Dynamics of branes in DFT

Transcription

1 Dynamics of branes in DFT Edvard Musaev Moscow Inst of Physics and Technology based on works with Eric Bergshoeff, Chris Blair, Axel Kleinschmidt, Fabio Riccioni Dualities Corfu, 2018

2 Web of (some) branes Extended objects Web of branes M: P M2 M5 KK6 IIA: D0 D2 D4 D6 IIB: D1 D3 D5 D7 NS: P 0 (1,6) F normal exotic NS KK IIB: IIA: M: 0 (1,7) S-duality; T-duality; Reduction; Edvard Musaev (Phystech) Dynamics of branes 1 / 21

3 The end Thank you Summary T-duality orbits NS5(5 0 2) KK5(5 1 2) Q(5 2 2) R(5 3 2) R (5 4 2) D0 D1 D2 D3 D4 D5 D6 D7 D8 D9 Effective actions, which depending on orientation project down to actions for normal branes Probably some non-geometric effects for D-branes Wess-Zumino terms for α = 3, 4 branes (linear) Edvard Musaev (Phystech) Dynamics of branes 2 / 21

4 Dualities and fluxes Strange objects T-duality orbit of NS branes NS5 : world-volume transverse T 9 KK5 : world-volume transverse special T 8 (1) Q : R : R : [deboer, Shigemori] Edvard Musaev (Phystech) Dynamics of branes 3 / 21

5 Dualities and fluxes Strange objects T-duality orbit of D-branes D0 : D1 : D2 : D3 : D4 : D5 : D6 : D7 : D8 : D9 : (2)! T-duality changes dimension of a D-brane Edvard Musaev (Phystech) Dynamics of branes 4 / 21

6 Dualities and fluxes Potentials Potentials Dp-brane: (p+1)-form C (p+1) Edvard Musaev (Phystech) Dynamics of branes 5 / 21

7 Dualities and fluxes Potentials Potentials Dp-brane: (p+1)-form C (p+1) NS5-brane magnetic dual of the string, B μ1 μ 6 B μν H μνρ = 3 [μ B νρ] 10 H (3) (3) B μ1 μ 6 H μ1 μ = 7 [μ1 B μ2 μ ] 7 7 H (7) KK5-monopole magnetic dual of the graviton, A μ1 μ 7 ν A m z = e m z f mn z = 2e z z [m e n] z 10 f (2) z A m1 m 6 z z f m1 m 7 z z = 8 [m1 A m2 m 7 z] z f (8) z (4) Edvard Musaev (Phystech) Dynamics of branes 5 / 21

8 Dualities and fluxes Potentials Exotic potentials brane, Q a bc -flux, interacts with B μ1 μ 8 ν 1 ν 2 β 34 Q a 34 = a β Q (1) 34 B α 1 α Q α 1 α = 9 [α1 B α 2 α 7 34] 34 Q (9) 34 (5) brane, R abc -flux, interacts with B μ1 μ 9 ν 1 ν 2 ν 3 β 34 R 234 = 2 β R (0) B α 1 α R α 1 α = 10 [α1 B α 2 α 7 234] R (10) (6) In general 5 p 2-brane interacts with B (6+p,p) -potential Bergshoeff, Kleinschmidt, Ricconi Edvard Musaev (Phystech) Dynamics of branes 6 / 21

9 Dualities and fluxes Potentials The problem Construct a T-covariant action reproducing the background of the DFT monopole when coupled to DFT Write an invariant Wess-Zumino term Edvard Musaev (Phystech) Dynamics of branes 7 / 21

10 Dualities and fluxes Red pill and blue pill DFT monopole A solution of DFT defined by the harmonic function H(z, y 1, y 2, y 3 ) = 1 + and identification between the doubled coordinates X M = (x z, x i, x a, x z, x i, x a ) and parameters (z, y i, z, ỹ i ) h z 2 + δ ij y i y j (7) (x z, x 1, x 2, x 3 ) = (z, y 1, y 2, y 3 ), NS5-brane, (x z, x 1, x 2, x 3 ) = ( z, y 1, y 2, y 3 ), KK5-brane, (x z, x 1, x 2, x 3 ) = ( z, ỹ 1, y 2, y 3 ), Q-brane, (8) (x z, x 1, x 2, x 3 ) = ( z, ỹ 1, ỹ 2, y 3 ), R-brane, (x z, x 1, x 2, x 3 ) = ( z, ỹ 1, ỹ 2, ỹ 3 ), R -brane, Non-geometric backgrounds depend on dual coordinates The same for xft (recall Ray s talk) [Berman, Rudolph, Bakhmatov, Kleinschmidt, EtM, Otsuki] Edvard Musaev (Phystech) Dynamics of branes 8 / 21

11 Branes a=-2 Embedding of the brane x 0 x 1 x 2 x 3 x 4 x 5 y 1 y 2 y 3 y 4 ỹ 1 ỹ 2 ỹ 3 ỹ 4 (9) world-volume transverse directions Coordinates of the space-time = scalar fields on the brane no issues with doubled geometry, section condition etc doubled number of scalar fields on the brane projection condition which leaves only half of them (to keep SUSY) Edvard Musaev (Phystech) Dynamics of branes 9 / 21

12 Branes a=-2 The action S DBI = g μν α X μ β X ν + H MN ˆDα Y M ˆD β Y N d 6 ξe h 2d ab 1 + e 2d h ab 1/2 ( λ brane C) 2 e d (det h) 1/4 λ brane G αβ 1 + e 2d (det h) 1/2 ( λ brane C) 2, where (10) G αβ = 2 [α c β] + C αβ (11) is a worldvolume field strength with the following pullback of RR fields: C αβ = (C μν (B μν + 12 A μ M A N ν Γ MN )C + ) 2A M μ Γ M C ν [α X μ β] X ν + ( 2Γ M C μ 1 ) A N μ Γ N C [α X μ ˆD β] Y M Γ MNC ˆD [α Y M ˆD β] Y N (12) Edvard Musaev (Phystech) Dynamics of branes 10 / 21

13 Branes a=-2 The action Keeping only NS-NS fields X M = (x m, x m ): S (NS) DBI = d 6 ξe det(h 2d ab ) det ( g μν α X μ β X ν + Π MN D α X M D β X N), Kaluza-Klein-like projections D α X M = α X M + α X μ A μ M (13) Π MN = H MN h ab H MP H NQ k P a k Q b, h ab = H MN k M a k M b (14) The section constraint (orientation): η MN k M a k N b = 0, (15) Allowed number of non-zero Killing vectors is half of the total Edvard Musaev (Phystech) Dynamics of branes 11 / 21

14 Branes a=-2 Invariant Wess-Zumino term The exotic potentials B (6+p,p) are all combined into D μ1 μ 6,MNKL, X M = (x a, x a ), a 1, 2, 3, 4 (16) The invariant charge T MNKL = k a M k b N k c K k d L ε abcd (17) The invariant Wess-Zumino term S WZ = d 6 ξt MNKL D (6)MNKL (18) Edvard Musaev (Phystech) Dynamics of branes 12 / 21

15 Branes a=-2 T-duality frames η MN k M a k N b = 0, & T MNKL 0 (19) Killing vectors k M a = (k m a ; k am ) satisfying the constraint define the embedding world-volume transverse directions x 0 x 1 x 2 x 3 x 4 x 5 y 1 y 2 y 3 y 4 ỹ 1 ỹ 2 ỹ 3 ỹ 4 NS5 k k k k KK5 k k k k Q k k k k R k k k k R k k k k (20) localization direction k Killing direction Edvard Musaev (Phystech) Dynamics of branes 13 / 21

16 Branes a=-2 Invariant Wess-Zumino term All couplings to the exotic potential appear from the single term D μ1 μ = B μ1 μ 6, D μ1 μ 6,1 234 = A μ1 μ 6 1 1, D μ1 μ 6,12 34 = B μ1 μ , D μ1 μ 6,123 4 = B μ1 μ , (21) D μ1 μ 6,1234 = B μ1 μ , To gauge covariantize the world-volume WZ expression one adds w-v gauge fields Bergshoeff, Riccioni Edvard Musaev (Phystech) Dynamics of branes 14 / 21

17 Branes a=-1 D-brane covariant potentials D-brane potentials C (p+1) can be combined: χ = 10 p=0 C m1 m p Γ m 1 m p 0 with {Γ M, Γ N } = 2η MN ; vacuum: Γ m 0 = 0 (22) Invariant interaction S wz = d 10 ξ Q Γ M1 M 10 χ dx M 1 dx M 10 (23) The vectors k M α = (k m α, k αm ) and the charge must satisfy η MN k α M k β N = 0, Γ M k M α Q p+1 = 0 (24) Edvard Musaev (Phystech) Dynamics of branes 15 / 21

18 Branes a=-1 Dynamics DBI action written in the same variables ( S D = d 10 ξe d det h αβ 1 4 det H MN ˆ a X M ˆ ) b X N +, (25) where one needs the projected derivatives ˆ a X M = a X M (h 1 ) αβ k α M k β N H NK a X K, h αβ = k M α k N β H MN (26) η MN k M a k N b = 0 Other approaches: Asakawa, Sasa, Watamura, Albertsson, Dai, Kao, Lin Edvard Musaev (Phystech) Dynamics of branes 16 / 21

19 Branes a=-1 Embedding of branes D0 k k k k k k k k k k D1 k k k k k k k k k k D2 k k k k k k k k k k D3 k k k k k k k k k k D4 k k k k k k k k k k D5 k k k k k k k k k k D6 k k k k k k k k k k D7 k k k k k k k k k k D8 k k k k k k k k k k D9 k k k k k k k k k k! Depending on the choice of k s one gets different D-branes!? D-branes can localize in dual space Edvard Musaev (Phystech) Dynamics of branes 17 / 21

20 Branes a=-3 Higher exotic branes: T g s 3 D7 0 (0,7) 3 1 (0,6) 3 2 (0,5) 3 3 (0,4) 3 4 (0,3) 3 5 (0,2) 3 6 (0,1) 3 7 (0,0) 3 0 (1,7) 3 1 (1,6) 3 2 (1,5) 3 3 (1,4) 3 4 (1,3) 3 5 (1,2) 3 6 (1,1) 3 7 (1,0) 3 0 (2,7) 3 1 (2,6) 3 2 (2,5) 3 3 (2,4) 3 4 (2,3) 3 5 (2,2) 3 6 (2,1) 3 7 (2,0) 3 These interact with exotic potentials of the type E (p,p+q,p+q+r) E m1 m p }{{},n 1 n q }{{},k 1 k r }{{} eg 4 (1,3) 3 E m1 m 5 uxyz,uxyz,u (27) Edvard Musaev (Phystech) Dynamics of branes 18 / 21

21 Branes a=-3 Higher exotic branes: T g s 3 0 (0,7) 3 1 (0,6) 3 2 (0,5) 3 3 (0,4) 3 4 (0,3) 3 5 (0,2) 3 6 (0,1) 3 7 (0,0) 3 D7 0 (1,7) 3 1 (1,6) 3 2 (1,5) 3 3 (1,4) 3 4 (1,3) 3 5 (1,2) 3 6 (1,1) 3 7 (1,0) 3 0 (2,7) 3 1 (2,6) 3 2 (2,5) 3 3 (2,4) 3 4 (2,3) 3 5 (2,2) 3 6 (2,1) 3 7 (2,0) 3 T-duality covariant potential is E MN E m 1m 2 = ε m 1 m 10 E m3 m 10,n 1 n p Γ n 1n p 0 p E m 1 q = ε m 1m 10 E m2 m 10,n 1 n p,qγ n 1n p 0 p E q1 q 2 = ε m 1m 10 E m1 m 10,n 1 n p,q 1 q 2 Γ n 1n p 0 p Edvard Musaev (Phystech) Dynamics of branes 18 / 21

22 Branes a=-3 Higher exotic branes: T g s 3 D7 0 (0,7) 3 1 (0,6) 3 2 (0,5) 3 3 (0,4) 3 4 (0,3) 3 5 (0,2) 3 6 (0,1) 3 7 (0,0) 3 0 (1,7) 3 1 (1,6) 3 2 (1,5) 3 3 (1,4) 3 4 (1,3) 3 5 (1,2) 3 6 (1,1) 3 7 (1,0) 3 0 (2,7) 3 1 (2,6) 3 2 (2,5) 3 3 (2,4) 3 4 (2,3) 3 5 (2,2) 3 6 (2,1) 3 7 (2,0) 3 The covariant Wess-Zumino term d 8 ξ Q M1 M 10 E M 1M 2 ˆdX M 3 ˆdX M 10 (27) Edvard Musaev (Phystech) Dynamics of branes 18 / 21

23 Branes a=-3 Higher exotic branes: T g s 3 0 (0,7) 3 1 (0,6) 3 2 (0,5) 3 3 (0,4) 3 4 (0,3) 3 5 (0,2) 3 6 (0,1) 3 7 (0,0) 3 D7 0 (1,7) 3 1 (1,6) 3 2 (1,5) 3 3 (1,4) 3 4 (1,3) 3 5 (1,2) 3 6 (1,1) 3 7 (1,0) 3 0 (2,7) 3 1 (2,6) 3 2 (2,5) 3 3 (2,4) 3 4 (2,3) 3 5 (2,2) 3 6 (2,1) 3 7 (2,0) 3 More interesting is the structure E m 1m 2 = ε m 1 m 10 E m3 m 10,n 1 n p Γ n 1n p 0 p E m 1 q = ε m 1m 10 E m2 m 10,n 1 n p,qγ n 1n p 0 p E q1 q 2 = ε m 1m 10 E m1 m 10,n 1 n p,q 1 q 2 Γ n 1n p 0 p Edvard Musaev (Phystech) Dynamics of branes 18 / 21

24 The end The message One is able to construct a single action for several branes, related by T-duality For D-branes this suggests localization in dual space Structure of T-dualities between branes repeats when going down in powers of g s Edvard Musaev (Phystech) Dynamics of branes 19 / 21

25 The end What s now? Discussion Prove microscopically, that D-branes (do not) localize in the dual space, calculate instanton corrections Field theories on worldvolume (especially for D-branes) Generalize stuff for exceptional field theories and U-dualities What s the use of all that? Tadpole cancellation conditions for flux compactifications (Bianchi identities), support for internal space String behavior on such backgrounds: non-commutativity and non-associativity Little string theories from NS five-branes Speculations around AdS/CFT correspondence Edvard Musaev (Phystech) Dynamics of branes 20 / 21

26 Thank you for your attention!