Conformal quantum field theory and subfactors Yasuyuki Kawahigashi ∗ Department of Mathematical Sciences University of Tokyo, Komaba, Tokyo, 153-8914, JAPAN e-mail: yasuyuki@ms.u-tokyo.ac.jp Abstract We survey a recent progress on algebraic quantum field theory in connection to subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory. 1 Introduction Algebraic quantum field theory is an operator algebraic approach to quantum field theory. Here we review methods of Haag-Kastler nets of operator algebras on a spacetime with emphasis on recent progresses in low dimensions in connection to subfactor theory and modular invariants. In algebraic quantum field theory, we have a family of operator algebras parameterized by regions in a certain spacetime. Each algebra represents a system of physical quantities observable in the corresponding region. Representation theory of such a family of operator algebras has turned out to be quite interesting mathematically. (See [24] for a general theory of algebraic quantum field theory.) A natural “spacetime” for such a formulation is a 4-dimensional Minkowski space, but in this article, we will concentrate on one-dimensional compactified “spacetime”, S 1 . (One way to get this situation naturally is making a tensor product decomposition of a theory of 2-dimensional spacetime. Such a one-dimensional theory is often called a chiral theory.) A one-dimensional theory has caught much attention recently and provides a rich source of mathematical problems and insight. 2 Conformal nets and representation theory Now our “spacetime” is one-dimensional circle S 1 and a region in this spacetime is an interval which means a non-empty, non-dense, open, and connected set in S 1 . We study * Supported in part by the Grants-in-Aid for Scientific Research, JSPS.