Emergent locality, large N matrices & entanglement
Keywords: Physics.

Broadly speaking, this thesis concerns itself with diverse routes to understanding the reconciliation of gravity and quantum mechanics. Pervasive to the various topics explored lies the idea that spacetime is best understood as an emergent concept. Indeed, many past inquiries regarding the microscopic nature of the fabric of spacetime ended in deep paradoxes, the most famous being how the dynamics of black holes is to be understood in a way compatible with unitary time evolution. Fundamental to the emergent spacetime paradigm is the conclusion that the general relativistic notion of spacetime itself fails to even make sense at the smallest of distance scales. Instead, it should be replaced with a collection of non-geometric degrees of freedom. Much in the same way statistical mechanics exposed the macroscopic notions of heat and temperature as average quantities describing the motion of millions of millions of atomic particles, gravity should perhaps best be conceived as the hydrodynamics of these non-geometric variables. The articulation of the holographic principle in \cite{susskind1995world, hooft1993dimensional} sharpened these somewhat philosophical musings, appealing to the Bekenstein-Hawking entropy of black holes. In a nutshell, should one try to estimate the upper bound on the number of degrees of freedom needed to describe the physics in some finite patch of spacetime, one might imagine cramming in more and more "stuff" into said region, rendering the dynamics as complicated as possible. Eventually, the mass of the constituents reaches it equivalent Schwarzschild radius and forms a black hole. Black holes carry an associated entropy, proportional not to their volume but to their bounding area (or its higher/lower-dimensional analogues). This entropy can in turn be used to estimate the dimension of the Hilbert space one might like to associate to this region. Given the number of degrees of freedom scales not as the volume but rather as the area, it is clear the underlying theory cannot be a quantum field theory local to this patch, like those describing all other known fundamental forces. The advent of the BFSS Matrix theory \cite{banks1997m} and the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence \cite{Maldacena:1997re} provided concrete examples of the holographic principle. The AdS/CFT duality posits a non-gravitational quantum mechanical theory, not so different from those familiar in particle physics, may be completely recast as a gravitational one, in one higher dimension. While this suspersymmetric theory is a local quantum field theory formulated on the boundary of the dual Anti-de Sitter spacetime, how these degrees of freedom encode the low-energy \textit{bulk local} physics remains a difficult question. Significant progress was made with the discovery of the Ryu-Takayangi (RT) formula, which one might consider a generalization of the Bekenstein-Hawking entropy \cite{Ryu:2006bv}. It equates the entanglement entropy of a subregion of the space on which the CFT lives with the area of the bulk minimal surface ending on the boundary of said region, plus the entanglement entropy of bulk fields across this minimal surface \cite{Faulkner:2013ana}. This quickly led to important insights on bulk reconstruction and refined the duality to a subregion-subregion duality. Indeed, entanglement wedge reconstruction states that from this restricted CFT region, we can reconstruct the bulk operators contained within the entanglement wedge (a slight generalization of the region bounded by the minimal area surface described above). While the RT formula has taught us tremendous amounts about the holographic dictionary, it is not enough to address several important points. In particular, it fails to describe the emergence of bulk local physics on scales below the AdS-radius. One intuitive way to see this is to recall the renormalization group perspective on the holographic dimension. If we imagine flowing further into the bulk, integrating out more and more CFT modes, we are eventually left with only the zero-modes. This defines a matrix quantum mechanics, living on a point. With its $N^2$ degrees of freedom, we expect it to describe a whole AdS-worth of space. Clearly, there is no sense in which we can further partition the space on which the boundary theory now lives. As such, the RT formula cannot describe sub-regions below the AdS scale. We face an analogous situation in BFSS matrix theory, our only theory of quantum gravity in flat space. BFSS being "nothing more" than the quantum mechanics of matrices, there is similarly no obvious notion of subsystems mapping onto sub-regions of the emergent spacetime. Even the very notion of a subsystem is difficult to describe, an important observation which motivated much of the work in Chapter IV. Chapter I used the simplest model of holography involving a single matrix, and reproduced the entanglement entropy of the bulk fields starting from a partitioning of purely matrix degrees of freedom. Intimately connected to the holographic renormalization program and attempts at finite-patch holography is the $T \bar{T}$-deformation of field theories \cite{McGough:2016lol, Gorbenko:2018oov, Shyam:2017znq}. \cite{Zamolodchikov:2004ce} first described the deformation of conformal field theories by the irrelevant and (in flat space) well-defined composite operator $T \bar{T}$ built out of the theory's stress tensor. \cite{McGough:2016lol} related this deformation to holography, showing that at large central charge, the deformed theory seemed to describe the holographic gravitational dual contained "inside" a region, whose size was set by the amount the seed had been deformed by. One important piece of evidence came from analysing the deformed theory's thermodynamic properties and comparing to those of a BTZ black hole in a finite, radially cutoff patch of AdS. It has been argued the $T \bar{T}$-theory is the field theory with "residual gravity" modes, which have not decoupled due to the finite size of the patch. \cite{Gorbenko:2018oov} defined an impressive new deformation aimed at constructing a de-Sitter space dual starting from an AdS-dual CFT. The work of \cite{Dubovsky:2017cnj, Dubovsky:2018bmo} further explored such a a connection to quantum gravity, by realizing the $T \bar{T}$ deformation of 2d QFTs in flat space as a coupling to a variant of 2d Jackiw-Teitelboim (JT) gravity. Chapters V and VI aim to generalize this work in two directions: first, by reproducing the $J \bar{T}$ deformation (a flow generated by a composite operator built out of a $U(1)$ current and the stress-tensor) via a coupling to a mixture of topological gravity and gauge theory, and secondly, by proposing a curved space $T \bar T$ flow using the connection to 3d gravity