Large spin operators in string/gauge theory duality M. Kruczenski Purdue University Based on: arXiv: (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) mesons String picture π, ρ,... Quark model Fund. strings ( Susy, 10d, Q.G. ) QCD [ SU(3) ] Large N-limit [SU(N)] Effective strings qq Strong coupling q Lowest order: sum of planar diagrams (infinite number) More precisely: (‘t Hooft coupl.)

AdS/CFT correspondence (Maldacena) Gives a precise example of the relation between strings and gauge theory. Gauge theory N = 4 SYM SU(N) on R 4 A μ, Φ i, Ψ a Operators w/ conf. dim. String theory IIB on AdS 5 xS 5 radius R String states w/ fixed 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 S 3 Tr( X X…Y X X Y ) | ↑ ↑…↓ ↑ ↑ ↓ › operator conf. of spin chain mixing matrix op. on spin chain 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 AdS 5 (Gubser, Klebanov, Polyakov) θ = ω t

Generalization to higher twist operators In flat space such solutions are easily found in conf. gauge:

Spiky strings in AdS: Beccaria, Forini, Tirziu, Tseytlin

Spiky strings in flat space Quantum case Classical: Quantum:

Strings rotating on AdS 5, in the field theory side are described by operators with large spin. Operators with large spin in the SL(2) sector Spin chain representation s i non-negative integers. Spin S=s 1 +…+s L Conformal dimension 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 u k (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 n w. We choose n w =-1 for u 0. Solution?

d a Define and We get on the cut: Consider i -i C z

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 are taken to zero. - recovers scaling ●  Particular limit: pp-wave type scaling In string theory: this limit is seen when zooming near the boundary of AdS. Spiky string solution in this background the same as spiky string solution in AdS in the limit: solutions near the boundary of AdS – S is large

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:

How to get this pp-wave scaling at weak coupling ? can get it from 1-loop BA 2-cut solution This is leading order strong coupling in while

●  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?