Collective T-duality transformations and non-geometric spaces

Transcription

1 Collective T-duality transformations and non-geometric spaces Erik Plauschinn LMU Munich ESI Vienna

2 based on... This talk is based on :: T-duality revisited On T-duality transformations for the three-sphere and some work in progress [arxiv: ] [arxiv: ] [arxiv:15xx.xxxxx]

3 motivation :: dualities I Duality :: two different theories describe the same physics.

4 motivation :: dualities I Duality :: two different theories describe the same physics. They play an important role in understanding string theory. M-theory Type IIA T Type IIB S Het E 8 E 8 T Het SO(32) S Type I

5 motivation :: dualities II New solutions can be found by applying duality transformations. known solution duality new solution

6 motivation :: dualities II New solutions can be found by applying duality transformations. known solution duality new solution In particular, consider string theory :: geometric space T-duality non-geometric space

7 motivation :: dualities II New solutions can be found by applying duality transformations. known solution duality new solution In particular, consider string theory :: geometric space T-duality non-geometric space

8 motivation :: goal Main goal :: understand better non-geometric spaces

9 motivation :: goal study (collective) T-duality transformations Main goal :: understand better non-geometric spaces

10 outline 1. motivation 2. non-geometric spaces 3. collective t-duality 4. examples 5. summary

11 outline 1. motivation 2. non-geometric spaces 3. collective t-duality 4. examples 5. summary

12 outline 1. motivation 2. non-geometric spaces a) t-duality b) non-geometry c) questions 3. collective t-duality 4. examples 5. summary

13 outline 1. motivation 2. non-geometric spaces a) t-duality b) non-geometry c) questions 3. collective t-duality 4. examples 5. summary

14 t-duality :: circle Consider the torus partition function of the bosonic string (with = 1 + i 2 ) Z( 1, 2 )=Tr H e 2 2H e +2 1P. For a compactification on a circle with radius R, it reads Z(, ) = 1 p ( ) 2 X m,n2z exp + i apple m 2 0 R + Rn 0 2! exp apple i m 2 0 R Rn 0 2!. This partition function is invariant under the replacement R! 0 R.

15 t-duality :: circle Consider the torus partition function of the bosonic string (with = 1 + i 2 ) Z( 1, 2 )=Tr H e 2 2H e +2 1P. For a compactification on a circle with radius R, it reads Z(, ) = 1 p ( ) 2 X m,n2z exp + i apple m 2 0 R + Rn 0 2! exp apple i m 2 0 R Rn 0 2!. This partition function is invariant under the replacement R! 0 R.

16 t-duality :: torus T-duality can be extended from the circle to toroidal backgrounds. R (G,B) For constant metric G and Kalb-Ramond field B, the partition function is invariant under O(D,D) transformations of H = G BG 1 B +BG 1 G 1 B G 1!.

17 t-duality :: curved background For general CFT backgrounds, with non-constant metric and B-field, one considers Z 1 h S = 4 0 G ij dx i ^?dx j + ib ij dx i ^ dx j + 0 R?1i. Gauging an isometry and integrating out the gauge field, gives the Buscher rules G = 1 G, G a = B a G, B a = G a G, G ab = G ab G a G b B a B b G, B ab = B ab G a B b B a G b G. The dual theory is conformal, if the dilaton is shifted as = log r det G det G. Buscher , 1987, 1988

18 t-duality :: summary T-duality :: circle compactification R! 1/R, toroidal background curved background O(D,D) transformation, Buscher rules.

19 outline 1. motivation 2. non-geometric spaces a) t-duality b) non-geometry c) questions 3. collective t-duality 4. examples 5. summary

20 non-geometry Non-geometry :: apply successive T-duality to a three-torus with H-flux.

21 non-geometry H abc T c! fab c T b! Qa bc T a! R abc

22 non-geometry :: h-flux 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 +1, y y +1, z z +1. The H-flux reads H xyz = N.

23 non-geometry :: h-flux 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 +1, y y +1, z z +1. The H-flux reads gauge tr. H xyz = N. (G, B)(x + 1) (G, B)(x) x

24 non-geometry :: h-flux 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 +1, y y +1, z z +1. The H-flux reads H xyz = N.

25 non-geometry :: f-flux 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 +1,z Ny), y y +1, z z +1. The geometric flux arises via 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

26 non-geometry :: f-flux 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 +1,z Ny), y y +1, z z +1. The geometric flux arises via e x = dx, e y = dy, e z = dz + Nxdy, (G, B)(x + 1) 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

27 non-geometry :: f-flux 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 +1,z Ny), y y +1, z z +1. The geometric flux arises via 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

28 non-geometry :: q-flux 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 (Ny) 2 (dx2 + dz 2 ), Ny B xz = 1+(Ny) 2, y y +1, z z +1. The non-geometric flux reads Q y xz = N. This space is locally geometry, but globally non-geometric. Hellermann, McGreevy, Williams Dabholkar, Hull Hull

29 non-geometry :: q-flux 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 (Ny) 2 (dx2 + dz 2 ), Ny B xz = 1+(Ny) 2, y y +1, z z +1. The non-geometric T-duality flux reads (G, B)(y + 1) Q y xz = N. (G, B)(y) This space is locally geometry, but globally non-geometric. y Hellermann, McGreevy, Williams Dabholkar, Hull Hull

30 non-geometry :: q-flux 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 (Ny) 2 (dx2 + dz 2 ), Ny B xz = 1+(Ny) 2, y y +1, z z +1. The non-geometric flux reads Q y xz = N. This space is locally geometry, but globally non-geometric. Hellermann, McGreevy, Williams Dabholkar, Hull Hull

31 non-geometry :: r-flux 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

32 non-geometry :: topological t-duality T-duality (toroidal) has also been studied in a more mathematical approach. three-torus with H-flux 1 T-duality twisted torus = S 1 E T 2 2 T-dualities non-commutative torus 3 T-dualities non-associative torus Bouwknegt, Evslin, Mathai Mathai, Rosenberg Bouwknegt, Hannabuss, Mathai

33 non-geometry :: topological t-duality T-duality (toroidal) has also been studied in a more mathematical approach. three-torus with H-flux 1 T-duality twisted torus = S 1 E T 2 2 T-dualities non-commutative torus 3 T-dualities non-associative torus Bouwknegt, Evslin, Mathai Mathai, Rosenberg Bouwknegt, Hannabuss, Mathai what is the precise relation to string theory?

34 non-geometry :: summary Summary :: non-geometry arises through a chain of T-duality transformations. H abc T c! fab c T b! Qa bc T a! R abc flux background "twisted torus" T-fold non-associative

35 outline 1. motivation 2. non-geometric spaces a) t-duality b) non-geometry c) questions 3. collective t-duality 4. examples 5. summary

36 open questions Questions :: 1. Why are three T-dualities (three-torus) not allowed? 2. Can other examples be constructed? 3. "What is the origin of non-geometry?"

37 outline 1. motivation 2. non-geometric spaces 3. collective t-duality 4. examples 5. summary

38 collective t-duality :: main idea Main idea :: 1. Identify a global symmetry of the theory. 2. Gauge this symmetry. 3. Integrate-out the gauge field. gauged theory gauging AS =0 original theory dual theory Buscher , 1987, 1988

39 collective t-duality :: main idea Main idea :: 1. Identify a global symmetry of the theory. 2. Gauge this symmetry. 3. Integrate-out the gauge field. gauged theory da =0 gauging AS =0 original theory dual theory Buscher , 1987, 1988

40 collective t-duality :: sigma-model Consider the sigma-model action for the NS-NS sector of the closed string S = h i G ij dx i ^?dx j + 0 R?1 i 2 0 Z 1 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 & 1991

41 collective t-duality :: gauged action Following Buscher s procedure, one first gauges isometries of the action (ignore dilaton) bs = i 2 0 i 2 0 Z h 1 2 G ij(dx i + k i A ) ^?(dx j + k j A ) 1 3! H ijk dx i ^ dx j ^ dx k (v + d ) ^ A k [ v ] + f A ^ A i. Hull, Spence & 1991 Alvarez, Alvarez-Gaume, Barbon, Lozano The local symmetry transformations and constraints take the form ˆ X i = k i, ˆ A = d A f, ˆ = k( v ) f, L k[ v ] = f v, 3 k[ f ] v = k k k H. EP

42 collective t-duality :: original model The original model is recovered via the equations of motion for 0=dA 1 2 f A ^ A. The gauged action can then be rewritten in terms of bs = i 2 0 Z DX i = dx i + k A i as h i G ij DX i ^?DX j + 0 R?1 1 3! H ijk DX i ^ DX j ^ DX k. Ignoring technical details, one replaces DX i! dy i and obtains the ungauged action.

43 collective t-duality :: dual model I The dual model is obtained via the equations of motion for A A = G DG 1 D i? DG 1 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 Z Z 1 Š = hǧ + 0 i 4 0 R?1i 2 0 A and reads where, with M = G DG 1 D Ǧ = G + invertible, T k M 1 M 1 DG 1 +M 1 DG 1 +M 1 k ^?, Ȟ = H d " k T +M 1 DG 1 +M 1 M 1 M 1 DG 1 ^ # k.

44 collective t-duality :: dual model II The coordinates X i and fields parametrize an enlarged target-space. The enlarged metric Ǧ and field strength Ȟ have null-eigenvectors 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 1 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.

45 collective t-duality :: summary The T-duality transformation rules are obtained via Buscher s procedure of 1. gauging isometries in the sigma-model action, 2. integrating-out the gauge field, 3. performing a change of coordinates. The following restrictions arise in this procedure :: from gauge invariance (recall that k H = dv ) L k[ v ] = f v, k[ f ] v = 1 3 k k k H. from solving equations of motion (recall that G = k G i ij k j ) det G 6= 0, det G DG 1 D 6= 0. The change of coordinates is performed using null-eigenvectors.

46 outline 1. motivation 2. non-geometric spaces 3. collective t-duality 4. examples 5. summary

47 outline 1. motivation 2. non-geometric spaces 3. collective t-duality 4. examples a) three-torus b) three-sphere 5. summary

48 outline 1. motivation 2. non-geometric spaces 3. collective t-duality 4. examples a) three-torus b) three-sphere 5. summary

49 torus :: setting Consider a three-torus with H-flux specified as follows ds 2 = R 2 1 dx R 2 2 dx R 2 3 dx 3 2, X i ' X i + `s, H = hdx 1 ^ dx 2 ^ dx 3, h 2 ` 1 s Z. The Killing vectors of interest are abelian and can be chosen as k 1 1, k 2 2, k 3 3. The one-forms v are defined via k H = dv.

50 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )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 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as

51 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )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 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as

52 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )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 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as

53 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )X 3 dx 2, 2 R. T k M Ǧ = G + 1 M 1 DG 1 k +M 1 DG 1 +M 1 ^? The constraints for gauging the sigma-model are trivially satisfied. The geometry of the dual background is determined from the quantities G = R 2 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as = d + v

54 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )X 3 dx 2, 2 R. T k M Ǧ = G + 1 M 1 DG 1 k +M 1 DG 1 +M 1 ^? The constraints for gauging the sigma-model are trivially satisfied. = G + R 2 1 dx 1 T 1 R 2 1 The geometry of the dual background is determined from the quantities R 2 1! R 2 ^? 1 dx 1 G = R 2 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as = d + v

55 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )X 3 dx 2, 2 R. T k M Ǧ = G + 1 M 1 DG 1 k +M 1 DG 1 +M 1 ^? The constraints for gauging the sigma-model are trivially satisfied. = G + R 2 1 dx 1 The geometry of the dual background is determined from the quantities = G R G = R dx 1 ^?dx 1 + 1, = d + v, R1 2 D =0, k = R 2 1 dx 1, The metric and field strength are obtained as T 1 R R 2 1! ^? R 2 ^? 1 dx 1 M = G = R 2 1. = d + v

56 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )X 3 dx 2, 2 R. T k M Ǧ = G + 1 M 1 DG 1 k +M 1 DG 1 +M 1 ^? The constraints for gauging the sigma-model are trivially satisfied. = G + R 2 1 dx 1 The geometry of the dual background is determined from the quantities = G R G = R dx 1 ^?dx , = d + v, R 2 ^? 1 M = G = R1 2. D =0, = 1 k = 1 2 dx 1, R1 2 ^? + R2 2 dx 2 ^?dx 2 + R3 2 dx 3 ^?dx 3 The metric and field strength are obtained as T 1 R R 2 1! R 2 ^? 1 dx 1 = d + v

57 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )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 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as

58 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )X 3 dx 2, 2 R. Ȟ = H d " k T +M 1 DG 1 +M 1 The constraints for gauging the sigma-model are trivially satisfied. M 1 M 1 DG 1 ^ # k The geometry of the dual background is determined from the quantities G = R 2 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as

59 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )X 3 dx 2, 2 R. Ȟ = H d " k T +M 1 DG 1 +M 1 The constraints for gauging the sigma-model are trivially satisfied. M 1 M 1 DG 1 ^ # k = H d " R 2 1 dx 1 T R The geometry of the dual background is determined from the quantities R 2 1! ^ R 2 1 dx 1 # G = R 2 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as

60 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )X 3 dx 2, 2 R. Ȟ = H d " k T +M 1 DG 1 +M 1 The constraints for gauging the sigma-model are trivially satisfied. M 1 M 1 DG 1 ^ # k = H d " R 2 1 dx 1 The geometry of the dual background is determined from the quantities h i G = R= 1 2, H + d dx 1 ^= d + v, D =0, k = R 2 1 dx 1, The metric and field strength are obtained as T R R 2 1! ^ R 2 1 dx 1 # M = G = R 2 1.

61 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )X 3 dx 2, 2 R. Ȟ = H d " k T +M 1 DG 1 +M 1 The constraints for gauging the sigma-model are trivially satisfied. M 1 M 1 DG 1 ^ # k = H d " R 2 1 dx 1 The geometry of the dual background is determined from the quantities h i G = R= 1 2, H + d dx 1 ^= d + v, D =0, =0 k = R1 2 dx 1, The metric and field strength are obtained as T R R 2 1! ^ R 2 1 dx 1 # M = G = R 2 1. d = d(d + v) =hdx 2 ^ dx 3

62 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )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 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as

63 torus :: one t-duality I Consider one T-duality along k 1 1. The corresponding one-form reads v = h X 2 dx 3 h(1 )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 1, = d + v, D =0, k = R 2 1 dx 1, M = G = R 2 1. The metric and field strength are obtained as Ǧ = 1 R 2 1 ^? + R 2 2 dx 2 ^?dx 2 + R 2 3 dx 3 ^?dx 3, Ȟ =0.

64 torus :: one t-duality II As expected, the dual background is a twisted torus (with =1) ďs 2 = 1 R 2 1 d + hx 2 dx R 2 2 dx R 2 3 dx 3 2, Ȟ =0.

65 torus :: two t-dualities I Consider two collective T-dualities along k 1 1 and k 2 2. The constraints on gauging the sigma-model imply (for 2 R ) v 1 = h X 2 dx 3 h(1 )X 3 dx 2, v 2 = h(1 + )X 3 dx 1 + h X 1 dx 3. The geometry of the dual background is determined from R 2 G = R2 2, = 0 +hx 3 D = hx 3, k 0 = d 1 + v 1, d 2 + v 2 R 2 1 dx 1 R2 2. dx2

66 torus :: two t-dualities II The metric of the enlarged target space (in the basis {dx 1,dX 2,dX 3, 1, 2 }) reads Ǧ IJ = 1 0 R1 hx R1 2 hx 3 R2 hx R2 2 hx R R2 2 hx 3 0 R2 2 0 R1 2 hx R1 2 1, C A = R 2 1R 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 C A

67 torus :: two t-dualities III Performing a similar analysis for the field strength and adjusting the notation, one finds ďs 2 = 1 R 2 1 R2 2 + hx 3 2 h R 2 1 d R 2 2 d 2 Ȟ = h R2 1R 2 2 hx 3 2 hr1 2R2 2 + hx 3 i 2 2 d 1 ^ d 2 ^ dx 3. 2 i + R 2 3 dx 3 2, This is the familiar T-fold background.

68 torus :: three t-dualities Finally, consider three collective T-dualities along k 1 1, 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 for H=0 is characterized by ďs 2 = 1 R 2 1 d R 2 2 d R 2 3 d 3 2, Ȟ =0.

69 torus :: three t-dualities Finally, consider three collective T-dualities along k 1 1, 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 ] = for f H=0 v is characterized by k[ f ďs 2 = ] v = 1 3 k k k H R1 2 d 1 + R 2 2 d R 2 3 d 3 2, Ȟ =0.

70 torus :: three t-dualities Finally, consider three collective T-dualities along k 1 1, 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 for H=0 is characterized by ďs 2 = 1 R 2 1 d R 2 2 d R 2 3 d 3 2, Ȟ =0.

71 torus :: summary The formalism for collective T-duality introduced above works as expected. three-torus with H-flux 1 T-duality twisted torus 2 T-dualities T-fold three-torus with H=0 3 T-dualities torus with R 1/R Answer 1 :: three T-dualities not allowed due to gauging constraint.

72 outline 1. motivation 2. non-geometric spaces 3. collective t-duality 4. examples a) three-torus b) three-sphere 5. summary

73 sphere :: setting Consider a three-sphere with H-flux, specified by ds 2 = R 2 h sin 2 (d 1 ) 2 + cos 2 (d 2 ) 2 +(d ) 2 i, 1,2 =0...2, H = h 2 2 sin cos d 1 ^ d 2 ^ d, = This model is conformal if h =4 2 R 2. The isometry algebra is 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,

74 sphere :: setting Consider a three-sphere with H-flux, specified by 0 h 1 0 i 1 ds 2 = R 2 sin 2 (d 1 ) 2 + cos 2 (d 2 ) 2 +(d ) 2, 1,2 =0...2, K 1 = 1 1 A, K1 = 1 +1 A, H = h = sin cos d ^ d 2 ^ d, 2. K 2 = 1 sin( 1 2 ) sin( 1 2 ) tan A, K2 = 1 +sin( ) sin( ) tan A, 2 2 cos( 1 2 ) cos( ) This model is conformal if h =4 2 R 2. K 3 = cos( 1 2 ) cot cos( 1 2 ) tan A, K3 = 2 sin( 1 2 ) + cos( ) cot cos( ) tan +sin( ) 1 A. The isometry algebra is 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,

75 sphere :: setting Consider a three-sphere with H-flux, specified by ds 2 = R 2 h sin 2 (d 1 ) 2 + cos 2 (d 2 ) 2 +(d ) 2 i, 1,2 =0...2, H = h 2 2 sin cos d 1 ^ d 2 ^ d, = This model is conformal if h =4 2 R 2. The isometry algebra is 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,

76 sphere :: one t-duality Consider one T-duality along K1. In this case, all constraints are satisfied: constraints from gauging the sigma-model the matrices and are invertible G M X 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 Bouwknegt, Evslin, Mathai incomplete in Alvarez, Alvarez-Gaume, Barbon, Lozano

77 sphere :: two t-dualities I For two collective T-dualities, consider the commuting Killing vectors K 1 and K1. The constraints for this model are almost satisfied: constraints from gauging the sigma-model the matrices and are invertible G M X X The dual model, via the above formalism, takes a form similar to the T-fold Ǧ = R 2 (d ) (d 1) 2 R 2 sin 2 + h R 2 cos2 + 1 R 2 cos 2 + (d 2)2 h 4 2 R 2 2 cos4 sin 2, Ȟ = 8h 2 h R 4 sin cos 16 2 R 4 sin 2 + h 2 cos 2 2 d ^ d 1 ^ d 2.

78 sphere :: two t-dualities I For two collective T-dualities, consider the commuting Killing vectors K 1 and K1. The constraints for this model are almost satisfied: constraints from gauging the sigma-model the matrices and are invertible G M X X det G = R4 16 sin2 (2 ) The dual model, via the above formalism, takes a form similar to the T-fold Ǧ = R 2 (d ) (d 1) 2 R 2 sin 2 + h R 2 cos2 + 1 R 2 cos 2 + (d 2)2 h 4 2 R 2 2 cos4 sin 2, Ȟ = 8h 2 h R 4 sin cos 16 2 R 4 sin 2 + h 2 cos 2 2 d ^ d 1 ^ d 2.

79 sphere :: two t-dualities I For two collective T-dualities, consider the commuting Killing vectors K 1 and K1. The constraints for this model are almost satisfied: constraints from gauging the sigma-model the matrices and are invertible G M X X The dual model, via the above formalism, takes a form similar to the T-fold Ǧ = R 2 (d ) (d 1) 2 R 2 sin 2 + h R 2 cos2 + 1 R 2 cos 2 + (d 2)2 h 4 2 R 2 2 cos4 sin 2, Ȟ = 8h 2 h R 4 sin cos 16 2 R 4 sin 2 + h 2 cos 2 2 d ^ d 1 ^ d 2.

80 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 ) R 2 h(d 1) 2 + tan 2 (d 2) 2i, H =0. With dual dilaton = log R 2 cos +, this is again a conformal model.

81 sphere :: three t-dualities Consider finally a non-abelian T-duality along, and. K 1 K 2 K 3 constraints from gauging the sigma-model the matrices G and M are invertible satisfied only for H=0 EP - work in progress The dual model for H=0 is obtained as (with 0 and 1,2 =0,...,2 ) Ǧ = 4 R2 d ^?d + R R4 16 i hd 1 ^?d 1 +sin 2 ( 1 ) d 2 ^?d 2, Ȟ = R apple 2 +3 R4 16 sin( 1) d ^ d 1 ^ d 2. Alvarez, Alvarez-Gaume, Barbon, Lozano Curtright, Uematsu, Zachos

82 sphere :: summary In the formalism for T-duality introduced above, for a conformal model one finds: three-sphere with H-flux 1 T-duality S 1 fibered over S 2 2 T-dualities non-compact, geometric three-sphere with H=0 3 T-dualities S 2 fibered over a ray Answer 2 :: T-duality for the (conformal) three-sphere leads to geometric spaces.

83 outline 1. motivation 2. non-geometric spaces 3. collective t-duality 4. examples 5. summary

84 summary I A formalism of collective T-duality has been developed :: highlighting the underlying structure (enlarged target space), and giving rise to new constraints. Collective T-duality has been studied for two examples :: three-torus three-sphere known results reproduced no non-geometric backgrounds

85 summary II For two collective T-duality transformations it was found :: three-torus with H-flux (not conformal) 2 T-dualities non-geometric three-sphere with H-flux (conformal) 2 T-dualities geometric Answer 3 :: non-geometry might be related to non-conformality