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Modern physics rests on two basic frameworks, quantum theory and general relativity. Quantum gravity aims to unify these two frameworks into one consistent theory. One can expect that such a formulation delivers in particular a novel understanding of space and time as quantum objects.

I will give an introduction to some basic concepts in quantum gravity research and present possible models of quantum space time.

We examine the interplay of symmetry and topological order in 2+1D topological phases of matter. We define the topological symmetry group, characterizing symmetry of the emergent topological quantum numbers, and describe its relation with the microscopic symmetry of the physical system.

We then derive a general classification of symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are anti-unitary. We develop a general algebraic theory of extrinsic defects (fluxes) associated with elements of the symmetry group, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter with symmetry. We also examine the promotion of the global symmetry to a local gauge invariance, wherein the extrinsic defects are turned into deconfined quasiparticle excitations, which results in a different topological phase.

I will discuss recent work studying universal properties of gravity in anti-de Sitter (AdS) spacetime from the perspective of conformal field theory (CFT). After reviewing relevant aspects of the AdS/CFT correspondence, I will demonstrate that all CFTs in three or more dimensions possess a spectrum of operators consistent with long-distance locality and Newtonian gravity in AdS. In generalizing these results to two-dimensional CFTs, I will then show that operators with large scaling dimension create classical backgrounds in the limit of large central charge. This result can be directly related to eigenstate thermalization in 2d CFTs and black hole geometries in 3d AdS. I will conclude by discussing methods for going beyond this large central charge limit in 2d and for studying black holes in higher dimensions.