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Correlation functions ofthe solitonic string 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
Magnon 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, [Minahan and Zarembo ’02]
In the string theory the magnon operator corresponds to a solitonic string rotating on , which is called the giant magnon. Consider the action for string moving in In the string world sheet In the target space
Conserved charges (in the infinite size limit ) The dispersion relation Giant Magnon where and This dispersion relation is exactly the same as one in the spin chain model in the large ‘t Hooft coupling limit The typical structure of the magnon’s dispersion relation in the infinite size limit is described by & = infinite and =finite [Hofman & Maldacena ’06]
Conserved charges 2) spike (another solution in the different parameter regime) The dispersion relation & = infinite =finite where Spike in the target space
2. Correlation functions of magnon Notice The conformal field theory (CFT) is usually characterized by the conformal dimension of all primary operators and the structure constant included in the three-point correlation functions, because higher point functions maybe 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.
Solitonic string on the Poincare AdS The Euclidean AdS metric in the Poincare patch The string action on is given by 1) AdS part
: modular parameter of the cylinder In AdS space, the string propagates as a point particle. we can find Notice that we do not use the Virasoro constraints.
1) part The equations of motion are reduced to 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, thesetwo additional integration constants are irrelevant in determining the dispersion relation.
Boundary conditions for fixing two integration constants 1) which plays an important role to determine the size of magnon and spike. 2) which guarantees that even the angle difference is finite while the energy and the angular momentum are infinite. In the infinite size limit( ) After imposing these boundary conditions, we finally obtain
I. Two-point function of Magnon The JSWproposed that ( ) -> the two point correlation function of heavy operator in gauge theory is proportional to the string partition function at the saddle point. JSW: Janik, Surowka and Wereszczynski,arXiv:1002.4613 Following the JSW procedure, after convolving the semiclassical propagator with the wave function of the state that we are interested in, we obtain ( which is nothing but the Virasoro constraints )
Two-point function Energy of the giant magnon the conformal dim. of magnon Using the definitions of the conserved charges, we can reproduce the dispersion relation of the magnon in the large ‘t Hooft coupling regime
2. Three-point function of Magnon [Costa, Monteiro, Santos, Zoakos , JHEP 1011 (2010) 141 [arXiv:1008.1070]] Now, calculate the three-point correlation function between two heavy operators and one marginal scalar operator Following the AdS/CFT correspondence, the SUGRA field dual to the marginal scalar operator is the dilaton (massless scalar) .
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
* 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 the marginally 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
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 Two-point correlation function Three-point correlation function in the RG analysis
2. Single spike which is described by the solitonic string rotating on in the different parameter range Two-point correlation function with Three-point correlation function in the RG analysis
4. Finite size effect on the 3-pt correlation function the 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
Two-point correlation function Three-point correlation function for , with This result coincides with the RG calculation
JSW: Janik, Surowka, Wereszczynski CMSZ : Costa, Monteiro, Santos, Zoakos 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