Geometric and computational aspects of chiral topological quantum matter

In this thesis, we study chiral topological phases of 2+1 dimensional quantum matter. Such phases are abstractly characterized by their non-vanishing chiral central charge $c$, a topological invariant which appears as the coefficient of a gravitational Chern-Simons (gCS) action in bulk, and of corresponding gravitational anomalies at boundaries. The chiral central charge is of particular importance in chiral superfluids and superconductors (CSF/Cs), where $U(1)$ particle-number symmetry is broken, and $c$ is, in some cases, the only topological invariant characterizing the system. However, as opposed to invariants which can be probed by gauge fields in place of gravity, the concrete physical implications of $c$ in the context of condensed matter physics is quite subtle, and has been the subject of ongoing research and controversy. The first two parts of this thesis are devoted to the physical interpretation of the gCS action and gravitational anomalies in the context of CSF/Cs, where they are of particular importance, but have nevertheless remained poorly understood. We then turn to a seemingly unrelated aspect of chiral topological phases - their computational complexity. The infamous $sign\ problem$ leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing an efficient classical simulation. The possibility of $intrinsic$ sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of chiral topological phases, both bosonic and fermionic, defined by the requirement that $e^{2\pi i c/24}$ is $not$ the topological spin of an anyon.