The AdS^2_{\theta}/CFT_1 Correspondence and Noncommutative Geometry I: A QM/NCG Correspondence

Abstract

A consistent QM/NCG duality is put forward as a model for the AdS^2/CFT_1 correspondence. This is a duality/correspondence between 1) the dAFF conformal quantum mechanics (QM) on the boundary (which is only "quasi-conformal" in the sense that there is neither an SO(1,2)-invariant vacuum state nor there are strictly speaking primary operators), and between 2) the noncommutative geometry of AdS^2_{\theta} in the bulk (which is only "quasi-AdS" in the sense of being only asymptotically AdS^2). The Laplacian operators on noncommutative AdS^2_{\theta} and commutative AdS^2 have the same spectrum and thus their correlators are conjectured to be identical. These bulk correlation functions are found to be correctly reproduced by appropriately defined boundary quantum observables in the dAFF quantum mechanics. Moreover, these quasi-primary operators on the boundary form a subalgebra of the operator algebra of noncommutative AdS^2_{\theta}.