Topological quantum many-body systems on the lattice are characterized by having the property that their low energy effective theories are TQFTs. Levin–Wen models are classes of spin systems on the 2D lattice whose low energy effective theories are Turaev–Viro TQFTs. The problem (from a mathematician's perspective) is that low energy effective theories are not at all well-defined! This leads to the question: for systems that (supposedly) exhibit topological order, how can we see the emergent TQFT directly on the lattice in the infinite volume limit? We will discuss our recently proposed approach to mathematically formalize the ideas of topological holography in terms of boundary algebras, and explain how this provides a solution for systems with local topological order.

I will present a computation of tree-level superstring measures on the moduli spaces of genus-zero super Riemann surfaces with Neveu–Schwarz (NS) and Ramond punctures. The answer in the NS case is not new, but it is done using first principles, i.e., exclusively complex algebraic supergeometry and, in particular, the super Mumford isomorphism. The answer in the Ramond case is totally new, but we do not quite have it. This is joint work with S. Cacciatori and S. Grushevsky: published in the NS case and in-progress in the Ramond case.

Turaev–Viro (TV) invariants are 3-manifold invariants, defined for a given fixed integer $r$ and $2r$-th root of unity. Chen and Yang extended the definition of TV-invariants to pseudo 3-manifolds and introduced a volume conjecture for TV-invariants which states that for the case of $r$-th roots of unity where $r$ is odd and $M$ is hyperbolic, the TV invariants of $M$ grow exponentially and determine the volume of $M$.

The Witten–Reshetikhin–Turaev (WRT) 3-manifold invariants (also known as the Chern–Simons 3-manifold invariants), are defined for a given fixed integer $r$, and a $2r$-th root of unity. The existence of such invariants were predicted by Witten in his work on Chern–Simons gauge theory and topological quantum field theory. They were constructed by Reshetikhin and Turaev by using representation theory and Kirby calculus. Later, Lickorish gave a skein theoretic definition. These invariants were also originally defined for closed orientable 3-manifolds, but were recently extended to link complements. Furthermore, Belletti, Detcherry, Kalfagianni, and Yang provided an explicit formula relating the TV invariant to the WRT invariant of link complements in a closed orientable 3-manifold and used this formula to prove the TV volume conjecture for octahedral link complements in the connected sums of $S^2 \times S^1$ called fundamental shadow links.

In contrast, fully augmented links are links in $S^3$ whose complements have nice geometric properties. For instance, Agol and Thurston showed that fully augmented links can be decomposed into totally geodesic, right-angled ideal polyhedra. In this talk, we will present a geometric description of the relationship between octahedral fully augmented links and fundamental shadow links and we will outline an alternative proof, using the colored Jones polynomial, to prove the TV volume conjecture for octahedral fully augmented links with no half-twists. This is joint work with Emma McQuire and Jessica Purcell.