On Induction for Twisted Representations of Conformal Nets

Abstract

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 $\mathcal {B}$.