On Induction for Twisted Representations of Conformal Nets

For a given finite index inclusion of strongly additive conformal nets $$\mathcal {B}\subset \mathcal {A}$$ and a compact group $$G < {{\,\mathrm{Aut}\,}}(\mathcal {A}, \mathcal {B})$$ , we consider the induction and the restriction procedures for twisted representations. Let $$G' < {{\,\mathrm{Aut}\,}}(\mathcal {B})$$ be the group obtained by restricting each element of G to $$\mathcal {B}$$ . We introduce two induction procedures for $$G'$$ -twisted representations of $$\mathcal {B}$$ , which generalize the $$\alpha ^{\pm }$$ -induction for DHR endomorphisms. One is defined with the opposite braiding on the category of $$G'$$ -twisted representations as in $$\alpha ^-$$ -induction. The other is also defined with the braiding, but additionally with the G-equivariant structure on the Q-system associated with $$\mathcal {B}\subset \mathcal {A}$$ and the action of G. We derive some properties and formulas for these induced endomorphisms in a similar way to the case of ordinary $$\alpha $$ -induction. We also show the version of $$\alpha \sigma $$ -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 $$\mathcal {A}$$ . 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 $$\mathcal {B}$$ .