We made the following study on algebraic quantum foeld theory.
We have studied entropy of local conformal nets of von Neumann algebras with Longo. 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.