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Conformal Perturbation Theory and LLM Geometries


This dissertation will focus on various aspects of the AdS/CFT correspondence. Each new result can be thought of as doing at least one of three things: 1) providing support of the duality, 2) using the duality to learn about quantum gravity, and 3) helping to further develop our understanding of the duality. The dissertation is divided into two parts, each dealing with a different physical system.

In the first part, we derive universal results for near conformal systems, which we have perturbed. In order to do this, we start by looking at the conformal correlation functions and compute the corrections that arise when he hit the system with a new operator. We were able to analyze what happens to the dual gravitational system under such circumstances and see that our answers agree, providing support for the AdS/CFT conjecture. These universal results also provided a previously lacking interpretation of the universality of energy found in a quenching your system between the perturbed and unperturbed set-ups. In order to perform these computations, we put our CFT on a cylinder, which happens to be the boundary of global AdS. This provided an IR regulator and we found that the remaining divergences were of the same form as one expects in dimensional regularization. Following along these same lines, we further analyzed the divergence structure of correlators in conformal perturbation theory. We found that on the plane, the logarithmic divergences that show up can be understood in terms of resonant behavior in time dependent perturbation theory, for a transition between states

that is induced by an oscillatory perturbation on the cylinder.

In part two, we restrict to the set of LLM geometries, which are the set of 1/2 BPS

solutions to IIB supergravity. In our first work, we analyzed limitations of the duality, showing that boundary expectation values are not enough to determine the classical bulk geometry. Next, we used this system in order to learn about quantum gravity. We first were able to show that a quantum superposition of states with a well defined spacetime topology leads to a new state with a different topology. From this, we were able to prove that for this set of states there cannot exist a quantum topology measuring operator, bringing to doubt whether such an operator can exist in quantum gravity more generally. Finally, we were able to advance our understanding of the dictionary itself by reinterpreting these results in terms of the language of quantum error correction, showing that questions about topology perhaps only make sense within a particular (code) subspace of states.

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