Catherine Meusburger

Quantum deformation of two four-dimensional spin foam models

We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the EPRL model. The q-deformed models are based on the representation theory of two copies of Uq(su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models we give a denition of the quantum EPRL intertwiners, study their convergence and braiding properties and construct an amplitude for the four-simplexes. We nd that both of the resulting models are convergent.

Poisson structure and symmetry in the Chern–Simons formulation of (2 + 1)-dimensional gravity

In the formulation of (2 + 1)-dimensional gravity as a Chern?Simons gauge theory, the phase space is the moduli space of flat Poincar? 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 Poincar? 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.

Algebraic Quantization of the Closed Bosonic String

Abstract: The gauge invariant observables of the closed bosonic string are quantized in four space-time dimensions by constructing their quantum algebra in a manifestly covariant approach, respecting all symmetries of the classical observables. The quantum algebra is the kernel of a derivation on the universal enveloping algebra of an infinite-dimensional Lie algebra. The search for Hilbert space representations of this algebra is separated from its construction, and postponed.

Mapping class group actions in Chern–Simons theory with gauge group G⋉g*

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.

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.

q-Deformation of Lorentzian spin foam models

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.

Gauge fixing in (2+1)-gravity: Dirac bracket and spacetime geometry

We consider (2+1)-gravity with a vanishing cosmological constant as a constrained dynamical system. By applying Diracs gauge fixing procedure, we implement the constraints and determine the Dirac bracket on the gauge-invariant phase space. The chosen gauge fixing conditions have a natural physical interpretation and specify an observer in the spacetime. We derive explicit expressions for the resulting Dirac brackets and discuss their geometrical interpretation. In particular, we show that specifying an observer with respect to two point particles gives rise to conical spacetimes, whose deficit angle and time shift are determined, respectively, by the relative velocity and minimal distance of the two particles.

Gauge Fixing and Classical Dynamical r-Matrices in ISO(2, 1)-Chern–Simons Theory

We apply the Dirac gauge fixing procedure to Chern–Simons theory with gauge group ISO(2, 1) on manifolds

GAUGE FIXING AND QUANTUM GROUP SYMMETRIES IN (2+1)-GRAVITY

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