Abstract
In this chapter we apply the inverse scattering and the algebraic Bethe ansatz to reduce the evaluation of form factors and correlation functions to the calculation of a product of Bethe states. In particular we present a method to compute correlation functions of spin operators located at arbitrary sites of the spin chain. We will focus our analysis on the SU(2) sector of N=4 supersymmetric Yang-Mills at weak-coupling. At one-loop we provide a systematic treatment of the apparent divergences arising from the algebra of the elements of the monodromy matrix of an homogeneous spin chain. Beyond one-loop the analysis can be extended through the map of the long-range Bethe ansatz to an inhomogeneous spin chain.
Taking this point of view, there is a possibility afforded of a satisfactory, that is, of a useful theory [...], never coming into opposition with the reality, and it will only depend on national treatment to bring it so far into harmony with action, that between theory and practice there shall no longer be that absurd difference which an unreasonable theory, in defiance of common sense, has often produced, but which, just as often, narrow-mindedness and ignorance have used as a pretext for giving way to their natural incapacity
Carl Von Clausewitz, On War [1], Book II, Chapter II
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Notes
- 1.
We impose the trace condition on the two-magnon state rather than on both states, because in this later case the correlation function becomes zero. From the CFT point of view this happens because the one-excitation state is not a new primary operator but a descendent of the vacuum.
- 2.
We want to thank N. A. Slavnov for discussions about this subject.
- 3.
It is important to stress here that this “norm” cannot be computed using the Gaudin determinant (5.2.45) because it assumes the fulfilling of the Bethe equation and \(\left| \xi \right\rangle \) is not a Bethe state.
- 4.
Because of periodicity it is unnecessary to write the explicit expression for C.
- 5.
From the point of view of the AdS/CFT correspondence, this can be seen as the degeneration of the torus that uniformized the magnon dispersion relation at weak-coupling when one of the periods becomes infinitely large, thus forbidding us the access to the crossing transformation.
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Nieto, J.M. (2018). Two-Points Functions and ABA. In: Spinning Strings and Correlation Functions in the AdS/CFT Correspondence. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-96020-3_6
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