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Large spin operators in string/gauge theory duality M. Kruczenski Purdue University Based on: arXiv:0905.3536 (L. Freyhult, A. Tirziu, M.K.) Miami 2009 Summary ● Introduction String / gauge theory duality (AdS/CFT) Classical strings and their dual field theory operators: Folded strings and twist two operators. Spiky strings and higher twist operators. Quantum description of spiky strings in flat space. ● Spiky strings in Bethe Ansatz Mode numbers and BA equations at 1-loop ● Solving the BA equations 1 cut at all loops and 2 cuts at 1-loop. AdS-pp-wave limit. ● Conclusions and future work Extending to all loops we find a precise matching with the results from the classical string solutions. String/gauge theory duality: Large N limit (‘t Hooft) String picture Fund. strings ( Susy, 10d, Q.G. ) mesons π, ρ, ... Quark model QCD [ SU(3) ] qq q q Strong coupling Large N-limit [SU(N)] Effective strings 2 More precisely: N , gYM N fixed (‘t Hooft coupl.) Lowest order: sum of planar diagrams (infinite number) 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 R4 IIB on AdS5xS5 radius R String states w/ E Aμ , Φi, Ψa Operators w/ conf. dim. gs g ; R / ls ( g 2 YM N , g R 2 YM 2 YM N fixed N) 1/ 4 λ large → string th. λ small → field th. Can we make the map between string and gauge theory precise? Nice idea (Minahan-Zarembo, BMN). Relate to a phys. system, e.g. for strings rotating on S3 › Tr( X X…Y X X Y ) operator mixing matrix H 2 4 | ↑ ↑…↓ ↑ ↑ ↓ conf. of spin chain op. on spin chain J j 1 1 S j S j 1 4 Ferromagnetic Heisenberg model ! For large number of operators becomes classical and can be mapped to the classical string. It is integrable, we can use BA to find all states. Rotation on AdS5 (Gubser, Klebanov, Polyakov) Y Y Y Y Y Y R 2 1 2 2 2 3 2 4 sinh 2 ; [ 3] 2 5 2 6 2 cosh 2 ; t ds cosh dt d sinh d 2 2 2 2 2 2 [ 3] E S ln S , ( S ) θ=ωt O Tr S , x z t Generalization to higher twist operators O[ 2 ] Tr S O[ n ] Tr S / n S / n S / n S / n In flat space such solutions are easily found in conf. gaug x A cos[(n 1) ] A (n 1) cos[ ] y A sin[(n 1) ] A (n 1) sin [ ] Spiky strings in AdS: n E S ln S , ( S ) 2 O Tr S / n S / n S / n S / n Beccaria, Forini, Tirziu, Tseytlin Spiky strings in flat space Quantum case Classical: Quantum: x A cos[(n 1) ] A (n 1) cos[ ] y A sin[(n 1) ] A (n 1) sin [ ] Strings rotating on AdS5, in the field theory side are described by operators with large spin. Operators with large spin in the SL(2) sector Spin chain representation si non-negative integers. Spin Conformal dimension S=s1+…+sL E=L+S+anomalous dim. Again, the matrix of anomalous dimensions can be thought as a Hamiltonian acting on the spin chain. At 1-loop we have It is a 1-dimensional integrable spin chain. Bethe Ansatz S particles with various momenta moving in a periodic chain with L sites. At one-loop: We need to find the uk (real numbers) For large spin, namely large number of roots, we can define a continuous distribution of roots with a certain density. It can be generalized to all loops (Beisert, Eden, Staudacher E = S + (n/2) f() ln S Belitsky, Korchemsky, Pasechnik described in detail the L=3 case using Bethe Ansatz. Large spin means large quantum numbers so one can use a semiclassical approach (coherent states). Spiky strings in Bethe Ansatz BA equations Roots are distributed on the real axis between d<0 and a>0. Each root has an associated wave number nw. We choose nw=-1 for u<0 and nw=n-1 for u>0. Solution? i Define d and We get on the cut: Consider z C -i a We get Also: Since we get We also have: Finally, we obtain: Root density We can extend the results to strong coupling using the all-loop BA (BES). We obtain In perfect agreement with the classical string result. We also get a prediction for the one-loop correction. Two cuts-solutions and a pp-wave limit When S is finite (and we consider also R-charge J) the simplest solution has two cuts where the roots are distributed with a density satisfying: where, as before: The result for the density is (example): Here n=3, d=-510, c=-9.8, b=50, a=100, S=607, J=430 It is written in terms of elliptic integrals. ● Particular limit: 1 – cut solution - can be obtained when parameters zero. - recovers scaling are taken to ● Particular limit: pp-wave type scaling In string theory: this limit is seen when zooming near the boundary of AdS. solutions near the boundary of AdS – S is large Spiky string solution in this background the same as spiky string solution in AdS in the limit: In pictures: z Spiky string in global AdS Periodic spike in AdS pp-wave If we do not take number of spikes to infinity we get a single spike: while This is leading order strong coupling in How to get this pp-wave scaling at weak coupling ? can get it from 1-loop BA 2-cut solution ● Take while keeping fixed. pp-wave scaling: 1-loop anomalous dimension complicated function of only 3 parameters If are also large: 1-loop anomalous dimension simplifies: Conclusions We found the field theory description of the spiky strings in terms of solutions of the BA equations. At strong coupling the result agrees with the classical string result providing a check of our proposal and of the all-loop BA. Future work Relation to more generic solutions by Jevicki-Jin found using the sinh-Gordon model. Relation to elliptic curves description found by Dorey and Losi and Dorey. Pp-wave limit for the all-loops two cuts-solution. Semiclassical methods?