A non - rational CFT with central charge 1

Two dimensional conformal field theories with central charge one are discussed. After a short review of theories based on one free boson, a different CFT is described, which is obtained as a limit of minimal models. We will be concerned with unitary two dimensional conformal field theories. Such theories describe, for example, critical points of equilibrium statistical mechanics systems in two dimensions, or vacua of string theory around which a perturbative expansion of string amplitudes can be computed. Since the discovery by Belavin, Polyakov and Zamolodchikov of a discrete series of CFTs, the minimal models [1], much effort has been spent in trying to classify all CFTs. An important parameter in this problem is the central charge c of a CFT, which appears in the two point function of the stress tensor T (ζ) on the complex plane T (ζ)T (ζ ′) = c/2 (ζ − ζ ′) 4. (1) Another useful notion is that of a rational CFT. Roughly speaking, a CFT is rational when the algebra generated by all conserved charges is large enough for the Hilbert space to decompose into only finitely many representations. Here we are only considering unitary CFTs which makes the classification problem somewhat easier, as for example logarithmic CFTs [2] cannot occur. Nonetheless it turned out to be a very difficult problem. A very brief summary of the state of affairs could take the following form • c < 1: All unitary CFTs with central charge less than one are known, they are the minimal models of [1] whose different modular invariants were given in [3]. Their central charges take the values c = 1 − 6 p(p + 1). (2) with p = 2, 3, 4,. . .. For p = 2 on has c = 0 and the theory is trivial in the sense that its only state is the vacuum. p = 3 gives c = 1/2 and one obtains the Ising model. For p → ∞ the central charge approaches one. All these theories are rational CFTs.