Towards new non-geometric backgrounds

Transcription

1 Towards new non-geometric backgrounds Erik Plauschinn University of Padova Ringberg

2 this talk is based on... This talk is based on T-duality revisited [arxiv:30.494], and on some work in progress [arxiv:407.xxxx].

3 introduction :: motivation Non-geometric backgrounds

4 introduction :: motivation Non-geometric backgrounds :: why?

5 introduction :: motivation Non-geometric backgrounds :: why? Have non-commutative or non-associative features. Are part of the string-theory landscape. Provide uplifts for gauged supergravities. Can help with moduli stabilization & cosmology.

6 introduction :: main example What is a non-geometric background?

7 introduction :: main example What is a non-geometric background? apply T-duality to a three-torus: T H c abc! c T b fab! bc T a Qa! R abc flux background "twisted torus" T-fold non-associative Shelton, Taylor, Wecht

8 introduction :: main example H abc T c! fab c T b! Qa bc T a! R abc

9 introduction :: h-flux background H abc T c! fab c T b! Qa bc T a! R abc Consider string theory compactified on a three-torus with H-flux: The geometry is determined by ds 2 = dx 2 + dy 2 + dz 2, B yz = Nx, x x +, y y +, z z +. The H-flux reads H xyz = N.

10 introduction :: h-flux background H abc T c! fab c T b! Qa bc T a! R abc Consider string theory compactified on a three-torus with H-flux: The geometry is determined by ds 2 = dx 2 + dy 2 + dz 2, B yz = Nx, x x +, y y +, z z +. The H-flux reads gauge tr. H xyz = N. (G, B)(x + ) (G, B)(x) x

11 introduction :: h-flux background H abc T c! fab c T b! Qa bc T a! R abc Consider string theory compactified on a three-torus with H-flux: The geometry is determined by ds 2 = dx 2 + dy 2 + dz 2, B yz = Nx, x x +, y y +, z z +. The H-flux reads H xyz = N.

12 introduction :: f-flux background H abc T c! fab c T b! Qa bc T a! R abc After a T-duality in the z-direction, one arrives at a twisted torus: The geometry is determined by ds 2 = dx 2 + dy 2 +(dz + Nxdy) 2, B =0, (x, z) (x +,z Ny), y y +, z z +. The geometric flux reads e x = dx, e y = dy, e z = dz + Nxdy,! z xy = N/2, [e x,e y ]= N e z. Scherk, Schwarz Dasgupta, Rajesh, Sethi Kachru, Schulz, Tripathy, Trivedi

13 introduction :: f-flux background H abc T c! fab c T b! Qa bc T a! R abc After a T-duality in the z-direction, one arrives at a twisted torus: The geometry is determined by ds 2 = dx 2 + dy 2 +(dz + Nxdy) 2, B =0, (x, z) (x +,z Ny), y y +, z z +. The geometric flux reads e x = dx, e y = dy, e z = dz + Nxdy, (G, B)(x + ) SL(2, Z) (G, B)(x)! z xy = N/2, [e x,e y ]= N e z. x Scherk, Schwarz Dasgupta, Rajesh, Sethi Kachru, Schulz, Tripathy, Trivedi

14 introduction :: f-flux background H abc T c! fab c T b! Qa bc T a! R abc After a T-duality in the z-direction, one arrives at a twisted torus: The geometry is determined by ds 2 = dx 2 + dy 2 +(dz + Nxdy) 2, B =0, (x, z) (x +,z Ny), y y +, z z +. The geometric flux reads e x = dx, e y = dy, e z = dz + Nxdy,! z xy = N/2, [e x,e y ]= N e z. Scherk, Schwarz Dasgupta, Rajesh, Sethi Kachru, Schulz, Tripathy, Trivedi

15 introduction :: q-flux background H abc T c! fab c T b! Qa bc T a! R abc After a second T-duality in the x-direction, one arrives at a T-fold: The geometry is determined by ds 2 = dy 2 + +(Ny) 2 (dx2 + dz 2 ), Ny B xz = +(Ny) 2, x x +, z z +. The non-geometric flux reads Q y xz = N. This space is locally geometry, but globally non-geometric. Hellermann, McGreevy, Williams Dabholkar, Hull Hull

16 introduction :: q-flux background H abc T c! fab c T b! Qa bc T a! R abc After a second T-duality in the x-direction, one arrives at a T-fold: The geometry is determined by ds 2 = dy 2 + +(Ny) 2 (dx2 + dz 2 ), Ny B xz = +(Ny) 2, x x +, z z +. The non-geometric T-duality flux reads (G, B)(y + ) Q y xz = N. (G, B)(y) This space is locally geometry, but globally non-geometric. y Hellermann, McGreevy, Williams Dabholkar, Hull Hull

17 introduction :: q-flux background H abc T c! fab c T b! Qa bc T a! R abc After a second T-duality in the x-direction, one arrives at a T-fold: The geometry is determined by ds 2 = dy 2 + +(Ny) 2 (dx2 + dz 2 ), Ny B xz = +(Ny) 2, x x +, z z +. The non-geometric flux reads Q y xz = N. This space is locally geometry, but globally non-geometric. Hellermann, McGreevy, Williams Dabholkar, Hull Hull

18 introduction :: r-flux background H abc T c! fab c T b! Qa bc T a! R abc After formally applying a third T-duality, one obtains an R-flux background: The geometry is not even locally defined. The non-geometric R-flux is obtained by raising the index of the Q-flux Q y xz! R xyz = N. This background gives rise to a non-associative structure. Bouwknegt, Hannabuss, Mathai Shelton, Taylor, Wecht Ellwood, Hashimoto Blumenhagen, EP & Lüst - 200

19 introduction :: more examples But ::

20 introduction :: more examples But :: what about other examples? The torus is the mainly (and only) studied background. Other and better examples are needed! Consider the three-sphere.

21 introduction :: goal Goal :: Construct new non-geometric backgrounds. Plan :: Revisit (collective) T-duality. Review the three-torus. Consider the three-sphere.

22 outline. introduction 2. collective t-duality 3. three-torus 4. three-sphere 5. discussion

23 outline. introduction 2. collective t-duality 3. three-torus 4. three-sphere 5. discussion

24 t-duality :: sigma-model action To study T-duality for the three-sphere, a non-abelian version might be needed.

25 t-duality :: sigma-model action To study T-duality for the three-sphere, a non-abelian version might be needed. de la Ossa, Quevedo Giveon, Rocek Sfetsos Alvarez, Alvarez-Gaume, [Barbon,] Lozano & 994 Consider the sigma-model action for the NS-NS sector of the closed string S = 4 0 h i G ij dx i ^?dx j + 0 R? i 2 0 Z 3! H ijk dx i ^ dx j ^ dx k. This action is invariant under global transformations X i = k i (X) if L k G =0, k H = dv, L k =0. In general, the isometry algebra is non-abelian [k,k ] L = f k. Hull, Spence & 99

26 t-duality :: gauged action Following Buscher s procedure, the gauged sigma-model action is found as bs = 2 0 i 2 0 i 2 0 Z h 2 G ij(dx i + k i A ) ^?(dx j + k j A ) 3! H ijk dx i ^ dx j ^ dx k (v + d ) ^ A + 2 k [ v ] + f A ^ A i. Hull, Spence & 99 Alvarez, Alvarez-Gaume, Barbon, Lozano - 994

27 t-duality :: gauged action Following Buscher s procedure, the gauged sigma-model action is found as bs = 2 0 i 2 0 i 2 0 Z h 2 G ij(dx i + k i A ) ^?(dx j + k j A ) 3! H ijk dx i ^ dx j ^ dx k (v + d ) ^ A + 2 k [ v ] + f A ^ A i. Hull, Spence & 99 Alvarez, Alvarez-Gaume, Barbon, Lozano The local symmetry transformations take the form ˆ X i = k i, ˆ A = d A f, ˆ = k( v ) f. EP (with F. Rennecke) - 204

28 t-duality :: gauged action Following Buscher s procedure, the gauged sigma-model action is found as bs = 2 0 i 2 0 i 2 0 Z h 2 G ij(dx i + k i A ) ^?(dx j + k j A ) 3! H ijk dx i ^ dx j ^ dx k (v + d ) ^ A + 2 k [ v ] + f A ^ A i. Hull, Spence & 99 Alvarez, Alvarez-Gaume, Barbon, Lozano EP (with F. Rennecke) - 204

29 t-duality :: gauged action Following Buscher s procedure, the gauged sigma-model action is found as bs = 2 0 i 2 0 i 2 0 Z h 2 G ij(dx i + k i A ) ^?(dx j + k j A ) 3! H ijk dx i ^ dx j ^ dx k (v + d ) ^ A + 2 k [ v ] + f A ^ A i. Hull, Spence & 99 Alvarez, Alvarez-Gaume, Barbon, Lozano This gauging is subject to the following constraints L k[ v ] = f v, k[ f ] v = 3 k k k H. EP (with F. Rennecke) - 204

30 t-duality :: recovering the original model The original model is recovered via the equations of motion for 0=dA 2 f A ^ A. The gauge action can then be rewritten in terms of bs = 4 0 i 2 0 Z DX i = dx i + k i A h i G ij DX i ^?DX j + 0 R? 3! H ijk DX i ^ DX j ^ DX k. as Ignoring technical details, one replaces DX i! dy i and obtains the ungauged action.

31 t-duality :: obtaining the dual model I The dual model is obtained via the equations of motion for A A = G DG D + i?dg k + i?, where G = k i G ij k j, = d + v, D = k[ v ] + f, k = k i G ij dx j. The action of the dual sigma-model is found by integrating-out Š = 4 0 hǧ + 0 R?i i 2 0 Z Ȟ, A and reads where, with M = G DG D Ǧ = G + invertible, T k M M DG +M DG +M k ^?, Ȟ = H + 2 d " k T +M DG +M M M DG ^ # k.

32 t-duality :: obtaining the dual model II Consider an enlarged target-space parametrized by coordinates X i and. The enlarged metric Ǧ and field strength Ȟ have null-eigenvectors (and isometries) ň Ǧ =0, ň Ȟ =0, ň = k + The dual metric and field strength are obtained via a change of coordinates T I A = k 0, Ǧ D AB =(T T Ǧ T ) AB = 0 0, 0 G Ȟ ABC = ȞIJK T I A T J B T K C, Ȟ ibc =0.

33 t-duality :: summary The T-duality transformation rules are obtained via Buscher s procedure of. gauging isometries in the sigma-model action, 2. integrating-out the gauge field, 3. performing a change of coordinates. The possible gaugings are restricted by (recall that k H = dv ) L k[ v ] = f v, k[ f ] v = 3 k k k H. The change of coordinates is performed using null-eigenvectors ň Ǧ IJ ň J =0, Ȟ IJK ň K =0.

34 outline. introduction 2. collective t-duality 3. three-torus 4. three-sphere 5. discussion

35 torus :: setting Consider a three-torus with H-flux specified as follows ds 2 = R 2 dx 2 + R 2 2 dx R 2 3 dx 3 2, X i ' X i + `s, H = hdx ^ dx 2 ^ dx 3, h 2 ` s Z. The Killing vectors (in the basis k = 0 0 {@,@ 2,@ 3 } A, k 2 = ) are abelian and can be chosen as 0 0 A, k 3 = 0 0 A. The one-forms v (defined via k H = dv ), up to exact terms take the form v = h X 2 dx 3 h 2 X 3 dx 2, + 2 =, v 2 = h X 3 dx h 2 X dx 3, + 2 =, v 3 = h X dx 2 h 2 X 2 dx, + 2 =.

36 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as

37 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as

38 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as

39 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. Ǧ = G + k T M M DG +M DG +M k ^? The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as = d + v

40 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. Ǧ = G + = G + k T R 2 dx M M DG +M DG +M T R 2 k ^? The geometry of the dual background is determined from the quantities R 2! R 2 ^? dx G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as = d + v

41 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. Ǧ = G + = G + k T R 2 dx M M DG +M DG +M T R 2 k ^? The geometry of the dual background is determined from the quantities R 2! R 2 ^? dx G = R 2, = G R = d + v, 2 dx ^?dx + R D =0, k = R 2 dx 2, ^? M = G = R 2. The metric and field strength are then computed as = d + v

42 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. Ǧ = G + = G + k T R 2 dx The geometry of the dual background is determined from the quantities G = R 2, = G R = d + v, 2 dx ^?dx + R D =0, k = R 2 dx 2, = R 2 M M DG +M DG +M The metric and field strength are then computed as T R R 2! ^? R 2 ^? dx ^? + R 2 2 dx 2 ^?dx 2 + R 2 3 dx 3 ^?dx 3 k ^? M = G = R 2. = d + v

43 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as

44 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. Ȟ = H + 2 d " k T +M DG +M M M DG ^ # k The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as

45 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. Ȟ = H + 2 d " k T +M DG +M M M DG ^ # k " R = H d dx T! 0 + R 2 R 2 ^ dx # 0 The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, The metric and field strength are then computed as R 2 M = G = R 2.

46 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. Ȟ = H + 2 d " k T +M DG +M M M DG ^ # k " R = H d dx T! 0 + R 2 R 2 ^ dx # 0 The geometry of the dual background is determined from the quantities G = R 2, h = d i + v, = H + d dx ^ D =0, k = R 2 dx, The metric and field strength are then computed as R 2 M = G = R 2.

47 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. Ȟ = H + 2 d " k T +M DG +M M M DG ^ # k " R = H d dx T! 0 + R 2 R 2 ^ dx # 0 The geometry of the dual background is determined from the quantities G = R 2, h = d i + v, = H + d dx ^ D =0, k = R 2 dx, =0 The metric and field strength are then computed as R 2 M = G = R 2. d = d(d + v) =hdx 2 ^ dx 3

48 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as

49 torus :: one t-duality I Consider one T-duality along k The corresponding one-form reads v = h X 2 dx 3 h( )X 3 dx 2, 2 R. The constraints for gauging the sigma-model are trivially satisfied. The geometry of the dual background is determined from the quantities G = R 2, = d + v, D =0, k = R 2 dx, M = G = R 2. The metric and field strength are then computed as Ǧ = R 2 ^? + R 2 2 dx 2 ^?dx 2 + R 2 3 dx 3 ^?dx 3, Ȟ =0.

50 torus :: one t-duality II As expected, the dual background is a twisted torus (with =) ďs 2 = R 2 d + hx 2 dx R 2 2 dx R 2 3 dx 3 2, Ȟ =0.

51 torus :: two t-dualities I Consider two collective T-dualities along k and k 2 2. The constraints on gauging the sigma-model imply (for 2 R ) v = h X 2 dx 3 h( )X 3 dx 2, v 2 = h( + )X 3 dx + h X dx 3. The geometry of the dual background is determined from ( ), 2{, 2} G = D = R R2 2, = 0 +hx 3 hx 3, k 0 = d + v, d 2 + v 2 R 2 dx R2 2. dx2

52 torus :: two t-dualities II The metric of the enlarged target space (in the basis {dx,dx 2,dX 3,, 2 }) reads Ǧ IJ = 0 R hx R 2 hx 3 R2 hx R2 2 hx R R2 2 hx 3 0 R2 2 0 R 2 hx R 2, C A = R 2 R hx 3 2. Performing then a change of basis one finds T I A = hx hx C A Ǧ AB =(T T Ǧ T ) AB = R R R 2. C A

53 torus :: two t-dualities III Performing a similar analysis for the field strength and adjusting the notation, one finds ďs 2 = R 2 R2 2 + hx 3 2 h R 2 d 2 + R 2 2 d 2 2 i + R 2 3 dx 3 2, Ȟ = h R2 R 2 2 hx 3 2 hr 2R2 2 + hx 3 i 2 2 d ^ d 2 ^ dx 3. This is the familiar T-fold background.

54 torus :: three t-dualities Finally, consider three collective T-dualities along k k 2 2 and k 3 3. The constraints on gauging the sigma-model require the H-flux to be vanishing k k k H =0 H =0. The dual model is characterized by ďs 2 = R 2 d 2 + R 2 2 d R 2 3 d 3 2, Ȟ =0.

55 torus :: three t-dualities Finally, consider three collective T-dualities along k k 2 2 and k 3 3. The constraints on gauging the sigma-model require the H-flux to be vanishing k k k H =0 H =0. The dual L k[ model v ] = is fcharacterized v by k[ f ďs 2 = 2 ] v = 3 k k k H R 2 d + R 2 2 d R 2 3 d 3 2, Ȟ =0.

56 torus :: three t-dualities Finally, consider three collective T-dualities along k k 2 2 and k 3 3. The constraints on gauging the sigma-model require the H-flux to be vanishing k k k H =0 H =0. The dual model is characterized by ďs 2 = R 2 d 2 + R 2 2 d R 2 3 d 3 2, Ȟ =0.

57 torus :: summary The formalism for T-duality introduced above works as expected. three-torus with H-flux T-duality twisted torus 2 T-dualities T-fold three-torus with H=0 3 T-dualities torus with R /R

58 torus :: twisted torus Bonus :: T-duality for the twisted torus with H-flux leads to twisted T-folds. Ǧ = + R R 2 fx 3 2 R 2 dx ^?dx + R 2 2 ^? + R 2 3 dx 3 ^?dx 3, Ȟ = f R2 R 2 2 R R 2 fx R R 2 fx dx ^ ^ dx 3, d = hdx ^ dx 3. This is a background with geometric and non-geometric flux.

59 outline. introduction 2. collective t-duality 3. three-torus 4. three-sphere 5. discussion

60 sphere :: setting Consider a three-sphere with H-flux, specified by ds 2 = R 2 h sin 2 (d ) 2 + cos 2 (d 2 ) 2 +(d ) 2 i,,2 =0...2, H = h 2 2 sin cos d ^ d 2 ^ d, = This model is conformal if h =4 2 R 2. The isometry algebra is (with,, 2{, 2, 3} ) so(4) = su(2) su(2), and the Killing vectors satisfy [K, K ] L = K, [K, K ] L =0, K 2 = K 2 = R2 4. [ K, K ] L = K,

61 sphere :: setting Consider a three-sphere with H-flux, specified by 0 h 0 i ds 2 = R 2 sin 2 (d ) 2 + cos 2 (d 2 ) 2 +(d ) 2,,2 =0...2, K = A, K = + A, H = h = sin cos d 0 ^ d 2 ^ d, 2. K 2 = sin( 2 ) sin( 2 ) tan A, K2 = +sin( + 2 ) sin( + 2 ) tan A, 2 cos( h =4 2 ) 2 R 2 2 cos( + 2 ) 0 0 K 3 = cos( 2 ) cos( 2 ) tan A, K3 = + cos( + 2 ) cos( + 2 ) tan A. 2 2 sin( 2 ) +sin( + 2 ) This model is conformal if. The isometry algebra is (with,, 2{, 2, 3} ) [K, K ] L = K, [ K, K ] L = K, so(4) = su(2) su(2), and the Killing vectors satisfy [K, K ] L =0, K 2 = K 2 = R2 4.

62 sphere :: setting Consider a three-sphere with H-flux, specified by ds 2 = R 2 h sin 2 (d ) 2 + cos 2 (d 2 ) 2 +(d ) 2 i,,2 =0...2, H = h 2 2 sin cos d ^ d 2 ^ d, = This model is conformal if h =4 2 R 2. The isometry algebra is (with,, 2{, 2, 3} ) so(4) = su(2) su(2), and the Killing vectors satisfy [K, K ] L = K, [K, K ] L =0, K 2 = K 2 = R2 4. [ K, K ] L = K,

63 sphere :: one t-duality Consider one T-duality along K. In this case, all constraints are satisfied: constraints from gauging the sigma-model the matrix X G = k i G ij k j is invertible X The dual model is characterized by the metric and H-flux Ǧ = R2 4 h (d ) 2 +sin 2 ( )(d ) 2i + 4 R 2 ^?, Ȟ =sin d ^ d ^, This metric describes a circle fibered over a two-sphere. d = h sin d ^ d. 6 2 Bouwknegt, Evslin, Mathai incomplete in Alvarez, Alvarez-Gaume, Barbon, Lozano - 993

64 sphere :: two t-dualities I For two collective T-dualities, consider the commuting Killing vectors K and K. The constraints for this model are almost satisfied: constraints from gauging the sigma-model X the matrix G = k i G ij k j is invertible X det G = R4 6 sin2 (2 ) The dual model, via the above formalism, takes a form similar to the T-fold Ǧ = R 2 (d ) 2 + (d ) 2 R 2 sin 2 + h R 2 cos2 + R 2 cos 2 + (d 2)2 h 4 2 R 2 2 cos4 sin 2, Ȟ = 8h 2 h R 4 sin cos 6 2 R 4 sin 2 + h 2 cos 2 2 d ^ d ^ d 2.

65 sphere :: two t-dualities II But, when starting from a conformal model with h =4 2 R 2, the background becomes G = R 2 (d ) 2 + R 2 h(d ) 2 + tan 2 (d 2) 2i, H =0. With dual dilaton = log R 2 cos +, this is again a conformal model.

66 sphere :: three t-dualities Consider finally a non-abelian T-duality along K, K 2 and K 3. The constraints from gauging the sigma-model imply H=0, and the matrix G = k i G ij k j is invertible X The dual model is obtained as (with 0 and,2 =0,...,2 ) Ǧ = 4 R2 d ^?d + R R4 6 i hd ^?d +sin 2 ( ) d 2 ^?d 2, Ȟ = R4 6 2 apple 2 +3 R4 6 sin( ) d ^ d ^ d 2. partially in Alvarez, Alvarez-Gaume, Barbon, Lozano - 993

67 sphere :: summary In the formalism for T-duality introduced above, for a conformal model one finds: three-sphere with H-flux T-duality S fibered over S 2 2 T-dualities non-compact, geometric three-sphere with H=0 3 T-dualities S 2 fibered over a ray

68 outline. introduction 2. collective t-duality 3. three-torus 4. three-sphere 5. discussion

69 discussion :: summary A formalism for collective T-duality transformations was developed Restrictions on allowed transformations arise. Reduction of the duality group? For the three-torus with H-flux, known results have been reproduced, and a twisted T-fold has been obtained. For the three-sphere with H-flux, new geometric backgrounds have been obtained, but their global structure is not clear.

70 discussion :: outlook For two collective T-duality transformations it was found :: three-torus with H-flux (not conformal) 2 T-dualities non-geometric compact three-sphere with H-flux (conformal) 2 T-dualities geometric non-compact Thus, the origin of non-geometry remains unclear