arXiv subjects

Catherine Meusburger

Publications and source records attributed to Catherine Meusburger.

At least 19 recordsLinked to original sources

arXiv2025

Categorical 4-manifold invariants from trisection diagrams

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 B\"arenz and Barrett defined by a pivotal functor from a spherical fusion category into a modular fusion category.

math.QA↗

arXiv2024

A geometrical description of untwisted 3d Dijkgraaf-Witten TQFT with defects

We give a simple, geometric and explicit construction of 3d untwisted Dijkgraaf-Witten theory with defects of all codimensions. It is given as a symmetric monoidal functor from a defect cobordism category into the category of finite-dimensional complex vector spaces. The objects of this category are oriented stratified surfaces and its morphisms are equivalence classes of stratified cobordisms, both labelled with higher categorical data. This TQFT is constructed in terms of geometric quantities such as fundamental groupoids and bundles and requires neither state sums on triangulations nor diagrammatic calculi for higher categories. It is obtained from a functor that assigns to each defect surface a representation of a gauge groupoid and to each defect cobordism a fibrant span of groupoids and an intertwiner between the groupoid representations at its boundary. It is constructed by homotopy theoretic methods and allows for an explicit computation of examples. In particular, we show how the 2d part of this defect TQFT gives a simple description of defects of all codimensions in Kitaev's quantum double model.

math.QA↗

arXiv2023

State sum models with defects based on spherical fusion categories

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.

math.QA↗

arXiv2023

Categorical generalisations of quantum double models

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.

math.QA↗

arXiv2021

Hopf algebra gauge theory on a ribbon graph

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.

math.QA↗

arXiv2021

Lightlike and ideal tetrahedra

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.

math.GT↗

arXiv2021

Mapping class group actions from Hopf monoids and ribbon graphs

We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $\mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g \geq 1$ with $n \geq 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over $H$. They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where $\mathcal{C}$ is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter-Drinfeld module structure.

math.QA↗

arXiv2017

3-dimensional defect TQFTs and their tricategories

We initiate a systematic study of 3-dimensional `defect' topological quantum field theories, that we introduce as symmetric monoidal functors on stratified and decorated bordisms. For every such functor we construct a tricategory with duals, which is the natural categorification of a pivotal bicategory. This captures the algebraic essence of defect TQFTs, and it gives precise meaning to the fusion of line and surface defects as well as their duality operations. As examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into our framework, and how they can be extended to defect TQFTs.

math.QA↗

arXiv2017

AdS Poisson homogeneous spaces and Drinfel'd doubles

The correspondence between Poisson homogeneous spaces over a Poisson-Lie group $G$ and Lagrangian Lie subalgebras of the classical double $D({\mathfrak g})$ is revisited and explored in detail for the case in which ${\mathfrak g}=D(\mathfrak a)$ is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group $\mathrm{SL}(2,R)\cong\mathrm{SO}(2,1)$, namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on ${sl}(2,R)$ and as a coisotropic one for the others. We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space $\mathrm{AdS}_3$ and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical $r$-matrices for ${so}(2,2)$, while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on $\mathrm{AdS}_3$ that arise from two Drinfel'd double structures on $\mathrm{SO}(2,2)$. The first one realises $\mathrm{AdS}_3$ as a quotient of $\mathrm{SO}(2,2)$ by the Poisson-subgroup $\mathrm{SL}(2,R)$, while the second one, the non-commutative spacetime of the twisted $κ$-AdS deformation, realises $\mathrm{AdS}_3$ as a coisotropic Poisson homogeneous space.

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arXiv2017

Kitaev lattice models as a Hopf algebra gauge theory

We prove that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D(H). This shows that Kitaev models are a special case of the older and more general combinatorial models. This equivalence is an analogue of the relation between Turaev-Viro and Reshetikhin-Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections. We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories. We first prove that the triangle operators of a Kitaev model form a module algebra over a Hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism. Both algebras can be viewed as the algebra of functions on gauge fields in a Hopf algebra gauge theory. The isomorphism between them induces an algebra isomorphism between their subalgebras of invariants, which are interpreted as gauge invariant functions or observables. It also relates the curvatures in the two models, which are given as holonomies around the faces of the lattice. This yields an isomorphism between the subalgebras obtained by projecting out curvatures, which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models.

math.QA↗

arXiv2014

Twisted (2+1) $κ$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes

We construct the full quantum algebra, the corresponding Poisson-Lie structure and the associated quantum spacetime for a family of quantum deformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter (AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to a Drinfel'd double structure on the isometry algebras that are motivated by their role in (2+1)-gravity. The construction includes the cosmological constant $Λ$ as a deformation parameter, which allows one to treat these cases in a common framework and to obtain a twisted version of both space- and time-like $κ$-AdS and dS quantum algebras; their flat limit $Λ\to 0$ leads to a twisted quantum Poincaré algebra. The resulting non-commutative spacetime is a nonlinear $Λ$-deformation of the $κ$-Minkowski one plus an additional contribution generated by the twist. For the AdS case, we relate this quantum deformation to two copies of the standard (Drinfel'd-Jimbo) quantum deformation of the Lorentz group in three dimensions, which allows one to determine the impact of the twist.

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arXiv2014

A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant

We show that the Drinfel'd double associated to the standard quantum deformation $sl_η(2,R)$ is isomorphic to the (2+1)-dimensional AdS algebra with the initial deformation parameter $η$ related to the cosmological constant $Λ=-η^2$. This gives rise to a generalisation of a non-commutative Minkowski spacetime that arises as a consequence of the quantum double symmetry of (2+1) gravity to non-vanishing cosmological constant. The properties of the AdS quantum double that generalises this symmetry to the case $Λ\neq 0$ are sketched, and it is shown that the new non-commutative AdS spacetime is a nonlinear $Λ$-deformation of the Minkowskian one.

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arXiv2014

Generalised shear coordinates on the moduli spaces of three-dimensional spacetimes

We introduce coordinates on the moduli spaces of maximal globally hyperbolic constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are derived from the parametrisation of the moduli spaces by the bundle of measured geodesic laminations over Teichmüller space of S and can be viewed as analytic continuations of the shear coordinates on Teichmüller space. In terms of these coordinates the gravitational symplectic structure takes a particularly simple form, which resembles the Weil-Petersson symplectic structure in shear coordinates, and is closely related to the cotangent bundle of Teichmüller space. We then consider the mapping class group action on the moduli spaces and show that it preserves the gravitational symplectic structure. This defines three distinct mapping class group actions on the cotangent bundle of Teichmüller space, corresponding to different values of the curvature.

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arXiv2013

Recovering the geometry of a flat spacetime from background radiation

We consider globally hyperbolic flat spacetimes in 2+1 and 3+1 dimensions, in which a uniform light signal is emitted on the $r$-level surface of the cosmological time for $r\to 0$. We show that the frequency of this signal, as perceived by a fixed observer, is a well-defined, bounded function which is generally not continuous. This defines a model with anisotropic background radiation that contains information about initial singularity of the spacetime. In dimension 2+1, we show that this observed frequency function is stable under suitable perturbations of the spacetime, and that, under certain conditions, it contains sufficient information to recover its geometry and topology. We compute an approximation of this frequency function for a few simple examples.

gr-qc↗

arXiv2013

Drinfel'd doubles for (2+1)-gravity

All possible Drinfel'd double structures for the anti-de Sitter Lie algebra so(2,2) and de Sitter Lie algebra so(3,1) in (2+1)-dimensions are explicitly constructed and analysed in terms of a kinematical basis adapted to (2+1)-gravity. Each of these structures provides in a canonical way a pairing among the (anti-)de Sitter generators, as well as a specific classical r-matrix, and the cosmological constant is included in them as a deformation parameter. It is shown that four of these structures give rise to a Drinfel'd double structure for the Poincaré algebra iso(2,1) in the limit where the cosmological constant tends to zero. We explain how these Drinfel'd double structures are adapted to (2+1)-gravity, and we show that the associated quantum groups are natural candidates for the quantum group symmetries of quantised (2+1)-gravity models and their associated non-commutative spacetimes.

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arXiv2012

Gray categories with duals and their diagrams

The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray category data. These can be viewed as a generalisation of ribbon diagrams. The Gray categories present two types of duals, which are extended to functors of strict tricategories with natural isomorphisms, and correspond directly to symmetries of the diagrams. It is shown that these functors can be strictified so that the symmetries of a cube are realised exactly. A new condition on Gray categories with duals called the spatial condition is defined. A class of diagrams for which the evaluation for spatial Gray categories is invariant under homeomorphisms is exhibited. This relation between the geometry of the diagrams and structures in the Gray categories proves useful in computations and has potential applications in topological quantum field theory.

math.QA↗

arXiv2012

Gauge fixing in (2+1)-gravity with vanishing cosmological constant

We apply Dirac's gauge fixing procedure to (2+1)-gravity with vanishing cosmological constant. For general gauge fixing conditions based on two point particles, this yields explicit expressions for the Dirac bracket. We explain how gauge fixing is related to the introduction of an observer into the theory and show that the Dirac bracket is determined by a classical dynamical r-matrix. Its two dynamical variables correspond to the mass and spin of a cone that describes the residual degrees of freedom of the spacetime. We show that different gauge fixing conditions and different choices of observers are related by dynamical Poincaré transformations. This allows us to locally classify all Dirac brackets resulting from the gauge fixing and to relate them to a set of particularly simple solutions associated with the centre-of-mass frame of the spacetime.

gr-qc↗

arXiv2012

Gauge fixing and classical dynamical r-matrices in ISO(2,1)-Chern-Simons theory

We apply Dirac's gauge fixing procedure to Chern-Simons theory with gauge group ISO(2,1) on manifolds RxS, where S is a punctured oriented surface of general genus. For all gauge fixing conditions that satisfy certain structural requirements, this yields an explicit description of the Poisson structure on the moduli space of flat ISO(2,1)-connections on S via the resulting Dirac bracket. The Dirac bracket is determined by classical dynamical r-matrices for ISO(2,1). We show that the Poisson structures and classical dynamical r-matrices arising from different gauge fixing conditions are related by dynamical ISO(2,1)-valued transformations that generalise the usual gauge transformations of classical dynamical r-matrices. By means of these transformations, it is possible to classify all Poisson structures and classical dynamical r-matrices obtained from such gauge fixings. Generically these Poisson structures combine classical dynamical r-matrices for non-conjugate Cartan subalgebras of iso(2,1).

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