Aspects of emergent geometry in the AdS/CFT context

Abstract

We study aspects of emergent geometry for the case of orbifold superconformal field theories in four dimensions, where the orbifolds are Abelian within the AdS/CFT proposal. In particular, we show that the realization of emergent geometry starting from the $\mathcal{N}=4$ super Yang-Mills (SYM) theory in terms of a gas of particles in the moduli space of vacua of a single D3 brane in flat space gets generalized to a gas of particles on the moduli space of the corresponding orbifold conformal field theory (a gas of D3 branes on the orbifold space). Our main purpose is to show that this can be analyzed using the same techniques as in the $\mathcal{N}=4$ SYM case by using the method of images, including the measure effects associated to the volume of the gauge orbit of the configurations. This measure effect gives an effective repulsion between the particles that makes them condense into a nontrivial vacuum configuration, and it is exactly these configurations that lead to the geometry of $X$ in the $\mathrm{AdS}\ifmmode\times\else\texttimes\fi{}X$ dual field theory.