Global Vertex Algebras on Riemann Surfaces

Conformal field theory is intimately connected to the theory of
vertex algebras and the geometry of Riemann surfaces. In this
thesis a new algebro-geometric structure called global vertex
algebra is defined on Riemann surfaces which is supposed to be a
natural generalization of vertex algebras. In order to define this
structure a formal calculus of fields on Riemann surfaces is
constructed. The basic objects in vertex algebra theory are fields.
They are defined as formal Laurent series with possibly infinite
principal part. The coefficients are endomorphisms. As an example
for such a structure the global vertex algebra of bosons of
Krichever-Novikov type will be constructed. At the beginning of
this thesis the formal calculus of classical vertex algebras is
introduced from the viewpoint of distributions in complex analysis.
Furthermore a graphical calculus for the computation of correlation
functions of primary fields associated to affine Kac-Moody algebras
is introduced.