AdS/CFT correspondence in operator formalism

Abstract

In this paper we study the AdS/CFT correspondence in the operator formalism without assuming the GKPW relation. We explicitly show that the low energy spectrum of the large N limit of CFT, which is realized by a strong coupling gauge theory, is identical to the spectrum of the free gravitational theory in the global AdS spacetime under some assumptions which are expected to be valid. Thus, two theories are equivalent for the low energy region under the assumptions. Using this equivalence, the bulk local field is constructed and the GKPW relation is derived.

Aspects of the first law of complexity

“…where P µ are the momentum generators, and s µ 1 µ 2 ···µ l lm is a symmetric traceless tensor, e.g., see [101].…”

“…Hence, to ensure proper orthogonality, one requires such time integration. A more traditional approach would associate the creation operators to the states created by the boundary operator and its descendants in the Euclidean theory, e.g., see [101]. The present construction (in particular eq.…”

Localized states in global AdS space

“…where Φ(ρ, t, Ω) is the scalar bulk field dual to the scalar primary CFT operator O ∆ . This approach was already followed in appendix B of [12] and extended to an arbitrary number of dimensions in [19].. We reproduce it here in any number of CFT dimensions. The basic idea is to use the properly normalized classical solution to the wave equation in global AdS [17,18,20]…”

Classical limit of large N gauge theories with conformal symmetry

“…In [5], instead of assuming such dictionaries or existence of a bulk dual, we studied the (low energy) spectrum of generic large N gauge theories with conformal symmetry, in the leading order in the large N limit, and found that it is identical to the spectrum of the Einstein gravity theory in global AdS d+1 in the free theory limit, following some earlier works [6,4,7]. Here, "generic" means that the theories satisfy the following two properties.…”

“…4 These properties are highly expected for the large N strongly coupled gauge theories. 5 With this explicit identification of the spectrum of CF T to the spectrum of the gravity on AdS space, we derived the GKPW relation.…”