Spiky Strings in the SL(2) Bethe Ansatz
This presentation is the property of its rightful owner.
Sponsored Links
1 / 20

Spiky Strings in the SL(2) Bethe Ansatz PowerPoint PPT Presentation


  • 80 Views
  • Uploaded on
  • Presentation posted in: General

Spiky Strings in the SL(2) Bethe Ansatz. M. Kruczenski. Purdue University. Based on: arXiv:0905.3536 (L. Freyhult, A. Tirziu, M.K.). Quantum Theory and Symmetries 6 , 2009. Summary. ● Introduction. String / gauge theory duality ( AdS/CFT ) Classical strings and field theory operators:

Download Presentation

Spiky Strings in the SL(2) Bethe Ansatz

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Spiky strings in the sl 2 bethe ansatz

Spiky Strings in the SL(2) Bethe Ansatz

M. Kruczenski

Purdue University

Based on: arXiv:0905.3536

(L. Freyhult, A. Tirziu, M.K.)

Quantum Theory and Symmetries 6, 2009


Spiky strings in the sl 2 bethe ansatz

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 their

field theory interpretation


Spiky strings in the sl 2 bethe ansatz

● Spiky strings in flat space

Quantum description.

●Spiky strings in Bethe Ansatz

Mode numbers and BA equations at 1-loop

● Solving the BA equations

Resolvent

● Conclusions and future work

Extending to all loops we a find precise matching

with the results from the classical string solutions.


Spiky strings in the sl 2 bethe ansatz

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) ]

q q

Large N-limit [SU(N)]

q

q

Effective strings

Strong coupling

More precisely:

(‘t Hooft coupl.)

Lowest order: sum of planar diagrams (infinite number)


Spiky strings in the sl 2 bethe ansatz

AdS/CFT correspondence(Maldacena)

Gives a precise example of the relation between

strings and gauge theory.

Gauge theory

N = 4 SYM SU(N) on R4

Aμ , Φi, Ψa

Operators w/ conf. dim.

String theory

IIB on AdS5xS5

radius R

String states w/

λ large → string th.

λ small → field th.

fixed


Spiky strings in the sl 2 bethe ansatz

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 )| ↑ ↑…↓ ↑ ↑ ↓ ›

operatorconf. of spin chain

mixing matrixop. 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.


Spiky strings in the sl 2 bethe ansatz

Rotation on AdS5(Gubser, Klebanov, Polyakov)

θ = ω t


Spiky strings in the sl 2 bethe ansatz

Generalization to higher twist operators

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


Spiky strings in the sl 2 bethe ansatz

Spiky strings in AdS:

Beccaria, Forini,

Tirziu, Tseytlin


Spiky strings in the sl 2 bethe ansatz

Spiky strings in flat spaceQuantum case

Classical:

Quantum:


Spiky strings in the sl 2 bethe ansatz

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+…+sL

Conformal dimension E=L+S+anomalous dim.


Spiky strings in the sl 2 bethe ansatz

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)


Spiky strings in the sl 2 bethe ansatz

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(l) 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 the sl 2 bethe ansatz

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?


Spiky strings in the sl 2 bethe ansatz

i

-i

Define

and

We get on the cut:

Consider

d a

z

C


Spiky strings in the sl 2 bethe ansatz

We get

Also:

Since we get


Spiky strings in the sl 2 bethe ansatz

We also have:

Finally, we obtain:


Spiky strings in the sl 2 bethe ansatz

Root density


Spiky strings in the sl 2 bethe ansatz

We can extend the results to strong coupling using

the 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.


Spiky strings in the sl 2 bethe ansatz

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.

Semiclassical methods?


  • Login