Local conformal nets arising from framed vertex operator algebras
Chiral conformal quantum field theory, where the unit circle serves as ``spacetime'', has been the subject of intense mathematical studies for many years. There are two approaches using infinite-dimensional algebraic systems. Either one starts from a local net of von Neumann algebras or from a vertex operator algebra. However, one rarely finds a comparison of the two approaches in the literature though both theories have mathematical objects in common such as the Virasoro algebra. The article fills the gap by explaining the relationship between the two. In particular, it provides a local net of von Neumann algebras corresponding to the ``moonshine'' vertex operator algebra of Frenkel-Lepowsky-Meurman. The net is shown to be trivial in the sense that the vacuum sector is the only irreducible Doplicher-Haag-Roberts sector. Its gauge group is known to be the Monster group, the largest among the (26) sporadic finite simple groups. Previously developed tools are employed such as \(\alpha\)-induction and complete rationality to study extensions of local nets.
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