Research Abstract of Yasuyuki Kawahigashi
[A] Classification of group actions on von Neumann algebras (19871990)
We have worked on classification problems of group actions on
injective factors. We have classified certain types of oneparameter
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 (19901998)
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 alphainduction 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 nondegeneracy 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 wellunderstood.
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.
List of selected papers

[1] Oneparameter automorphism groups of the injective
II_{1} factor arising from the irrational rotation C^{*}algebra,
Amer. J. Math. 112 (1990), 499524.

[2] (with C. E. Sutherland, M. Takesaki)
The structure of the automorphism group of an injective factor
and the cocycle conjugacy of discrete abelian group actions,
Acta Math. 169 (1992), 105130.

[3] (with M. Takesaki)
Compact abelian group actions on injective factors,
J. Funct. Anal. 105 (1992), 112128.

[4] On flatness of Ocneanu's connections on the Dynkin diagrams
and classification of subfactors,
J. Funct. Anal. 127 (1995), 63107.

[5] (with D. E. Evans)
Orbifold subfactors from Hecke algebras,
Commun. Math. Phys. 165 (1994), 445484.

[6] Centrally trivial automorphisms and an analogue of Connes's chi(M)
for subfactors,
Duke Math. J. 71 (1993), 93118.

[7] (with D. E. Evans)
"Quantum symmetries on operator algebras",
Oxford University Press, 1998.

[8] (with J. Böckenhauer, D. E. Evans)
On alphainduction, chiral generators and modular invariants for subfactors,
Commun. Math. Phys. 208 (1999), 429487.

[9] (with R. Longo, M. Müger)
Multiinterval subfactors and modularity of representations in conformal field theory,
Commun. Math. Phys. 219 (2001), 631669.

[10] (with J. Böckenhauer, D. E. Evans)
Chiral structure of modular invariants for subfactors,
Commun. Math. Phys. 210 (2000), 733784.

[11] (with R. Longo)
Classification of local conformal nets: Case c < 1,
Ann. of Math. 160 (2004), 493522.

[12] (with R. Longo)
Classification of twodimensional local conformal nets with c < 1
and 2cohomology vanishing for tensor categories,
Commun. Math. Phys. 244 (2004), 6397.

[13] (with R. Longo)
Local conformal nets arising from framed vertex operator algebras,
Adv. Math. 206 (2006), 729751.

[14] (with S. Carpi, R. Hillier, R. Longo)
Spectral triples and the superVirasoro algebra,
Commun. Math. Phys. 295 (2010), 7197.

[15] (with S. Carpi, R. Hillier, R. Longo, F. Xu)
N=2 superconformal nets,
Commun. Math. Phys. 336 (2015), 12851328.

[16]
(with S. Carpi, R. Longo, M. Weiner)
From vertex operator algebras to conformal nets and back,
Mem. Amer. Math. Soc. 254 (2018), no. 1213, vi+85 pp.
Selected inivited talks
 Subfactor theory and its applications 
Operator algebras and quantum field theory 
Annual meeting of Mathematical Society of Japan (Plenary talk),
Meiji University (Japan), March 2002.
 Conformal field theory and operator algebras
International Congress on Mathematical Physics 
ICMP 2006 (Plenary talk), Rio de Janeiro (Brazil), August 2006.
 Quantum field theory and operator algebras
5th Asian Mathematical Conference
(Plenary talk), Kuala Lumpur (Malaysia), June 2009.
 Superconformal field theory and operator algebras
Seminal Interactions between
Mathematics and Physics, Rome (Italy), September 2010.
(slides)
 Conformal field theory, vertex operator algebras and operator algebras
ICM 2018, Rio de Janeiro (Brazil), August 2018 (Sections: "Analysis and Operator Algebras", "Mathematical Physics").
(video)