The sl_3 web algebra

In this paper we use Kuperberg’s sl3-webs and Khovanov’s sl3-foams to define a new algebra KS, which we call the sl3-web algebra. It is the sl3 analogue of Khovanov’s arc algebra. We prove that KS is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that KS is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K0 (WS )Q(q) , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that KS is a graded cellular algebra.info:eu-repo/semantics/publishedVersio

Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams

We show how Feynman amplitudes of standard QFT on flat and homogeneous space can naturally be recast as the evaluation of observables for a specific spin foam model, which provides dynamics for the background geometry. We identify the symmetries of this Feynman graph spin foam model and give the gauge-fixing prescriptions. We also show that the gauge-fixed partition function is invariant under Pachner moves of the triangulation, and thus defines an invariant of four-dimensional manifolds. Finally, we investigate the algebraic structure of the model, and discuss its relation with a quantization of 4d gravity in the limit where the Newton constant goes to zero.Comment: 28 pages (RevTeX4), 7 figures, references adde

An Invitation to Higher Gauge Theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a 'tangent 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra that governs 11-dimensional supergravity.Comment: 60 pages, based on lectures at the 2nd School and Workshop on Quantum Gravity and Quantum Geometry at the 2009 Corfu Summer Institut

Trace as an alternative decategorification functor

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with arXiv:1405.5920 by other author

The Supermembrane with Central Charges on a G2 Manifold

We construct the 11D supermembrane with topological central charges induced through an irreducible winding on a G2 manifold realized from the T7/Z2xZ2xZ2 orbifold construction. The hamiltonian H of the theory on a T7 target has a discrete spectrum. Within the discrete symmetries of H associated to large diffeomorphisms, the Z2xZ2xZ2 group of automorphisms of the quaternionic subspaces preserving the octonionic structure is relevant. By performing the corresponding identification on the target space, the supermembrane may be formulated on a G2 manifold, preserving the discretness of its supersymmetric spectrum. The corresponding 4D low energy effective field theory has N=1 supersymmetry.Comment: Reviewed version. spectral propertis discussed, two more sections added, 27 pages,Late

Sui Generis Rights on Folklore Viewed from a Property Rights Perspective

Aussi publié dans: Ejan MACKAAY, « Sui generis rights on foklore viewed from a property rights perspective », dans Kilian BIZER, Matthias LANKAU et Gerald SPINDLER (dir.), Sui generis Rechte zum Schutz traditioneller kultureller Ausdrucksweisen - Interdisziplinäre Perspektiven, Göttingen, Universitätsverlag Göttingen, 20132, p. 139, en ligne: http://resolver.sub.uni-goettingen.de/purl?isbn-978-3-86395-064-4 (consulté le 5 avril 2017)This paper looks at sui generis rights claimed for the protection of folklore. Since rights should not be created in any which way if one is to avoid privileges and rent-seeking, it is important to be clear about design constraints stemming from such rights being species of property rights, adapted to deal with the particular content of information structures that need special encouragement or protection. Examination of the logic of property rights in general and of intellectual property rights in particular reveals that intellectual property rights are sought because of their decentralised incentive and information effects, but that they need to be circumscribed because of the monopolistic effects they entail. The trouble with monopoly is that whilst it is in place, one does not realise the creativity that is prevented from emerging. All intellectual property rights reflect compromises of these contradictory tendencies and as a result, more and stronger intellectual property rights are not necessarily better from a general welfare point of view. The forms of sui generis rights proposed for folklore appear modelled on copyright, but with the removal of several key features that define the equilibrium inherent in copyright: no originality requirement; no known creation date or creators; indefinite duration. Folklore kept secret is altogether taken out of commerce. As a result, these rights strike a balance very much more to the monopoly side of the spectrum than do existing intellectual property rights and hence risk severely constraining creativity. This may seem like an acceptable constraint given the objective of preservation, but one must realise that it will affect the future carriers of the protected information. Faced with severe restrictions on ways they can improve their lives within the protected setting, they may well opt for the exit option and head for greener pastures. This would severely strain efforts to preserve whatever the sui generis rights aim to protect. Information lock-up may not be the most promising formula for preservation