# $\mathbb Z/2$-orbifolds in Virasoro representations, CFTs, VOAs

Suppose that ${\rm Vir}_c$ is a rational Virasoro algebra with central charge $c$. Then ${\rm Vir}_c$ has finitely many irreducible modules $M_h$, parametrised by the highest weights $h$. Furthermore the decomposition $M_i\otimes M_j = \sum_h N_{ij}^h M_h$ has a combinatorial description, and the decomposition is usually referred to as the fusion rules.

Suppose that a group $G$ acts on the representation $W$ of $V$. How (and when) is the orbifold $W^G_{\rm orb}$ defined?

Following the book Conformal Field Theory (CFT) by Di-Francesco--Mathieu--Senechal: An example of a CFT is a representation $W = \sum M_h\otimes \overline{M_{h'}}$ of ${\rm Vir}_c\otimes\overline{{\rm Vir}_c}$. The partition function $Z$ is the corresponding sum of characters $\sum \chi_h\otimes\overline{\chi_{h'}}$. The orbifold $Z^{\langle G\rangle}_{\mathop{orb}}$ is defined as $\frac{1}{\lvert G\rvert}\sum_{(a,b)}Z_{a,b}$, where $(a,b)$ is a commuting pair of elements from $G$ and $Z_{a,b}$ is the partition function of a conformal field theory with suitably twisted boundary conditions. A representation is modular if its partition function is invariant under the action of $SL_2(\mathbb Z)$. Generally, orbifolding seems to be described as a method of getting a modular representation from another modular representation. Is there instead a combinatorial way to find orbifolds, or a description of the method on the modules?

On the other hand, suppose that group $G$ acts on a VOA $V$. Its 'orbifold' $V^G$ (in the papers I've seen) is the set of $G$-fixed points in $V$. A conformal VOA $V$ carries a representation of ${\rm Vir}_c$. I've heard that $V$ describes the left-handed part of a CFT (i.e. only the modules $M_h\otimes 1$ in the chiral representation above). Can the Virasoro-orbifolding be translated to VOAs? (Or to any other representation of ${\rm Vir}_c$?)

It turns out from the fusion rules that the highest weights $H$ of ${\rm Vir}_c$ always admit a $\mathbb Z/2$-grading $t$ such that if $M_h$ is a constituent of $M_i\otimes M_j$ then $t(h) = t(i) + t(j)$. Set $\tau(M_h) = (-1)^{t(h)}M_h$, and set $W = \sum_{h\in H}M_h\otimes\overline{M_h}$. The $\tau$-fixed module $\sum_{t(h)=0} M_h$ looks very different to the orbifold $W^{\langle\tau\rangle}_{\rm orb}$ of $W$: possibly $W^{\langle\tau\rangle}_{\rm orb} = W$ (for $c = 1/2$) or certain modules occur in $W^{\langle\tau\rangle}_{\rm orb}$ with multiplicity $2$ (for $c = 4/5$). This is an example of an orbifolding I would like to understand. From Miyamoto's q-alg/9710038, for example, it looks like some of these $\mathbb Z/2$-orbifolds play cool roles in the VOAs.

-