, A. Kehagias, hep-th/ What is the meaning of the odd-dimensional phase space? Derivation from M-brane sigma model See also:d. Mylonas, P. Schupp, R.Szabo, arxiv: Generalization to higher dimensional exceptional field theory? 29