arXiv subjects

Robert Oeckl

Publications and source records attributed to Robert Oeckl.

65 records · Page 4Linked to original sources

arXiv2002

Structure Theorem for Covariant Bundles on Quantum Homogeneous Spaces

The natural generalization of the notion of bundle in quantum geometry is that of bimodule. If the base space has quantum group symmetries one is particularly interested in bimodules covariant (equivariant) under these symmetries. Most attention has so far been focused on the case with maximal symmetry -- where the base space is a quantum group and the bimodules are bicovariant. The structure of bicovariant bimodules is well understood through their correspondence with crossed modules. We investigate the ``next best'' case -- where the base space is a quantum homogeneous space and the bimodules are covariant. We present a structure theorem that resembles the one for bicovariant bimodules. Thus, there is a correspondence between covariant bimodules and a new kind of ``crossed'' modules which we define. The latter are attached to the pair of quantum groups which defines the quantum homogeneous space. We apply our structure theorem to differential calculi on quantum homogeneous spaces and discuss a related notion of induced differential calculus.

math.QA↗

arXiv2002

Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants

We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold (<=3, <=4, any). Ordinary LGT is recovered if the category is the (symmetric) category of representations of a compact Lie group. In the weak coupling limit we recover discretized BF-theory in terms of a coordinate free version of the spin foam formulation. We work on general cellular decompositions of the underlying manifold. In particular, we are able to formulate LGT as well as spin foam models of BF-type with quantum gauge group (in dimension <=4) and with supersymmetric gauge group (in any dimension). Technically, we express the partition function as a sum over diagrams denoting morphisms in the underlying category. On the LGT side this enables us to introduce a generalized notion of gauge fixing corresponding to a topological move between cellular decompositions of the underlying manifold. On the BF-theory side this allows a rather geometric understanding of the state sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we recover. The construction is extended to include Wilson loop and spin network type observables as well as manifolds with boundaries. In the topological (weak coupling) case this leads to TQFTs with or without embedded spin networks.

hep-th↗

arXiv2002

The Quantum Geometry of Supersymmetry and the Generalized Group Extension Problem

We examine the notion of symmetry in quantum field theory from a fundamental representation theoretic point of view. This leads us to a generalization expressed in terms of quantum groups and braided categories. It also unifies the conventional concept of symmetry with that of exchange statistics and the spin-statistics relation. We show how this quantum group symmetry is reconstructed from the traditional (super) group symmetry, statistics and spin-statistics relation. The old question of extending the Poincare group to unify external and internal symmetries (solved by supersymmetry) is reexamined in the new framework. The reason why we should allow supergroups in this case becomes completely transparent. However, the true symmetries are not expressed by groups or supergroups here but by ordinary (not super) quantum groups. We show in this generalized framework that supersymmetry remains the most general unification of internal and space-time symmetries provided that all particles are either bosons or fermions. Finally, we demonstrate with some examples how quantum geometry provides a natural setting for the construction of super-extensions, super-spaces, super-derivatives etc.

hep-th↗

arXiv2001

Spin Foam Diagrammatics and Topological Invariance

We provide a simple proof of the topological invariance of the Turaev-Viro model (corresponding to simplicial 3d pure Euclidean gravity with cosmological constant) by means of a novel diagrammatic formulation of the state sum models for quantum BF-theories. Moreover, we prove the invariance under more general conditions allowing the state sum to be defined on arbitrary cellular decompositions of the underlying manifold. Invariance is governed by a set of identities corresponding to local gluing and rearrangement of cells in the complex. Due to the fully algebraic nature of these identities our results extend to a vast class of quantum groups. The techniques introduced here could be relevant for investigating the scaling properties of non-topological state sums, being proposed as models of quantum gravity in 4d, under refinement of the cellular decomposition.

gr-qc↗

arXiv2001

The dual of non-Abelian Lattice Gauge Theory

Non-Abelian Lattice Gauge Theory in Euclidean space-time of dimension d>=2 whose gauge group is any compact Lie group is related to a Spin Foam Model by an exact strong-weak duality transformation. The group degrees of freedom are integrated out and replaced by combinatorial expressions involving irreducible representations and intertwiners of the gauge group. This transformation is available for the partition function, for the expectation value of observables (spin networks), and for the correlator of centre monopoles which is a ratio of partition functions in the original model and an ordinary expectation value in the dual formulation.

hep-lat↗

arXiv2001

The Quantum Geometry of Spin and Statistics

Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin-statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose-Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach.

hep-th↗

arXiv2001

The dual of pure non-Abelian lattice gauge theory as a spin foam model

We derive an exact duality transformation for pure non-Abelian gauge theory regularized on a lattice. The duality transformation can be applied to gauge theory with an arbitrary compact Lie group G as the gauge group and on Euclidean space-time lattices of dimension d >= 2. It maps the partition function as well as the expectation values of generalized non-Abelian Wilson loops (spin networks) to expressions involving only finite-dimensional unitary representations, intertwiners and characters of G. In particular, all group integrations are explicitly performed. The transformation maps the strong coupling regime of non-Abelian gauge theory to the weak coupling regime of the dual model. This dual model is a system in statistical mechanics whose configurations are spin foams on the lattice.

hep-th↗

arXiv2000

Braided Quantum Field Theory

We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for $n$-point functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have non-trivial over- and under-crossings. We demonstrate the power of our approach by applying it to $ϕ^4$-theory on the quantum 2-sphere. We find that the basic divergent diagram of the theory is regularised.

hep-th↗

arXiv2000

Untwisting Noncommutative R^d and the Equivalence of Quantum Field Theories

We show that there is a duality exchanging noncommutativity and non-trivial statistics for quantum field theory on R^d. Employing methods of quantum groups, we observe that ordinary and noncommutative R^d are related by twisting. We extend the twist to an equivalence for quantum field theory using the framework of braided quantum field theory. The twist exchanges both commutativity with noncommutativity and ordinary with non-trivial statistics. The same holds for the noncommutative torus.

hep-th↗

arXiv1998

Twisting of quantum differentials and the Planck scale Hopf algebra

We show that the crossed modules and bicovariant different calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisation of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra $\C[x]\bicross_{\hbar,\grav}\C[p]$. This is a quantum group which has an $\hbar\to 0$ limit as the functions on a classical but non-Abelian group and a $\grav\to 0$ limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality like self-duality. The noncommutative geometry turns out to be singular when $\grav\to 0$ and is therefore not visible in flat space quantum mechanics alone.

math.QA↗

arXiv1998

Classification of Differential Calculi on U_q(b+), Classical Limits, and Duality

We give a complete classification of bicovariant first order differential calculi on the quantum enveloping algebra U_q(b+) which we view as the quantum function algebra C_q(B+). Here, b+ is the Borel subalgebra of sl_2. We do the same in the classical limit q->1 and obtain a one-to-one correspondence in the finite dimensional case. It turns out that the classification is essentially given by finite subsets of the positive integers. We proceed to investigate the classical limit from the dual point of view, i.e. with ``function algebra'' U(b+) and ``enveloping algebra'' C(B+). In this case there are many more differential calculi than coming from the q-deformed setting. As an application, we give the natural intrinsic 4-dimensional calculus of kappa-Minkowski space and the associated formal integral.

math.QA↗