A braided quasi quantum group from strings in a magnetic field : two applications of noncommutative geometry
The operators that realize a discrete symmetry group on a physical system describing a particle in a magnetic field furnish a projective representation of the group on the particle states. Promoting the magnetic background to a topologically non trivial field, one can still derive the magnetic translation operators from the path integral approach in their general form. For that, one has to introduce some basic differential geometrical techniques, for example the concept of a complex line bundle with connection, the wave functions seen as sections thereof as well as holonomies along trajectories. It is interesting to ask whether such a construction can be extended to the case of a propagating string, making use of similar geometrical techniques. In the higher dimensional string case, the magnetic field is a 3-form, the so called Kalb-Ramond field strength. To derive the operators that lift the group action to the states, one has to construct the magnetic amplitudes in terms of the locally given fields. Once constructed, the translation operators turn out to generate a braided quasi quantum group which is the twisted version of the Drinfel'd quantum double of a finite group algebra. The 3-cocycle providing the twist is related to the Kalb-Ramond field by a series of de Rham, Cech and group cohomological equations finding their natural framework in a tricomplex. As a Hopf algebra, the algebra generated by the magnetic translations comes equipped with some algebraic ingredients, that find their physical counterpart in operations on magnetic amplitudes. For example, the coproduct provides the representation of a string interaction process given by a world-sheet having the shape of a pair of pants. A braided Hopf algebra allows for a representation of the braid group on the tensor category of its modules, which becomes as such a braided monoidal category. This allows to identify the Hilbert spaces of string states with objects of the category. It is rather amazing that such a non-trivial algebraic structure pops up almost automatically by the requirement of gauge invariance and the commutation of translations with propagation.