Gauge-gravity duality in low dimensions
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Abstract
This thesis studies formal aspects of the gauge/gravity duality in the context of ensemble averaged holography. The former states that a (quantum)gravitational theory on a d-dimensional space can be identified with a quantum gauge theory on the boundary of this space—which is (d − 1)-dimensional. In ensemble average holography, the boundary theory is not a single theory but an ensemble of theories. That is to say, to match boundary quantities—such as partition functions—to bulk quantities we have to take an average over the ensemble of boundary theories. Here we focus on d = 2 and d = 3.
For d = 2, we study the established duality between Jackiw-Teitelboim (JT) gravity—a gravitational theory in two-dimensions which includes a scalar field—and random matrices. The latter can be seen as an ensemble of Hamiltonians on the one-dimensional boundary. We approach the problem from a topological gravity point of view in the sense that we express partition functions of JT gravity in terms of topological gravity correlators which obey the Korteweg-de Vries (KdV) hierarchy—a infinite system of differential equations. This approach is applied in particular to deformations of JT gravity, i.e. JT gravity with a more general potential, to map solutions of the KdV hierarchy to classes of JT deformations. This is explicitly achieved for a particular JT deformation, namely the one that introduces conical defects in the bulk manifold. We formulate generalisations to JT partition functions that when evaluated at specific values of their arguments give back the known JT partition functions and study their low temperature expansions with a focus on the ones that correspond to conical defects.
For d = 3, we focus on ensemble averages of Narain conformal field theories (CFTs) and their bulk duals. We start with a treatment of the former, by discussing how one can calculate ensemble averages of partition functions of such CFTs using the Siegel-Weil formula and generalise this to supersymmetric versions of Narain CFTs. Then, we introduce two classes of Z_2 Narain orbifolds called “factorisable” and “non-factorisable” and calculate the ensemble averages of their partition functions as well. Next, we treat symmetric product orbifolds of Narain CFTs and calculate ensemble averages of partition functions and correlators of twists fields.
Having discussed the technical details of Narain CFTs, orbifolds and their ensemble averages, we move on to study their potential bulk duals. Starting with the known Narain-U(1) gravity correspondence, we propose a bulk dual for the ensemble average of the symmetric product orbifold of Narain CFTs which can reproduce parts of the boundary ensemble averaged partition functions and correlators. Also, we generalise this to the supersymmetric case (at least for the partition functions). Finally, we speculate on ensemble averages of products of arbitrary CFTs and exemplify these ideas using products of Narain CFTs and the “factorisable” and “non-factorisable” orbifolds.