q form field and Hodge duality on brane

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1 Lanzhou University (=²ŒÆ) With C.E. Fu, H. Guo and S.L. Zhang [PRD 93 (206) ] 4th International Workshop on Dark Matter, Dark Energy and Matter-Antimatter Asymmetry December 29-3, 206, December 3, 206 NTHU, Hsinchu, Taiwan

2 Outline p-brane, q-form field, localization Hodge duality in bulk Motivation 2 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane 3

3 p-brane, q-form field, localization Hodge duality in bulk Motivation p-brane, q-form field, localization Hodge duality in bulk Motivation

4 p brane p-brane, q-form field, localization Hodge duality in bulk Motivation (in D = (p + 2) dimensions) The line element for a Randall-Sundrum brane is ds 2 = e 2A(y) η µν dx µ dx ν + dy 2 = e 2A(z)( η µν dx µ dx ν + dz 2). ()

5 p-brane, q-form field, localization Hodge duality in bulk Motivation Localization of gravity on brane Recover GR (Newtonian potential) on brane (massless graviton) Correction to Newtonian potential (massive gravitons) Localization of bulk matter fields on brane Realize the Standard model on brane (massless modes) Probe extra dimensions (massive KK particles)

6 p-brane, q-form field, localization Hodge duality in bulk Motivation q form field X M M 2 M q A q form field X M M 2 M q is a totally antisymmetric field X M M 2...M q = X [M M 2...M q]. (X [q] ) (2) Its field strength is expressed as Y M M 2 M q+ = [M X M2 M q+ ]. (Y [q+] = dx [q] ) (3) Examples: 0-form: Scalar field φ -form: Vector field A M, ( q dx M A M ) 2-form: Kalb-Ramond field 2 B MN, ( dx M dx N B MN ) 3 The action for a charged particle moving in an electromagnetic potential 2 The KR field (NS-NS B-field) appears, together with the metric tensor and dilaton, as a set of massless excitations of a closed string. 3 The action for a string coupled to the Kalb-Ramond field. This term in the action implies that the fundamental string of string theory is a source of the NS-NS B-field, much like charged particles are sources of the electromagnetic field.

7 p-brane, q-form field, localization Hodge duality in bulk Motivation Hodge duality in bulk: q form (p q) form in p + 2 dimensions A massless q form field is dual to a (p q) form field: g Ỹ M M p q+ = (q + )! εm M p q+ N N q+ Y N N q+. (4) S bulk,q = S bulk,p q. (5) Example: 0 form 3 form in 5 dimensions (p = 3)

8 p-brane, q-form field, localization Hodge duality in bulk Motivation Localization of q form fields on RS brane (Susskind et al) The bulk action for a massless q form field is S bulk,q = Y M M 2 M q+ Y M M 2(q + )! 2 M q+, (6) M where M d D x g. Susskind et al gave the following ansatz [JHEP 005 (200) 03] 4 X µ µ q (x, y) = ˆX µ µ q (x), (zero mode) (7a) X µ µ q z(x, y) = 0, (gauge condition) (7b) so that Y [q+] = Ŷ [q+]. Thus, the brane action reads S brane = Ŷ µ µ q+ Ŷ µ µ 2(q + )! q+ dye 2(q+ (D )/2)k y. (8) M 4 N. Kaloper, E. Silverstein, and L. Susskind, JHEP 005 (200) 03.

9 p-brane, q-form field, localization Hodge duality in bulk Motivation The localization of a q form field on RS brane requires the convergence of the y integral, or q < D 3 2 = p 2. (9) So the ansatz (7) implies that only 0-forms (scalars) can be localized on the brane in D = 5 dimensions. The above result claims that scalars can be bound to a RS brane, but higher q form fields cannot 5. This conflicts with the Hodge duality between 0-form and 3-form fields in a 5-dimensional bulk. 5 N. Kaloper, E. Silverstein, and L. Susskind, Gauge symmetry and localized gravity in M-theory, JHEP 005 (200) 03, hep-th/

10 p-brane, q-form field, localization Hodge duality in bulk Motivation Resolution (Duff and Liu) The resolution of the paradox was given by Duff and Liu 6 : choose different ansatzs for small and large q { ˆX µ X µ µ q = µ q (x), q < D 3 2 ˆX µ µ q (x)e (2q D+)k y, q D 3 (0a) 2 X µ µ q z = 0. (0b) With this choice, Hodge duality on the brane for the massless modes can be kept. However, the Hodge duality for massive KK modes cannot be obtained. 6 M. Duff and J.T. Liu, Hodge duality on the brane, PLB 508 (200) 38

11 p-brane, q-form field, localization Hodge duality in bulk Motivation Motivation Is there other resolution? For which, the KK decomposition of the q form field is suitable for any q, Hodge duality for the massless modes on the brane is also kept, and more important, Hodge duality for massive KK modes can be obtained. Our resolution: make a general KK decomposition without gauge choice X µ µ q (x µ, z) = n X µ µ q z(x µ, z) = n ˆX µ µ q (x µ ) U (z)ea A(z), (a) ˆX µ µ q (x µ ) U 2 (z)ea 2A(z).(b) where a = a 2 = (2q p)/2.

12 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane Localiztion and Hodge duality for q form field on brane

13 Localiztion for q form field on brane Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane The action for a massless q form field is S = Y M M 2 M q+ Y M M 2(q + )! 2 M q+, (2) M and the EoMs are µ ( g Y µ µ 2...µ q+ ) + z ( g Y zµ 2...µ q+ ) = 0, (3a) µ ( g Y µ µ 2...µ qz ) = 0, (3b)

14 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane Substituting the KK decomposition () into the above equaions, we get ( ) µ ĝ Ŷ µ µ 2...µ q+ ĝ where µ ( ĝ Ŷ µ µ 2...µ q λ = e (a +p 2q)A (q + ) U λ 2 = q e (a +p 2q)A (q + ) U + λ ˆX µ 2...µ q+ + λ 2 Ŷ µ 2...µ q+ ) ) + λ 3 µ ( ĝ ˆX µ µ 2...µ q = 0, (4) = 0, (5) z ( e (p 2q)A z ( U e a A )), (6) ( ) z U 2 e (a 2+p 2q)A, (7) λ 3 = z(u e a A ) q U 2 e a 2A. (8)

15 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane Substituting the general KK decomposition () into the bulk action for the q form field, we have S q = S q,n, (9) n where the effective action for the n level KK modes is S q,n = Ŷ µ µ q+ 2(q + )! Ŷ µ µ q+ M ( Ŷ µ µ q 2q! + m ) 2 n q + ˆX µ µ q. (20) M Here we have assumed that U,2 (z) satisfy the following orthonormality conditions dz U ) U(n = δ nn, (2a) dz U ) (q + )2 2 U(n 2 = q 2 δ nn. (2b)

16 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane Equations of motion for U and U 2 are QQ U (z) = m2 n U (z), (22) Q Q U 2 (z) = m2 n U 2 (z), (23) with the operator Q given by Q = z + p 2q 2 A (z), which indicates that m 2 n 0: There is no KK modes with negative eigenvalue. Solutions of the zero modes U (0),2 (m 0 = 0): U (0) (z) = N e +(p 2q)A/2, (24) U (0) 2 (z) = N 2 e (p 2q)A/2, (25)

17 Massless KK modes Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane Their effective action reads ( S q,0 = I () 00 M Ŷ µ µ q+ (0) Ŷ µ (0) µ q+ + I (2) 00 Ŷ µ µ q (0) Ŷ µ (0) µ q ), (26) where I () q,00 = N 2 I (2) q,00 = N2 2 dz e (p 2q)A, (27) dz e (p 2q)A. (28) It is clear that only one of the zero modes can be localized on the RS brane, q form or (q ) form zero mode.

18 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane For the dual (p q) form field X ν ν 2 ν p q, one has Ĩ () q,00 p q,00 = I (2) q +, Ĩ (2) p q,00 = (p q + ) I () q,00. So, if there is a localized q form zero mode, there must be a localized (p q ) form one for its dual field. If there is a localized (q ) form zero mode, there must be a localized (p q) form one.

19 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane Hodge duality on the brane for massless modes Substituting the KK decompositions of the q and (p q) form fields into the Hodge duality in the bulk, we obtain the Hodge duality on the brane ĝ Ŷ µ µ p q (0) (x µ ) = ĝ Ŷ µ µ p q+ (0) (x µ ) = (q + )! εµ µ p q ν ν q+ Ŷ ν (0) ν q+ (x µ ), ε µ µ p q+ ν ν q Ŷ ν (0) q! ν q (x µ ). The Hodge duality on the brane just suggests that a massless q-form field is dual to a (p q )-form one, or a massless (q )-form field to a (p q)-form one. It can be shown that the corresponding effective actions are the same.

20 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane Massive KK modes and a new duality

21 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane The effective action for each n level KK modes of the bulk q-form field is S q,n = 2(q + )! 2q! M M which is for two kinds of KK modes: Ŷ µ µ q+ Ŷ µ µ q+ ( Ŷ µ µ q + m n q + ˆX µ µ q ) 2, a massive n-level q-form mode with mass m n and a massless n-level (q )-form mode.

22 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane The effective action for each n level KK modes of the (p q)-form field is S p q,n = 2(p q + )! 2(p q)! M M Ŷ µ µ p q+ ( Ŷ µ µ p q m n p q + Ŷ µ µ p q+ ˆX µ µ p q ) 2. The above effective action is also for two kinds of KK modes: a massive n-level (p q)-form mode with mass m n and a massless n-level (p q )-form mode.

23 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane Substituting the field decompositions into the bulk Hodge duality (4), we obtain the following dual relation on the brane between two groups of n level KK modes: µ µ p q Ŷ m n ˆX µ µ p q p q+ = εµ µ p qν ν q+ (q+)! ĝ Ŷ ν ν q+, (30) Ŷ µ µ p q+ = εµ µ p q+ν ν q q! (Ŷ ν ĝ ν q + m n ˆX ν q + ν q ). (3) With these relations it can be shown that the following two effective actions are equal: S q,n = S p q,n. (32)

24 Localiztion for q form field on brane Massless KK modes and Hodge duality on the brane The duality between two group KK modes: one is an n-level massive q-form KK mode with mass m n and an n-level massless (q )-form mode, another is an n-level (p q)-form mode with the same mass m n and an n-level massless (p q )-form mode.

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26 Thanks for your attention!

27 Hodge duality on the brane for massless modes It can be shown that the corresponding effective actions are the same: S q,0 = Ŷ µ µ q+ 2(q + )! (0) Ŷ µ (0) µ q+, M = µ µ p q (0) S p q,0 = Ŷ (0) Ŷ µ 2(p q)! µ p q, or S q,0 = 2q! M = S p q,0 = 2(p q + )! M Ŷ µ µ q (0) Ŷ µ (0) µ q, M Ŷ µ µ p q+ (0) Ŷ (0) µ µ p q+.

28 The effective action for each n level KK modes of the bulk q-form field is S q,n = Ŷ µ µ q+ 2(q + )! Ŷ µ µ q+ M ( Ŷ µ µ q 2q! + m ) 2 n q + ˆX µ µ q, M which is gauge invariant under the following gauge transformation: ˆX µ µ q ˆX µ µ q + [µ ˆΛ µ2 µ q], (33) ˆX µ 2 µ q ˆX µ 2 µ q m n q + ˆΛ µ2 µ q. (34)

29 Then we can fix the gauge by choosing µ ˆX µ µ q = 0. (35) The above effective action is for two kinds of KK modes: a massive n-level q-form mode with mass m n and a massless n-level (q )-form mode.

30 On the other hand, we find that the effective potentials of the q form and its dual (p q) form fields have the following relationships: Ṽ p q, (z) = V q,2 (z), Ṽ p q,2 (z) = V q, (z). (36) And there are some relationships between the KK modes: Ũ = Ũ q q + U 2, (37a) 2 = p q + p q U. (37b)