  This thesis is divided into two parts, where in the first part we investigate
the computation of Virasoro 1-point blocks on the torus in the framework of
Zamolodchikov's recursion relation. It is widely accepted that this recursion
relation contains unphysical poles in the central charge $c$. At each order we
conjecture how the pole free expressions depend on the internal and external
conformal dimensions and central charge, and propose how to compute it
numerically. In this thesis, we have calculated the pole free expression up to
order 4. In the second part we introduce a conformal field theory with an extra
symmetry, described by highest weight representations of the affine Lie algebra
$\hat{\mathfrak{sl}}_2$. At level 1, we determined a universal
$\mathfrak{sl}_2$ basis-independent 'null operator', which generates null
vectors in the usual sense. The 'null operators' are generalized objects and
can be applied to any state of the horizontal representation, yielding null
vectors and are therefore independent from the choice of horizontal
representations.
