Related papers: Mapping-class groups of 3-manifolds in canonical q…
The {\em topological symmetry group} of an embedding $\Gamma$ of an abstract graph $\gamma$ in $S^3$ is the group of automorphisms of $\gamma$ which can be realized by homeomorphisms of the pair $(S^3, \Gamma)$. These groups are motivated…
We consider an inhomogeneous quantum supergroup which leaves invariant a supersymmetric particle algebra. The quantum sub-supergroups of this inhomogeneous quantum supergroup are investigated.
The Reshetikhin - Turaeve approach to topological invariants of three - manifolds is generalized to quantum supergroups. A general method for constructing three - manifold invariants is developed, which requires only the study of the…
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum $sl(2)$ were obtained by the last three authors in arXiv:1202.3553 . They are invariants of $3$-manifolds together with a cohomology class which…
Each rule $f$ that assigns a vector $f(G)$ to an $(n+1)$-graph $G$ determines a class (or property) of $n$-manifold invariants. An invariant $v=v(M)$ is in this class if, for any triangulated manifold $|G|=M$, one has that $v(M)$ is a…
Recent developments concerning canonical quantisation and gauge invariant quantum mechanical systems and quantum field theories are briefly discussed. On the one hand, it is shown how diffeomorphic covariant representations of the…
We identify and study a class of hyperbolic 3-manifolds (which we call Macfarlane manifolds) whose quaternion algebras admit a geometric interpretation analogous to Hamilton's classical model for Euclidean rotations. We characterize these…
The article is devoted to the investigation of groups of diffeomorphisms and loops of manifolds over ultra-metric fields of zero and positive characteristics. Different types of topologies are considered on groups of loops and…
Spin foam models, loop quantum gravity and group field theory are discussed as quantum gravity candidate theories and usually involve a continuous Lie group. We advocate here to consider quantum gravity inspired models with finite groups,…
We propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and…
We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within $SU(2)$ quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering…
We define some new invariants for 3-manifolds using the space of taut codim-1 foliations along with various techniques from noncommutative geometry. These invariants originate from our attempt to generalise Topological Quantum Field…
Spin layer groups are the crystallographic symmetry groups with a periodic plane, and their symmetry operations are inherited from three-dimensional (3D) spin space groups. However, the direct application of 3D symmetry groups to…
We propose and in some cases prove a precise relation between 3-manifold invariants associated with quantum groups at roots of unity and at generic $q$. Both types of invariants are labeled by extra data which plays an important role in the…
This survey focuses on the computational complexity of some of the fundamental decision problems in 3-manifold theory. The article discusses the wide variety of tools that are used to tackle these problems, including normal and almost…
We make a case for the unique relevance of Cartan geometry for gauge theories of gravity and supergravity. We introduce our discussion by recapitulating historical threads, providing motivations. In a first part we review the geometry of…
On basis of generalized 6j-symbols we give a formulation of topological quantum field theories for 3-manifolds including observables in the form of coloured graphs. It is shown that the 6j-symbols associated with deformations of the…
We show that R-matricies of all simple quantum groups have the properties which permit to present quantum group twists as transitions to other coordinate frames on quantum spaces. This implies physical equivalence of field theories…
We construct a space of quantum states and an algebra of quantum observables, over the set of all metrics of arbitrary but fixed signature, defined on a manifold. The construction is diffeomorphism invariant, and unique up to natural…
The phenomenology for the deep spatial geometry of loop quantum gravity is discussed. In the context of a simple model of an atom of space, it is shown how purely combinatorial structures can affect observations. The angle operator is used…