Bosonic ghostbusting - vertex algebras and quantum groups

The aim of this project has been to investigate into the representation theory of the bosonic ghost vertex algebra and to construct a quantum group with an equivalent category of modules. The bosonic ghost vertex algebra violates many properties of other vertex algebras which simplify the representation theory and this means one cannot directly apply many of the methods commonly used in these simpler cases. A key component in many of the methods employed is the screening operator. The vertex algebra itself is the kernel of the screening operator acting on a lattice vertex algebra. The module containing the screening operator characterises a Nichols algebra which is crucial in the construction of the corresponding quantum group. An equivalence of representation theory is with respect to a certain level of structure. The consensus is that a sensible choice of module category for vertex algebras should admit a braided tensor structure. We realised that the notion of a dual module vertex algebras possess leads to a generalised duality called Grothendieck-Verdier structure on the module category. Therefore we construct the equivalence at the level of ribbon Grothendieck-Verdier structure, a braided monoidal category with a twist and Grothendieck-Verdier dual. We develop a rigorous framework for constructing functors from a given category to a vertex algebra module category. We work through all the details of an equivalence for lattice vertex algebras and the bosonic ghost vertex algebra with their Hopf algebra counterparts. Finally, we construct the candidate quantum group and prove the ribbon Grothendiec-Verdier equivalence to the bosonic ghost vertex algebra, up to one equation involving inter-twining operators. A more general equivalence between vertex algebras which are the kernel of screenings and quantum groups constructed from the corresponding Nichols algebra is still conjectural but more evidence and tools are now availabl