2006 Fiscal Year Final Research Report Summary
Operator algebras and mathematical physics
| Project/Area Number |
16340045
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | University of Tokyo |
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Principal Investigator |
KAWAHIGASHI Yasuyuki University of Tokyo, Graduate School of Mathematical Sciences, Professor (90214684)
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| Co-Investigator(Kenkyū-buntansha) |
OZAWA Namtaka University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor (60323466)
IZUMI Masaki Kyoto University, Graduate School of Sciences, Professor (80232362)
KISHIMOTO Akitaka Hokkaido University, Graduate School of Sciences, Professor (00128597)
HIAI Fumio Tohoku University, Graduate School of Information Science, Professor (30092571)
KOSAKI Hideki Kyushu University, Graduate School of Mathematics, Professor (20186612)
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| Project Period (FY) |
2004 – 2006
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| Keywords | operator algebra / mathematical physics / quantum field theory / dynamical systems / von Neumann algebras / subfactor / free probability / operator inequality |
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| Research Abstract |
We have studied entropy of local conformal nets of von Neumann algebras. It is defined in terms of the coefficients in the expansion of the logarithm of the trace of the "heat kernel" semigroup. In analogy with study on the asymptotic density distribution of the Laplacian eigenvalues of a manifold, we regard these coefficients as noncommutative geometric invariants of infinitely many degrees of freedom. Under a natural modularity assumption, the leading term of the entropy, noncommutative area, is proportional to the central charge and the first order correction, noncommutative Euler characteristic, is proportional to the logarithm of the global index of the net. We have also studied their relations to black hole entropy.
We have made a construction of local conformal nets of von Neumann algebras analogous to the one of framed vertex operator algebras with Longo. As an example, we have obtained a local conformal net corresponding to the moonshine vertex operator algebra. We have also shown that the automorphism group of this local conformal net is indeed the Monster group, as expected.
We completely classified irreducible, but possibly non-local extensions of the Virasoro net with central charge less than 1. By general theory of Longo and Rehren, this amounts to a complete classification of algebraic boundary CFT with central charge less than 1 satisfying the Haag duality.
We have studied operator algebraic approach to super conformal field theory. It is known that the super Virasoro algebras have discrete series of representations for central charges less than 3/2. We have realized the super Virasoro nets of operator algebras for these cases as coset nets and obtained a classification result by studying their extensions. Together with general theory we have established, we also use the classification technique of modular invariants given by Gannon and Walton.
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