Large spin operators in string/gauge theory duality

M. Kruczenski

Purdue University

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

M iam iM iam iM iam iM iam i 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 AnsatzMode numbers and BA equations at 1-loop

Solving the BA equations1 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

, , ...

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, BMN). Relate to a phys. system, e.g. for strings rotating on S3

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 on 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

( )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:

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 S=s1+…+sLConformal 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 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 detail theL=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

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.

0

0.5

1

1.5

2

2.5

3

3.5

-500 -400 -300 -200 -100 100

Particular limit: 1 – cut solution- can be obtained when parameters are taken to zero. - recovers scaling

Particular limit: pp-wave type scalingIn 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 stringin global AdS

Periodic spike in AdS pp-wave

If we do not take number of spikes to infinity we get asingle 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 spikystrings 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-Jinfound 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?