In this paper the fixed-point Wilson action for the critical O(N ) model in D = 4 − ǫ dimensions is written down in the ǫ expansion to order ǫ 2 . It is obtained by solving the fixed-point Polchinski Exact Renormalization Group equation (with anomalous dimension) in powers of ǫ. This is an example of a theory that has scale and conformal invariance despite having a finite UV cutoff. The energy-momentum tensor for this theory is also constructed (at zero momentum) to order ǫ 2 . This is done by solving the Ward-Takahashi identity for the fixed point action. It is verified that the trace of the energy-momentum tensor is proportional to the violation of scale invariance as given by the exact RG, i.e., the β function.The vanishing of the trace at the fixed point ensures conformal invariance. Some examples of calculations of correlation functions are also given.

In earlier work, the evolution operator for the exact RG equation was mapped to a field theory in Euclidean AdS. This gives a simple way of understanding AdS/CFT. We explore aspects of this map by studying a simple example of a Schrödinger equation for a free particle with time-dependent mass. This is an analytic continuation of an ERG like equation. We show for instance that it can be mapped to a harmonic oscillator. We show that the same techniques can lead to an understanding of dS/CFT too.

Recently, a method was described for deriving Holographic RG equation in AdSD+1 space starting from an Exact RG equation of a D-dimensional boundary CFT [22]. The evolution operator corresponding to the Exact RG equation was rewritten as a functional integral of a D + 1 dimensional field theory in AdSD+1 space. This method has since been applied to elementary scalars and composite scalars in the O(N) model [34]. In this paper, we apply this technique to the conserved vector current and the energy momentum tensor of a boundary CFT, the O(N) model at a fixed point. These composite spin one and spin two operators are represented by auxiliary fields and extend into the bulk as gauge fields and metric perturbations. We obtain, at the free level, the (gauge fixed) Maxwell and Einstein actions. While the steps involved are motivated by the AdS/CFT correspondence, none of the steps logically require the AdS/CFT conjecture for their justification.