String / gauge theory duality and ferromagnetic spin chains M. Kruczenski Princeton Univ. In collaboration w/ Rob Myers, David Mateos, David Winters Arkady Tseytlin, Anton Ryzhov
Summary Introduction mesons,,... q String picture Quark model q q q Fund. strings ( Susy, 0d, Q.G. ) QCD Large N-limit Strong coupling Effective strings
AdS/CFT N = 4 SYM II B on AdS 5 xs 5 S 5 : X +X + X 6 = R AdS 5 : Y +Y + -Y 5 -Y 6 =-R deform Strings? AdS/CFT Known examples QCD
Add quarks We get models where the spectrum of qq bound states can be computed in the strong coupling regime Strings from gauge theory Problem: Compute scaling dimension (or energy) of states of a large number of particles in N =4 SYM Equivalent to solving Heisenberg spin chain (Minahan and Zarembo). Use effective action for spin waves: is the same as the string action AdS/CFT predicts!.
Introduction String theory ) Quantum field theory: Relativistic theory of point particles ) String theory: Relativistic theory of extended objects: Strings Why? Original motivation: Phenomenological model for hadrons (proton, neutron,pions, rho, etc.)
Regge trajectories M (GeV) 7 6 5 4 3 a 3 a 4 0 0 3 4 5 6 7 J Simple model of rotating strings gives 5 a 6 6 5 4 3 0 3 4 5 6 E J improvement m Strings thought as fundamental m
Theoretical problems Tachyons Taking care by supersymmetry Quantum mechanically consistent only in 0 dim. Unified models? Including gravity 5 types of strings
What about hadrons? Instead: bound states of quarks. mesons: qq baryons: qqq Interactions: SU(3); q = ; A = quarks gluons Coupling constant small at large energies (00 GeV) but large at small energies. No expansion parameter. Confinement V=k r (color) electric flux=string? V=- k/r
Idea ( t Hooft) Take large-n limit, q= ; A = N N x N N, g YM N = λ fixed ( t Hooft coupling) /N: perturbative parameter. Planar diagrams dominate (sphere) Next: /N corrections (torus) + /N 4 (-handles) + Looks like a string theory Can be a way to derive a string descriptions of mesons
AdS/CFT correspondence (Maldacena) Gives a precise example of the relation between strings and gauge theory. Gauge theory String theory N = 4 SYM SU(N) on R 4 A, i, a Operators w/ conf. dim. IIB on AdS 5 xs 5 radius R String states w/ 4 s YM s YM g = g ; R / l = ( g N ) / E = R YM N, λ = g N fixed large small string th. field th.
Mesons (w/ D. Mateos, R. Myers, D. Winters) We need quarks (following Karch and Katz) 3+ bdy z=0 q q D-brane q q bound state=string z ds = = dt + d x r + d z z So, in AdS/CFT, a meson is a string rotating in 5 dim.!
Meson spectrum (N = 4 is conformal Coulomb force) m q Coulomb E = m α / J q Regge (numerical result) J / ( g N) / 4 YM The cases J=0, ½, are special, very light, namely tightly bound. (E b ~ m q )
For J=0,/, we can compute the exact spectrum (in t Hooft limit and at strong coupling) scalars (M/M 0 ) = (n+m+) (n+m+), m 0 scalar (M/M 0 ) = (n+m+) (n+m+), m scalar (M/M 0 ) = (n+m+) (n+m+3), m scalar (M/M 0 ) = (n+m) (n+m+), m vector (M/M 0 ) = (n+m+) (n+m+), m 0 fermion (M/M 0 ) = (n+m+) (n+m+), m 0 fermion (M/M 0 ) = (n+m+) (n+m+3), m 0 n 0 ; there is a mass gap of order M 0 for m q 0 M L mq = = < < mq for gym N > > R g N 0 YM
Confining case (w/ D. Mateos, R. Myers, D. Winters) Add quarks to Witten s confining bkg. Spectrum is numerical For m q =0 there is a massless meson. (M =0) Goldstone boson of chiral symmetry breaking M m For m q 0 GMOR- relation φ = f q φ ψψ Rot. String (w/ Vaman, Pando-Zayas, Sonnenschein) reproduces improved model : m m
We can compute meson spectrum at strong coupling. In the confining case results are similar to QCD. How close are we to QCD? 5-dim Ideal sit. E In practice E QCD confinement M KK 4-dim QCD quarks gluons mesons glueballs 5-dim M KK 4-dim dim. red.
Can we derive the string picture from the field theory? (PRL 93 (004) M.K.) Study known case: N = 4 SYM Take two scalars X = + i ; Y= 3 + i 4 O = Tr(XX Y..Y X), J X s, J Y s, J +J large Compute -loop conformal dimension of O, or equiv. compute energy of a bound state of J particles of type X and J of type Y (but on a three sphere) R 4 S 3 xr E
Large number of ops. (or states). All permutations of Xs and Ys mix so we have to diag. a huge matrix. Nice idea (Minahan-Zarembo). Relate to a phys. system Tr( X X Y X X Y ) operator conf. of spin chain mixing matrix op. on spin chain J j+ j= λ r r H = S S 4 j 4π Ferromagnetic Heisenberg model!
Ground state (s) Tr( X X X X X X ) Tr( Y Y Y Y Y Y ) First excited states k π n i k l = e......, k = ; ( J = J + J ) l J λ ε( k) = ( + cos k) k 0 J λn J More generic (low energy) states: Spin waves (BMN)
Other states, e.g. with J =J Spin waves of long wave-length have low energy and are described by an effective action in terms of two angles, : direction in which the spin points. Seff. = J dσ dτ cosθ τ φ λ dσ dτ σθ + θ σ φ 3π J [ ( ) sin ( ) ] Taking J large with /J fixed: classical solutions
According to AdS/CFT there is a string description particle: X(t) string: X(,t) We need S 3 : X +X +X 3 +X 4 = R J J CM: J Rot: J Action: S[ (,t), (,t) ], which, for large J is: (agrees w/ f.t.) λ Seff. = J dσ dτ cos θ τ φ ( σ ) + sin ( σ ) 3π J [ θ θ φ ]
Suggests that (, ) = (, ) namely that S is the position of the string Examples point-like
Strings as bound states Fields create particles: X x, Y y Q.M. : ψ = cos(θ /) exp(i φ / ) x + sin(θ /) exp(- i φ /) y We consider a state with a large number of particles i= J each in a state v i = ψ (θ i, φ i ). (Coherent state) Can be thought as created by O = Tr (v v v 3 v n )
x θ = 0 Ordered set of points looks like a string y θ = π/ S Internal space Strings are useful to describe states of a large number of particles (in the large N limit)
Extension to higher orders in the field theory: O( ) We need H (Beisert et al.) J λ r r H = S S 4π 4 j j= j+ J + λ 3 r r r r + J + 8 S S S S 8π j j+ j j= j= J j+ At next order we get second neighbors interactions
We have to define the effective action more precisely n n n 3 n 4 n 5 n i Look for states ψ such that r ψ S ψ / / n ; n j j j = From those, find ψ such that ψ H ψ = ( ) =minimum E n j S = p q& H( p, q) = cos θ φ i t i E( ni ) n i = (sin θ i cos ϕ i,sinθ i sin ϕ i,cos θ i ) i
After doing the calculation we get: (Ryzhov, Tseytlin, M.K.) S eff. J = d σ d τ cosθ φ J d σ d τ τ 8J λ 3J 4 3 4 ( n) ( n) σ 4 σ λ ( n) σ Agrees with string theory!
Rotation in AdS 5? (Gubser, Klebanov, Polyakov) 3 4 Y + Y + Y + Y Y Y = R 5 6 sinh ; [ 3] ρ Ω cosh ρ ; t ds = cosh ρ dt + dρ + sinh ρ dω [ ] 3 = t λ E S + S S π ln, ( ) ( Φ S Φ ) O = Tr, x = z + t + +
Verification using Wilson loops (MK, Makeenko) The anomalous dimensions of twist two operators can also be computed by using the cusp anomaly of light-like Wilson loops (Korchemsky and Marchesini). In AdS/CFT Wilson loops can be computed using surfaces of minimal area in AdS 5 (Maldacena, Rey, Yee) z The result agrees with the rotating string calculation.
Generalization to higher twist operators (MK) 0 0 E S n + S S λ π ln, ( ) O ( S/ n S / n S / n S/ n Φ Φ ΦK Φ ) = Tr + + + + λ λ S = dt (cosh ρ j dt + ) θ ρ π & 4 ln sin j 8π j θ j+ θ j
Conclusions AdS/CFT provides a unique possibility of analytically understanding the low energy limit of non-abelian gauge theories (confinement) Two results: Computed the masses of quark / anti-quark bound states at strong coupling. Showed a way in which strings directly emerge from the gauge theory.