Wheeler-DeWitt Equation in AdS/CFT Correspondence

We discuss a quantum extension of the holographic RG flow equation obtained previously from the classical Hamiltonian constraint in the bulk AdS supergravity. The WheelerDeWitt equation is proposed to generate the extended RG flow and to produce 1/N subleading corrections systematically. Our formulation in five dimensions is applied to the derivation of the Weyl anomaly of boundary N = 4 SU(N) super-Yang-Mills theory beyond the large N limit. It is shown that subleading 1/N corrections arising from fields in AdS5 supergravity agree with those obtained recently by Mansfield et al. using their Schrödinger equation, thereby guaranteeing to reproduce the exact form of the boundary Weyl anomaly after summing up all of the KK modes. Since the proposal of the AdS/CFT correspondence by Maldacena [1], a lot of effort has been made to test his conjecture. Among several others, Henningson and Skenderis derived the boundary theory Weyl anomaly in the largeN limit by evaluating the tree-level on-shell action of the bulk AdS supergravity [2]. (See also [3].) The same result was also obtained by Mansfield and Nolland [4], where they constructed a functional Schrödinger equation for the partition function on AdS space and solved it at the tree-level. Recently the Schrödinger equation was fully considered to obtain a subleading correction to the large N result, and the exact form of the boundary Weyl anomaly has been successfully derived [5],[6]. On the other hand, an alternative formulation of the bulk theory was given by de Boer, Verlinde and Verlinde [7], who gave attention to the fact that the radial flow in AdS space transverse to boundary directions corresponds to the renormalization group (RG) flow at the boundary. The classical Hamiltonian constraint arising from the reparameterization invariance of the bulk supergravity with respect to (w.r.t.) the radial direction was shown to be cast into the Callan-Symanzik RG flow equation on the boundary space. In this formulation, the derivation of the anomaly, at least in the large N limit, was seen to be performed more simply and quickly than with the other methods. The purpose of this article is to extend the classical Hamiltonian constraint to a quantum equation on the bulk AdS and to present a generalized holographic RG flow equation, which systematically produces subleading corrections in the 1/N expansion. Obviously, the Wheeler-DeWitt equation is a promising candidate for it and suggests that the subleading correction arises from terms with second-order functional derivatives w.r.t. fields of AdS supergravity, as happens in the ordinary WKB method of quantum mechanics. We demonstrate that the Wheeler-DeWitt equation applied to the AdS5/CFT4 case works indeed to derive the exact 1/N correction to the Weyl anomaly of the boundary N = 4 SU(N) super-Yang-Mills (SYM) theory. We start with a brief review of the tree-level (large N) calculation of the boundary Weyl anomaly via the holographic RG flow equation [7],[8],[9], in which the temporal gauge is used for the metric in (d+ 1) dimensions, ds = gab dx dx = dr + gμν(x, r) dx dx , (μ, ν = 1, · · · , d) (1) where gμν is dynamical in the bulk but supposed to tend to the form of an AdSd+1 metric asymptotically at the boundary r → ∞ [10], gμν(x, r) = e 2r/l [ ĝμν(x)− l d− 2 ( R̂μν − 1 2(d− 1) ĝμν )