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Marginal Deformations and Penrose limits with continuous spectrum

Marginal Deformations and Penrose limits with continuous spectrum. Toni Mateos Imperial College London. Universitat de Barcelona, December 22, 2005. Introduction. In general, CFTs are isolated points in space of couplings.  ( g i ) = 0 ! fixes all g i.

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Marginal Deformations and Penrose limits with continuous spectrum

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  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 ( gi ) = 0 ! fixes all gi • Susy )-functions » anomalous dimensions, • # anomalous dimensions < # marginal couplings )continuous families of CFTs • e.g. N=4 , # exactly marginal deformations = 3C - deformation, breaks SU(3) flavor !U(1) £ U(1) • other SCFTs, like Klebanov-Witten (T1,1) also admit •  - defomations, SU(2) £ SU(2) flavor !U(1) £ U(1)

  3. Introduction • How to construct them? General case not known. Continuous families of CFTs = Continuous familes of AdS5£ X5 solutions • Lunin and Maldacena: simple way out for  – deformations. • If SCFT has U(1) £ U(1) AdS5£ X5, isom( X5 ) ¾U(1) £ U(1) 8d solution, with SL(2,R) £ SL(3,R) duality group U SL(2,R) acts on • Original solution regular ) final solution regular, if

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

  5. Introduction N= 4  - deformed:

  6. see also C. Ahn, J.F. Vazquez-Portitz, hep-th/0505168 • see also R. Mello Koch, J. Murugan, S. Smolic, M. Smolic, hep-th/0505227 Contents Part 0 : Exactly Marginal Deformations Lunin, Maldacena, hep-th/0502086 Part I : Marginal deformations of 3d FTs with AdS4 duals J. Gauntlett, S. Lee, T. M., D. Waldram hep-th/0505207 Part II : Penrose Limit of -deformation of 4dN = 4 SYM T. M. hep-th/0505243

  7. Part I: AdS4 and 3d Field Theories 8d solution, with SL(2,R) £ SL(3,R) duality group Supergravity Part I: Field theory • Are there similar exactly marginal deformations of 3d CFTs ? • At least at the level of supergravity, method generalises • to AdS solutions of D=11. AdS4£ Y7 , Isom(Y7) ¾U(1)3 U SL(2,R) acts on • FT on M2 branes much less understood (strongly coupled IR) • Solid proposals have been made for some cases

  8. Part I: AdS4 and 3d Field Theories M2 C(Y7) Y7 • Weak G2 : N = 1!N = 1 ( very non-trivial !) N =0, if U(1)3 • Sasaki-Einstein: N = 2 N= 2, if U(1)4 • Tri-Sasaki:N = 3!N= 1 I.a. Supergravity ds2 = dr2 + r2 ds2(Y7) L= 0 • Susy after deformation:

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

  10. Part I: AdS4 and 3d Field Theories I.a. Supergravity • Deformation procedure simplified. Pick 3 U(1)'s and...

  11. Part I: AdS4 and 3d Field Theories I.a. Supergravity • Deformation of AdS4£ Yp,q (Sasaki-Einstein, 2 ! 2)

  12. Part I: AdS4 and 3d Field Theories I.a. Supergravity • Deformation of AdS4£ S7squashed (weak G2, 1 ! 1)

  13. Part I: AdS4 and 3d Field Theories Q(1,1,1) M(3,2) N(1,1) I.b. Field Theory • Like in QCD !: empirical data in IR)UV lagrangian probe particles / M-branes in AdS5£ { Q(1,1,1), M(3,2), N(1,1) } X • Moduli space = C(Y7) •  of baryons=energy of M5 wrapping 5-cycles X • Spectrum of chiral operators = KK spectrum Y7 X

  14. Part I: AdS4 and 3d Field Theories •  = 2 chiral primary • Global symmetry !U(1)3 • N!N I.b. Field Theory How to identify -deformation without NC open string intuition? • Look for a superpotential with: X • Unique answer for N=4 and T1,1 in 4d. W ( · ! * ) = cos WN=4 + i sin  Tr (123+132) X X X • Unique answer for the 3d known susy cases.

  15. Part I: AdS4 and 3d Field Theories X • U(1)3 preserving: unique I.b. Field Theory • AdS4£ Q(1,1,1), N=2 • Global Symmetry: SU(2)3£ U(1)R • Gauge Symmetry: SU(N)3 • Baryons: det A, det B, det C ! A = B = C = 1/3 • Chiral primaries: Tr (ABC )k , •  = k, in (k+1,k+1,k+1) of SU(2)3 • W = Tr (ABC ABC) in the (3,3,3) of SU(2)3

  16. Part I: AdS4 and 3d Field Theories I.b. Field Theory Summary • Extension of -deformations for 3d CFTs via AdS4 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! )

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

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

  19. Part II: Penrose Limit of deformed N=4 • Penrose Limit ! IIB configuration: NS-NS: G , B ,  R-R: F5 , F3 • Number of supersymmetries = 16 + 4 = 20 • Covariantly constant null Killing v + null potentials • ) generalised super-GS action ( quantisable! ) • U(1) £ U(1)½S5 of -deformation still isometry (y2 , y4)

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

  21. Part II: Penrose Limit of deformed N=4 Quantisation of Bosonic Sector • n = 0, decoupling of planes y1y2 and y3y4  = 0 : Landau problem Vacuum with 1 discrete degeneracy X

  22. If  0 : Landau + spring y2 y1

  23. If  0 : Landau + spring Vacuum unique! Spectrum continuous!

  24. Part II: Penrose Limit of deformed N=4 y2 y1

  25. Part II: Penrose Limit of deformed N=4 y2 y1 v2» y1 • it takes energy to speed up / climb the wall • constraint system (2nd class) ! Dirac bracket quantisation

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

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

  28. Part II: Penrose Limit of deformed N=4 Summary • Exploration of new phenomena of SCFTs via AdS • (chiral ring, spectrum of anomalous dimensions) • New exactly solvable (and physically motivated) • string theory backgrounds ( F3 , F5 , H3 ) • Half-way between flat-space and pp-wave • ( Modified Landau Problem ) • Predictions for  of dual operators • String Theory analysis directly applicable • to other cases; e.g. Penrose limit of AdS5£ T1,1 - End -

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