Chanyong Park (CQUeST) @ 35th Johns Hopkins Workshop (Budapest, 22-24 June 2011)

Based on Phys. Rev. D 83, 126004 (2011) arXiv : 1104.1896 arXiv : 1105.3279 collaborated with B.H. Lee and X. Bai

Plan 1. Review of the solitonic string

2. Correlation functions of magnon

3. Correlation functions of other solutions

4. Finite size effect on the 3-pt correlation function

5. Conclusion and discussion

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1.

1. Review of magon and spike

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1.

Magnon in the gauge theory (spin chain model)

Consider a gauge invariant (heavy) scalar operator which can be interpreted as a magnon in the spin chain

model. The anomalous dimension of magnon

In the large ‘t Hooft coupling limit,

Magnon

2sin1 2

2

pJ

[Minahan and Zarembo ’02]5

In the string theory the magnon operator corresponds to a solitonic stringrotating on , which is called the giant magnon.2S

In the string world sheetIn the target space

Consider the action for string moving in

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The dispersion relation

where

& = infinite and =finite

Conserved charges (in the infinite size limit )

Giant Magnon2S

2sin

pJE

[Hofman & Maldacena ’06]

p

and

The typical structure of the magnon’s dispersion relation in the infinite size limit is described by

This dispersion relation is exactly the same as one in the spin chain model in the large ‘t Hooft coupling limit

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2) spike (another solution in the different parameter regime)

Conserved charges

The dispersion relation

where & = infinite

=finite

Spike in the target space8

Notice

The conformal field theory (CFT) is usually characterized by the conformaldimension of all primary operators and the structure constant included in the three-point correlation functions, because higher point functions may be determined by using the operator product expansion (OPE).

- After finding an integrable structure in N=4 SYM theory, there were great progresses in calculating the spectra (the anomalous dimensions) of various operators.

- On the contrary, although the structure constant describing the interaction can be evaluated in the weak coupling limit by computing the Feynman diagrams, at the strong coupling there still remain many things to be done.

From now on, we will investigate the three-point correlation function of two heavy operators (magnon or spike) and one light (marginal) operator.

2. Correlation functions of magnon

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Solitonic string on the Poincare AdS

The Euclidean AdS metric in the Poincare patch

The string action on is given by

1) AdS part

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we can find

: modular parameter of the cylinder

Notice that we do not use the Virasoro constraints.

In AdS space, the string propagates as a point particle.

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The equations of motion are reduced to

1) part2S

where and are two integration constants.

Notice that there are two additional integration constants, which determine the position of magnon. Because the dispersion relation is described by the conserved quantities which contain one derivative, these two additional integration constants are irrelevant in determining the dispersion relation.

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Boundary conditions for fixing two integration constants

1)which plays an important role to determine the size of magnon and spike.

In the infinite size limit ( )

2)

which guarantees that even the angle difference is finite while the energy and the angular momentum are infinite.

After imposing these boundary conditions, we finally obtain

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The JSW proposed that ( ) -> the two point correlation function of heavy operator in gauge theory is proportional to the string partition function at the saddle point.

I. Two-point function of Magnon

( which is nothing but the Virasoro constraints )

Following the JSW procedure, after convolving the semiclassical propagator with the wave function of the state that we are interested in, we obtain

JSW : Janik, Surowka and Wereszczynski, arXiv:1002.4613

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Two-point function

Using the definitions of the conserved charges, we can reproduce the dispersion relation of the magnon in the large ‘t Hooft coupling regime

Energy of the giant magnon

the conformal dim. of magnon

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Now, calculate the three-point correlation functionbetween two heavy operators and one marginalscalar operator

2. Three-point function of Magnon

Following the AdS/CFT correspondence, the SUGRA field dual to the marginal scalar operator is the dilaton (massless scalar) .

[Costa, Monteiro, Santos, Zoakos , JHEP 1011 (2010) 141 [arXiv:1008.1070]]

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Finally, we obtain

The CFT result is

- the powers in the denominator are fixed by the global conformal transformation

- the structure constant is not determined by conformal symmetry

By comparing above two results, we can determine the structure constant

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* The structure constant in the gauge theory

[Costa, Monteiro, Santos, Zoakos , arXiv:1008.1070]

From the RG analysis, it was shown that the structure constant of themarginally deformed theory can be determined by

: coupling between two op. and one marginal op. : the anomalous dimension of two op.

For two heavy op. (magnon) and one marginal op., from the dispersion relation of magnon we can find

in the large coupling limit

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3. Correlation functions of other solutions

using the same method, we calculated the correlation functions of various Solitonic strings.

1. Dyonic magnon

which is described by the solitonic string rotating on 3S

Two-point correlation function

Three-point correlation function

in the RG analysis

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2. Single spike

which is described by the solitonic string rotating on in the different parameter range

Two-point correlation function

Three-point correlation function

2S

with

in the RG analysis

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4. Finite size effect on the 3-pt correlation functionthe finite size effect of the giant magnon ~ the wrapping effects in the spin chain model

Consider the case of

The conserved charges for the giant magnon

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Two-point correlation function

Three-point correlation function for ,

This result coincides with the RG calculation

with

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4. Conclusion and discussion

- Using the [JSW] & (CMSZ) prescription, we calculated the two- and three-point correlation functions of various solitonic string solutions

- Checked that these prescriptions are working well.

- Showed that the correlation functions in the string and gauge theory are perfectly matched, which is another evidence of the AdS/CFT correspondence.

- Calculated the finite size effect on the three-point function of the giant

magnon

JSW : Janik, Surowka, WereszczynskiCMSZ : Costa, Monteiro, Santos, Zoakos

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Thank you !