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Generalized Orbifold Construction for Conformal Nets | | Marcel Bischoff
; | | Date: |
31 Jul 2016 | | Abstract: | Let $mathcal{B}$ be a conformal net. We give the notion of a proper action
of a finite hypergroup acting by vacuum preserving unital completely positive
(so-called stochastic) maps, which generalizes the proper actions of finite
groups. Taking fixed points under such an action gives a finite index subnet
$mathcal{B}^K$ of $mathcal{B}$, which generalizes the $G$-orbifold.
Conversely, we show that if $mathcal{A}subset mathcal{B}$ is a finite
inclusion of conformal nets, then $mathcal{A}$ is a generalized orbifold
$mathcal{A}=mathcal{B}^K$ of the conformal net $mathcal{B}$ by a unique
finite hypergroup $K$. There is a Galois correspondence between intermediate
nets $mathcal{B}^Ksubset mathcal{A} subset mathcal{B}$ and subhypergroups
$Lsubset K$ given by $mathcal{A}=mathcal{B}^L$. In this case, the fixed
point of $mathcal{B}^Ksubset mathcal{A}$ is the generalized orbifold by the
hypergroup of double cosets $Lackslash K/ L$.
If $mathcal{A}subset mathcal{B}$ is an finite index inclusion of
completely rational nets, we show that the inclusion $mathcal{A}(I)subset
mathcal{B}(I)$ is conjugate to a Longo--Rehren inclusion. This implies that if
$mathcal{B}$ is a holomorphic net, and $K$ acts properly on $mathcal{B}$,
then there is a unitary fusion category $mathcal{F}$ which is a
categorification of $K$ and $mathrm{Rep}(mathcal{B}^K)$ is braided equivalent
to the Drinfel’d center $Z(mathcal{F})$. More generally, if $mathcal{B}$ is
completely rational conformal net and $K$ acts properly on $mathcal{B}$, then
there is a unitary fusion category $mathcal{F}$ extending
$mathrm{Rep}(mathcal{B})$, such that $K$ is given by the double cosets of the
fusion ring of $mathcal{F}$ by the Verlinde fusion ring of $mathcal{B}$ and
$mathrm{Rep}(mathcal{B}^K)$ is braided equivalent to the M"uger centralizer
of $mathrm{Rep}(mathcal{B})$ in $Z(mathcal{F})$. | | Source: | arXiv, 1608.0253 | | Services: | Forum | Review | PDF | Favorites | |
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