Higher Categorical Tools for Quantum Phases of Matter

I discuss some analogies between the Haag-Kastler approach to QFT and quantum statistical mechanics of lattice systems. As an illustrative example, I consider the interpretation of the Hall conductance of gapped 2d lattice systems as an obstruction to gauging a global symmetry of a gapped state. I argue that in order to define a proper analog of the net of algebras of observables one needs to study a category of subsets of the lattice equipped with a natural Grothendieck topology.

I will describe models with on-site symmetry that permutes anyons with non-trivial mutual statistics, and show that the action of this symmetry on the boundary can effectively be that of a non-invertible symmetry such as Kramers-Wannier duality. I will sketch some implications of this for anomalies in non-invertible symmetries. Finally, I will introduce a construction (based on idempotent completion) that allows us to realize all possible anyon permuting symmetries of a given topological order in an on-site way.

We use 3d defect TQFTs and state sum models with defects to give a gauge theoretical formulation of Kitaev's quantum double model (for a finite group) and (untwisted) Dijkgraaf-Witten TQFT with defects. This leads to a simple description in terms of embedding quivers, groupoids and their representations. Defect Dijkgraaf-Witten TQFTs is then formulated in terms of spans of groupoids and their representations. This is work in progress with João Faría-Martins, University of Leeds.

The state-sum invariants of 4d manifolds obtained from spherical fusion 2-categories due to Douglas-Reutter offer an exciting entrypoint to the study of 4d TQFTs. In this talk we will argue that these invariants arise from a TQFT, obtained by filling the trivial 4d TQFT with a defect foam. Such construction is known as a generalised orbifold, the Turaev-Viro-Barrett-Westbury (i.e. 3d state-sum) models are also known to arise in this way from the defects in the trivial 3d TQFT (a result by Carqueville-Runkel-Schaumann). Advantages of this point of view offer e.g. realisations of state-spaces, examples of domain walls and commuting-projector realisations of (3+1)-dimensional topological phases. Based on a joint project with Nils Carqueville and Lukas Müller.

Zesting is a construction that takes a (2+1)D topological order and produces a new one by changing the fusion rules of its anyons. We'll discuss properties of zesting from a physical and computational point of view and explain how the theory produces some closely related families of topological orders, like Kitaev's 16-fold way and modular isotopes. Time permitting we'll cover a generalization of zesting to symmetry-enriched topological order and comment on connections to fusion 2-categories.

The braiding statistics of point particles in 2-dimensional topological phases are given by representations of the braid groups. One approach to the study of generalised particles in topological phases, loop particles in 3-dimensions for example, is to generalise (some of) the several different realisations of the braid group. In this talk I will construct for each manifold M its motion groupoid $Mot_M$, whose object class is the power set of M. I will discuss several different, but equivalent, quotients on motions leading to the motion groupoid. In particular that the quotient used in the construction $Mot_M$ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows -- worldlines (e.g. monotonic `tangles'). I will also give a construction of a mapping class groupoid $MCG_M$ associated to a manifold M with the same object class. For each manifold M I will construct a functor $F \colon Mot_M \to MCG_M$, and prove that this is an isomorphism if $\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of M is trivial. In particular there is an isomorphism in the physically important case $M=[0,1]^n$ with fixed boundary, for any $n\in\mathbb{N}$. I will discuss several examples throughout.

In this talk we explore the symmetry in 1+1d fermionic systems from a category theory perspective. We argue that It requires additional structure than a tensor category over sVect, and rather captured by a fusion category equipped with a braided central functor from a specific braided fusion category, dependent on the chiral central charge. The same data defines a 1-morphism in the 4-category of braided tensor categories, which lead us to a 4d-3d-2d picture via cobordism hypothesis. This talk is based on an ongoing work with K. Inamura.