[A] Classification of group actions on von Neumann algebras (1987-1990)
We have worked on classification problems of group actions on injective factors. We have classified certain types of one-parameter automorphism groups of the hyperfinite II_{1} factor in [1], actions of discrete abelian groups on injective factors in [2], and actions of compact abelian groups on injective factors in [3]. Paper [3] has been a source for many recent classification papers on group actions and realted topics.
[B] Subfactor theory (1990-1998)
We have studied theory of subfactors initiated by V. F. R. Jones. We have clarified basics of paragroup theory of Ocneanu and introduced the orbifold construction for subfactors in [4], [5]. We have also studied the structure of the automorphism groups of subfactors in [6]. These studies have been compiled into a book [7], which has been cited for more than 170 times on MathSciNet.
[C] Conformal field theory and operator algebras (since 1998)
We have been studying operator algebraic approach to conformal field theory. We have unified the theory of alpha-induction studied by Longo, Rehren and Xu, and Ocneanu's graphical calculus, and obtained various fundamental results in [8], [10], which have been used in a wide context. We have introduced a key notion of complete rationality, given its operator algebraic characterization, and proved non-degeneracy of braiding in [9]. This was the first such achievement in a general setting of conformal field theory.
We have classified operator algebraic chiral conformal field theories with central charge less than 1 in [11]. This is the first definite classification result in the history of 50 years of algebraic quantum field theory and our classification list contains an example which does not seem to be constructed with other known methods. We have also obtained a related classification result for full conformal field theory in [12].
We have constructed an operator algebraic counterpart of the Moonshine vertex operator algebra and shown that its automorphism group is indeed the Monster group in [13].
We have given a construction of spectral triples, "noncommutative manifolds" in noncommutative geometry, from a superconformal field theory in [14]. This has given a new relation between superconformal field theory and noncommutative geometry. In [15], we further studied noncommutative geometric aspects of N=2 superconformal field theory, and through computations of the index pairing, have found a new connection between subfactor theory and noncommutative geometry. On the way, we have also found a new proof of the character formulas of the N=2 super Virasoro algebras, which have not been well-understood.
We have given a general construction for a local conformal net from a (strongly local) vertex operator algebra and a backward construction in [16]. This gives the first definite connection between the two mathematical approaches to conformal field theory which has been sought for more than 15 years.