Classification of operator algebraic conformal field theories.pdf

a r X i v : m a t h / 0 2 1 1 1 4 1 v 3 [ m a t h .O A ] 1 6 A p r 2 0 0 3 Classification of operator algebraic conformal field theories Yasuyuki Kawahigashi ? Department of Mathematical Sciences University of Tokyo, Komaba, Tokyo, 153-8914, Japan e-mail: yasuyuki@ms.u-tokyo.ac.jp Abstract We give an exposition on the current status of classification of operator algebraic conformal field theories. We explain roles of complete rationality and α-induction for nets of subfactors in such a classification and present the current classification result, a joint work with R. Longo, for the case with central charge less than 1, where we have a complete classification list consisting of the Virasoro nets, their simple current extensions of index 2, and four exceptionals. Two of the four exceptionals appear to be new. 1 Introduction Classification of conformal field theory is obviously one of the most important and exciting problems in mathematical physics today. Here we review the current status of the classification theory of 1-dimensional operator algebraic conformal field theories. First, we explain what we mean by “1-dimensional operator algebraic conformal field theories” and how they naturally appear in the setting of 1 + 1-dimensional conformal field theory. We are now in the framework of algebraic quantum field theory in the sense of [21]. In the algebraic quantum field theory, we consider a net of von Neumann algebras {B(O)} where O is a certain bounded region in the Minkowski space, and the physical idea is that the von Neumann algebra B(O) is generated by physical quantities observable in the region O. From a physical viewpoint, the most natural setting is that we have a 4-dimensional Minkowski space and O is a double cone, which is defined to be a set of the form (x+ V+) ∩ (y + V?), where V± = {z = (z0, z1, z2, z3) ∈ R 4 | z20 ? z 2 1 ? z 2 2 ? z 2 3 0,±z0 0}. We say a net because the index set of double cones is directed with respect to in- clusions. Th