Abstract: Local fields of a conformal field theory (CFT) are precisely the observable quantities which can be inserted in correlation functions, and by the state-field correspondence the space of local fields of a CFT can be identified with the state space of the theory. In particular, the space of local fields carries the actions of the symmetries of the theory, notably a representation of two commuting copies of the Virasoro algebra which account for the effect of conformal transformations.

In this talk we focus on the specific example of the Fock space of local fields of a free bosonic CFT, consisting roughly speaking of (normal ordered) polynomials in the gradient of the Gaussian Free Field (GFF). We review the Sugawara construction, which equips the Fock space with a representation of Virasoro algebras, and we review how, by recursive OPE coefficient extraction, one assigns correlation functions to Fock space local fields in simply connected planar domains, with either Dirichlet or Neumann boundary conditions for the GFF. The talk is meant to be a review of some known results in CFT which serve as a background for a subsequent talk by David Adame-Carrillo on probabilistically constructed local fields of the discrete Gaussian free field.