Spiky Strings in the SL(2) Bethe Ansatz

M. Kruczenski

Purdue University

Based on: arXiv:0905.3536(L. Freyhult, A. Tirziu, M.K.)

Q uantum Theory and Sym m etries 6Q uantum Theory and Sym m etries 6Q uantum Theory and Sym m etries 6Q uantum Theory and Sym m etries 6 , 2009

Summary

Introduction

String / gauge theory duality (AdS/CFT )

Classical strings and field theory operators:folded strings and twist two operators

Spiky strings and higher twist operators

Classical strings moving in AdS and theirfield theory interpretation

Spiky strings in flat spaceQuantum description.

Spiky strings in Bethe AnsatzMode numbers and BA equations at 1-loop

Solving the BA equationsResolvent

Conclusions and future work

Extending to all loops we a find precise matching with the results from the classical string solutions.

String/gauge theory duality: Large N limit (‘t Hooft)

mesons

String picture

, , ...

Quark model

Fund. strings

( Susy, 10d, Q.G. )

QCD [ SU(3) ]

Large N-limit [SU(N)]

Effective strings

q q

Strong coupling

q q

Lowest order: sum of planar diagrams (infinite number)

N g N fixedYM→ ∞ =, λ 2More precisely: (‘t Hooft coupl.)

AdS/CFT correspondence (Maldacena)

Gives a precise example of the relation betweenstrings and gauge theory.

Gauge theory

N = 4 SYM SU(N) on R4

A , i, a

Operators w/ conf. dim.

String theory

IIB on AdS5xS5

radius RString states w/ ∆ E

R=

∆

g g R l g Ns YM s YM= =2 2 1 4; / ( ) /

N g NYM→ ∞ =, λ 2 fixedlarge string th.small field th.

Can we make the map between string and gaugetheory precise? 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

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.

H S Sj jj

J

= − ⋅

+

=∑

λπ4

1

42 11

r r

Rotation in AdS 5? (Gubser, Klebanov, Polyakov)

Y Y Y Y Y Y R12

22

32

42

52

62 2+ + + − − = −

sinh ; [ ]2

3ρ Ω cosh ;2 ρ t

ds dt d d2 2 2 2 23

2= − + +cosh sinh [ ]ρ ρ ρ Ω

( )

E S S S

O Tr x z tS

≅ + → ∞

= ∇ = ++ +

λπ ln , ( )

,Φ Φ= t

Generalization to higher twist operators (MK)

( )O TrnS n S n S n S n

[ ]/ / / /= ∇ ∇ ∇ ∇+ + + +Φ Φ Φ ΦK( )O Tr S

[ ]2 = ∇ +Φ Φ

x A n A n

y A n A n

= − + −= − + −

+ −

+ −

cos[( ) ] ( ) cos[ ]

sin[( ) ] ( ) sin [ ]

1 1

1 1

σ σσ σ

In flat space such solutions are easily found in conf. gauge

Spiky strings in AdS:

( )

E Sn

S S

O Tr S n S n S n S n

≅ +

→ ∞

= ∇ ∇ ∇ ∇+ + + +

2

λπ ln , ( )

/ / / /Φ Φ Φ ΦK

–2

–1

0

1

2

–2 –1 1 2

–2

–1

0

1

2

–2 –1 1 2

Beccaria, Forini, Tirziu, Tseytlin

Spiky strings in flat space Quantum case

x A n A n

y A n A n

= − + −= − + −

+ −

+ −

cos[( ) ] ( ) cos[ ]

sin[( ) ] ( ) sin [ ]

1 1

1 1

σ σσ σ

Classical:

Quantum:

Operators with large spin SL(2) sector

Spin chain representation

si non-negative integers.

Spin S=s1+…+sLConformal dimension E=L+S+anomalous dim.

Bethe Ansatz

S particles with various momenta moving in a periodicchain with L sites. At one-loop:

We need to find the uk (real numbers)

For large spin, namely large number of roots, wecan 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(λ) ln S

Belitsky, Korchemsky, Pasechnik described in detailL=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 anda>0. Each root has an associated wave number nw. We choose nw=-1 for u<0 and nw=n-1 for u>0.Solution?

d aDefine

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 usingthe all-loop BA (BES).

We obtain (see Alin’s talk)

In perfect agreement with the classical string result.We also get a prediction for the one-loop correction.

Conclusions

We found the field theory description of the spikystrings in terms of solutions of the BA ansatz equations.At strong coupling the results 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-Jinfound using the sinh-Gordon model.Relation to elliptic curves description found by

Dorey and Losi and Dorey.Semiclassical methods?