1. BRANE SOLUTIONS AND RG FLOW UNIVERSIDADE FEDERAL DE CAMPINA GRANDE September 2006 FRANCISCO A. BRITO

2. BRANE SOLUTIONS AND RG FLOW INTRODUCTION i) Compactification - Factorizable - Non-factorizable (phenomenology d=4) * Other interests (BTZ black holes, gravity in 2d string theory, and sugra 10 and 11 to lower dimensions > 4) ii) Dualidade gauge/gravity (e.g. AdS/CFT) - gravity duals (brane solutions): D - dimensions - RG flow of a dual field theory: (D-1) - dimensions

3. BRANE SOLUTIONS AND RG FLOW BOSONIC STRINGS SUPERSTRINGS COMPACTIFICATIONS OF SIX DIM D = 26 D = 10

4. BRANE SOLUTIONS AND RG FLOW BOSONIC STRINGS SUPERSTRINGS COMPACTIFICATIONS OF SIX DIM D = 26 D = 10 M 10 = M 4 X K 6 “factorizable geometry” Compact 6-manifold Our four dim universe

5. BRANE SOLUTIONS AND RG FLOW OUR UNIVERSE ON A 3-BRANE Randall & Sundrum, (1999) AN ALTERNATIVE TO COMPACTIFICATION 3-BRANE r NON-COMPACT DIMENSION M 4 ½ AdS 5 NON-FACTORIZABLE “WARPED GEOMETRY”

6. BRANE SOLUTIONS AND RG FLOW AdS 5 METRIC , = 0, 1, 2, 3 (brane world-volume indices) e 2A(r) ≡ warp factor ds 5 2 = e 2A(r)   dx  dx + dr 2

7. BRANE SOLUTIONS AND RG FLOW THE Randall-Sundrum SCENARIO r A (r) r e 2A (r) SOLUTION: |  5 | = 12 k 2 = σ 2 / 12 A = - k |r|

8. BRANE SOLUTIONS AND RG FLOW GRAVITY FLUCTUATIONS

9. BRANE SOLUTIONS AND RG FLOW GRAVITY FLUCTUATIONS

10. BRANE SOLUTIONS AND RG FLOW GRAVITY FLUCTUATIONS H  (r) = m 2  (r) H = Q + Q Q =  r + 3  r A(r) _ 2

11. BRANE SOLUTIONS AND RG FLOW GRAVITY FLUCTUATIONS SOLUTION: Zero Mode: m = 0 H  (r) = m 2  (r) H = Q + Q Q =  r + 3  z A(r) _ 2 H  o = 0 ) Q  o = 0 )  o  e 3/2 A(r)

12. BRANE SOLUTIONS AND RG FLOW GRAVITY FLUCTUATIONS SOLUTION: Zero Mode: m = 0 H  (r) = m 2  (r) H = Q + Q Q =  r + 3  r A(r) _ 2 H  o = 0 ) Q  o = 0 )  o  e 3/2 A(r)

13. BRANE SOLUTIONS AND RG FLOW r  o  e -3/2 k |r| Localization of Gravity!

14. BRANE SOLUTIONS AND RG FLOW GRAVITY FLUCTUATIONS SOLUTION: r Zero Mode: m = 0 Localization of gravity! H  (r) = m 2  (r) H = Q + Q Q =  r + 3  r A(r) _ 2 H  o = 0 ) Q  o = 0 )  o  e 3/2 A(r)  o  e -3/2 k |r|

15. BRANE SOLUTIONS AND RG FLOW z V(z) Massive modes

16. z V(z) BRANE SOLUTIONS AND RG FLOW Massive modes

17. BRANE SOLUTIONS AND RG FLOW z V(z) KK modes Massive modes

18. BRANE SOLUTIONS AND RG FLOW z V(z) Massive modes

19. BRANE SOLUTIONS AND RG FLOW Massive modes Correction of Newtonian Potential!

20. BRANE SOLUTIONS AND RG FLOW GRS SCENARIO Massive gravity: metastable gravity Gregory, Rubakov & Sibiryakov (2000)

21. BRANE SOLUTIONS AND RG FLOW GRS SCENARIO Massive gravity: metastable gravity Gregory, Rubakov & Sibiryakov (2000)

22. BRANE SOLUTIONS AND RG FLOW GRS SCENARIO Flat brane embeded into 5d Minkowski bulk: infinite volume! No zero modes rcrc rcrc σ < 0 σ > 0 0 A r

23. BRANE SOLUTIONS AND RG FLOW ASYMMETRIC BRANES Brito & Gomes (work in progress) Finite volume massive modes

24. BRANE SOLUTIONS AND RG FLOW U (R) ~ 1 / R L L log R 1 2 R >> R c R << R c

25. BRANE SOLUTIONS AND RG FLOW LOCALLY LOCALIZED GRAVITY Karch & Randall (2001) ds 2 = e A(r) g  dx  dx + dr 2 - - Λ > 0 - Λ = 0 - Λ < 0 - dS 4 M4M4 AdS 4 Λ → four dimensional - cosmological constant

26. BRANE SOLUTIONS AND RG FLOW LOCALLY LOCALIZED GRAVITY r A (r) AdS 4 (Local localization)

27. BRANE SOLUTIONS AND RG FLOW A = -k |r| M4M4 LOCALLY LOCALIZED GRAVITY r A (r) AdS 4 (Local localization)

28. BRANE SOLUTIONS AND RG FLOW LOCALLY LOCALIZED GRAVITY r A (r) A = -k |r| M4M4 dS 4 “No global issues !” e. g. infinite volume AdS 4 (Local localization)

29. BRANE SOLUTIONS AND RG FLOW SCHROEDINGER POTENTIAL z V (z) AdS 4

30. BRANE SOLUTIONS AND RG FLOW SCHROEDINGER POTENTIAL z V (z) M4M4 AdS 4

31. BRANE SOLUTIONS AND RG FLOW SCHROEDINGER POTENTIAL z V (z) M4M4 AdS 4 dS 4

32. BRANE SOLUTIONS AND RG FLOW SCHROEDINGER POTENTIAL z V (z) AdS 4 Quase-zero mode emerges M4M4 dS 4 (Massive) GRAVITY LOCALIZATION : A LOCAL EFFECT !!

33. BRANE SOLUTIONS AND RG FLOW GEOMETRIC TRANSITIONS & LOCALLY LOCALIZED GRAVITY Brito, Bazeia & Gomes (2004) Λ = L -2 [ σ (T) 2 – σ* ] - - 4 dim cosmological constant Brane tension depending on temperature T σ

34. BRANE SOLUTIONS AND RG FLOW dS 4 M4M4 AdS 4 Susy Breaking Λ = 0 - Λ < 0 - Λ > 0 - 0 T* ∞ critical temperature T GEOMETRIC TRANSITIONS & LOCALLY LOCALIZED GRAVITY

35. BRANE SOLUTIONS AND RG FLOW SUPERGRAVITY ACTION 5 dim cosmological constant → critical points W - superpotential ; Cvetic et al (2000) Brito & Cvetic (2001) Bazeia, Brito & Nascimento (2003)

36. BRANE SOLUTIONS AND RG FLOW SUPERGRAVITY ACTION CONTENTS TURNED ON Supergravity multiplet: (e a m,  i m ) Scalar super multiplet: ( ,  i m )  S = 0   im im  eam eam  ;; ; UNDER SUSY TRANSFORMATIONS!!!!

37. BRANE SOLUTIONS AND RG FLOW THE SUSY FLOW EQUATIONS  = 0   n = 0 ds 2 = a 2 (r)   dx  dx + dr 2 KILLING EQUATIONS ) ) (  i ) ’ = ± 3 g i j  j W g i j - metric definied on moduli space energy scale (AdS/CFT) or Skenderis & Townsend (1999) Freedman et al (1999) Kallosh & Linde (2000) Cvetic & Behrndt (2000)

38. BRANE SOLUTIONS AND RG FLOW THE SUSY FLOW EQUATIONS CRITICAL POINTS  i (r →∞) =  i * ) (  i ) ’ = 0 )  j W (  i * ) = 0 )

39. BRANE SOLUTIONS AND RG FLOW THE SUSY FLOW EQUATIONS CRITICAL POINTS  i (r →∞) =  i * ) (  i ) ’ = 0 )  j W (  i * ) = 0  W * * Flow )

40. BRANE SOLUTIONS AND RG FLOW RG EQUATION X

41. BRANE SOLUTIONS AND RG FLOW RG EQUATION where a – energy scale  i - couplings RG EQUATION ON THE FIELD THEORY SIDE

42. BRANE SOLUTIONS AND RG FLOW RG EQUATION where

43. BRANE SOLUTIONS AND RG FLOW RG EQUATION where Restrictions on W?

44. BRANE SOLUTIONS AND RG FLOW SPECIAL GEOMETRIES Thus we find Assuming perturbation as ; c i = constant

45. BRANE SOLUTIONS AND RG FLOW SPECIAL GEOMETRIES STABLE CRITICAL POINT i) SUGRA D = 5 Not good for localizing gravity! ) UV FIXED POINT (QFT) QFT on AdS boundary r e 2 A ( r) IRUV AdS 5 solution: a (r) = e k r UNSTABLE IR > 0 r →∞ a →∞  i → 0 ;

46. BRANE SOLUTIONS AND RG FLOW SPECIAL GEOMETRIES ii) GRAVITY LOCALIZATION < 0 AdS5 solution: a (r) = e -k r  i = c i a | | : “IR FIXED POINT” STABLE CRITICAL POINT r →∞ a → 0  i → 0 ; r e 2 A ( r)

47. BRANE SOLUTIONS AND RG FLOW SPECIAL GEOMETRIES STABLE CRITICAL POINT r →∞ a → 0  i → 0 ; INTRODUCING A BRANE: a (r) = e –k |r| zero mode  o  e -k|r| Two copies of AdS 5 pasted together LOCALIZATION OF GRAVITY!! (Massless) r e 2 A ( r)

48. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM AND “BENT” BRANES: Freedman et al. (2004) Bazeia et al. (2006) Brito, Bazeia, Losano (work in progress) NEW DEVELOPMENTS “fake sugra”

49. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM AND “BENT” BRANES: Freedman et al. (2004) Bazeia et al. (2006) Brito, Bazeia, Losano (work in progress) NEW DEVELOPMENTS “BENT” BRANE GEOMETRIES

50. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM AND “BENT” BRANES: Freedman et al. (2004) Bazeia et al. (2006) Brito, Bazeia, Losano (work in progress) NEW DEVELOPMENTS “BENT” BRANE GEOMETRIES

51. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM AND “BENT” BRANES: NEW DEVELOPMENTS EQUATIONS OF MOTION

52. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM i) MINKOWSKI BRANES: FIRST ORDER EQUATIONS

53. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM FIRST ORDER EQUATIONS ii) “BENT” BRANES:

54. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM FIRST ORDER EQUATIONS ii) “BENT” BRANES: CONSTRAINTS

55. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM FIRST ORDER EQUATIONS ii) “BENT” BRANES:

56. BRANE SOLUTIONS AND RG FLOW FIRST ORDER FORMALISM iii) BETA FUNCTION

57. BRANE SOLUTIONS AND RG FLOW EXAMPLES  r A r i)

58. BRANE SOLUTIONS AND RG FLOW EXAMPLES  r A r i)

59. BRANE SOLUTIONS AND RG FLOW  r EXAMPLES ii) A r

60. BRANE SOLUTIONS AND RG FLOW ii) EXAMPLES  r A r

61. BRANE SOLUTIONS AND RG FLOW CONCLUSIONS i) D=4 is phenomenologically motivated ii) Infinite volume implies no zero modes iii) Warp factor regarded as energy scale on dual theory iv) Bent branes may give a dual gravitational description of RG flows in susy field theories in a curved spacetime v) Theories in AdS spaces exhibit improved infrared behavior

62. Th e E n d