|Beschreibung:||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
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.