In a physical system with conformal symmetry, observables depend on cross-ratios, measures of distance invariant under global conformal transformations (conformal geometry for short). We identify a quantum information-theoretic mechanism by which the conformal geometry emerges at the gapless edge of a 2+1D quantum many-body system with a bulk energy gap. We introduce a novel pair of information-theoretic quantities (c,n) that can be defined locally on the edge from the wavefunction of the many-body system, without prior knowledge of any distance measure. We posit that, for a topological groundstate, the quantity c is stationary under arbitrary variations of the quantum state, and study the logical consequences. We show that stationarity, modulo an entanglement-based assumption about the bulk, implies (i) c is a non-negative constant that can be interpreted as the total central charge of the edge theory. (ii) n is a cross-ratio, obeying the full set of mathematical consistency rules, which further indicates the existence of a distance measure of the edge with global conformal invariance. Thus, the conformal geometry emerges from a simple assumption on groundstate entanglement. The stationarity of c is equivalent to a vector fixed-point equation involving n, making our assumption locally checkable. If time permits, we discuss a class of modular flow on a disk, which creates only edge excitations. We intuitively explain why Virasoro algebra can be revealed from a single wavefunction by analyzing such modular flows.

In the 2nd lecture, we dig into the underlying logic of entanglement bootstrap. Illustrative examples are aimed to be simple but nontrivial. The following will be included: (1) We explain a few basic uses of strong subadditivity and quantum Markov states; we explain why axiom A0 is crucial to protect coherence. We derive the information convex set of the sphere as an application. (2) Related to the information convex set of the annulus, we explain the definition of quantum dimensions, why the vacuum has the smallest entropy, and why a certain "merged state" has the maximum entropy. (3) We classify immersed annuli on a sphere and explain why some puzzles of figure-8 annulus are not solved in naive ways. (4) In the context the reference state has a 0-form symmetry, we sketch a way to create a symmetry defect line, which (in some models) permutes anyons. This lecture is given using an ipad and is, thus, flexible. We also discuss topics from questions (feedbacks) during the 1st (and 2nd) lecture. See https://www.london-tqft.co.uk for details.

Topological quantum field theory can emerge in gapped many-body quantum systems at low energies. In 2+1D systems, anyons can emerge, and in 3+1D, emergent excitations, including point-particles and loops-like excitations, possibly knotted or linked. In this lecture, we introduce an ongoing effort to understand (in fact, derive) laws of the emergent theory in 2+1D, 3+1D, (and higher D) gapped systems from a few axioms about the entanglement of a many-body ground state wave function. This research program, referred to as entanglement bootstrap, is an approach independent of quantum field theory, and it uses nontrivial quantum information and topology ideas. We explain the axioms and key concepts. We sketch the proof of several main theorems, including the definition of superselection sectors (anyons in 2+1D, point and loop excitations in 3+1D), the fusion spaces, and their constraints. We explain why immersion (i.e., local embedding) is valuable for, e.g., putting systems on closed space manifolds and what we hope to learn next. (see https://www.london-tqft.co.uk for more details)