Arthur Bartels, Christopher L. Douglas, André Henriques
We describe a coordinate-free perspective on conformal nets, as functors from intervals to von Neumann algebras. We discuss an operation of fusion of intervals and observe that a conformal net takes a fused interval to the fiber product of von Neumann algebras. Though coordinate-free nets do not a priori have vacuum sectors, we show that there is a vacuum sector canonically associated to any circle equipped with a conformal structure. Our main result is that a conformal net has finite index if and only if the category of sectors of the net is a fusion category. As a corollary of this characterization of finiteness, we give a new proof that the net associated to the loop group of the special unitary group has finite index. This is the first in a series of papers constructing a 3-category of conformal nets, defects, sectors, and intertwiners.
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http://arxiv.org/abs/1302.2604
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