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 \(\mathcal {B}\).

对于给定的有限索引，包含强加性共形网络\（\ mathcal {B} \ subset \ mathcal {A} \）和一个紧凑组\（G <{{\，\ mathrm {Aut} \，}}（\ mathcal {A}，\ mathcal {B}）\），我们考虑了扭曲表示的归纳和限制程序。令\（G'<{{\，\ mathrm {Aut} \，}}（\ mathcal {B}）\）是通过将G的每个元素限制为\（\ mathcal {B} \）获得的组。我们介绍了\（\ mathcal {B} \）的\（G'\）扭曲表示的两种归纳程序，它们概括了DHR同态性的\（\ alpha ^ {\ pm} \）归纳。在以下类别上定义了相反的编织线\（G'\）扭曲表示形式，如\（\ alpha ^-\）-归纳法。另一个也通过编织来定义，但是另外还定义了Q系统上与\（\ mathcal {B} \ subset \ mathcal {A} \）和G的作用相关的G-等价结构。我们以与普通\（\ alpha \）归纳的情况相似的方式，得出了这些诱导的同构的一些性质和公式。我们还显示了设置的\（\ alpha \ sigma \）-倒数公式的版本。特别是，我们表明每个G扭曲表示作为正负诱导同态的子对象。此外，我们构造了一个相对的编织算子，并表明该构造在\（\ mathcal {A} \）的G扭曲表示类别中给出了编织。结果，我们证明了我们的归纳程序提供了一种方式来捕获\（\ mathcal {B} \）上的代数结构形式的G扭曲表示的类别。