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

TL;DR

This thesis advances the computation of torus one-point blocks in 2D CFT by addressing unphysical poles in the recursion relation and introduces a universal null operator in affine Lie algebra representations, enhancing understanding of null vectors.

Contribution

It provides a pole-free recursive method for Virasoro blocks on the torus and introduces a universal null operator in affine Lie algebra representations, independent of specific states.

Findings

Computed pole-free expressions up to order 4 in the recursion relation.

Introduced a universal null operator for affine Lie algebra representations.

Demonstrated the null operator generates null vectors universally.

Abstract

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.