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AdS/CFT: Introduction

AdS/CFT: Introduction. Jorge Casalderrey-Solana LBNL. The AdS/CFT Correspondence. 4 dimensional N =4 Super Symmetric SU(N c ) Yang-Mills theory (N c  ∞) is dual to Type II B Super String theory in an AdS S 5 background. J. Maldacena 1998. N =4 Super Symmetric Yang-Mills.

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AdS/CFT: Introduction

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  1. AdS/CFT:Introduction Jorge Casalderrey-Solana LBNL

  2. The AdS/CFT Correspondence 4 dimensional N=4 Super Symmetric SU(Nc) Yang-Mills theory (Nc  ∞) is dual to Type II B Super String theory in an AdSS5 background. J. Maldacena 1998

  3. N=4 Super Symmetric Yang-Mills SU(Nc) Gauge Theory with N=4 Super Charges Field Content: Am Gauge field (2) Adjoint in SU(Nc) laa Gaugino (4 2) ; a=1..4, a=1,2 Xi Scalar (6) ; i=1..6 Rigid Symmetries: SU(4)R acting on index a, i (similar to SU(Nf)) Poincare (translation + Lorentz) Scale invariance (b=0) SO(2,4) Large Nc limit l 0 (t’Hooft) Nc ∞ l=g2Nc fixed g  0 l ∞ (strong coupling)

  4. Xm(t,s) s t String Theory String Xm(t,s) is a 2D surface Fundamental length scale l2s=a’ First quatization  2D field theory Motion of C.O.M Operators amn┼amn Vibrational modes Create vibrational/translational states At scales l>>ls the string modes are quantum fields The masses of the states (E(p=0)) proportional to l-1s Super Strings add also a fermionic Ym(t,s) The theory is only defined at D=10 dimensions

  5. Types of closed String Theory Two types of world-sheets Non-oriented (Moebius strip)  Type I Type II A Two ways of quantizing Oriented  Type II Type II B Low Energy (Massless) modes self dual 5D field strength gravitino graviton dilaton dilatino String interactions lead to coupling between fields Low energy effective action D=10 Super Gravity Newton constant

  6. Dirichlet Branes Polchinski (97) : p+1 dimensional hyperplanes charged under the close string fields. Open strings can end in those planes Nc Nc  Nc open strings when e=0symmetry U(Nc) ~ SU(Nc)  U(1) related to shifts of the stack Valid for gsNc<<1 e D3 branes Low energy effective action of the open strings (3+1)d N=4 super-symmetric SU(Nc) Yang-Mills D3 branes + bulk N=4 SUSY+ SUGRA at low energy they decouple

  7. p-branes Solution to the classical SUGRA equations of motions p-dim “charged” “black holes” quantized charge electric 3-branes  couple to magnetic Extremal case SUGRA valid if : R>> ls (gs Nc >>1 ) => Large actions gs<< 1 => no string loops

  8. Near Brane Limit Two kinds of low energy modes (observer at r= ∞ ) low frequency SUGRA modes in the bulk (r>>R) Modes of arbitrary energy close to the source redshifted energy The two modes are decoupled w<<R-1 => SUGRA modes don’t interact with brane Modes close to r=0 can’t reach the asymptotic region Close to the brane, AdS5 S5 Isometry group SO(2,4) SO(6) ~ SU(4)

  9. Duality D-branes and p-branes are supposed to be the same object in different regions. Bulk Near Brane SUGRA N=4 SYM SUGRA ? Same symmetries as N=4 SYM Conjecture: the string dynamics near the brane describes N=4 SYM at strong coupling Concrete realization: GKPW prescription Source for OBoundary value of field  Field  Operator correspondence

  10. O (x) O (y)c= Computation of Correlation Functions Strong coupling (classical SUGRA) limit: 1) Find the dual field  to the operator O 2) Solve the classical equations of motion for  with a fixed value at the boundary 0 3) Compute the SUGRA action of the classical solution SSUGRA[0] is a functional of 0 4) In the classical limit, the string partition function is given by the classical action 5)

  11. Back up

  12. N=4 SYM Lagrangian

  13. N=4 SUSY Transformations

  14. SUGRA Action(bosonic part)

  15. Field Operator

  16. Old fashion charged black hole The metric of a black hole of mass M electric charge q and magnetic charge p is The horizon is at D(r)=0 Extremal black hole The two horizons collapse and there is not “no escape” region

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