On Induction for Twisted Representations of Conformal Nets
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Annales Henri Poincar´ e
On Induction for Twisted Representations of Conformal Nets Ryo Nojima Abstract. For a given finite index inclusion of strongly additive conformal nets B ⊂ A and a compact group G < Aut(A, B), we consider the induction and the restriction procedures for twisted representations. Let G < Aut(B) be the group obtained by restricting each element of G to B. We introduce two induction procedures for G -twisted representations of B, which generalize the α± -induction for DHR endomorphisms. One is defined with the opposite braiding on the category of G -twisted representations as in α− -induction. The other is also defined with the braiding, but additionally with the G-equivariant structure on the Q-system associated with B ⊂ A and the action of G. We derive some properties and formulas for these induced endomorphisms in a similar way to the case of ordinary α-induction. We also show the version of ασ-reciprocity formula for our setting. In particular, we show that every G-twisted representation is obtained as a subobject of both plus and minus induced endomorphisms. Moreover, we construct a relative braiding operator and show that this construction gives the braiding in the category of G-twisted representations of A. As a consequence, we show that our induction procedures give a way to capture the category of G-twisted representations in terms of algebraic structures on B.
1. Introduction In the Haag–Kastler framework of quantum field theory, a chiral components of 2D conformal field theory are described by a conformal net on the unit circle S 1 . A conformal net A is defined to be a map I → A(I) from the set of open intervals of S 1 to that of von Neumann algebras. These von Neumann algebras are considered as algebras of observables and required to satisfy certain axioms. We have a natural notion of representations of A, and the representation theory plays an important role in the study of conformal nets. By the Doplicher–Haag– Roberts theory [10,11], it turns out that every representation is equivalent to
R. Nojima
Ann. Henri Poincar´e
a localized transportable endomorphism (called a DHR endomorphism) and Rep(A) has a structure of a braided C*-tensor category. For a given conformal net, we can consider its extensions and subnets. Let B a conformal net and A an extension. This gives us a net of subfactors {B(I) ⊂ A(I)}. In the article [23], general theory of nets of subfactors has been developed. By applying their results, the extension A is completely characterized by a commutative Q-system (or standard C*-Frobenius algebra objects) Θ = (θ, w, x) in Rep(B). We also have induction and restriction procedures for representations of B and A. The induction procedure is called the α-induction, and the restriction procedure is called the σ-restriction, respectively. The notions of α-induction and σ-restriction were first introduced in [23] and studied with examples in [30], and then further developed in [5–7]. For a DHR endomorphism λ of B, αλ± is given as an extension of λ to A. The endomorphism αλ±
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