Modave Lectures on Applied AdS/CFT with Numerics

TL;DR

This paper introduces applied AdS/CFT techniques with numerical methods, demonstrating their use through a holographic superfluid case study and presenting new results on superfluid density and sound speed.

Contribution

It provides an accessible introduction to applied AdS/CFT with numerics and includes novel numerical and analytical results on superfluid properties.

Findings

Numerical evidence and proof that superfluid density equals particle density.

Confirmation of the conformal field theory prediction for sound speed saturation.

Application of numerics to solve differential equations in holographic models.

Abstract

These lecture notes are intended to serve as an introduction to applied AdS/CFT with numerics for an audience of graduate students and others with little background in the subject. The presentation begins with a poor man's review of current status of quantum gravity, where AdS/CFT correspondence is believed to be the well formulated quantum gravity in the anti-de Sitter space. Then we present the basic ingredients in applied AdS/CFT and introduce the relevant numerics for solving differential equations into which the bulk dynamics collapses. To demonstrate how to apply AdS/CFT with numerics, we take the zero temperature holographic superfluid as a concrete example for case study. In passing, we also present some new results, which include the numerical evidence as well as an elegant analytic proof for the equality between the superfluid density and particle density, namely $\rho_s=\rho$, and the saturation to the predicted value $\frac{1}{\sqrt{2}}$ by conformal field theory for the sound speed in the large chemical potential limit.