1 Marginal Deformations and Penrose limits with continuous spectrum Toni Mateos Imperial College London Universitat de Barcelona, December 22, 2005

2 Introduction In general, CFTs are isolated points in space of couplings  ( g i ) = 0 ! fixes all g i  - deformation, breaks SU(3) flavor ! U(1) £ U(1) other SCFTs, like Klebanov-Witten (T 1,1 ) also admit  - defomations, SU(2) £ SU(2) flavor ! U(1) £ U(1) Susy )  -functions » anomalous dimensions, # anomalous dimensions < # marginal couplings ) continuous families of CFTs e.g. N =4, # exactly marginal deformations = 3 C

3 Continuous families of CFTs = Continuous familes of AdS 5 £ X 5 solutions Lunin and Maldacena: simple way out for  – deformations. If SCFT has U(1) £ U(1) 8d solution, with SL(2,R) £ SL(3,R) duality group How to construct them? General case not known. SL(2,R) acts on Original solution regular ) final solution regular, if U Introduction AdS 5 £ X 5, isom( X 5 ) ¾ U(1) £ U(1)

4 Applicable to any FT with U(1) £ U(1) global symmetry (even non CFT !) (even non SUSY ! ) Deformation of Lagrangian is simple to obtain: · ! * Sugra side very simple, and for finite  e.g. N =4 : Introduction stringy SL(2,Z) X If N  0 ) final N  depends on number of Killing spinors invariant under U(1) £ U(1).

5 Introduction N = 4  - deformed:

6 Contents Part II : Penrose Limit of  -deformation of 4d N = 4 SYM T. M. hep-th/ Part I : Marginal deformations of 3d FTs with AdS 4 duals J. Gauntlett, S. Lee, T. M., D. Waldram hep-th/ Part 0 : Exactly Marginal Deformations see also C. Ahn, J.F. Vazquez-Portitz, hep-th/ see also R. Mello Koch, J. Murugan, S. Smolic, M. Smolic, hep-th/ Lunin, Maldacena, hep-th/

7 Part I: AdS 4 and 3d Field Theories At least at the level of supergravity, method generalises to AdS solutions of D=11. AdS 4 £ Y 7, Isom(Y 7 ) ¾ U(1) 3 8d solution, with SL(2,R) £ SL(3,R) duality group SL(2,R) acts on Are there similar exactly marginal deformations of 3d CFTs ? FT on M2 branes much less understood (strongly coupled IR) Solid proposals have been made for some cases Part I: Supergravity Field theory U

8 I.a. Supergravity M2 C(Y 7 ) Y7Y7 ds 2 = dr 2 + r 2 ds 2 (Y 7 ) Susy after deformation: L    = 0 Part I: AdS 4 and 3d Field Theories Sasaki-Einstein: N = 2 N  = 0, if U(1) 3 N  = 2, if U(1) 4 Tri-Sasaki: N = 3 ! N  = 1 ( very non-trivial !) Weak G 2 : N = 1 ! N  = 1

9 Proposed field theories Scanning for SUSY marginal deformations… [Fabbri, Fre, Gualtieri, Reina, Tomasiello, Zaffaroni, Zampa]

10 I.a. Supergravity Part I: AdS 4 and 3d Field Theories Deformation procedure simplified. Pick 3 U(1)'s and...

11 I.a. Supergravity Part I: AdS 4 and 3d Field Theories Deformation of AdS 4 £ Y p,q (Sasaki-Einstein, 2 ! 2)

12 I.a. Supergravity Part I: AdS 4 and 3d Field Theories Deformation of AdS 4 £ S 7 squashed (weak G 2, 1 ! 1)

13 Q(1,1,1)M(3,2)N(1,1) Moduli space = C(Y 7 ) X X X Spectrum of chiral operators = KK spectrum Y 7  of baryons = energy of M5 wrapping 5-cycles Part I: AdS 4 and 3d Field Theories Like in QCD !: empirical data in IR ) UV lagrangian I.b. Field Theory probe particles / M-branes in AdS 5 £ { Q(1,1,1), M(3,2), N(1,1) }

14 I.b. Field Theory How to identify  -deformation without NC open string intuition? Look for a superpotential with: Unique answer for N =4 and T 1,1 in 4d. W ( · ! * ) = cos  W N =4 + i sin  Tr (  1  2  3 +  1  3  2 )  = 2 chiral primary Global symmetry ! U(1) 3 N ! N  Part I: AdS 4 and 3d Field Theories Unique answer for the 3d known susy cases. XXX X

15 I.b. Field Theory Part I: AdS 4 and 3d Field Theories AdS 4 £ Q(1,1,1), N =2 Global Symmetry: SU(2) 3 £ U(1) R Gauge Symmetry: SU(N) 3 W  = Tr (ABC ABC) in the (3,3,3) of SU(2) 3 Chiral primaries: Tr (ABC ) k,  = k, in (k+1,k+1,k+1) of SU(2) 3 U(1) 3 preserving: unique X Baryons: det A, det B, det C !  A =  B =  C = 1/3

16 Summary Extension of  -deformations for 3d CFTs via AdS 4 duals. Prediction N = (3, 2, 1) ! N  = (1, 2 / 0, 1) Operators identified without open string theory. Possibility of studying modification of chiral ring, new branches… Discussion in paper about non-susy deformations. ( see paper! ) Part I: AdS 4 and 3d Field Theories I.b. Field Theory

17 Part II: Penrose Limit of deformed N =4 New exact results SCFT $ string theory ? Spectrum of chiral operators S5S5 S5S5 SO(6) · ! * add complicated phases e i   [Berenstein, Leigh, Jejjala] N = 4 : N = 1 :

18 Focus on huge discrete degeneracy Expectations: N = 4 : N = 1 : All states with zero charge under U(1) £ U(1) ! not affected Vacuum should be unique : Other exchanges: Part II: Penrose Limit of deformed N =4 · ! * add complicated phases e i  

19 Covariantly constant null Killing  v + null potentials ) generalised super-GS action ( quantisable! ) U(1) £ U(1) ½ S 5  of  -deformation still isometry (y 2, y 4 ) Part II: Penrose Limit of deformed N =4 Penrose Limit ! IIB configuration: NS-NS: G , B ,  R-R: F 5, F 3 Number of supersymmetries = = 20

20 For n  0 (stringy modes) : Quantisation of Bosonic Sector For n  0 (particle-like modes): Part II: Penrose Limit of deformed N =4

21 n = 0, decoupling of planes y 1 y 2 and y 3 y 4 Quantisation of Bosonic Sector Vacuum with 1 discrete degeneracy  = 0 : Landau problem Part II: Penrose Limit of deformed N =4 X

22 If   0 : Landau + spring y2y2 y1y1

23 Vacuum unique! Spectrum continuous!

24 y2y2 y1y1 Part II: Penrose Limit of deformed N =4

25 y2y2 y1y1 v 2 »  y 1 it takes energy to speed up / climb the wall constraint system (2 nd class) ! Dirac bracket quantisation

26 Field Theory Interpretation Charge under p y, Charge under U(1) £ U(1) if uncharged )   f(  ) X if charged )   f(  ) ) more energy (  ) X Part II: Penrose Limit of deformed N =4, Departure from

27 Field Theory Interpretation Part II: Penrose Limit of deformed N =4 (J,0,0) (J,J,J)

28 Summary Exploration of new phenomena of SCFTs via AdS (chiral ring, spectrum of anomalous dimensions) New exactly solvable (and physically motivated) string theory backgrounds ( F 3, F 5, H 3 ) Half-way between flat-space and pp-wave ( Modified Landau Problem ) Predictions for  of dual operators Part II: Penrose Limit of deformed N =4 - End - String Theory analysis directly applicable to other cases; e.g. Penrose limit of AdS 5 £ T 1,1