W-algebras arising as chiral algebras of conformal field theory

The presence of infinite dimensional chiral symmetries is central to the study of two dimensional conformal field theories. These symmetries, called chiral algebras, are formed by the purely holomorphic and purely anti-holomorphic fields, respectively. How the field content of a conformal field theory (CFT) is organised into representations of the chiral algebras is encoded in the fusion ring. The task of solving and classifying general conformal field theories can be split into two separate problems: to find chiral algebras of conformal field theory and to determine possible fusion rings of chiral algebras. This programme has been most successful for the Virasoro minimal series [1], where the fields fall into finitely many representations of the Virasoro algebra. This has subsequently enabled the complete characterisation of the field content of such theories [2]. But considerable progress has also been made in constructing other chiral algebras of CFT. I present here a survey of these results.