Putting String Theory to the Test with AdS/CFT

Transcription

1 Putting String Theory to the Test with AdS/CFT Leopoldo A. Pando Zayas University of Iowa Department Colloquium

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3 L = 1 4g 2 Ga µνg a µν + j G a µν = µ A a ν ν A a µ + if a bc Ab µa c ν, D µ = µ + it a A a µ q j (iγ µ D µ + m j ) q j, The Lagrangian is known: Theorist s dream Confinement is experimentally there but with no theoretical explanation Goal: The spectrum of particles Physics Fermat s Last Problem: Easy to state, believed to be true, solution with deep implications

4 What is confinement? We never see an isolated quark The confinement hypothesis

5 What is confinement? We never see an isolated quark The confinement hypothesis: Color flux with E L. 0123

6 Area Law for the Wilson loop (probability amplitude that the quark travels around such contour): W (C) exp ( ka(c)) with k > 0. Time C Space

7 A Millenium Prize Problem Yang-Mills Existence and Mass Gap: Prove that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on R 4 and has a mass gap > 0. Existence includes establishing axiomatic properties at least as strong as those cited in [Osterwalder, Schrader, Streater and Wightman]. Theory/

8 Regge trajectories are best explained by a string J M 2

9 The t Hooft limit: Universality of strings In any theory of Hermitian matrices Generalize QCD: SU(3) SU(N) N, g 2 N = λ fixed, O(N) nonlinear sigma model can be solved in the t Hooft limit: Asymptotic freedom and a mass gap is generated nonperturbatively

10 M ij double line notation Planar diagram is the leading contribution at large N Corrections are organized in g string = 1/N. This is a string theory! ln Z = C h N 2 2h f(g 2 N)

11 How can two theories be the same? Baby examples Differential equation polynomial equation Fourier transform (position space versus momentum space) Harmonic oscillator in quantum mechanics: Analytic (differential equations) creation and annihilation. One expects to get some insight into the problem.

12 IIB superstrings on AdS 5 S 5 with flux is equivalent to N = 4 Supersymmetric Super Yang-Mills. No proof but a huge amount of evidence is available: An equivalence of partition functions AdS 5 S 5

13 AdS/CFT : Two equivalent ways of describing a stack of D-branes

14 AdS/CFT Parameters: g 2 Y M = 4π g string N = S 5 F 5 R S 5 = R AdS5 = (g 2 Y M N)1/4 l s Same superconformal symmetry SO(2, 4) SO(6) SU(2, 2 4) States on AdS = Operators in the CFT Single Particle state single Trace T rf X... Multi-Particle Multitrace T r(f X...X)T r(f...x..x) Chiral primaries (protected, BPS) Supergravity modes

15 Program: Can we? Toward more realistic situations: N = 1 supersymmetric theories Constructing supergravity solutions Flavor in the fundamental representation Beyond the classical regime: Regge trajectories revisited, Wilczek s spectroscopy, RHIC: The black hole dual of strongly coupled Quark Gluon Plasma Use the learn about string theory

16 What can we reliably claim? Too early for numerical comparison: Asymptotic freedom is now the problem Start with qualitative behavior and work towards quantitative understanding General behavior

17 Hagedorn density of states for hadronic matter Based on the study of elastic pp cross sections, in the 1965 (pre QCD) Hagedorn established that the hadronic density of matter is (bootstrap condition, Hagedorn density of states) ρ(e) exp(ae) In the 1980 s it was discovered that string theory shows very generically a Hagedorn density of states [speculations about a relation to gauge theories].

18 Thermodynamics == Partition Function Genus expansion: The intuition maker [ t Hooft, Polyakov, Maldacena] Z string = e 2Φ 0 Z 0 + Z 1 + e 2Φ 0 Z Z string = N 2 Z 0 + N 0 Z N 2 Z Conformal Theories: Main Contribution is N 2. Confining Theories: Main Contribution is N 0. To compute Z 1 you need string theory

19 A generic property of holographic duals of confining theories Confinement == Wilson Area Law End of Space. Wilson Loop shows confining behavior. Q Q o r0 r T s = 1 2πα g tt(r 0 ) g tt (r 0 ) 0 and g tt (r 0 ) has a minimum

20 Hagedorn Temperature from Gauge/Gravity Duality From the partition function: β H = 2π/( 3T 1/2 s ) Density for generic confining theories with gravity duals (Phys.Rev.Lett.91:111602,2003) S = 2π 3 Correct Λ QCD dependence. E T 1/2 s.

21 Beyond the classical approximation Instead of phenomenological models we now have the right string models. Gauge Theory State Glueballs Mesons of heavy quarks Baryons of heavy quarks Dibaryons Mesons of light quarks k strings String Theory Configuration Spinning Folded Closed String Spinning open strings ending at boundary Strings attached to a baryonic vertex Strings attached to wrapped branes Spinning open strings ending on D7 Wrapped branes with flux States in gauge theory and their corresponding classical configuration in the string theory

22 Regge trajectories for Glueballs Closed spinning string 0123 r 0 r

23 Closed spinning strings in confining theories Conditions for confinement in gauge/gravity: g 00 has a nonzero minimum at some point r 0. Typical Regge trajectories E 2 = 4 πt s S or J = α 2 t (1) Problems This trajectory has zero intercept. It is strictly linear.

24 The soft Pomeron: UA8 Collaboration Experimental suggestion α(t) = t + α t 2, α = ± GeV 4 Positive nonvanishing intercept. Positive curvature α > 0.

25 Semiclassical quantization Compute how the classical energy changes (similar to Lüscher term): e E = dσ Ψ H(δX) Ψ = sum of zero point energies New feature of confining backgrounds? [Compared to strings in flat space] [ 2 τ 2 σ + m 2 0 cos 2 (ωσ) ] δτ i = 0. (2)

26 Nonlinear ( ) 3 E E Class = π 2 m 0 4l = z 0 J α(t) = α α t + β t Positive Nonvanishing intercept Positive curvature α (t) > 0 Dual Models with Mandelstam Analyticity: [ ] T α(t) = α(0) + γ 1/2 T t (3)

27 What are? L The flux tube bounding k quarks in the fundamental and k antiquarks: ( ) T k π k = sin. T f N Analytic results: Seiberg-Witten theory, (2+1)-models. Lattice results: Teper, Del Debbio, Lucini.

28 in the AdS/CFT correspondence 0123 R k S 3 S 2 A brane with flux along its worldvolume. Classical (solitonic) configuration. String theory reproduces the sin law: How about the fine structure?.

29 The Lüscher Term Quadratic fluctuations (fermions), zero point energy Lüscher term. Unofficial main contribution from massless dof V = T k L π 1 12 L +...

30 Technical details Hadronic Density from String Theory, with D. Vaman, Phys. Rev. Lett. 91 (2003) , hep-th/ Revisited in the Gauge/String, with J. Sonnenschein and D. Vaman, Nucl.Phys.B682:3-44,2004, hep-th/ for Mesons in the Holographic Dual of Large-N c QCD, with M. Kruczenski, J. Sonnenschein and D. Vaman, JHEP 0506:046,2005, hep-th/ Wilson Loop, Regge Trajectory and Hadron Masses in a Yang-Mills Theory from Semiclassical Strings, with F. Bigazzi, A. Cotrone and L. Martucci, Phys. Rev. D71 (2005) , hep-th/ On the Universality Class of Certain String Theory Hadrons, with G. Bertoldi, F. Bigazzi, A. Cotrone and C. Núñez, Nucl.Phys.B700:89-139,2004, hep-th/

31 : We have the Bohr energy levels down. A precise restatement of Regge theory: string amplitudes, Froissart bound. The full spectrum of QCD and gauge theories. A unification of ideas