AdS/CFT Correspondence and Quotient Space Geometry

TL;DR

This paper explores a generalized AdS/CFT correspondence where the bulk space is a quotient manifold, revealing how geometric singularities influence the boundary CFT spectrum and confirming the scalar two-point function matches conformal invariance expectations.

Contribution

It introduces a quotient space version of AdS/CFT, analyzing geometric effects on the holographic principle and boundary operator dimensions, with explicit calculations of correlation functions.

Findings

Singular boundary structures affect the physical spectrum.

Conformal dimensions increase with larger quotient groups.

Scalar two-point functions match conformal invariance predictions.

Abstract

We consider a version of the $AdS_{d+1}/CFT_{d}$ correspondence, in which the bulk space is taken to be the quotient manifold $AdS_{d+1} /\Gamma$ with a fairly generic discrete group $\Gamma$ acting isometrically on $AdS_{d+1}$. We address some geometrical issues concerning the holographic principle and the UV/IR relations. It is shown that certain singular structures on the quotient boundary ${\bf S}^{d}/\Gamma$ can affect the underlying physical spectrum. In particular, the conformal dimension of the most relevant operators in the boundary CFT can increase as $\Gamma$ becomes ``large''. This phenomenon also has a natural explanation in terms of the bulk supergravity theory. The scalar two-point function is computed using this quotient version of the AdS/CFT correspondence, which agrees with the expected result derived from conformal invariance of the boundary theory.