Representations of Conformal Nets, Universal C*-Algebras and K-Theory
arXiv:1202.2543 · doi:10.1007/s00220-012-1561-5
Abstract
We study the representation theory of a conformal net A on the circle from a K-theoretical point of view using its universal C*-algebra C*(A). We prove that if A satisfies the split property then, for every representation πof A with finite statistical dimension, π(C*(A)) is weakly closed and hence a finite direct sum of type I_\infty factors. We define the more manageable locally normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C*_ln(A) is a direct sum of n type I_\infty factors. Its ideal K_A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C*(A) with finite statistical dimension act on K_A, giving rise to an action of the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.
v2: we added some comments in the introduction and new references. v3: new authors' addresses, minor corrections. To appear in Commun. Math. Phys. v4: minor corrections, updated references
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- How to add a boundary condition
- Snowmass White Paper: The Quest to Define QFT
- The K-homology of nets of C*-algebras
- Super-KMS functionals for graded-local conformal nets
- Rotational KMS states and type I conformal nets
- Local-entire cyclic cocycles for graded quantum field nets
- From vertex operator superalgebras to graded-local conformal nets and back
- Loop groups and noncommutative geometry