A remark on the AdS/CFT correspondence and the renormalization group flow

Abstract

The correspondence between the four-dimensional SU (N ), N = 4 SYM taken at large N and the type II B SUGRA on the AdS 5 ×S 5 background is considered. We argue that the classical equations of motion in the SUGRA picture can be interpreted as that of the renormalization group on the SYM side. In fact, when the D3-brane is slightly excited higher derivative terms in the field theory on its world-volume deform it form the conformal N = 4 SYM limit. We give arguments in favor of that the deformation goes in the way set by the SUGRA equations of motion. Concrete example of the s-wave dilaton is considered. Having in mind those facts, in this note we try to give a geometric interpretation of the renormalization group flow in the SYM theory. Our work is based on the proposed duality [10] between the four-dimensional large N SYM theory and type IIB SUGRA on a background which we describe below. Concretely, we argue that the classical equation of motion of the dilaton in the bulk SUGRA theory is nothing but the renormalization group equation for the SYM "coupling constant" on the D-brane world-volume [11]. As we discuss below, this fact is sensible if the dilaton -"coupling constant" -is excited and, hence, is a function of the four-dimensional coordinates. While if the dilaton is an arbitrary constant it remains to be the one at any energy scale. In the both cases the movement in the direction transversal to the D-brane is the renormalization group transformation [10,12] in the field theory on its world-volume.In fact, it is widely believed that super-strings suggest the regularization of field theory [13]. What is new in the D-brane case is that the regularization of their world-volume field theories can happen at much smaller energy scale than the Plank one, if some particular double scaling limit is taken [11,14,15]. The regularized theory should be considered as a vacuum in that of super-strings [10]. For after the regularization we are missing information 1

Energy and the AdS/CFT correspondence

“…We also aim to integrate such a Legendre transform into the generalized prescription Eq. (29) in such a way that there exists an exact agreement between the resulting AdS/CFT prescription and the quantization that we develop in this paper.…”

“…On the other hand, we will show that the constraints for which irregular modes propagate in the bulk are fixed, in a natural way, by the generalized AdS/CFT prescription Eqs. (29,31), and thus they are not fixed 'by hand', as it happens in the usual formalism. In the cases for which only regular modes propagate in the bulk, we will show that the Legendre transformation Eq.…”

“…In particular, we will show that the new constraints for which irregular modes propagate in the bulk are fixed in a natural way by the generalized AdS/CFT prescription Eqs. (29)(30)(31), and thus they need no longer to be fixed 'by hand', as it is the case when considering the usual formalism. Another point that we will show is that such new constraints are the same for which the divergent local terms of the on-shell action cancel out.…”

“…This provides strong evidence in support of our conjectures that the canonical energy is the natural one to be considered in the context of the AdS/CFT correspondence, and that the generalized prescription Eqs. (29)(30)(31) is needed in order to map to the border all the information contained in the bulk. We will show that the constraints Eqs.…”

“…Further refinements include the addition of local counterterms to the action [11] [28] which cancel the divergences arising when going to the boundary. We also mention the work on holographic renormalization group flows from deformations on AdS which extends the AdS/CFT correspondence to non-conformal field theories [29] [30][31] [32]. In this work, we concentrate on the formulation of the scalar field theory on AdS spaces in both minimally and non-minimally coupled cases.…”