Catherine Meusburger

We use Gay and Kirby's description of 4-manifolds in terms of trisections and trisection diagrams to define a new 4-manifold invariant. The algebraic data are an indecomposable finite semisimple bimodule category over a pair of spherical fusion categories, equipped with a bimodule trace, and a pivotal functor from another spherical fusion category into the spherical fusion category of its bimodule endofunctors and natural transformations between them. The 4-manifold invariant has a simple description in terms of a diagrammatic calculus for this data, in which the three spherical fusion categories correspond to the three colours of the trisection diagram. It includes the Hopf algebraic 4-manifold invariants by Chaidez, Cotler and Cui, which arise when the bimodule category is the category of finite-dimensional complex vector spaces. We also recover the 4-manifold invariants of Bärenz and Barrett defined by a pivotal functor from a spherical fusion category into a modular fusion category.

We show that every involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category gives rise to invariants of oriented surfaces defined in terms of ribbon graphs. For every ribbon graph this yields an object in the category, defined up to isomorphism, that depends only on the homeomorphism class of the associated surface. This object is constructed via (co)equalisers and images and equipped with a mapping class group action. It can be viewed as a categorical generalisation of the ground state of Kitaev's quantum double model or of a representation variety for a surface. We apply the construction to group objects in cartesian monoidal categories, in particular to simplicial groups as group objects in SSet and to crossed modules as group objects in Cat. The former yields a simplicial set consisting of representation varieties, the latter a groupoid whose sets of objects and morphisms are obtained from representation varieties.

We define a Turaev-Viro-Barrett-Westbury state sum model of triangulated 3-manifolds with surface, line and point defects. Surface defects are oriented embedded 2d PL submanifolds and are labeled with bimodule categories over spherical fusion categories with bimodule traces. Line and point defects form directed graphs on these surfaces and labeled with bimodule functors and bimodule natural transformations. The state sum is based on generalised 6j symbols that encode the coherence isomorphisms of the defect data. We prove the triangulation independence of the state sum and show that it can be computed in terms of polygon diagrams that satisfy the cutting and gluing identities for polygon presentations of oriented surfaces. By computing state sums with defect surfaces, we show that they detect the genus of a defect surface and are sensitive to its embedding. We show that defect lines on defect surfaces with trivial defect data define ribbon invariants for the centre of the underlying spherical fusion category.

We show that every involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category gives rise to invariants of oriented surfaces defined in terms of ribbon graphs. For every ribbon graph this yields an object in the category, defined up to isomorphism, that depends only on the homeomorphism class of the associated surface. This object is constructed via (co)equalisers and images and equipped with a mapping class group action. It can be viewed as a categorical generalisation of the ground state of Kitaev's quantum double model or of a representation variety for a surface. We apply the construction to group objects in cartesian monoidal categories, in particular to simplicial groups as group objects in SSet and to crossed modules as group objects in Cat. The former yields a simplicial set consisting of representation varieties, the latter a groupoid whose sets of objects and morphisms are obtained from representation varieties.

We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra that generalizes the complex numbers. Equivalently, tetrahedra with lightlike faces are determined by a pair of edge lengths and their duals by a pair of dihedral angles. We prove that the dual tetrahedra are precisely the generalized ideal tetrahedra introduced by Danciger. Finally, we compute the volumes of both types of tetrahedra as functions of their edge lengths or dihedral angles, obtaining generalizations of the Milnor–Lobachevsky volume formula of ideal hyperbolic tetrahedra.

We generalise gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are replaced by linearised analogues. Starting from physical considerations, we derive an axiomatic definition of Hopf-algebra gauge theory, including locality conditions under which the theory for a general ribbon graph can be assembled from local data in the neighbourhood of each vertex. For a vertex neighbourhood with n incoming edge ends, the algebra of non-commutative "functions" of connections is dual to a two-sided twist deformation of the n-fold tensor power of the gauge Hopf algebra. We show these algebras assemble to give an algebra of functions and gauge-invariant subalgebra of "observables" that coincide with those obtained in the combinatorial quantisation of Chern-Simons theory, thus providing an axiomatic derivation of the latter. We then discuss holonomy in a Hopf algebra gauge theory and show that for semisimple Hopf algebras this gives, for each path in the embedded graph, a map from connections into the gauge Hopf algebra, depending functorially on the path. Curvatures -- holonomies around the faces canonically associated to the ribbon graph -- then correspond to central elements of the algebra of observables, and define a set of commuting projectors onto the subalgebra of observables on flat connections. The algebras of observables for all connections or for flat connections are topological invariants, depending only on the topology, respectively, of the punctured or closed surface canonically obtained by gluing annuli or discs along edges of the ribbon graph.

We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra that generalizes the complex numbers. Equivalently, tetrahedra with lightlike faces are determined by a pair of edge lengths and their duals by a pair of dihedral angles. We prove that the dual tetrahedra are precisely the generalized ideal tetrahedra introduced by Danciger. Finally, we compute the volumes of both types of tetrahedra as functions of their edge lengths or dihedral angles, obtaining generalizations of the Milnor-Lobachevsky volume formula of ideal hyperbolic tetrahedra.