Characters of representations of chiral algebras are important tools in conformal field theory. They are a special chase of chiral torus 1-point functions (namely those where the vacuum has been inserted) and their modular properties famously give rise to the Verlinde formula. In this talk we will generalise from vacuum insertions to insertions from any irreducible representation in the example of the su2 WZW models at non-negative integral level and explore their modular properties.

Two-dimensional conformally invariant quantum field theories (CFTs for short) form a sprawling network of ideas connecting many areas of physics and mathematics. A particularly celebrated class are the rational CFTs. These are essentially characterised by having a completely reducible representation theory and only a finite number of inequivalent irreducible representations. Rational CFTs exhibit a number of extraordinary features, foremost being the Verlinde formula which determines correlation functions from certain transformation properties of the CFTs characters. Logarithmic CFTs by contrast are almost maximally awful in that their representation theory is necessarily not completely reducible and need not have finitely many inequivalent irreducible representations. I will present recent results on such logarithmic CFTs and argue that suitable generalisations of rational features exist, at least in certain cases. So things are not as bad as one might fear.

Given some chiral conformal field theory (also known as a vertex operator algebra in the mathematics literature), a natural but highly non-trivial task is to classify its representation theory. In this talk, I will use some well known examples of conformal field theories, such as the Virasoro minimal models, to show how certain hard questions in representation theory can be neatly rephrased as comparatively easy questions in the theory of symmetric polynomials. After a brief overview of the theory of symmetric polynomials, I will show how they can be used to classify irreducible representations.