Relating DFT to N=2 gauged supergravity

Transcription

1 Relating DFT to N=2 gauged supergravity Erik Plauschinn LMU Munich Chengdu

2 based on... This talk is based on :: Relating double field theory to the scalar potential of N=2 gauged supergravity (with R. Blumenhagen and A. Font) [arxiv: ]

3 motivation :: dualities

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

5 motivation :: dualities 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

6 motivation :: non-geometry 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 :: goal Non-geometry :: applications in string phenomenology and cosmology. Main goal here :: understand better non-geometric spaces

8 motivation :: goal Non-geometry :: applications in string phenomenology and cosmology. Main goal here :: understand better non-geometric spaces study T-duality transformations study double field theory

9 outline 1. motivation 2. t-duality 3. double field theory 4. dft on cy3 5. conclusion

10 outline 1. motivation 2. t-duality 3. double field theory 4. dft on cy3 5. conclusion

11 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.

12 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!.

13 t-duality :: curved background I 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

14 t-duality :: curved background II When studying collective T-duality transformations, constraints become apparent :: isometries k (can be non-abelian [k,k ] L = f k ) L k G =0, k H = dv, L k =0, gauge invariance L k[ v ] = f v, k[ f ] v = 1 3 k k k H. EP

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

16 outline 1. motivation 2. t-duality 3. double field theory 4. dft on cy3 5. conclusion

17 dft :: idea Main idea :: make T-duality into a manifest symmetry of an action.

18 dft :: idea Main idea :: make T-duality into a manifest symmetry of an action. O(D,D) Siegel Hull, Zwiebach Hohm, Hull, Zwiebach

19 dft :: idea Main idea :: make T-duality into a manifest symmetry of an action. O(D,D) 2D dimensions Siegel Hull, Zwiebach Hohm, Hull, Zwiebach

20 dft :: basics DFT is defined on a space with a doubled number of coordinates ( i =1,...,D ) X I =( x i,x i ) x i x i ordinary coordinates, winding coordinates. The metric and B-field are arranged into a generalized metric H IJ = G ij G ik B kj B ik G kj G ij B ik G kl B lj!. The dilaton is combined with the metric G into e 2 d = p det Ge 2.

21 dft :: vielbeins The generalized metric can be expressed using generalized vielbeins E A I as follows H IJ = E A I S AB E B J E A I = ea i 0 e a i e k a B ki S AB = s ab 0 0 s ab s ab = e a i G ij e b j For later reference, the O(D,D) invariant form used here reads IJ = 0 i j i j 0.

22 dft :: symmetries An action for DFT is obtained by invoking symmetries :: 1. invariance under local diffeomorphisms ( x i,x i )! ( x i + i (X),x i + i (X)), 2. invariance under global O(D,D) transformations. Closure of the diffeomorphism algebra requires the strong i i B i i B =0. Different actions exist equivalent modulo total derivatives and the strong constraint.

23 dft :: action ns-ns In the flux formulation of DFT, the action for the NS-NS sector reads S NS NS = 1 2apple 2 10 Z d D Xe 2d apple 1 F ABC F A0 B 0 C 0 4 SAA0 BB0 1 CC SAA S BB0 S CC0 + F A F A 0 AA0 S AA AA0 BB0 CC0 The field strengths F are defined using the generalized Weitzenböck connection F A = B BA +2E A I d, F ABC =3 [ABC], ABC = E A I (@ I E B J )E CJ. Aldazabal, Baron, Marques, Nunez Geissbühler Grana, Marques

24 dft :: fluxes Expectation values for F ABC are commonly denoted as F abc = H abc F c c ab = F ab F bc bc a = Q a F abc = R abc geometric geometric non-geometric non-geometric H-flux F-flux Q-flux R-flux

25 dft :: action r-r In the R-R sector, the action is expressed through the field strengths G = X n 1 n! G(n) i 1...i n e a1 i 1...e an i n a 1...a n 0i. Gauge potentials C can be defined as G = /r C, /r = A D A 1 2 A F a 1 6 ABC F ABC. The action (type IIB, relevant for the following) takes the form S RR = 1 2apple 2 10 Z d D X apple SAA S BB0 S CC0 G ABC G A 0 B 0 C 0. Rocen, West Hohm, (Kwak,) Zwiebach Geissbühler Jeon, Lee, Park

26 dft :: summary Summary :: DFT is defined on a doubled space and is O(D,D) invariant. The strong constraint has to be imposed. Actions for the NS-NS and R-R sector have been given. Geometric & non-geometric fluxes can be incorporated.

27 outline 1. motivation 2. t-duality 3. double field theory 4. dft on cy3 5. conclusion

28 outline 1. motivation 2. t-duality 3. double field theory 4. dft on cy3 a) ansatz b) re-writing c) reduction 5. conclusion

29 dft on cy3 :: motivation Known result :: Compactification of DFT on a T 6 gives d =4 N =4 gauged sugra. with fluxes Aldazabal, Baron, Marques, Nunez Geissbühler

30 dft on cy3 :: motivation Known result :: Compactification of DFT on a T 6 gives d =4 N =4 gauged sugra. with fluxes Aldazabal, Baron, Marques, Nunez Geissbühler Question :: What happens for a Calabi-Yau three-fold with fluxes? Blumenhagen, Font, EP

31 dft on cy3 :: calabi-yau Properties of Calabi-Yau three-folds :: The holomorphic three-form is closed, d =0. The Kähler form J is closed, dj =0. The first and fifth co-homology are trivial (no one- or five-forms).

32 dft on cy3 :: fluxes Fluxes F A and F ABC are treated as perturbations around the Calabi-Yau geometry. For the vielbein, this is realized via the expansion E A I = E A I + E A I + O E 2, E A I 1. Calabi-Yau fluxes For the field strengths, the expansion results in F ABC = F ABC + F ABC + O E 2, F A = F A + F A + O E 2, F Calabi-Yau (no potential).

33 dft on cy3 :: technical details Technical details :: 1. Interested in scalar potential in from type IIB DFT. d =4 2. Only the Calabi-Yau part is considered (relevant for potential). 3. Flux-contribution only up to O(E 2 ). 4. Impose strong constraint for background! G ij ( x, x) =g ij (x), Set F A =0. 6. Consider constant fluxes F IJK. 7. Use Bianchi identities (see below) for F ABC F ABC! 0.

34 dft on cy3 :: relevant terms The NS-NS terms relevant for the scalar potential are then the following 1 L NS NS = e 2 F IJK F I 0 J 0 K 0 4 HII0 JJ0 1 KK HII H JJ0 H KK usual dilaton constant fluxes usual metric and B-field no contribution to potential This expression can be expanded

35 dft on cy3 :: relevant terms L NS NS = e 2 The NS-NS terms relevant for the scalar potential are then the following 12 H ijk H i0 j 0 k 0 g ii0 g jj0 g kk0 +3F i jk F i0 j 0 k 0 g ii 0 g jj0 g kk0 +3Q i jk Q i 0 j0 k 0 g ii0 g jj 0 g kk 0 + R ijk R i0 j 0 k 0 g ii 0 g jj 0 g kk 0 +6F m ni F n mi 0 g ii0 +6Q m ni Q n mi 0 g ii 0 L NS NS = e 2 F IJK F I 0 J 0 K HII0 JJ0 KK0 1 6 H mni Q i 0 mn g ii0 6 F i mn R mni0 g ii 0 12 HII 0 H JJ0 H KK H ijk = H ijk +3F m [ij B mk] +3Q [i mn B mj B nk] + R mnp B m[i B nj B pk] F i jk = F i jk +2Q [j mi B mk] + R mni B m[j B nk] Q k ij = Q k ij + R mij B mk R ijk = R ijk usual dilaton constant fluxes usual metric and B-field no contribution to potential L RR = 1 12 G ijk G i0 j 0 k 0 g ii 0 g jj0 g kk0 This expression can be expanded G ijk = F (3) ijk H ijk C (0) 3F m [ij C(2) m k] Q [i mn C (4) jk]mn Rmnp C (6) mnp ijk Blumenhagen, Gao, Herschmann, Shukla

36 dft on cy3 :: relevant terms The NS-NS terms relevant for the scalar potential are then the following 1 L NS NS = e 2 F IJK F I 0 J 0 K 0 4 HII0 JJ0 1 KK HII H JJ0 H KK usual dilaton constant fluxes usual metric and B-field no contribution to potential This expression can be expanded

37 dft on cy3 :: main difficulty Main difficulty :: how to perform the dimensional reduction of terms L NS NS =...F i jk F i0 j 0 k 0 g ii 0g jj0 g kk if the metric of a CY3 is in general not known?

38 outline 1. motivation 2. t-duality 3. double field theory 4. dft on cy3 a) ansatz b) re-writing c) reduction 5. conclusion

39 dft on cy3 :: twisted differential Fluxes can be treated as operators acting on p-forms H ^ : p-form! (p + 3)-form F : p-form! (p + 1)-form Q : p-form! (p 1)-form R x : p-form! (p 3)-form H ^ = 1 3! H ijk dx i ^ dx j ^ dx k, F = 1 2! F k ij dx i ^ dx j ^ k, Q = 1 2! Q i jk dx i ^ j ^ k, R x = 1 3! Rijk i ^ j ^ k. These operators can be combined into a twisted differential (acting on multi-forms) D = d H ^ F Q R x. Requiring D to be nil-potent ( D 2 =0) gives the Bianchi identities for the fluxes. Aldazabal, Camara, Font, Ibanez Villadoro, Zwirner Shelton, Taylor, Wecht

40 dft on cy3 :: twisted differential Fluxes can be treated as operators acting on p-forms H ^ : p-form 0=H! (p m[ab + F3)-form m cd], F : p-form 0=F! (p m [bc + F1)-form d a]m + H m[ab Q md c], H ^ = 1 3! H ijk dx i ^ dx j ^ dx k, F = 1 2! F k ij dx i ^ dx j ^ k, Q : p-form 0=F! (p m [cd] [ab] 1)-form Q m 4 F [c m[a Q d]m b] Q + H= mab 1 2! Q R mcd i jk, dx i ^ j ^ k, 0=Q [bc m Q a]m d + R m[ab F c] md, R x : p-form! (p 3)-form R x = 1 3! Rijk i ^ j ^ k. 0=R m[ab Q cd] m, 0=R mn[a F b] mn, 0=R amn H bmn F a mnq b mn, These operators can be combined into a twisted differential (acting on multi-forms) 0=Q [a mn H b]mn, D = d H ^ F Q R x. 0=R mnl H mnl. Shelton, Taylor, Wecht Robbins, Wrase Requiring D to be nil-potent ( D 2 =0) gives the Bianchi identities for the fluxes. Aldazabal, Camara, Font, Ibanez Villadoro, Zwirner Shelton, Taylor, Wecht

41 dft on cy3 :: twisted differential Fluxes can be treated as operators acting on p-forms H ^ : p-form! (p + 3)-form F : p-form! (p + 1)-form Q : p-form! (p 1)-form R x : p-form! (p 3)-form H ^ = 1 3! H ijk dx i ^ dx j ^ dx k, F = 1 2! F k ij dx i ^ dx j ^ k, Q = 1 2! Q i jk dx i ^ j ^ k, R x = 1 3! Rijk i ^ j ^ k. These operators can be combined into a twisted differential (acting on multi-forms) D = d H ^ F Q R x. Requiring D to be nil-potent ( D 2 =0) gives the Bianchi identities for the fluxes. Aldazabal, Camara, Font, Ibanez Villadoro, Zwirner Shelton, Taylor, Wecht

42 dft on cy3 :: re-writing I For B=0 and only H-flux present, the NS-NS Lagrangian becomes L NS NS = e 2 12 H ijk H i0 j 0 k 0 g ii 0 g jj0 g kk0. Using the Hodge star operator, this can be expressed as?l NS NS = e 2 2 H ^?H. That expression can be evaluated using special geometry.

43 dft on cy3 :: re-writing II For B=0 and only F-flux present, the NS-NS Lagrangian becomes L NS NS = e 2 4 F i jk F i0 j 0 k 0 g ii 0g jj0 g kk0 +2F m ni F n mi 0 g ii0. Using Bianchi identities (in a complex basis), this can be rewritten as?l NS NS = e 2 apple 1 2 (F J) ^?(F J) (F ) ^?(F ) ( ^ F J) ^?( ^ F J). That expression depends on J and, and can be evaluated using special geometry.

44 dft on cy3 :: re-writing III For the cases B=0 and only Q-flux B=0 and only R-flux similar expressions are obtained.

45 dft on cy3 :: general result I For B=0 and all fluxes present (subject to BIs), the NS-NS Lagrangian can be written?l NS NS = e 2 apple 1 2 ^? ^? 1 4 ^ ^? ^ 1 4 ^ ^? ^. The appearing multi-forms are defined as = D e ij, = D.

46 dft on cy3 :: b-field To include a non-vanishing B-field, replace in all computations H ijk! H ijk = H ijk +3F m [ij B mk] +3Q mn [i B mj B nk] + R mnp B m[i B nj B pk], F i jk! F i jk = F i jk +2Q mi [j B mk] + R mni B m[j B nk], ij Q k! Q ij k = Q ij k + R mij B mk, R ijk! R ijk = R ijk.

47 dft on cy3 :: b-field L NS NS = e 2 H ijk H i0 j 12 0 k 0 g ii0 g jj0 g kk0 +3F i jk F i0 j 0 k 0 g ii 0 g jj0 g kk0 H ijk! H ijk +3Q = H jk ijk +3F m [ij B mk] +3Q mn [i B mj B nk] + R mnp i Q i 0 j0 k 0 g ii0 g jj 0 g kk 0 + R ijk R i0 j 0 k 0 g ii 0 gb jjm[i 0 g kk B0 nj B pk], To include a non-vanishing B-field, replace in all computations F i jk! F i jk +6F = F i jk m ni +2Q F n mi mi [j 0 gb ii0 mk] + R mni +6Q B m[j B ni mi m nk] Q, 0 n g ii 0 ij Q k! Q ij k = 6Q H ij k mni + QR mij mn i 0 Bg ii0 mk, 6 F i mn R mni0 g ii 0 R ijk! R ijk = R ijk. H ijk = H ijk +3F m [ij B mk] +3Q mn [i B mj B nk] + R mnp B m[i B nj B pk] F i jk = F i jk +2Q mi [j B mk] + R mni B m[j B nk] Q ij k = Q ij k + R mij B mk R ijk = R ijk L RR = 1 12 G ijk G i0 j 0 k 0 g ii 0 g jj0 g kk0 G ijk = F (3) ijk H ijk C (0) 3F m [ij C(2) m k] Q [i mn C (4) jk]mn Rmnp C (6) mnp ijk Blumenhagen, Gao, Herschmann, Shukla

48 dft on cy3 :: b-field To include a non-vanishing B-field, replace in all computations H ijk! H ijk = H ijk +3F m [ij B mk] +3Q mn [i B mj B nk] + R mnp B m[i B nj B pk], F i jk! F i jk = F i jk +2Q mi [j B mk] + R mni B m[j B nk], ij Q k! Q ij k = Q ij k + R mij B mk, R ijk! R ijk = R ijk.

49 dft on cy3 :: b-field To include a non-vanishing B-field, replace in all computations H ijk! H ijk = H ijk +3F m [ij B mk] +3Q mn [i B mj B nk] + R mnp B m[i B nj B pk], F i jk! F i jk = F i jk +2Q mi [j B mk] + R mni B m[j B nk], ij Q k! Q ij k = Q ij k + R mij B mk, R ijk! R ijk = R ijk. For the twisted differential, this implies D! D = d H ^ F Q R x = e B D e B 1 2 Q i mn B mn dx i + R imn B mn i. For the multi-forms this means! = e B D e B+iJ,! = e B D e B.

50 dft on cy3 :: general result II For the R-R sector, introduce the multi-form potential C = C (0) + C (2) + C (4) + C (6) + C (8) + C (10). The three-form flux can be written as G = F (3) H ^ C (0) F C (2) Q C (4) R xc (6) = F (3) + D C = F (3) + e B D e B C. The action in the R-R sector then becomes?l RR = 1 2 G ^?G.

51 dft on cy3 :: summary re-writing Summary :: The DFT action restricted to a CY3 has been re-written?l NS NS = e 2 apple 1 2 ^? ^? 1 4 ^ ^? ^ 1 4 ^ ^? ^?L RR = 1 2 G ^?G and depends only on known data = e B D e B+iJ, = e B D e B. Next :: Perform the dimensional reduction

52 outline 1. motivation 2. t-duality 3. double field theory 4. dft on cy3 a) ansatz b) re-writing c) reduction 5. conclusion

53 dft on cy3 :: reduction I The (relevant part of the) Lagrangian in ten dimensions depends only NS-NS fluxes through the differential D, R-R three-form flux F (3), the B-field and R-R gauge potentials B, C (2n), the Kähler form J, the holomorphic three-form. The dimensional reduction to d=4 can be performed using special geometry.

54 dft on cy3 :: reduction II The computation is standard only an outline and the final result will be given.

55 dft on cy3 :: co-homology I Odd co-homology of a Calabi-Yau three-fold :: basis {, } 2 H 3 (X ), =0,...,h 2,1, intersections Z X ^ =, expansion of = X F, period matrix N = F +2i Im(F )X Im(F )X X Im(F )X, "metric" M 1 = 1 Re N 0 1 Im N Im N 1. Re N 1

56 dft on cy3 :: co-homology II Even cohomology of a Calabi-Yau three-fold :: basis {! A } 2 H 1,1 (X ), { A } 2 H 2,2 (X ), A =1,...,h 1,1, {! A } = p g V dx6,! A, { A } = 1, A, A =0,...,h 1,1, intersections Z X! A ^ B = A B, apple ABC = Z X! A ^! B ^! C, expansion of J J = B + ij = J A! A, "metric" M 2 = Z X! h! A,? B! B i h! A,? B B i. h A,? B! B i h A,? B B i

57 dft on cy3 :: fluxes Fluxes are defined via the twisted differential D = q A! A + f A A, D = q A! A + f A A, D! A = f A + f A, D A = q A q A. The H- and R-flux are contained in f 0 = r, f 0 = r, q 0 = h, q 0 = h. The fluxes can be combined into a (2h 2,1 + 2) (2h 1,1 + 2) O = f A q A. f A q A matrix Grana, Louis, Waldram

58 dft on cy3 :: definitions Further definitions :: Kähler moduli e J =(! A A ) V 1, complex-structure moduli = V 2, R-R three-form flux F (3) = F, R-R potentials e B C =(! A A ) C, Õ = C O C T, matrices C =

59 dft on cy3 :: result Performing the dimensional reduction gives the following scalar potential V = 1 2 FT + C T O T M 1 F + O C + e 2 2 V 1 T O T M 1 O V 1 + e 2 2 V 2 T Õ M 2 ÕT V 2 e 2 4V V 2 T C O V 1 V T 1 + V 1 V1 T O T C T V 2. This is the scalar potential of N=2 gauged supergravity in d=4. D Auria, Ferrara, Trigiante

60 dft on cy3 :: summary Summary :: Compactifications of DFT on a CY3 have been studied. Expressions (explicitly involving the metric) of the form L NS NS =...F i jk F i0 j 0 k 0 g ii 0g jj0 g kk have been re-written in terms of J, and?. Dim. reduction gives the potential of N=2 gauged sugra.

61 outline 1. motivation 2. t-duality 3. double field theory 4. dft on cy3 5. conclusion

62 summary T-duality :: Different approaches to T-duality have been reviewed. DFT :: Double field theory is a T-duality invariant formulation. Reduction on a CY3 gives N=2 gauged supergravity. Applications for moduli stabilization, inflation,