Christoph Schweigert

Symmetries in the form of actions of groups and Lie algebras have been an important tool for the analysis of physical systems for a very long time. It has been realized only relatively recently that quantum field theories admit symmetries that are described by more general mathematical structures: fusion categories and their higher-categorical generalizations.The understanding of such higher categories and the construction of concrete examples raise many mathematical challenges which are at present addressed with a wide variety of mathematical tools, including homotopy theory, the theory of C^{*} -algebras, factorization algebras and operads.An improved mathematical understanding of higher categories has an immediate impact on the classification and characterization of quantum systems which can realize these categorical structures as symmetries. Particularly note-worthy applications of these symmetries include integrable systems with fusion category symmetries, defects in topological and conformal field theories, the characterization of three-dimensional topological order and systems in quantum information theory realizing quantum codes.The workshop Higher Structures from Symmetries in Quantum Field Theory, brought a wide range of mathematical researchers together, working on a broad range of topics, from category theory to condensed matter physics, and provided an extremely fruitful interaction across disciplines. For some leading researchers from neighbouring fields, this was their first visit to Oberwolfach.

We present a state sum construction that assigns a scalar to a skeleton in a closed oriented three-dimensional manifold. The input datum is the pivotal bicategory \mathbf{Mod}^{\mathrm{sph}}(\mathcal{A}) of spherical module categories over a spherical fusion category \mathcal{A} . The interplay of algebraic structures in this pivotal bicategory with moves of skeleta ensures that our state sum is independent of the skeleton on the manifold. We show that the bicategorical invariant recovers the value of the standard Turaev–Viro invariant associated to \mathcal{A} , thereby proving the independence of the Turaev–Viro invariant under pivotal Morita equivalence without recurring to the Reshetikhin–Turaev construction. A key ingredient for the construction is the evaluation of graphs on the sphere with labels in \mathbf{Mod}^{\mathrm{sph}}(\mathcal{A}) that we develop in this article. A central tool is Nakayama-twisted traces on pivotal bimodule categories, which we study beyond semisimplicity.

Recently, large degeneracy based on product eigenstates has been found in spin ladders, kagome-like lattices, and motif magnetism, connected to spin liquids, anyonic phases, and quantum scars. We unify these systems by a complete classification of product eigenstates of Heisenberg XXZ Hamiltonians with Dzyaloshinskii-Moriya interaction on general graphs in the form of Kirchhoff rules for spin supercurrents. By this, we construct spin systems with extensive degree of degeneracy linked to exotic condensates which could be studied in atomic gases and quantum spin lattices.

We develop a string-net construction of a modular functor whose algebraic input is a pivotal bicategory; this extends the standard construction based on a spherical fusion category. An essential ingredient in our construction is a graphical calculus for pivotal bicategories, which we express in terms of a category of colored corollas. The globalization of this calculus to oriented surfaces yields the bicategorical string-net spaces as colimits. We show that every rigid separable Frobenius functor between strictly pivotal bicategories induces linear maps between the corresponding bicategorical stringnet spaces that are compatible with the mapping class group actions and with sewing. Our results are inspired by and have applications to the description of correlators in two-dimensional conformal field theories. Contents 1 Introduction 474 2 Graphical calculus for pivotal bicategories 478 3 String-net models based on pivotal bicategories 511 4 Application: Universal correlators in RCFT 532 We thank Christian Blanchet, Nils Carqueville, Lukas Müller and Lukas Woike for helpful discussions.

The evaluation of graphs on 2-spheres is a central ingredient of the Turaev-Viro construction of three-dimensional topological field theories. In this article, we introduce a class of graphs, called extruded graphs, that is relevant for the Turaev-Viro construction with general defect configurations involving defects of various dimensions. We define the evaluation of extruded graphs and show that it is invariant under a set of moves. This ensures the computability and uniqueness of our evaluation.

The skein module for a d-dimensional manifold is a vector space spanned by embedded framed graphs decorated by a category A with suitable extra structure depending on the dimension d, modulo local relations which hold inside d-balls. For a full subcategory S of A, an S-admissible skein module is defined analogously, except that local relations for a given ball may only be applied if outside the ball at least one edge is coloured in S. In this paper we prove that admissible skein modules in any dimension satisfy excision, namely that the skein module of a glued manifold is expressed as a coend over boundary values on the boundary components glued together. We furthermore relate skein modules for different choices of S, apply our result to cylinder categories, and recover the relation to modified traces.

We discuss a string-net construction on 2 -framed surfaces, taking as algebraic input a finite, rigid tensor category, which is assumed to be neither pivotal nor semi-simple. It is shown that circle categories of our framed string-net construction essentially compute Drinfeld centers twisted by powers of the double dual functor.

We discuss a string-net construction on 2-framed surfaces, taking as algebraic input a finite, rigid tensor category, which is neither assumed to be pivotal nor semi-simple. It is shown that circle categories of our framed string-net construction essentially compute Drinfeld centers twisted by powers of the double dual functor.