We show how an observer could measure the non -local holonomy variables that

parametrise the ﬂat Lorentzian 3d manifolds arising as spacetimes in (2+1)-gravity.

We consider an observer who emits lightray s that return to him at a later time

and performs several realistic measurements associated with such retur ning ligh-

trays: the eigentime elapsed between the emission of the lightrays and their return,

the directions into which the light is emitted and fr om which it returns an d the

frequency shift between the emitted and returning lightray. We show how the

holonomy variables and hence the full geometry of these manifolds can be recon-

structed from these measurements in ﬁnite eigentime.

}, author = {Meusburger, Cathérine}, faupublication = {yes}, journal = {AMS/IP Studies in Advanced Mathematics}, pages = {261-276}, peerreviewed = {No}, title = {{Global} {Lorentzian} geometry of lightlike geodesics: what does an observer in (2+1) gravity see?}, volume = {50}, year = {2011} } @article{faucris.116758444, abstract = {We relate the geometrical construction of (2+1)-spacetimes via grafting to phase space and Poisson structure in the Chern-Simons formulation of (2+1)-dimensional gravity with vanishing cosmological constant on manifolds of topology ℝ times S , where S is an orientable two-surface of genus g>1. We show how grafting along simple closed geodesics λ is implemented in the Chern-Simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on S .We prove that this action is generated via the Poisson bracket by a gauge invariant observable associated to the holonomy of λ. We deduce a symmetry relation between the Poisson brackets of observables associated to the Lorentz and translational components of the holonomies of general closed curves on S and discuss its physical interpretation. Finally, we relate the action of grafting on the phase space to the action of Dehn twists and show that grafting can be viewed as a Dehn twist with a formal parameter θ satisfying θ = 0.}, author = {Meusburger, Cathérine}, doi = {10.1007/s00220-006-0037-x}, faupublication = {no}, journal = {Communications in Mathematical Physics}, pages = {735-775}, peerreviewed = {Yes}, title = {{Grafting} and poisson structure in (2+1)-gravity with vanishing cosmological constant}, volume = {266}, year = {2006} } @article{faucris.252112214, abstract = {We generalize 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 linearized analogs. 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 neighborhood of each vertex. For a vertex neighborhood 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 quantization 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.}, author = {Meusburger, Cathérine and Wise, Derek}, doi = {10.1142/S0129055X21500161}, faupublication = {yes}, journal = {Reviews in Mathematical Physics}, keywords = {Chern-Simons gauge theory; Hopf algebras; Moduli spaces of flat connections; ribbon graphs}, note = {CRIS-Team Scopus Importer:2021-03-19}, peerreviewed = {Yes}, title = {{Hopf} algebra gauge theory on a ribbon graph}, year = {2021} } @article{faucris.212491719, author = {Meusburger, Cathérine}, doi = {10.1007/s00220-017-2860-7}, faupublication = {yes}, journal = {Communications in Mathematical Physics}, pages = {413-468}, peerreviewed = {Yes}, title = {{Kitaev} {Lattice} {Models} as a {Hopf} {Algebra} {Gauge} {Theory}}, url = {https://arxiv.org/abs/1607.01144}, volume = {353:413}, year = {2017} } @article{faucris.272553297, abstract = {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.}, author = {Meusburger, Cathérine and Scarinci, Carlos}, doi = {10.1007/s10711-022-00687-6}, faupublication = {yes}, journal = {Geometriae Dedicata}, note = {CRIS-Team WoS Importer:2022-04-08}, peerreviewed = {Yes}, title = {{Lightlike} and ideal tetrahedra}, volume = {216}, year = {2022} } @article{faucris.265356699, abstract = {We show that any pivotal Hopf monoid H in a symmetric monoidal category C gives rise to actions of mapping class groups of oriented surfaces of genus g >= 1 with n >= 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 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.}, author = {Meusburger, Cathérine and Voß, Thomas}, doi = {10.4171/QT/158}, faupublication = {yes}, journal = {Quantum Topology}, month = {Jan}, note = {CRIS-Team WoS Importer:2021-10-22}, pages = {507-591}, peerreviewed = {Yes}, title = {{Mapping} class group actions from {Hopf} monoids and ribbon graphs}, volume = {12}, year = {2021} } @article{faucris.111431804, abstract = {We study the action of the mapping class group of an oriented genus g surface with n punctures and a disc removed on a Poisson algebra which arises in the combinatorial description of Chern-Simons gauge theory when the gauge group is a semidirect product G ⋉ g*. We prove that the mapping class group acts on this algebra via Poisson isomorphisms and express the action of Dehn twists in terms of an infinitesimally generated G-action. We construct a mapping class group representation on the representation spaces of the associated quantum algebra and show that Dehn twists can be implemented via the ribbon element of the quantum double D (G) and the exchange of punctures via its universal R-matrix. © 2004 Elsevier B.V. All rights reserved.}, author = {Meusburger, Cathérine and Schroers, Bernd}, doi = {10.1016/j.nuclphysb.2004.10.057}, faupublication = {no}, journal = {Nuclear Physics B}, pages = {569-597}, peerreviewed = {Yes}, title = {{Mapping} class group actions in {Chern}-{Simons} theory with gauge group {G} ⋉ g*}, volume = {706}, year = {2005} } @article{faucris.122152404, abstract = {We construct stationary flat three-dimensional Lorentzian manifolds with singularities that are obtained from Euclidean surfaces with cone singularities and closed one-forms on these surfaces. In the application to (2 + 1)-gravity, these spacetimes correspond to models containing massive particles with spin. We analyse their geometrical properties, introduce a generalised notion of global hyperbolicity and classify all stationary flat spacetimes with singularities that are globally hyperbolic in that sense. We then apply our results to (2 + 1)-gravity and analyse the causality structure of these spacetimes in terms of measurements by observers. In particular, we derive a condition on observers that excludes causality violating light signals despite the presence of closed timelike curves in these spacetimes. © 2012 Springer Science+Business Media B.V.}, author = {Barbot, Thierry and Meusburger, Cathérine}, doi = {10.1007/s10711-011-9692-y}, faupublication = {yes}, journal = {Geometriae Dedicata}, keywords = {Cone singularity; Euclidean surface; Flat spacetime; Global hyperbolicity; Minkowski space; Particle with spin}, pages = {23-50}, peerreviewed = {Yes}, title = {{Particles} with spin in stationary flat spacetimes}, volume = {161}, year = {2012} } @article{faucris.119311324, abstract = {We explicitly determine the symplectic structure on the phase space of Chern-Simons theory with gauge group G ⋉ g* on a three-manifold of topology ℝ × S, where S is a surface of genus g with n + 1 punctures. At each puncture additional variables are introduced and coupled minimally to the Chern-Simons gauge field. The first n punctures are treated in the usual way and the additional variables lie on coadjoint orbits of G ⋉ g *. The (n + 1)st puncture plays a distinguished role and the associated variables lie in the cotangent bundle of G ⋉ g *. This allows us to impose a curvature singularity for the Chern-Simons gauge field at the distinguished puncture with an arbitrary Lie algebra valued coefficient. The treatment of the distinguished puncture is motivated by the desire to construct a simple model for an open universe in the Chern-Simons formulation of (2 + 1)-dimensional gravity. © 2006 Elsevier B.V. All rights reserved.}, author = {Meusburger, Cathérine and Schroers, Bernd}, doi = {10.1016/j.nuclphysb.2006.01.014}, faupublication = {no}, journal = {Nuclear Physics B}, pages = {425-456}, peerreviewed = {Yes}, title = {{Phase} space structure of {Chern}-{Simons} theory with a non-standard puncture}, volume = {738}, year = {2006} } @article{faucris.262302200, author = {Meusburger, Cathérine}, doi = {10.3390}, faupublication = {yes}, journal = {Symmetry}, peerreviewed = {unknown}, title = {{Poisson}–{Lie} {Groups} and {Gauge} {Theory}}, volume = {13}, year = {2021} } @article{faucris.122172644, abstract = {In the formulation of (2+ 1)-dimensional gravity as a Chern-Simons gauge theory, the phase space is the moduli space of flat Poincaré group connections. Using the combinatorial approach developed by Fock and Rosly, we give an explicit description of the phase space and its Poisson structure for the general case of a genus g oriented surface with punctures representing particles and a boundary playing the role of spatial infinity. We give a physical interpretation and explain how the degrees of freedom associated with each handle and each particle can be decoupled. The symmetry group of the theory combines an action of the mapping class group with asymptotic Poincaré transformations in a nontrivial fashion. We derive the conserved quantities associated with the latter and show that the mapping class group of the surface acts on the phase space via Poisson isomorphisms.}, author = {Meusburger, Cathérine and Schroers, Bernd}, doi = {10.1088/0264-9381/20/11/318}, faupublication = {no}, journal = {Classical and Quantum Gravity}, pages = {2193-2233}, peerreviewed = {Yes}, title = {{Poisson} structure and symmetry in the {Chern}-{Simons} formulation of (2 + 1)-dimensional gravity}, volume = {20}, year = {2003} } @inproceedings{faucris.109336744, abstract = {We construct and analyse a quantum deformation of the Lorentzian EPRL model. The model is based on the representation theory of the quantum Lorentz group with real deformation parameter. We give a definition of the quantum EPRL intertwiner, study its convergence and braiding properties and construct an amplitude for the four-simplexes. We find that the resulting model is finite.}, author = {Fairbairn, Winston and Meusburger, Cathérine}, booktitle = {Proceedings of the 3rd Quantum Gravity and Quantum Geometry School}, date = {2011-02-28/2011-03-13}, faupublication = {yes}, pages = {PoS (QGQGS 2011)017}, peerreviewed = {unknown}, title = {q-{Deformation} of {Lorentzian} spin foam models}, venue = {Zakopane}, year = {2011} } @article{faucris.119308464, abstract = {We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the Engle, Pereira, Rovelli and Livine (EPRL) model. The q-deformed models are based on the representation theory of two copies of U (su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models, we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties, and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent. © 2012 American Institute of Physics.}, author = {Fairbairn, Winston and Meusburger, Cathérine}, doi = {10.1063/1.3675898}, faupublication = {yes}, journal = {Journal of Mathematical Physics}, peerreviewed = {Yes}, title = {{Quantum} deformation of two four-dimensional spin foam models}, volume = {53}, year = {2012} } @article{faucris.116759764, abstract = {We clarify the role of Drinfeld doubles and κ-Poincaré symmetries in quantized (2+1)-gravity and Chern-Simons theory. We discuss the conditions under which a given Hopf algebra symmetry is compatible with a Chern-Simons theory and determine this compatibility explicitly for the Drinfeld doubles and κ-Poincaré symmetries associated with the isometry groups of (2+1)-gravity. In particular, we show that κ-Poincaré symmetries with a timelike deformation are not directly associated with (2+1)-gravity. The association between these κ-Poincaré symmetries and Chern-Simons theory is possible only in the de Sitter case and the relevant Chern-Simons theory is physically inequivalent to (2+1)-gravity.}, author = {Meusburger, Cathérine}, doi = {10.1139/P08-076}, faupublication = {no}, journal = {Canadian Journal of Physics}, pages = {245-250}, peerreviewed = {Yes}, title = {{Quantum} double and κ-{Poincaré} symmetries in (2+1)-gravity and {Chern}-{Simons} theory}, volume = {87}, year = {2009} } @article{faucris.116762624, abstract = {Each of the local isometry groups arising in three-dimensional (3d) gravity can be viewed as a group of unit (split) quaternions over a ring which depends on the cosmological constant. In this paper we explain and prove this statement and use it as a unifying framework for studying Poisson structures associated with the local isometry groups. We show that, in all cases except for the case of Euclidean signature with positive cosmological constant, the local isometry groups are equipped with the Poisson-Lie structure of a classical double. We calculate the dressing action of the factor groups on each other and find, among others, a simple and unified description of the symplectic leaves of SU (2) and SL (2,R). We also compute the Poisson structure on the dual Poisson-Lie groups of the local isometry groups and on their Heisenberg doubles; together, they determine the Poisson structure of the phase space of 3d gravity in the so-called combinatorial description. © 2008 American Institute of Physics.}, author = {Meusburger, Cathérine and Schroers, Bernd}, doi = {10.1063/1.2973040}, faupublication = {no}, journal = {Journal of Mathematical Physics}, peerreviewed = {Yes}, title = {{Quaternionic} and {Poisson}-{Lie} structures in three-dimensional gravity: {The} cosmological constant as deformation parameter}, volume = {49}, year = {2008} } @article{faucris.121028864, abstract = {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 → 0. We show that the frequency shift 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 shift 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 shift function for a few simple examples. © 2013 Springer Basel.}, author = {Bonsante, Francesco and Meusburger, Cathérine and Schlenker, Jean-Marc}, doi = {10.1007/s00023-013-0300-6}, faupublication = {yes}, journal = {Annales Henri Poincaré}, pages = {1733-1799}, peerreviewed = {Yes}, title = {{Recovering} the {Geometry} of a {Flat} {Spacetime} from {Background} {Radiation}}, volume = {15}, year = {2014} } @inproceedings{faucris.122174404, abstract = {We consider an observer in a (2+l)-spacetime without matter and cosmological constant who measures spacetime geometry by emitting lightrays which return to him at a later time. We investigate several quantities associated with such Ughtrays: the return time, the directions into which light needs to be emitted to return and the frequency shift between the Ughtray at its emission and its return. We derive explicit expressions for these quantities as functions on the reduced phase space and show how they allow the observer to reconstruct the fidl geometry of the spacetime in finite eigentime. We comment on conceptual issues. In particidar, we clarify the relation between these quantities and Dirac observables and show that Wilson loops arise naturally in these quantities. © 2009 American Institute of Physics.}, author = {Meusburger, Cathérine}, booktitle = {Proceedings, 25th Max Born Symposium: The Planck Scale}, doi = {10.1063/1.3284381}, editor = {Jerzy Kowalski‐Glikman, R. Durka, M. Szczachor}, faupublication = {no}, isbn = {9780735407336}, keywords = {(2+l)-gravity; Lorentz geometry; Quantum gravity; Teichmilller space}, pages = {181-189}, peerreviewed = {unknown}, title = {{Spacetime} geometry in (2+l)-gravity via measurements with returning lightrays}, venue = {Wroclaw}, volume = {1196}, year = {2009} } @article{faucris.306934032, abstract = {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 labelled with bimodule categories over spherical fusion categories with bimodule traces. Line and point defects form directed graphs on these surfaces and labelled 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.}, author = {Meusburger, Cathérine}, doi = {10.1016/j.aim.2023.109177}, faupublication = {yes}, journal = {Advances in Mathematics}, keywords = {Bimodule categories; Defects; Diagrammatic calculus; Fusion categories; Spherical categories; State sum models}, note = {CRIS-Team Scopus Importer:2023-06-30}, peerreviewed = {Yes}, title = {{State} sum models with defects based on spherical fusion categories}, volume = {429}, year = {2023} } @article{faucris.119312424, abstract = {We relate three-dimensional loop quantum gravity to the combinatorial quantization formalism based on the Chern-Simons formulation for three-dimensional Lorentzian and Euclidean gravity with vanishing cosmological constant. We compare the construction of the kinematical Hilbert space and the implementation of the constraints. This leads to an explicit and very interesting relation between the associated operators in the two approaches and sheds light on their physical interpretation. We demonstrate that the quantum group symmetries arising in the combinatorial formalism, the quantum double of the three-dimensional Lorentz and rotation group are also present in the loop formalism. We derive explicit expressions for the action of these quantum groups on the space of cylindrical functions associated with graphs. This establishes a direct link between the two quantization approaches and clarifies the role of quantum group symmetries in three-dimensional gravity. © 2011 International Press.}, author = {Meusburger, Cathérine and Noui, Karim}, faupublication = {no}, journal = {Advances in Theoretical and Mathematical Physics}, pages = {1651-1716}, peerreviewed = {Yes}, title = {{The} {Hilbert} space of 3d gravity: {Quantum} group symmetries and observables}, url = {https://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=84858185698&origin=inward}, volume = {14}, year = {2010} } @article{faucris.122175724, abstract = {We quantise a Poisson structure on H, where H is a semidirect product group of the form G × g*. This Poisson structure arises in the combinatorial description of the phase space of Chern-Simons theory with gauge group G ⋉ g* on R × S, where S is a surface of genus g with n punctures. The quantisation of this Poisson structure is a key step in the quantisation of Chern-Simons theory with gauge group G × g*. We construct the quantum algebra and its irreducible representations and show that the quantum double D(G) of the group G arises naturally as a symmetry of the quantum algebra. © 2002 International Press.}, author = {Meusburger, Cathérine and Schroers, Bernd}, faupublication = {no}, journal = {Advances in Theoretical and Mathematical Physics}, pages = {1003-1042}, peerreviewed = {Yes}, title = {{The} quantisation of poisson structures arising in chern-simons theory with gauge group {G} x g*}, url = {https://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=11944255902&origin=inward}, volume = {7}, year = {2004} } @article{faucris.122188044, abstract = {We show how the constant curvature spacetimes of 3d gravity and the associated symmetry algebras can be derived from a single quantum deformation of the 3d Lorentz algebra sl (2, R). We investigate the classical Drinfel'd double of a "hybrid" deformation of sl (2, R) that depends on two parameters (η, z). With an appropriate choice of basis and real structure, this Drinfel'd double agrees with the 3d anti-de Sitter algebra. The deformation parameter η is related to the cosmological constant, while z is identified with the inverse of the speed of light and defines the signature of the metric. We generalise this result to de Sitter space, the three-sphere and 3d hyperbolic space through analytic continuation in η and z; we also investigate the limits of vanishing η and z, which yield the flat spacetimes (Minkowski and Euclidean spaces) and Newtonian models, respectively. © 2010 Elsevier B.V. All rights reserved.}, author = {Ballestreros, Angel and Herranz, Francisco J. and Meusburger, Cathérine}, doi = {10.1016/j.physletb.2010.03.043}, faupublication = {no}, journal = {Physics Letters B}, keywords = {Anti-de Sitter; Chern-Simons theory; Contraction; Cosmological constant; Deformation; Gravity; Hyperbolic; Spacetime}, pages = {375-381}, peerreviewed = {Yes}, title = {{Three}-dimensional gravity and {Drinfel}'d doubles: {Spacetimes} and symmetries from quantum deformations}, volume = {687}, year = {2010} } @article{faucris.119308684, abstract = {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→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.}, author = {Ballestreros, Angel and Herranz, Francisco J. and Meusburger, Cathérine and Naranjo, Pedro}, doi = {10.3842/SIGMA.2014.052}, faupublication = {yes}, journal = {Symmetry Integrability and Geometry-Methods and Applications}, keywords = {Anti-de sitter; Contraction; Cosmological constant; Deformation; Non-commutative spacetime; Poisson-lie groups; Quantum groups}, peerreviewed = {Yes}, title = {{Twisted} (2+1) κ-{AdS} algebra, {Drinfel}'d doubles and non-commutative spacetimes}, volume = {10}, year = {2014} }