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UNIVERSIDADE FEDERAL DE CAMPINA GRANDE

BRANE SOLUTIONS AND RG FLOW. FRANCISCO A. BRITO. UNIVERSIDADE FEDERAL DE CAMPINA GRANDE. September 2006. BRANE SOLUTIONS AND RG FLOW. INTRODUCTION. Compactification - Factorizable - Non-factorizable ( phenomenology d=4 )

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UNIVERSIDADE FEDERAL DE CAMPINA GRANDE

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

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

  4. M10 = M4 X K6 “factorizable geometry” BRANE SOLUTIONS AND RG FLOW D = 26 BOSONIC STRINGS SUPERSTRINGS D = 10 COMPACTIFICATIONS OF SIX DIM Our four dim universe Compact 6-manifold

  5. NON-FACTORIZABLE “WARPED GEOMETRY” BRANE SOLUTIONS AND RG FLOW • AN ALTERNATIVE TO COMPACTIFICATION Randall & Sundrum, (1999) NON-COMPACT DIMENSION 3-BRANE r M4 ½ AdS5 • OUR UNIVERSE ON A 3-BRANE

  6. ,  = 0, 1, 2, 3 (brane world-volume indices) e 2A(r)≡ warp factor BRANE SOLUTIONS AND RG FLOW • AdS5 METRIC ds52= e2A(r) dx dx + dr2

  7. A = - k |r| SOLUTION: |5| = 12 k2 = σ2 / 12 e 2A (r) A (r) r r BRANE SOLUTIONS AND RG FLOW • THE Randall-Sundrum SCENARIO

  8. BRANE SOLUTIONS AND RG FLOW • GRAVITY FLUCTUATIONS

  9. BRANE SOLUTIONS AND RG FLOW • GRAVITY FLUCTUATIONS

  10. H = Q+Q Q = r + 3r A(r) _ H  (r) = m2(r) 2 BRANE SOLUTIONS AND RG FLOW • GRAVITY FLUCTUATIONS

  11. H = Q+Q Q = r + 3z A(r) _ 2 SOLUTION: BRANE SOLUTIONS AND RG FLOW • GRAVITY FLUCTUATIONS H  (r) = m2(r) • Zero Mode: m = 0 H o = 0 Q o = 0 o  e 3/2 A(r) ) )

  12. H = Q+Q Q = r + 3r A(r) _ 2 BRANE SOLUTIONS AND RG FLOW • GRAVITY FLUCTUATIONS H  (r) = m2(r) • Zero Mode: m = 0 H o = 0 Q o = 0 o  e 3/2 A(r) SOLUTION: ) )

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

  14. H = Q+Q Q =  r + 3r A(r) _ 2 BRANE SOLUTIONS AND RG FLOW • GRAVITY FLUCTUATIONS H  (r) = m2(r) • Zero Mode: m = 0 H o = 0 Q o = 0 o  e 3/2 A(r) SOLUTION: ) ) o  e -3/2 k |r| Localization of gravity! r

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

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

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

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

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

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

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

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

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

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

  25. - - ds2= eA(r)gdx dx + dr2 ds2= eA(r)gdx dx + dr2 dS4 M4 AdS4 - Λ→ four dimensional - - - Λ = 0 Λ > 0 Λ < 0 cosmological constant BRANE SOLUTIONS AND RG FLOW Karch & Randall (2001) • LOCALLY LOCALIZED GRAVITY

  26. AdS4 (Local localization) A (r) r BRANE SOLUTIONS AND RG FLOW • LOCALLY LOCALIZED GRAVITY

  27. AdS4 (Local localization) A (r) M4 r BRANE SOLUTIONS AND RG FLOW • LOCALLY LOCALIZED GRAVITY A = -k |r|

  28. AdS4 (Local localization) dS4 BRANE SOLUTIONS AND RG FLOW • LOCALLY LOCALIZED GRAVITY A (r) M4 r A = -k |r| “No global issues !” e. g. infinite volume

  29. AdS4 V (z) z BRANE SOLUTIONS AND RG FLOW • SCHROEDINGER POTENTIAL

  30. AdS4 BRANE SOLUTIONS AND RG FLOW V (z) • SCHROEDINGER POTENTIAL z M4

  31. dS4 BRANE SOLUTIONS AND RG FLOW AdS4 V (z) • SCHROEDINGER POTENTIAL z M4

  32. BRANE SOLUTIONS AND RG FLOW AdS4 V (z) • SCHROEDINGER POTENTIAL dS4 z Quase-zero mode emerges M4 (Massive) GRAVITY LOCALIZATION : A LOCAL EFFECT !!

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

  34. - - - Λ = 0 Λ > 0 Λ < 0 T 0 ∞ T* critical temperature BRANE SOLUTIONS AND RG FLOW • GEOMETRIC TRANSITIONS & LOCALLY LOCALIZED GRAVITY Susy Breaking dS4 M4 AdS4

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

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

  37. energy scale (AdS/CFT)  = 0 n = 0 or BRANE SOLUTIONS AND RG FLOW Skenderis & Townsend (1999) Freedman et al (1999) Kallosh & Linde (2000) Cvetic & Behrndt (2000) • THE SUSY FLOW EQUATIONS ds2= a2 (r) dx dx + dr2 • KILLING EQUATIONS (i)’ = ± 3 g i j j W ) ) g i j - metric definied on moduli space

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

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

  40. X BRANE SOLUTIONS AND RG FLOW • RG EQUATION

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

  42. BRANE SOLUTIONS AND RG FLOW • RG EQUATION where

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

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

  45. r →∞  > 0 a →∞ i → 0 ; e 2 A ( r) r IR UV BRANE SOLUTIONS AND RG FLOW Not good for localizing gravity! • SPECIAL GEOMETRIES • i) SUGRA D = 5 STABLE CRITICAL POINT UV FIXED POINT (QFT) ) QFT on AdS boundary AdS5 solution: a (r) = e k r UNSTABLE IR

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

  47. a (r) = e –k |r| INTRODUCING A BRANE: r →∞ zero mode STABLE CRITICAL POINT i → 0 o  e-k|r| a → 0 ; e 2 A ( r) r LOCALIZATION OF GRAVITY!! (Massless) BRANE SOLUTIONS AND RG FLOW • SPECIAL GEOMETRIES Two copies of AdS5 pasted together

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

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

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

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