correlation functions of the solitonic string
DESCRIPTION
Correlation functions of the solitonic string. Chanyong Park (CQUeST) @ 35 th 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 - PowerPoint PPT PresentationTRANSCRIPT
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
3
1.
1. Review of magon and spike
4
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
6
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
7
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
9
Solitonic string on the Poincare AdS
The Euclidean AdS metric in the Poincare patch
The string action on is given by
1) AdS part
10
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.
11
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.
12
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
13
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
14
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
15
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]]
16
17
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
18
* 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
19
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
20
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
21
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
22
Two-point correlation function
Three-point correlation function for ,
This result coincides with the RG calculation
with
23
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
24
25
Thank you !