Holography and Infrared Conformality in Two Dimensions

This is a very brief review of some results from Refs. [2] and [3]. In holographic renormalization, we studied the RG flow of a 2d N = (4, 4) CFT perturbed by a relevant operator, flowing to a conformal fixed point in the IR. Here, the supergravity dual is displayed, and the computation of correlators is discussed. The sample stress-energy correlator given here provides an opportunity to explicitly compare Zamolodchikov’s Cfunction to the proposal for a “holographic C-function”. First, I will recall how to compute correlators holographically, even in the presence of domain walls (for a review, see [1]). As a simple analogy to keep in mind, consider a medieval castle where soldiers are practicing cannon-firing from atop the castle walls into the interior courtyard. Let us say that when the wall has height h0, a horizontally fired cannon ball hits the ground precisely in the middle of the courtyard. A priori the height of the wall h and the angle of firing θ would of course be independent boundary conditions, but by requiring that the cannon ball always hit the center of the courtyard, these two become related: h = h0/ cos 2 θ − r tan θ, for a circular wall of radius r, and neglecting air resistance. For an uneven castle wall, where the height of the wall varies as h(x) with the distance x around the circumference of the castle, the same relation relates θ(x) to h(x). Going back to AdS/CFT, the requirement that a bulk field φ vanishes in the deep interior relates a radial derivative at the boundary φ(1)(x) to the boundary value φ(0)(x). This relation encodes the desired boundary correlator