The starting point of this paper is that any two-dimensional conformal field theory is defined in terms of correlation functions on the complex plane. Normally, the 2-point correlation function has a power law singularity. Such behavior occurs when the arguments of two fields approach each other and the fields are eigenvectors of the generator of unitary scale transformations. Unitarity is essential for the power law. In non-unitary theories, however, logarithmic singularities may occur. In unitary theories the state space is a direct sum of irreducible representations of the Virasoro algebra, while in non-unitary theories the indecomposable summands may not be irreducible. The paper reviews the definition of bulk and conformal field theory with boundary suited for logarithmic singular 2-point correlators. The paper consists of four parts. Section 1 provides an Introduction. Section 2 defines conformal field theory on the complex plane with boundary. Section 3 investigates in detail conformal field theory notions using an algebraic setting of braided monoidal abelian categories. The Deligne product of abelian categories is also reviewed. Section 4 studies in detail one specific example logarithmic CFT, the \(W_{2,3}\)-model with zero central charge.
For the entire collection see [Zbl 1278.00022].