6d theories and nilpotent orbits in Lie algebras

Transcription

1 6d theories and nilpotent orbits in Lie algebras Alessandro Tomasiello ICTP, (mainly) based on with M. del Zotto, J.Hecman, C.Vafa; with J. Hecman, T. Rudelius; with N. Meareeya, T. Rudelius

2 Summary We will consider M5s at ADE singularities with M2s, this strategy led to [Aharony, Bergman, Jafferis, Maldacena 08] Described by 6d N =(, 0) theories They can be Higgsed to many other theories A case: IIA realization, AdS7 duals D, E cases: F-theory two punctures instead of the 3 for Class S in 4d in general, nilpotent elements!

3 Plan A case: IIA realization Conformal matter theories T-brane theories

4 Z N R R 4 /Z N =(, 0) () () a 2 N 3 Separate the M5s: M5 M5 R R 4 /Z

5 Analysis is more convenient if we reduce to IIA NS5 NS5 () () ( +, Aµ ) (q, ) (, tensor mult. + hypermult., b+ µ ) NS5 transverse motions dimhiggs = N L ( i+ i) F 2 i =x coincident NS5s = strong coupling point; CFT?

6 alternative realization: each D6 ends on a single D8 D6 s this suggests Higgs RG flows: [Gaiotto, Witten 08; Gaiotto, AT 4] N D8 stacs These D8 s can be thought of as Nahm poles for the D6 s. BPS equations on D6: Nahm equations z X =[X 2,X 3 ] X i t i z [t i,t j ]= ij t (2) () t + it 2 = 3 = partition [Young diagram]

7 Here is how to reconstruct the theory. s Start with partitions: R=6 L=5 Y...ransri r integrate : gauge groups. Y ri+ fi+ fi s ds2 3/ dz )/2 2 ds S2 2 2 ( [Apruzzi,Fazzi,Rosa,AT 3; Gaiotto, AT 4; Apruzzi,Fazzi,Passias,Rota 5;Cremonesi,AT 5] 0 0 R=6 L=5 0 Gravity dual: ds2 =

8 We expect a hierarchy of RG flows corresponding to the hierarchy of Y.d. Higgs moduli space dimension unhiggsed theory dim Higgs = N + 2 dim OL dim OR dimensions nilpotent orbits associated to partitions

9 II. From IIA to Conformal Matter SU(N) simplest example: M-theory R R 4 /Z IIA lift T-duality IIB = D7 s

10 F-theory allows to include more general gauge groups IIA lift M-theory G G G G G R R 4 / G directly??? T-duality nonperturbative D7 s

11 what is this lin? no longer just a tensor+hyper G G G G G in F-theory, we can blow up the singularity between two touching sevenbranes

12 E 6 (3) E 6 E 6 E 6 tensor multiplets E 6 E 6 E 6 E 6 E 6 this pattern also appeared in [Berhadsy, Johansen 96] [Aspinwall, Morrison 97] [Intriligator 97] E 6 (3) E 6 (3) E 8 E 8 = E-string (no gauge group) E 8 It also appears for M5 s near M9

13 E 6 (3) E 6 In M-theory: M5 Conjecture: 4 fractional M5 s [del Zotto, Hecman, AT, Vafa 4] R R 4 / E6 a discrete flux is created whenever a fractional M5 is crossed for a nice alternative explanation [Ohmori, Shimizu, Tachiawa, Yoneura 5, Tachiawa 5]

14 III. T-brane theories What about D8s? [del Zotto, Hecman, AT, Vafa 4] D8 s T-duality: D7 s but actually the D7 s fuse together: NS5 s = D6 s D7 s Nahm s equations Hitchin s equations Chain of intersecting curves, decorated by Hitchin poles = X + ix 2 t z z 0 residue is nilpotent T-brane So there should be 6d SCFTs T N ρ,ρ nilpotent ADE elements how do we find them?

15 Starting with a chain of conformal matter theories We wored out the web of RG flows; [Hecman, Rudelius, AT 6] using methods in [Hecman, Morrison, Rudelius, Vafa 6] it coincides precisely with the hierarchy of nilpotent elements! 0:[E 6 ] su 3 3 e 6 6 su 3 3 [E 6 ] A :[SU(6)] su 3 2 e 6 6 su 3 3 [E 6 ] 2A :[SO(7)] su 2 2 e 6 6 su 3 3 [E 6 ] 3A :[SU(2)] 2 [SU(3)] A 2 :[SU(3)] e 66 [SU(3)] e 66 su 3 3 [E 6 ] su 3 3 [E 6 ] A 2 + A :[SU(3)] e 65 [N f =] 3 [E 6 ] 2A 2 :[G 2 ] f 4 5 su 3 3 e 6 6[E6 ] A 2 +2A :[SU(2)] e 6 4 su 3 3 e 6 6 2A 2 + A :[SU(2)] f 4 4 su 3 3 e 6 6[E6 ] A 3 :[Sp(2)] so 0 4 su 3 3 e 6 6. A 3 + A :[SU(2)] so 9 4 su 3 3 e 6 6[E6 ] D 4 (a ): so 8 4 su 3 3 e 6 6[E6 ]

16 0 : [E6 ] su su 3 [E6 ] [E66] ] [E 0 0: :[E [E 6 ]6 ] Higgs moduli spaces A : [E6 ] A : [SU (6)] [E6 ] A : [SU (6)] [E6 ] dimh = dimh (c.m. chain) dimol we lose one tensor, gain 6 fundam. of 29 8 = [SU (6)]2 6 3 [Meareeya, Rudelius, AT 6] conjecture: do = su2 su2 [SO(7)]su dimor 2A: :[SO(7)] [E [E]6 ] 2A 2A : [SO(7)] su [E66 ] 2A : [SO(7)] [E6 ] e 6 3A: :[SU [SU(2)] (2)] [E [E 6] 3A 6] [SU (3)] 3A : [SU (2)] 2 [SU(3)] 6e 3 [E6 ] [SU (3)] 6 let s chec! difference of [anomaly arguments] 29nT + nh 3A : [SU (2)] [E6 ] 6 3 [E6 ] 3 susu 3 su [SU(3)] (3)] [E [E ] AA22: :[SU 6 ]6 A2 : [SU (3)] [SU 6 (3)] 3 [E6 ] [SUe(3)] 6 [SU (3)] nv [SUe(3)] 6 A2 : [SU (3)] 29 2 ( do = 24 [SU (3)] [SU (3)] e 5 f4 52) = 24 A22++AA: :[SU (3)] A A2 + A : [SU (3)] ff44 su 3 2A2 : [G22]] f [E [E66] ] 2A2 : [G2 ] [E6 ] f4 it always wors in fun ways A2 + A : [SU (3)] ff44 su A22 + A : [SU [SU(2)] (2)]f [E [E6 ]6 ] 2A2 + A : [SU (2)] [E6 ] 2A : [G ] [E ] f4 susu 3 3 [E [E 6]6] [N =] [Nf5=] f 3 [E6 ] [Nf =] 5 [Nf =] 3 [E6 ] AA 2A2A (2)] : [SU (2)] 2+ : [SU 2+ A2 + 2A : [SU (2)] so0 so0 2 AA 3 :3 [Sp(2)] : [Sp(2)]so A + 2A : [SU (2)] 4 3 A3 : [Sp(2)] soso susu e AA33++AA: :[SU 6]6]3 [SU6(2)] (2)]so [E [E 2A2 + A : [SU (2)] [E ] so0 A : [Sp(2)] 4 3 A3 + A : [SU (2)] [E6 ] so8 so8 sosu 3 ] 9 3 su D (a ) : ] su D4 (a ) :so [E [E 6 A3 + AD (a : [SU (2)] [E 6 ] [E6 ] [all dimensions ) : are quaternionic]

17 really! [all dimensions are quaternionic]

18 (2n) IIA brane diagram sometimes negative numbers of branes But F-theory succeeds! (2) g 2 (9) (2) (0) (0) (3) (9) (2) (0) (0)

19 Conclusions Many ways to Higgs M5s at singularities A ρ T N ρ,ρ N nilpotent ADE elements ρ nilpotent = pattern of D6s ending on D8s analytically nown AdS7 duals D E more exotic ingredients but similar overall structure! what implications for M5-dynamics? E 6 (3) E 6 E 6 (3) E 6 E 6 27