I have a lot of confusion about the AdS/CFT with holography. Does it show that strings and branes are excitations of fields on the conformal boundary of the universe (CFT)? Can someone explain the AdS/CFT holographic universe idea to me?

As far as I can tell, there is a formalism of the theory where strings and branes are fundamental (AdS), and the other formalism quantum fields are what is fundamental and strings/branes are excitations of the fields (CFT)...

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    $\begingroup$ AdS/CFT is a duality between theories, it's not there to demonstrate strings and branes as excitations of fields. If you have this confusion I'm guessing you don't know basic quantum field theory? $\endgroup$ – JamalS Nov 19 '17 at 23:15

The $AdS_n~\sim~CFT_{n-1}$ means the gravitational content in the interior of the anti-de Sitter spacetime is equivalent to the conformal field theory on the boundary. The equivalency is holographic because of the equivalency in the bulk spacetime and the boundary of one dimension less.

Now consider this with the corresponding $CFT_1$. The conformal field theory transforms under a conformal rescaling so under a transformation its action is invariant, up to boundary terms, under the transformation. In our case a conformal transformation is given by $$ t'~=~\lambda t,~\phi'~=~\lambda ^{-\Delta}\phi $$ where $\Delta$ is the scaling dimension of $\phi$, which is just its energy dimension classically.

Consider a Lagrangian with only the kinetic term and infer the dimension. To this end plug the transformed variables into the action $$ S'~=~\int dt' \frac{1}{2}\left(\frac{d\phi'}{dt'} \right)^2~=~\lambda^{-2\Delta-1}S. $$ which for the action invariant requires $\Delta = -\frac{1}{2}$. Any potential term $V~=~g\phi^n$ the transformation $g'~=~\lambda^m$ and thus contributes an action term $V'dt'~=~\lambda^{m~+~1~+~n/2}\phi^n$, which requires $ m~+~1~+~n/2~=~0$. The case for scale free $g$, or $m~=~0$, means $n~=~-2$. Consider Gauss's law, for a force or field $F~=~-dV/dt$. This defines a “charge” as this force integrated over the interval (volume) $q~=~\int_a^bFdt$ $=~V(b)~-~V(a)$. The potential field in this case is $V~=~0$, where the constant term evaluated is the charge. The charge is then not determined. The Lagrangian is then $$ L~=~\frac{1}{2}\left(\frac{d\phi}{dt} \right)^2~-~g\phi^{-2}. $$ I did a quick numerical solution that I include below. This is the hyperbolic circle around the Poincare disk. This elementary $CFT_1$ solution is then an illustration on how the boundary of the anti-de Sitter spacetime is equivalent to the interior dynamics.

enter image description here

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  • $\begingroup$ Thanks for answering. But I'm just a laymen so it's tough for me to understand the math. Are strings and branes still fundamental objects in the AdS/CFT, or are they excitations of more fundamental fields? Sorry to confuse $\endgroup$ – Nod Nov 20 '17 at 0:30

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