1. Large N c QCD Towards a Holographic Dual of David Mateos Perimeter Institute ECT, Trento, July 2004

2. Towards the Holographic Dual of Large N c QCD (hep-th/0311270) M. Kruczenski, DM, R. Myers and D. Winters The Holographic Life of the (hep-th/0404260) J. Barbon, C. Hoyos, DM and R. Myers Based on: Meson Spectroscopy in AdS/CFT with Flavour (hep-th/0304032) Related work: M. Kruczenski, DM, R. Myers and D. Winters

3. QCD = SU(N c ) Yang-Mills + N f quarks Mystery of strong interaction dynamics Dynamical Probes Phenomenology (mesons, baryons,…) (especially in ‘t Hooft limit: N c ! 1, N f fixed ) Gravity = Closed Strings N f D-brane probes = Open Strings

4. Plan AdS/CFT with Flavour QCD from D4/D6 N c =1, N f =1: S  SB and meson spectrum (goldstone boson) N c =1, N f >1: Holographic Vafa-Witten, and pions? Baryons N c finite:  0 physics (Barbon’s talk)

5. AdS/CFT Gauge Theory SU(N c ) SYM in Mink 4+1 Gluons + Adjoint Scalars and Fermions Fundamental Scalars and Fermions in Mink 3+1 String Theory Closed Strings in `AdS’ N f D6-branes 0123 4 567 N c D4-branes D6 `AdS’ Mink 4+1 =  AdS Interacting closed + open strings Decoupling Limit + N f D6 probes CorrespondenceHolographic QCD with Flavour MaldacenaAFM, KR, KK N f ¿ N c Nc ! 1Nc ! 1

6. N c D4: 0 1 2 3 4 _ _ _ _ _ N f D6: 0 1 2 3 _ 5 6 7 _ _ QCD from D4/D6 On D4-branes: 4+1 SU(N c ) SYM = Gluons + Adjoint Scalars and Fermions Compactify 4-direction with antiperiodic boundary conditions for Adjoint Fermions Witten Renders theory four-dimensional at E ¿ M KK = 1 / R KK Breaks SUSY: Gives masses to Adjoint Fermions (tree level) and Adjoint Scalars (one-loop) With D6-branes: N f flavours of 3+1 Fundamental Scalars + Dirac Fermions  L +  R Fundamental Scalars aquire one-loop mass Fundamental Fermions remain massless because protected by chiral U(1) A U(1) A So do we have QCD at E ¿ M KK ?

7. So do we have strictly QCD ? E M KK  QCD  QCD » M KK e # g YM (M KK ) N c 2 Decoupling g YM N c ¿ 1 2 String description has: Curvature » 1 g YM N c 2 g YM N c À 1 2 Supergravity approximation No, but many features are captured, and confining theory in its own right.

8. The Vacuum: D6-brane Embedding and S  SB N c D4: 0 1 2 3 4 _ N f =1: D6: 0 1 2 3 _ 5 6 7 _ _ D6-brane probe in D4-brane background: r,  Minimum energy  = const, r =r( ) 89-plane r (  )  D6  ) Asymptotically ( ! 1): r = a + b / + … mqmq Proportional Field/Operator Correspondence: r ( ) Vacuum embedding: Choose a and adjust b for regularity  D6 r Unbroken SUSY: r = m q  D6 r  D6 r Broken SUSY…

9. The Vacuum: D6-brane Embedding and S  SB mqmq mqmq m q ! 0 Finite bump remains S  SB r( )

10. The Vacuum: D6-brane Embedding and S  SB as m q ! 0 for large m q

11. Meson Spectrum Regular, normalizable fluctuations:  n » e i k ¢ x f n ( ), M n 2 = - k 2 S  SB Pseudo-scalar Goldstone boson, k 2 = 0:  r  D6 r  non-normalizable  normalizable =

12. Meson Spectrum Scalars Pseudo-scalars M2M2 M 2 KK Scale set by M KK GMOR (analytically!) Degeneracy?

13. Meson Spectrum M2M2 M 2 KK M2M2 Supersymmetry restoration

14. Holographic Vafa-Witten Theorem and N f >1 Spectrum QCD with N f flavours of mass m q >0 U(N f ) V is not spontaneously broken In brane picture U(N f ) V is gauge symmetry: Spontaneously Broken N f D6-branes  r Unbroken N f D6-branes  r Spectrum ¾ N f 2 massless pseudo-scalars Goldstone Bosons of U(N f ) V £ U(N f ) A ! U(N f ) V ? Probably not: Gauge theory contains Non-derivative interactions (from D6s action) Gauge theory reason? String mechanism: D=9+1 Gauge Invariance NA DBI = A DBI + [X i,X j ] F ij =  [i A j] + [A i,A j ]

15. Baryons: Constituent quarks vs Chiral Soliton D6 = Flavour Brane AdS Meson Baryon? D4 = Baryonic Brane Baryons: Constituent quarks vs Chiral Soliton BIon

16. Conclusions What is String Theory good for? Many things. As far as QCD is concerned, at the very least it provides a new, geometric way of looking at old problems.

18. Finite N c and Physics Massless quarks Physics depends only on  + N f  To linear order: Tadpole =  YM  N f  Calculable in pure D4 background (ie no D6) E D6 (  ) To quadratic order: Witten-Veneziano Exact agreement

19. Finite N c and Physics String corrections to propagator: Gauge theory String analog of de Rujula-Georgi-Glashow: Splits U(N f ) singlet from non-singlets… Provided -glueball coupling is non-zero at p=0 !Shown in SUGRA limit from D6 couplings