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A Short Guide to Orbifold Deconstruction

A Short Guide to Orbifold Deconstruction††This paper is a contribution to the
Special Issue on Moonshine and String Theory. The full collection is available
at https://www.emis.de/journals/SIGMA/moonshine.html

Peter BANTAY

P. Bantay

Institute for Theoretical Physics, Eötvös Lóránd University, H-1117 Budapest,
Pázmány Péter s. 1/A, Hungary<EMAIL_ADDRESS>
Received September 28, 2018, in final form March 27, 2019; Published online
April 09, 2019

We study the problem of orbifold deconstruction, i.e., the process of
recognizing, using only readily available information, whether a given
conformal model can be realized as an orbifold, and the identification of the
twist group and the original conformal model.

conformal symmetry; orbifold models

11F99; 13C05

##  1 Introduction

Orbifolding [14, 15, 16], i.e., the gauging of (discrete) symmetries of a
conformal model, is one of the most important procedures for obtaining new
models from known ones, especially since it respects such important properties
as the central charge, unitarity and rationality111It has even been argued
that all rational conformal models may be obtained either as GKO cosets or as
orbifolds thereof.. Unfortunately, the precise determination of the properties
of the orbifold could be quite involved in general, and effective techniques
are only known for special types, like holomorphic [1, 2, 12, 13] or
permutation orbifolds [3, 5, 8, 9, 20, 25].

The present note addresses the inverse problem: instead of trying to determine
the structure of an orbifold from the knowledge of the original model and the
action of the twist group, we ask for effective procedures to recognize
whether a conformal model could be obtained as an orbifold, and if so, to
identify the original model together with the relevant twist group. That such
procedures could exist is actually not that surprising, as simple current
extensions [4, 21] are nothing but an example of this for Abelian twist
groups. But the question is how the corresponding results could be extended to
more general orbifolds, and what kind of new structures emerge in the process.
We should stress that our approach in the present note is computational: we
aim to identify in a down-to-earth manner the relevant conformal models with
the help of some simple, mostly numerical data (like conformal weights,
quantum dimensions and fusion rules of the primary fields) [10], providing
explicit expressions for the characteristic quantities of the deconstructed
model. We have no doubt that the whole process could be described elegantly in
more abstract terms [19, 24], but the relevant techniques seem (at least to
us) less amenable to practical computations.

In the next section we’ll summarize those general features of orbifold
constructions that will form the basis of the deconstruction procedure. Then
we’ll turn to the description of the procedure itself, while the last section
provides an outlook on questions not addressed in this note. We stress that
our exposition is informal, presenting only the most important ideas and their
mutual interrelations, but omitting formal arguments. Most of these arguments
can be supplied readily, while some of them seem to require more effort, but
with hindsight one could argue that a strong supporting evidence of the ideas
to be presented is that they lead to a coherent algorithmic procedure giving
meaningful answers that agree with the correct ones in all known cases which
could be checked by alternative means. Should some of our assertions prove to
be wrong, one would expect that the whole procedure would lead to completely
meaningless results in most cases. And it goes without saying that in case of
Abelian orbifolds we just get back a suitable simple current extension [21].

##  2 Generalities on orbifolds

Consider a (unitary) conformal model whose chiral algebra ${\mathbb{V}}$ is a
well behaved (rational, $C_{2}$-cofinite, etc.) vertex operator algebra [18,
23, 26, 29]. Given a subgroup $G<\operatorname{Aut}({\mathbb{V}})$ of
automorphisms of ${\mathbb{V}}$, the $G$-orbifold [13, 14] is obtained by
identifying the states that can be transformed into each other by an element
of $G$. The Hilbert space of the orbifold is a direct sum of twisted modules
of the chiral algebra ${\mathbb{V}}$ (these consist of those operators which
are local with respect to ${\mathbb{V}}$ only up to an element of the twist
group $G$), while its chiral algebra is the fixed-point subalgebra
${\mathbb{V}}^{G}=\left\\{v\in{\mathbb{V}}\,|\,gv=v\textrm{ for all }g\in
G\right\\}$.

There is a natural action of $G$ on the set of all twisted modules under which
an element $h\in G$ sends a $g$-twisted module $M$ to a
$hgh^{\,\textrm{-}1}$-twisted module $h(M)$. This shows that the set of all
$g$-twisted modules, as $g$ runs over a given conjugacy class of the twist
group, is $G$-stable under this action, leading to a partition of the
orbifold’s Hilbert space into sectors labeled by the conjugacy classes of $G$,
each of which usually splits into several $G$-orbits. Since the twisted
modules inside a given $G$-orbit are all related by the action of an
automorphism of ${\mathbb{V}}$, it follows that they have pretty similar
properties (while still not being isomorphic), e.g., their conformal weights
$\mathtt{h}_{M}$, quantum dimensions $\mathtt{d}_{M}$ and trace functions [17,
29]

$\displaystyle
Z_{M}(\tau)=\operatorname{Tr}_{M}\big{\\{}\mathtt{e}^{2\pi\mathtt{i}\tau(L_{0}-\nicefrac{{c}}{{24}})}\big{\\}}$

all coincide. This means that it is enough to know these data for just one
representative module $M$ from the orbit.

The stabilizer $G_{M}=\left\\{h\in G\,|\,h(M)\cong M\right\\}$ of a
$g$-twisted module $M$ is the subgroup of those elements $h\in G$ for which
$h(M)$ is isomorphic to $M$; clearly, it is a subgroup of the centralizer
$\mathtt{C}_{G}\left(g\right)=\left\\{h\in G\,|\,gh=hg\right\\}$, and the
stabilizers of different modules belonging to the same $G$-orbit are all
conjugate to each other. Note that the length of the orbit of $M$, i.e., the
number of different modules contained in it, is just the index $[G:G_{M}]$ in
$G$ of its stabilizer. By the previous reasoning, we have on each twisted
module $M$ an action of its stabilizer $G_{M}$, which is usually not a linear
representation, but only a projective one, with associated $2$-cocycle
$\vartheta_{M}\in Z^{2}(G_{M},\mathbb{C})$. This projective representation
decomposes into homogeneous components corresponding to the irreducible
projective representations
$\xi_{p}\in\mathtt{Irr}\left(G_{M}|\vartheta_{M}\right)$ of the stabilizer,
and each such homogeneous component corresponds to a primary field of the
orbifold.

To recapitulate, twisted modules are organized into sectors labeled by
conjugacy classes of the twist group $G$, and each such sector splits into
orbits of $G$, with each orbit characterized by the stabilizer $G_{M}$ of one
representative module $M$ from the orbit and by the associated $2$-cocycle
$\vartheta_{M}$. In turn, each irreducible projective representation
$\xi_{p}\in\mathtt{Irr}\left(G_{M}|\vartheta_{M}\right)$ corresponds to a
primary field of the orbifold. We’ll call the set of primaries originating
from a particular $G$-orbit the block corresponding to that orbit. By the
above, the primaries of the orbifold are partitioned into blocks, each
corresponding to a $G$-orbit of twisted modules, and the primaries inside a
given block correspond to the (projective) irreducible representations of the
stabilizer of the orbit. We shall use the notation
$\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}$ to denote the $G$-orbit of
twisted modules corresponding to a block $\mathfrak{b}$. Note that
$|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|=\left[G:G_{M}\right]$ and
$|\mathfrak{b}|=|\mathtt{Irr}\left(G_{M}|\vartheta_{M}\right)|$ by the above,
for any $M\in\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}$.

If $M\in\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}$ is a $g$-twisted module
of ${\mathbb{V}}$ with $g\in G$ of order $n$, then the eigenvalues of the
operator $nL_{0}$ are integrally spaced on $M$, i.e., the difference of two
$L_{0}$ eigenvalues is an integer multiple of $\nicefrac{{1}}{{n}}$. We define
the order of a block $\mathfrak{b}$ as the smallest positive integer $n$ such
that the eigenvalues of the operator $nL_{0}$ are integrally spaced on any
$M\in\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}$. In particular, untwisted
modules, are characterized by the property that their $L_{0}$ eigenvalues are
integrally spaced, providing a simple criterion singling out untwisted modules
from twisted ones. This also means that a block corresponding to a $G$-orbit
of untwisted modules is characterized by the fact that the conformal weights
of its members differ by integers, i.e., its order is $1$.

In particular, the vacuum module belongs to the untwisted sector and is left
fixed by all elements of $G$, hence it forms in itself a $G$-orbit of length
$1$ whose stabilizer is the whole of $G$, with trivial associated cocycle
(because the twist group acts linearly on ${\mathbb{V}}$). It follows that the
block corresponding to this vacuum module (the vacuum block
$\mathfrak{b}_{\mathtt{{\scriptstyle 0}}}$) consists of primaries in one-to-
one correspondence with the irreducible representations of $G$, whose quantum
dimensions are integers (since they equal the dimension of the corresponding
irreducible representations), and whose conformal weights are integers too,
because the conformal weight of the vacuum module is zero (recall that we
consider unitary models).

Since the vacuum acts as the identity of the fusion product, the fusion of an
element of any given block $\mathfrak{b}$ with an element of the vacuum block
$\mathfrak{b_{\mathtt{{\scriptstyle 0}}}}$ will contain only primaries from
block $\mathfrak{b}$. This means that two primaries $p$ and $q$ belong to the
same block precisely when $N_{\alpha p}^{q}>0$ for at least one element
$\alpha\in\mathfrak{b_{\mathtt{{\scriptstyle 0}}}}$ of the vacuum block. In
particular, the vacuum block generates a closed subring of the fusion ring of
the orbifold, which is identical to the representation ring (Grothendieck
ring) of the twist group, since the fusion product of any two of its members
is the same as the tensor product of the corresponding irreducible
representations.

##  3 Orbifold deconstruction

Armed with the above, we can now attack the problem of orbifold
deconstruction, i.e., the identification of the original model and the twist
group from the sole knowledge of data related to the orbifold. We start from a
(unitary) conformal model for which we know the fusion coefficients
$N_{pq}^{r}$, conformal weights $\mathtt{h}_{p}$, quantum dimensions
$\mathtt{d}_{p}$ and chiral characters $\chi_{p}(\tau)$ of the primary fields,
and we wish to identify it as a non-trivial orbifold of some other conformal
model. Notice that one and the same model might have several deconstructions,
reflecting the fact that the same model can be realized as an orbifold in many
distinct ways; for example, if the twist group $G$ has a (non-trivial) normal
subgroup $N\triangleleft G$, then a $G$-orbifold can be obtained as a
$G/N$-orbifold of an $N$-orbifold. This indicates that there exists a whole
hierarchy of deconstructions, whose most interesting members are the maximal
ones leading to primitive models, i.e., models that cannot be obtained as a
non-trivial orbifold of some other model, hence cannot be deconstructed any
further.

The starting point of the deconstruction procedure is the observation, made at
the end of the previous section, that the vacuum block of an orbifold has very
special properties: it is a set of primaries closed under the fusion product,
and all its members have integer conformal weight and quantum dimension. For
an arbitrary conformal model, we’ll call a set of primaries with these
properties a twister. By the above, the vacuum block of an orbifold is a
twister, and we assume in the sequel that all twisters arise as the vacuum
block of a suitable orbifold realization of the model under study.222While we
have no formal argument supporting this assumption, all available
computational results corroborate it. For each twister there is a different
deconstruction of the model, and maximal deconstructions correspond to maximal
twisters not contained in any other twister. The trivial twister (that
contains the vacuum primary only) gives rise to a trivial deconstruction
resulting in the original model.

Let’s now consider the deconstruction with respect to a particular twister
$\mathfrak{g}$, assumed to be non-trivial. The sole knowledge of the twister
allows the determination of a host of important information about the twist
group. For example, because the quantum dimensions of the elements of
$\mathfrak{g}$ equal the dimensions of the corresponding irreducible
representations of the twist group, and since the order of a group equals, by
Burnside’s famous theorem [22, 28], the sum of the squared dimensions of its
irreducible representations, it follows that the order of the twist group
should equal the spread

$\displaystyle\left\|\mathfrak{g}\right\|=\sum_{\alpha\in\mathfrak{g}}\mathtt{d}_{\alpha}^{2}$

of the twister. Along the same lines, the number of conjugacy classes of the
twist group equals the size $|\mathfrak{g}|$ of the twister, i.e., the number
of its elements. Since the twister generates a subring of the fusion ring
isomorphic to the representation ring of the twist group, and the knowledge of
the representation ring determines the character table of the underlying group
[27], even the character table of the twist group and the size of its
conjugacy classes may be computed from this information, and this is usually
sufficient to identify the twist group up to isomorphism.333We note that,
while the representation ring does not determine the group uniquely up to
isomorphism (a famous example being that of the dihedral group
$\mathbb{D}_{4}$ of order $8$ and the group $\mathtt{Q}$ of unit quaternions,
with identical representation rings), the braided structure of the module
category of ${\mathbb{V}}$ allows one to determine the $\lambda$-ring
structure of the representation ring, and in particular the powers of the
conjugacy classes, leading to the possibility of identifying the twist group
unambiguously.

Once the twist group has been identified, the next problem is the
determination of the primary fields of the original model and their most
important attributes, like conformal weights, quantum dimensions and chiral
characters. We know that these primaries originate in the untwisted sector of
the orbifold, corresponding to untwisted modules organized into $G$-orbits,
and to each such orbit corresponds a block of the orbifold, whose elements are
in one-to-one correspondence with the projective irreducible representations
of the stabilizer subgroup of the orbit (more precisely, of one module from
it). These blocks, viewed as set of primaries of the orbifold, have the
characteristic property that the primaries $p$ and $q$ belong to the same
block precisely when $N_{\alpha p}^{q}>0$ for at least one element $\alpha$ of
the vacuum block. But the twister $\mathfrak{g}$ chosen for deconstruction is
precisely the vacuum block of the orbifold realization of our model, hence the
blocks my be recovered from the knowledge of $\mathfrak{g}$ alone: these are
the maximal sets $\mathfrak{b}$ of primaries such that $p,q\in\mathfrak{b}$
implies $N_{\alpha p}^{q}>0$ for at least one element $\alpha\in\mathfrak{g}$.

Actually, there is no need to compute all the blocks for deconstruction, since
only those corresponding to untwisted modules are needed for identifying the
deconstructed model. But we know that blocks corresponding to a $G$-orbit of
untwisted modules have order $1$, i.e., the conformal weights of their members
can differ only by integers, providing a simple criterion to single them out.

Having identified the blocks in the untwisted sector, i.e., the $G$-orbits of
(untwisted) modules of the deconstructed model, what remains is to determine
the properties of the primary fields corresponding to these modules. Since
they lie on the same $G$-orbit and are thus related by an automorphism of the
relevant chiral algebra, they have the same conformal weights, quantum
dimensions and chiral characters, so it is enough to determine these data for
just one of them, but one should keep in mind that they correspond to
different primaries of the deconstructed theory, hence their multiplicity (the
length $|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|$ of the corresponding
$G$-orbit) should be taken into account.

Since they are related by an element of $\operatorname{Aut}({\mathbb{V}})$,
the modules $M$ belonging to the $G$-orbit
$\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}$ corresponding to a block
$\mathfrak{b}$ all have the same conformal weight
$\mathtt{h}_{M}=\min\left\\{\mathtt{h}_{p}\,|\,p\in\mathfrak{b}\right\\}$,
quantum dimension $\mathtt{d}_{M}$ and chiral character (trace function)

$\displaystyle\operatorname{Tr}_{M}\big{\\{}\mathtt{e}^{2\pi\mathtt{i}\tau(L_{0}-\nicefrac{{c}}{{24}})}\big{\\}}=\sum_{p\in\mathfrak{b}}(\dim\xi_{p})\chi_{p}(\tau)=\frac{1}{|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|}\sum_{p\in\mathfrak{b}}\frac{\mathtt{d}_{p}}{\mathtt{d}_{M}}\chi_{p}(\tau),$

where $|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|$ denotes the length of
the $G$-orbit $\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}$. Note that this
last result would provide an explicit form of the chiral characters should we
know the quantum dimensions $\mathtt{d}_{M}$ and the orbit lengths
$|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|$. Unfortunately, their
determination could be tricky in general because the 2-cocycle
$\vartheta_{M}\in Z^{2}(G_{M},\mathbb{C})$ is usually non-trivial, and the
irreducible representations
$\xi_{p}\in\mathtt{Irr}\left(G_{M}|\vartheta_{M}\right)$ corresponding to the
primaries $p\in\mathfrak{b}$ are not ordinary representations, but only
projective ones. Nevertheless, exploiting general properties of projective
representations it is possible to set up a set of rules that allow their
unambiguous computation in most instances. In particular, $\mathtt{d}_{M}$ is
an algebraic integer $\geq 1$ such that the quantum dimensions
$\mathtt{d}_{p}$ are all integer multiples of
$|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|\mathtt{e}_{M}\mathtt{d}_{M}$
for $p\in\mathfrak{b}$, where $\mathtt{e}_{M}$ denotes the multiplicative
order of the cohomology class of $\vartheta_{M}$ (which is obviously the same
for all modules $M\in\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}$). These
observations are usually sufficient to pin down the precise values of
$\mathtt{d}_{M}$ and $|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|$.

Finally, to finish the identification of the deconstructed model, one needs to
find out its fusion rules. This can be accomplished by assigning to each block
$\mathfrak{b}$ of the untwisted sector a block-fusion matrix
$\mathtt{N}(\mathfrak{b})$ with matrix elements

$\displaystyle\mathtt{N}(\mathfrak{b})_{q}^{r}=\frac{1}{\left\|\mathfrak{g}\right\|}\sum_{p\in\mathfrak{b}}\frac{\mathtt{d}_{p}}{\mathtt{d}_{M}}N_{pq}^{r}.$

Such matrices __ form a ring, i.e., the product of any two of them may be
expressed as a sum

$\displaystyle\mathtt{N}(\mathfrak{a})\mathtt{N}(\mathfrak{b})=\sum_{\mathfrak{c}}\mathfrak{\mathsf{N}_{ab}^{c}}\mathtt{N}(\mathfrak{c})$

over the untwisted blocks, where the block-fusion coefficients
$\mathfrak{\mathsf{N}_{ab}^{c}}\in\mathbb{Z_{+}}$ are given by the sums

$\displaystyle\mathfrak{\mathsf{N}_{ab}^{c}}=\sum_{A\in\mathfrak{a}^{{\scriptscriptstyle\mathtt{o}}},\,B\in\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}}N_{AB}^{C}$
(3.1)

for any representative module
$C\in\mathfrak{c}^{{\scriptscriptstyle\mathtt{o}}}$. Note that these block-
fusion coefficients come near to provide the fusion rules of the deconstructed
model, but for one thing: they do not describe the fusion of the individual
modules, but only that of the direct sum of the modules contained in the
orbits corresponding to the individual blocks. This shouldn’t come as a
surprise since, after all, orbifolding amounts to identifying the modules
related by a symmetry, so that their individual properties are lost in the
process, except for those (like conformal weights and quantum dimensions)
which are the same for all modules on the same orbit. Nevertheless, this
aggregated version of the fusion rules, supplemented by the knowledge of the
conformal weights, quantum dimensions and chiral characters is enough in many
examples to identify uniquely the deconstructed model, at least in those
amenable to direct computations.444We should also point out that, since the
block-fusion matrices depend explicitly on the quantum dimensions
$\mathtt{d}_{M}$ but not on the orbit lengths
$|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|$, the integrality of the
block-fusion coefficients $\mathfrak{\mathsf{N}_{ab}^{c}}$ provides an extra
condition for determining the precise value of the dimensions in case they are
not fixed by the previous considerations.

##  4 Example: the 3-fold symmetric product
of the moonshine module

Consider the self-dual $c=72$ conformal model made up of three identical
copies of the moonshine module $\mathbb{V}^{\natural}$. Clearly, any
permutation of these identical copies is a symmetry, hence one may orbifold
this model with respect to the symmetric group $\mathbb{S}_{3}$ of degree $3$,
resulting in the $3$-fold symmetric product
$\mathbb{V}^{\natural}\wr\mathbb{S}_{3}$ of the moonshine module [6, 11]. The
properties of this symmetric product are well understood, since it is by
construction a permutation orbifold [5], and at the same time a holomorphic
orbifold [13], because the moonshine module is self-dual.

$\mathbb{V}^{\natural}\wr\mathbb{S}_{3}$ is known to have $8$ different
primaries [5], whose most important properties are summarized in the following
table (with
$\zeta=\mathtt{e}^{\mathtt{{\scriptscriptstyle\textrm{}}}\frac{2\pi\mathtt{i}}{3}}$
denoting a primitive third root of unity).

label | conformal weight | dimension | character
---|---|---|---
$\boldsymbol{~{}~{}1}_{{\scriptscriptstyle+}}$ | $0$ | $1$ | ${\displaystyle\frac{1}{6}}\left\\{J(\tau)^{3}+3J(\tau)J\left(2\tau\right)+2J\left(3\tau\right)\right\\}$
$~{}~{}\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $4$ | $1$ | ${\displaystyle\frac{1}{6}}\left\\{J(\tau)^{3}-3J(\tau)J\left(2\tau\right)+2J\left(3\tau\right)\right\\}$
$~{}~{}\boldsymbol{2}$ | $2$ | $2$ | ${\displaystyle\frac{1}{3}}\left\\{J(\tau)^{3}-J\left(3\tau\right)\right\\}$
$~{}~{}\boldsymbol{\uprho}_{0}$ | ${\displaystyle\frac{8}{3}}$ | $2$ | ${\displaystyle\frac{1}{3}}\left\\{J\left(\frac{\tau}{3}\right)+\zeta J\left(\frac{\tau+1}{3}\right)+\overline{\zeta}J\left(\frac{\tau+2}{3}\right)\right\\}$
$~{}~{}\boldsymbol{\uprho}_{1}$ | $4$ | $2$ | ${\displaystyle\frac{1}{3}}\left\\{J\left(\frac{\tau}{3}\right)+J\left(\frac{\tau+1}{3}\right)+J\left(\frac{\tau+2}{3}\right)\right\\}$
$~{}~{}\boldsymbol{\uprho}_{2}$ | ${\displaystyle\frac{10}{3}}$ | $2$ | ${\displaystyle\frac{1}{3}}\left\\{J\left(\frac{\tau}{3}\right)+\overline{\zeta}J\left(\frac{\tau+1}{3}\right)+\zeta J\left(\frac{\tau+2}{3}\right)\right\\}$
$~{}~{}\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}$ | ${\displaystyle\frac{3}{2}}$ | $3$ | ${\displaystyle\frac{1}{2}}J(\tau)\left\\{J\left(\frac{\tau}{2}\right)+J\left(\frac{\tau+1}{2}\right)\right\\}$
$~{}~{}\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $3$ | $3$ | ${\displaystyle\frac{1}{2}}J(\tau)\left\\{J\left(\frac{\tau}{2}\right)-J\left(\frac{\tau+1}{2}\right)\right\\}$
Table 1: Primaries of $\mathbb{V}^{\natural}\wr\mathbb{S}_{3}$.

It follows that the only primaries that can belong to a twister are among
$\boldsymbol{1}_{{\scriptscriptstyle+}}$,
$\boldsymbol{1}_{{\scriptscriptstyle-}}$, $\boldsymbol{2}$,
$\boldsymbol{\uprho}_{1}$ and $\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$,
since only these have simultaneously integer quantum dimension and conformal
weight.

With the notations
$\boldsymbol{\varXi}=\boldsymbol{1}_{{\scriptscriptstyle+}}\oplus\boldsymbol{1}_{{\scriptscriptstyle-}}$
and
$\boldsymbol{\text{$\varSigma$}}=\boldsymbol{2}\oplus\boldsymbol{\uprho}_{0}\oplus\boldsymbol{\uprho}_{1}\oplus\boldsymbol{\uprho}_{2}$,
the fusion rules read

| $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $~{}~{}\boldsymbol{2}$ | $~{}~{}\boldsymbol{\uprho}_{0}$ | $~{}~{}\boldsymbol{\uprho}_{1}$ | $~{}~{}\boldsymbol{\uprho}_{2}$ | $~{}~{}\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}$ | $~{}~{}\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$
---|---|---|---|---|---|---|---|---
$\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $~{}~{}\boldsymbol{2}$ | $~{}~{}\boldsymbol{\uprho}_{0}$ | $~{}~{}\boldsymbol{\uprho}_{1}$ | $~{}~{}\boldsymbol{\uprho}_{2}$ | $~{}~{}\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}$ | $~{}~{}\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$
$\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $~{}~{}\boldsymbol{2}$ | $~{}~{}\boldsymbol{\uprho}_{0}$ | $~{}~{}\boldsymbol{\uprho}_{1}$ | $~{}~{}\boldsymbol{\uprho}_{2}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}$
$\boldsymbol{2}$ | $\boldsymbol{2}$ | $\boldsymbol{2}$ | $\boldsymbol{2}\oplus\boldsymbol{\varXi}$ | $\boldsymbol{\uprho}_{1}\oplus\boldsymbol{\uprho}_{2}$ | $\boldsymbol{\uprho}_{0}\oplus\boldsymbol{\uprho}_{2}$ | $\boldsymbol{\uprho}_{0}\oplus\boldsymbol{\uprho}_{1}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$
$\boldsymbol{\uprho}_{0}$ | $\boldsymbol{\uprho}_{0}$ | $\boldsymbol{\uprho}_{0}$ | $\boldsymbol{\uprho}_{1}\oplus\boldsymbol{\uprho}_{2}$ | $\boldsymbol{\uprho}_{0}\oplus\boldsymbol{\varXi}$ | $\boldsymbol{2}\oplus\boldsymbol{\uprho}_{2}$ | $\boldsymbol{2}\oplus\boldsymbol{\uprho}_{1}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$
$\boldsymbol{\uprho}_{1}$ | $\boldsymbol{\uprho}_{1}$ | $\boldsymbol{\uprho}_{1}$ | $\boldsymbol{\uprho}_{0}\oplus\boldsymbol{\uprho}_{2}$ | $\boldsymbol{2}\oplus\boldsymbol{\uprho}_{2}$ | $\boldsymbol{\uprho}_{1}\oplus\boldsymbol{\varXi}$ | $\boldsymbol{2}\oplus\boldsymbol{\uprho}_{0}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$
$\boldsymbol{\uprho}_{2}$ | $\boldsymbol{\uprho}_{2}$ | $\boldsymbol{\uprho}_{2}$ | $\boldsymbol{\uprho}_{0}\oplus\boldsymbol{\uprho}_{1}$ | $\boldsymbol{2}\oplus\boldsymbol{\uprho}_{1}$ | $\boldsymbol{2}\oplus\boldsymbol{\uprho}_{0}$ | $\boldsymbol{\uprho}_{2}\oplus\boldsymbol{\varXi}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$
$\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\text{$\varSigma$}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}\oplus\boldsymbol{\text{$\varSigma$}}$
$\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}\oplus\boldsymbol{\text{$\varSigma$}}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}\oplus\boldsymbol{\text{$\varSigma$}}$
Table 2: Fusion rules of $\mathbb{V}^{\natural}\wr\mathbb{S}_{3}$.

It follows that there are precisely three non-trivial twisters: the maximal
twisters
$\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}},\boldsymbol{2}\\}$
and
$\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}},\boldsymbol{\uprho}_{1}\\}$,
both of spread $6$ and size $3$, and their intersection
$\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}}\\}$.
We note that the existence of two maximal twisters is related to the
automorphism of the fusion rules that exchanges the primaries $\boldsymbol{2}$
and $\boldsymbol{\uprho}_{1}$ while leaving all other primaries invariant.
Let’s take a closer look at the different deconstructions corresponding to
these cases.555An important feature of this example that simplifies
tremendously the analysis is that, because the relevant groups are pretty
small, all the cocycles are trivial, so one does only encounter ordinary
representations during the procedure.

The twister
$\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}}\\}$
consists of two simple currents corresponding to the one-dimensional
representations of $\mathbb{S}_{3}$, hence the deconstructed model will be a
simple current extension of $\mathbb{V}^{\natural}\wr\mathbb{S}_{3}$ that
could be determined alternatively using the techniques of [21]. Obviously, the
twist group is isomorphic to $\mathbb{Z}_{2}$. There are $6$ different blocks,
whose properties can be read off the following table (the multiplicity of a
block $\mathfrak{b}$ is the length
$|\mathfrak{b}^{{\scriptscriptstyle\mathtt{o}}}|$ of the associated orbit, cf.
Section 3).

block | order | multiplicity | dimension | trace function
---|---|---|---|---
$\left\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}}\right\\}$ | $1$ | $1$ | $1$ | ${\displaystyle{\textstyle\frac{1}{3}}}\left\\{J(\tau)^{3}+2J\left(3\tau\right)\right\\}$
$\left\\{\boldsymbol{2}\right\\}$ | $1$ | $2$ | $1$ | ${\textstyle\frac{1}{3}}\left\\{J(\tau)^{3}-J\left(3\tau\right)\right\\}$
$\left\\{\boldsymbol{\uprho}_{0}\right\\}$ | $1$ | $2$ | $1$ | ${\displaystyle{\textstyle\frac{1}{3}}}\left\\{J\left(\frac{\tau}{3}\right)+\zeta J\left(\frac{\tau+1}{3}\right)+\overline{\zeta}J\left(\frac{\tau+2}{3}\right)\right\\}$
$\left\\{\boldsymbol{\uprho}_{1}\right\\}$ | $1$ | $2$ | $1$ | ${\displaystyle{\textstyle\frac{1}{3}}}\left\\{J\left(\frac{\tau}{3}\right)+J\left(\frac{\tau+1}{3}\right)+J\left(\frac{\tau+2}{3}\right)\right\\}$
$\left\\{\boldsymbol{\uprho}_{2}\right\\}$ | $1$ | $2$ | $1$ | ${\displaystyle{\textstyle\frac{1}{3}}}\left\\{J\left(\frac{\tau}{3}\right)+\overline{\zeta}J\left(\frac{\tau+1}{3}\right)+\zeta J\left(\frac{\tau+2}{3}\right)\right\\}$
$\left\\{\boldsymbol{\upsigma}_{{\scriptscriptstyle+}},\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}\right\\}$ | $2$ | $1$ | $3$ | $J(\tau)J\left(\frac{\tau}{2}\right)$

The untwisted sector consists of $9$ modules arranged into $4$ orbits of
length $2$ and one fixed-point (the vacuum block). Inspecting their conformal
weights and trace functions, one recognizes that the deconstructed model is
nothing but the permutation orbifold $\mathbb{V}^{\natural}\wr\mathbb{A}_{3}$
of the moonshine module by the alternating group $\mathbb{A}_{3}$ of degree
$3$ (the commutator subgroup of $\mathbb{S}_{3}$); this is further
corroborated by the fusion rules computed using equation (3.1). We note that
there is a unique $\mathbb{Z}_{2}$-twisted module in this case, corresponding
to the block of order $2$.

For the twister
$\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}},\boldsymbol{2}\\}$,
the fusion rules are

| $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{2}$
---|---|---|---
$\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{2}$
$\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{2}$
$\boldsymbol{2}$ | $\boldsymbol{2}$ | $\boldsymbol{2}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}\oplus\boldsymbol{1}_{{\scriptscriptstyle-}}\oplus\boldsymbol{2}$

leading to the following character table for the twist group

| $\boldsymbol{\mathfrak{1}}$ | $\boldsymbol{\mathfrak{2}}$ | $\boldsymbol{\mathfrak{3}}$
---|---|---|---
$\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $1$ | $1$ | $1$
$\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $1$ | $-1$ | $1$
$\boldsymbol{2}$ | $2$ | $0$ | $-1$

from which can one infer that the twist group is isomorphic to
$\mathbb{S}_{3}$ in this case. Actually, this already follows from the fact
that it is a group of order $\left\|\mathfrak{g}\right\|=6$ with
$|\mathfrak{g}|=3$ different conjugacy classes, and all such groups are
isomorphic to $\mathbb{S}_{3}$.

There are $2$ blocks besides the vacuum block, and the following table
summarizes their most important properties.

block | order | multiplicity | dimension | trace function
---|---|---|---|---
$\left\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}},\boldsymbol{2}\right\\}$ | $1$ | $1$ | $1$ | $J(\tau)^{3}$
$\left\\{\boldsymbol{\upsigma}_{{\scriptscriptstyle+}},\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}\right\\}$ | $2$ | $3$ | $1$ | $J(\tau)J\left(\frac{\tau}{2}\right)$
$\left\\{\boldsymbol{\uprho}_{0},\boldsymbol{\uprho}_{1},\boldsymbol{\uprho}_{2}\right\\}$ | $3$ | $2$ | $1$ | $J\left(\frac{\tau}{3}\right)$

Since there is just one block in the untwisted sector (i.e., of order $1$), it
follows that the deconstructed model is self-dual, with trace function equal
to $J(\tau)^{3}$. From this we can conclude that the deconstructed model is
made up from $3$ identical copies of the moonshine module
$\mathbb{V}^{\natural}$.

For the twister
$\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}},\boldsymbol{\uprho}_{1}\\}$,
the situation is pretty similar to the previous one after one performs the
exchange $\boldsymbol{2}\leftrightarrow\boldsymbol{\uprho}_{1}$. The fusion
rules read

| $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\uprho}_{1}$
---|---|---|---
$\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{\uprho}_{1}$
$\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $\boldsymbol{\uprho}_{1}$
$\boldsymbol{\uprho}_{1}$ | $\boldsymbol{\uprho}_{1}$ | $\boldsymbol{\uprho}_{1}$ | $\boldsymbol{1}_{{\scriptscriptstyle+}}\oplus\boldsymbol{1}_{{\scriptscriptstyle-}}\oplus\boldsymbol{\uprho}_{1}$

leading once again to the character table

| $\boldsymbol{\mathfrak{1}}$ | $\boldsymbol{\mathfrak{2}}$ | $\boldsymbol{\mathfrak{3}}$
---|---|---|---
$\boldsymbol{1}_{{\scriptscriptstyle+}}$ | $1$ | $1$ | $1$
$\boldsymbol{1}_{{\scriptscriptstyle-}}$ | $1$ | $-1$ | $1$
$\boldsymbol{\uprho}_{1}$ | $2$ | $0$ | $-1$

showing that the twist group is isomorphic to $\mathbb{S}_{3}$ in this case
too. Once again, there are $3$ blocks: the vacuum block
$\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}},\boldsymbol{\uprho}_{1}\\}$,
and the blocks
$\\{\boldsymbol{\boldsymbol{\upsigma}_{{\scriptscriptstyle+}}},\boldsymbol{\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}}\\}$
and $\\{\boldsymbol{\uprho}_{0},\boldsymbol{2},\boldsymbol{\uprho}_{2}\\}$ of
respective orders $2$ and $3$. These share the properties of their
counterparts for the twister
$\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}},\boldsymbol{2}\\}$,
except for their trace functions, which read in this case

block | trace function
---|---
$\left\\{\boldsymbol{1}_{{\scriptscriptstyle+}},\boldsymbol{1}_{{\scriptscriptstyle-}},\boldsymbol{\uprho}_{1}\right\\}$ | $J(\tau)^{3}-2c_{1}J(\tau)-2c_{2}$
$\left\\{\boldsymbol{\upsigma}_{{\scriptscriptstyle+}},\boldsymbol{\upsigma}_{{\scriptscriptstyle-}}\right\\}$ | $J(\tau)J\left(\frac{\tau}{2}\right)$
$\left\\{\boldsymbol{\uprho}_{0},\boldsymbol{2},\boldsymbol{\uprho}_{2}\right\\}$ | $J\left(\frac{\tau}{3}\right)+c_{1}J(\tau)+c_{2}$

upon taking into account the replication identity666This illustrates the fact
that the existence of different maximal deconstructions of a permutation
orbifold is related to the existence of non-trivial replication identities
[7].

$\displaystyle
J\left(\frac{\tau}{3}\right)+J\left(\frac{\tau+1}{3}\right)+J\left(\frac{\tau+2}{3}\right)+J\left(3\tau\right)=J(\tau)^{3}-3c_{1}J(\tau)-3c_{2}$

for the modular invariant function $J(\tau)$, where $c_{1}=196884$ and
$c_{2}=21493760$.

Note that this second deconstruction differs markedly from the previous one,
although their twist groups are isomorphic. In particular, since the
polynomial $P(x)=x^{3}-2c_{1}x-2c_{2}$ has $3$ different roots, it follows
that the action of the twist group is not a permutation action in this case.
This exemplifies that one and the same conformal model (in our case
$\mathbb{V}^{\natural}\wr\mathbb{S}_{3}$) may be obtained as an orbifold of
quite different models, with different (possibly non-isomorphic) twist groups.

##  5 Summary and outlook

As described above, there is an effective algorithmic procedure for orbifold
deconstruction, i.e., the realization of a given conformal model as an
orbifold of some other model. Different deconstructions correspond to
different twisters of the model under study, where non-maximal deconstructions
lead to models that can be further deconstructed themselves, corresponding to
orbifolding in stages. Deconstructions corresponding to maximal twisters
(i.e., not contained in any other twister) are the ones that may lead to
primitive models that cannot be realized as non-trivial orbifolds.777It may
happen that orbifolding by stages breaks down, i.e., the orbifold of an
orbifold is not an orbifold of the original model, in which case the
corresponding maximal deconstruction is itself a non-trivial orbifold. This
phenomenon shows up, as put forward by one of the referees, in the 16-fold
tensor power of the Ising model, which has a maximal deconstruction to the
${\rm SO}(16)$ Wess–Zumino model at level 1, itself a
$\mathbb{Z}_{2}$-orbifold of the $E_{8}$ Wess–Zumino model of the same level.
The deconstruction algorithm provides us with a more-or-less unique
identification of the twist group and of the original model, up to some
ambiguities related to the projective realization of the twist group and to
the individual fusion rules of the untwisted modules. It should be emphasized
that this ambiguity is a finite one, which means that, as a last resort, one
can in principle go through a tedious case-by-case analysis of the allowed
possibilities to find out the correct one.

Besides its intrinsic interest, what could be the benefits of the
deconstruction procedure in the study of 2D CFT? There is one such benefit
that is more or less obvious: if we can identify both the deconstructed model
and the twist group unambiguously, then we get an example of an orbifold
construction where the result, being the input of the deconstruction
procedure, is known right from the start, making possible a thorough
investigation of the orbifolding process. Actually, the procedure can be
extended so as to give precise results not only about the modules in the
untwisted sector (the ones that are of interest for identifying the
deconstructed model), but also about the structure of all of the twisted
modules. This is particularly important, since the structure of $g$-twisted
modules does only depend on the conjugacy class of $g$ in
$\operatorname{Aut}({\mathbb{V}})$, hence the deconstruction of a $G$-orbifold
gives valuable information about the structure of all orbifolds (of the
deconstructed model) whose twist group contains elements that are conjugate in
$\operatorname{Aut}({\mathbb{V}})$ to some element of $G$.

Finally, let us note that the above ideas might be used in attempts to
classify rational conformal models. For one thing, the classification problem
can be reduced to that of primitive models (the ones that don’t have
nontrivial twisters), since all others are orbifolds of these, and the latter
can be classified by group theoretic means. On the other hand, even primitive
models can be grouped together if they arise from maximal deconstructions of
one and the same conformal model, i.e., if some of their orbifolds are
identical. An interesting problem is to develop simple criteria testing
whether two primitive models are related in this way, and to understand what
kind of common structures are responsible for such behavior.

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