We present a bicategorical state sum construction for 3-manifold invariants. Using the pivotal bicategory of spherical module categories over a spherical fusion category, we construct invariants that manifestly preserve Morita equivalence. Our main result shows that this bicategorical invariant recovers the standard Turaev–Viro invariant, thereby proving Morita invariance of Turaev–Viro invariants without appealing to the Reshetikhin–Turaev construction.

This is joint work with Jürgen Fuchs, David Jaklitsch, and Christoph Schweigert.

Topological phases of matter in (2+1)D should naturally form a 3-category, in which k-morphisms represent defects of codimension k. By the cobordism hypothesis, the 3-categories of (2+1)D topological order with a fixed anomaly are each equivalent to the 3-category of fusion categories enriched over a corresponding UMTC. In ongoing work with Fiona Burnell, we introduce algebraic techniques for concrete computations in 3-categories of (2+1)D topological order, including a tunneling approach to the classification of point defects which generalizes the use of braiding to identify anyon types. We then apply these techniques to compute ground state degeneracy and classify low energy excitations in a class of fracton-like (2+1)D topological defect networks.

I will present the super volumes of the moduli space of super Riemann surfaces. They will be defined using a family of finite measures on the moduli space of genus $g$ curves. These measures are in turn given by a construction analogous to the classical construction of the Weil–Petersson metric, using the extra data of a spin structure. The total measure gives the volume of the moduli space of super curves and can be calculated via a deep relationship with the KdV equation.

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.

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.