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This thesis is divided into two parts, where in the first part we investigatethe computation of Virasoro 1-point blocks on the torus in the framework ofZamolodchikov's recursion relation. It is widely accepted that this recursionrelation contains unphysical poles in the central charge $c$. At each order weconjecture how the pole free expressions depend on the internal and externalconformal dimensions and central charge, and propose how to compute itnumerically. In this thesis, we have calculated the pole free expression up toorder 4. In the second part we introduce a conformal field theory with an extrasymmetry, 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 nullvectors in the usual sense. The 'null operators' are generalized objects andcan be applied to any state of the horizontal representation, yielding nullvectors and are therefore independent from the choice of horizontalrepresentations.

The torus one-point block of 2d CFT and null vectors in $\hat{\mathfrak{sl_2}}$