V_{s+s_{a}}\ ,$ (3.8) $\displaystyle
I=\bigoplus_{s\in\widehat{\mathsf{\Gamma}}}\,I^{s}\ \in\
\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(W,V)\qquad\mbox{with}$
$\displaystyle\quad I^{s}:W_{s}\longrightarrow V_{s}\ .$ (3.9)

Starting from the equations (3.5), the same argument from Section 2.4 then
defines the orbifold version of the ADHM equations (2.44):

$\displaystyle\begin{split}\mu_{ab}^{\mathbbm{C}s}&:=B_{a}^{s+s_{b}}\,B_{b}^{s}-B_{b}^{s+s_{a}}B_{a}^{s}-\tfrac{1}{2}\,\varepsilon_{ab\bar{c}\bar{d}}\,\big{(}B_{\bar{c}}^{s-s_{d}-s_{c}\dagger}\,B_{\bar{d}}^{s-s_{d}\dagger}-B_{\bar{d}}^{s-s_{d}-s_{c}\dagger}\,B_{\bar{c}}^{s-s_{c}\dagger}\big{)}=0\
,\\\\[4.0pt]
\mu^{\mathbbm{R}s}&:=\sum_{a=1}^{4}\,\big{(}B_{a}^{s-s_{a}}\,B_{\bar{a}}^{s-s_{a}\dagger}-B_{\bar{a}}^{s\dagger}\,B_{a}^{s}\big{)}+I^{s}\,I^{s\dagger}=\zeta_{s}\,\mathbbm{1}_{k_{s}}\
,\end{split}$ (3.10)

for each $s\in{\widehat{\mathsf{\Gamma}}}$ and $1\leq a<b\leq 4$.

The $\mathcal{N}=(0,2)$ gauge theories on D1-branes at Calabi–Yau orbifolds of
$\mathbbm{C}^{4}$ are also obtained by orbifold projection and have been
studied by e.g. [67, 68, 69] (in the absence of D1–D9 strings). This theory is
anomaly-free with suitable Chern–Simons couplings to background R–R chiral
scalars. It consists of four types of chiral superfields on
$\mathbbm{R}^{1,1}$, transforming in the bifundamental representation of
$\mathsf{U}(k_{s+a})\times\mathsf{U}(k_{s})$ for $a\in\underline{\sf 4}$ and
$s\in{\widehat{\mathsf{\Gamma}}}$, whose lowest components are the linear maps
$B_{a}^{s}$. The choice of $k_{s}=n$ for all $s\in{\widehat{\mathsf{\Gamma}}}$
corresponds to a stack of $n$ regular D1-branes, while more general dimension
vectors $\vec{k}$ correspond to fractional D1-branes. The ADHM equations
(3.10) are equations for the Higgs branch of this $\mathcal{N}=(0,2)$ theory.

#### Moduli Spaces of Orbifold Instantons

The moduli space $\mathfrak{M}_{r,k}^{\mathsf{\Gamma}}$ of charge $k$
noncommutative $\mathsf{U}(r)$ instantons on the orbifold
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ stratifies into connected components

$\displaystyle\mathfrak{M}_{r,k}^{\mathsf{\Gamma}}=\bigsqcup_{|\vec{r}\,|=r\,,\,|\vec{k}\,|=k}\,\mathfrak{M}_{\vec{r},\vec{k}}$
(3.11)

according to the decompositions above, where the disjoint union runs over all
$\mathsf{\Gamma}$-representations $W$ and $V$ of dimensions $r$ and $k$,
respectively. The $\mathsf{\Gamma}$-action breaks the $\mathsf{U}(k)$ gauge
symmetry of $V$ to the subgroup
$\operatorname*{\scalerel*{\times}{\textstyle\sum}}_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathsf{U}(k_{s})$,
whose action on the ADHM variables is given by

$\displaystyle
g\cdot\big{(}B_{a}^{s}\,,\,I^{s}\big{)}_{\begin{subarray}{c}s\in{\widehat{\mathsf{\Gamma}}}\\\
a\in\underline{\sf
4}\end{subarray}}=\big{(}g_{s+s_{a}}\,B_{a}^{s}\,g_{s}^{-1}\,,\,g_{s}\,I^{s}\big{)}_{\begin{subarray}{c}s\in{\widehat{\mathsf{\Gamma}}}\\\
a\in\underline{\sf 4}\end{subarray}}\qquad\mbox{with}\quad
g=(g_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}\ \in\
\operatorname*{\scalerel*{\times}{\displaystyle\sum}}_{s\in\widehat{\mathsf{\Gamma}}}\,\mathsf{U}(k_{s})\
.$ (3.12)

Then $\mathfrak{M}_{\vec{r},\vec{k}}$ can be described as the space of
solutions to (3.10) modulo these gauge transformations.

The moduli space $\mathfrak{M}_{\vec{r},\vec{k}}$ admits an equivalent
holomorphic parametrization as a GIT quotient. For this, we note that the
equations $\mu^{\mathbbm{C}s}_{ab}=0$ from (3.10) arise as the complex moment
map equations $\mu_{ab}^{\mathbbm{C}}=0$ from (2.44) for the equivariant
decomposition
$\mu_{ab}^{\mathbbm{C}}=\bigoplus_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mu_{ab}^{\mathbbm{C}s}$.
Since the latter are equivalent to the equations $[B_{a},B_{b}]=0$ by the
identity (2.48), the equivariant decomposition (3.8) shows that the former can
be substituted with the holomorphic equations

$\displaystyle B_{a}^{s+s_{b}}\,B_{b}^{s}=B_{b}^{s+s_{a}}\,B_{a}^{s}\ ,$
(3.13)

for $1\leq a<b\leq 4$ and $s\in\widehat{\mathsf{\Gamma}}$.

On the other hand, the real moment map equations
$\mu^{\mathbbm{R}s}=\zeta_{s}\,\mathbbm{1}_{k_{s}}$ in (3.10) for generic
Fayet–Iliopoulos parameters $\zeta_{s}>0$ can be traded for a stability
condition: A set of maps (3.8) and (3.9) is _stable_ if there are no proper
$\mathsf{\Gamma}$-subrepresentations

$\displaystyle
S=\bigoplus_{s\in{\widehat{\mathsf{\Gamma}}}}\,S_{s}\otimes\rho_{s}^{*}$
(3.14)

of $V$ such that $B_{a}^{s}(S_{s})\subset S_{s+s_{a}}$ for $a\in\underline{\sf
4}$ and ${\rm im}(I^{s})\subset S_{s}$ for all
$s\in{\widehat{\mathsf{\Gamma}}}$.

The proof is similar to the stability proof of [37]: Let $P_{s}$ be the
orthogonal projection of $V_{s}$ to the orthogonal complement $S_{s}^{\perp}$
of the invariant subspace $S_{s}\subset V_{s}$, for each
$s\in{\widehat{\mathsf{\Gamma}}}$. Then $P_{s}\,I^{s}=0$ and
$P_{s+s_{a}}\,B_{a}^{s}\,P_{s}=P_{s+s_{a}}\,B_{a}^{s}$, so

$\displaystyle\begin{split}0\ \leq\
\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\,\zeta_{s}\dim_{\mathbbm{C}}S_{s}^{\perp}=\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathrm{Tr}\big{(}P_{s}\,\mu^{\mathbbm{R}s}\big{)}&=\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\
\sum_{a\in\underline{\sf
4}}\,\mathrm{Tr}\big{(}P_{s+s_{a}}\,B_{a}^{s}\,B_{\bar{a}}^{s\dagger}-B_{a}^{s}\,P_{s}\,B_{\bar{a}}^{s\dagger}\big{)}\\\\[4.0pt]
&=-\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\ \sum_{a\in\underline{\sf
4}}\,\big{\|}(\mathbbm{1}_{k_{s+s_{a}}}-P_{s+s_{a}})\,B^{s}_{a}\,P_{s}\big{\|}_{\textrm{\tiny
F}}^{2}\ \leq\ 0\ ,\end{split}$ (3.15)

which implies $S_{s}=V_{s}$ for all $s\in{{\widehat{\mathsf{\Gamma}}}}$. The
stability condition is equivalent to the condition that the actions of the
operators $B_{a}^{s}$ for $a\in\underline{\sf 4}$ and
$s\in{\widehat{\mathsf{\Gamma}}}$ on $I^{s^{\prime}}(W_{s^{\prime}})$ generate
the isotopical subspaces $V_{s^{\prime\prime}}$. Conversely, the orbit of any
set of stable maps $(B,I)$ under

$\displaystyle\mathsf{G}_{\vec{k}}:=\operatorname*{\scalerel*{\times}{\displaystyle\sum}}_{s\in\widehat{\mathsf{\Gamma}}}\,\mathsf{GL}(k_{s},\mathbbm{C})$
(3.16)

intersects the locus
$\bigoplus_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mu^{\mathbbm{R}s}=\bigoplus_{s\in{\widehat{\mathsf{\Gamma}}}}\,\zeta_{s}\,\mathbbm{1}_{k_{s}}$.

Then $\mathfrak{M}_{\vec{r},\vec{k}}$ can be described as the space of stable
maps $(B,I)$ which satisfy the holomorphic equations (3.13), modulo the
natural action of the complex gauge group $\mathsf{G}_{\vec{k}}$ of the
$\mathsf{\Gamma}$-module $V$. It follows that the real virtual dimension of
the moduli space $\mathfrak{M}_{\vec{r},\vec{k}}$ is given by

$\displaystyle{\rm
vdim}_{\mathbbm{R}}\,\mathfrak{M}_{\vec{r},\vec{k}}=\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\,\Big{(}2\,\sum_{a=1}^{4}\,k_{s}\,k_{s+s_{a}}+2\,r_{s}\,k_{s}-\sum_{1\leq
a<b\leq 4}\,k_{s}\,k_{s+s_{ab}}-2\,k_{s}^{2}\Big{)}\ .$ (3.17)

The moduli space $\mathfrak{M}_{\vec{r},\vec{k}}$ has a remaining symmetry
under framing rotations: The global colour symmetry $\mathsf{U}(r)_{\rm col}$
is broken to the centralizer
$\operatorname*{\scalerel*{\times}{\textstyle\sum}}_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathsf{U}(r_{s})_{\rm
col}$, which acts on the orbifold ADHM data as

$\displaystyle
h\cdot\big{(}B_{a}^{s}\,,\,I^{s}\big{)}_{\begin{subarray}{c}s\in{\widehat{\mathsf{\Gamma}}}\\\
a\in\underline{\sf
4}\end{subarray}}=\big{(}B_{a}^{s}\,,\,I^{s}\,h_{s}^{-1}\big{)}_{\begin{subarray}{c}s\in{\widehat{\mathsf{\Gamma}}}\\\
a\in\underline{\sf 4}\end{subarray}}\qquad\mbox{with}\quad
h=(h_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}\ \in\
\operatorname*{\scalerel*{\times}{\displaystyle\sum}}_{s\in\widehat{\mathsf{\Gamma}}}\,\mathsf{U}(r_{s})_{\rm
col}\ .$ (3.18)

Its maximal torus is

$\displaystyle\mathsf{T}_{\vec{a}}=\operatorname*{\scalerel*{\times}{\displaystyle\sum}}_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathsf{T}_{\vec{a}^{s}}\
,$ (3.19)

where $\mathsf{T}_{\vec{a}^{s}}$ is the maximal torus of
$\mathsf{U}(r_{s})_{\rm col}$.

In addition, $\mathfrak{M}_{\vec{r},\vec{k}}$ has an $\mathsf{SU}(4)$ symmetry
inherited from the holonomy group of the toric Calabi–Yau four-orbifold
$\mathbbm{C}^{4}/\mathsf{\Gamma}$. In particular, the action of the
(complexified) maximal torus $\mathsf{T}_{\vec{\epsilon}}$ is the evident
descent of (2.55).

###  3.2 Crepant Resolutions

While our constructions of orbifold instanton partition functions will hold
quite generally for any toric Calabi–Yau four-orbifold, a central role is
played by those orbifolds which admit a _crepant resolution_ [70]: Recall
that, for a singular variety $Y$, a proper algebraic map $\pi:X\longrightarrow
Y$ is a crepant resolution if $X$ is smooth and $\pi$ is a birational morphism
which preserves the canonical bundles, i.e. $K_{X}\simeq\pi^{*}K_{Y}$. We are
interested in the case where $Y=\mathbbm{C}^{4}/\mathsf{\Gamma}$, or more
generally $Y=\mathbbm{C}^{d}/\mathsf{\Gamma}$. The existence of a crepant
resolution then requires the group $\mathsf{\Gamma}$ to be a finite subgroup
of $\mathsf{SL}(d,\mathbbm{C})$ [71, 72].

Type II string theory on an orbifold is defined by imposing equivariance under
the action of the finite group $\mathsf{\Gamma}$. When all twisted NS–NS
sectors have age one, a crepant resolution is induced by vacuum expectation
values of scalar fields in these sectors. This allows one to continuously
smoothen the quotient singularities while preserving the Calabi–Yau
properties. The spectrum of the orbifold string theory is then the spectrum of
type II string theory on a compactification over a smooth Calabi–Yau space
obtained by blow-up of the orbifold singularities [73, 74], whose sizes are
controlled by the moduli of the scalar fields. In particular, the BPS states
can be considered in ‘orbifold’ and ‘large radius’ phases, which are related
by collapsing the compact cycles of the resolution [75]. For example,
partition functions of fractional D-branes on orbifolds can be related to
partition functions of D-branes wrapping compact cycles of a crepant
resolution through changes of variables and wall-crossing formulas.

It is well-known that the existence of a crepant resolution of
$\mathbbm{C}^{d}/\mathsf{\Gamma}$ depends dramatically on the dimension $d$ of
the orbifold [76]. In dimensions two and three such resolutions always exist
[77, 78], and in particular in dimension two they are also unique. In these
cases a crepant resolution is always provided by the Hilbert–Chow morphism
from the $\mathsf{\Gamma}$-Hilbert scheme $X={\sf
Hilb}^{\mathsf{\Gamma}}(\mathbbm{C}^{d})$ of $\mathsf{\Gamma}$-invariant zero-
dimensional subschemes $Z\subset\mathbbm{C}^{d}$ of length $|\mathsf{\Gamma}|$
whose global sections $H^{0}(\mathcal{O}_{Z})$ form the regular representation
$R$ of $\mathsf{\Gamma}$, which are called _$\mathsf{\Gamma}$ -clusters_ in
$\mathbbm{C}^{d}$; it can be parametrized by $\mathsf{\Gamma}$-invariant
ideals $J\subset\mathbbm{C}[z_{1},\dots,z_{d}]$ of codimension
$|\mathsf{\Gamma}|$, where $\mathsf{\Gamma}$ acts on the coordinate functions
$z_{1},\dots,z_{d}$. On the other hand, for $d\geq 4$ little is known: they
exist in rather special cases. A simple example for which no crepant
resolution exists is given by the action of the generator of
$\mathsf{\Gamma}=\mathbbm{Z}_{2}$ on $\mathbbm{C}^{4}$ by simultaneous
reflection $z_{a}\longmapsto-z_{a}$ for $a\in\underline{\sf 4}$. It is unknown
if _a priori_ there exists a crepant resolution given an orbifold group
action, and even when one does it is not necessarily given by the
$\mathsf{\Gamma}$-Hilbert scheme, which generally behaves badly and can be
singular.

In the type II string theory picture, the absence of a crepant resolution
means that compactification on $\mathbbm{C}^{d}/\mathsf{\Gamma}$ does not
yield the requisite moduli fields from the twisted NS–NS sectors. For the
worldvolume gauge theories of D-branes localized at the singularity of a
Calabi–Yau orbifold $\mathbbm{C}^{d}/\mathsf{\Gamma}$ for $d=2$ [79, 80] or
$d=3$ [81], the Higgs moduli space is a smooth Calabi–Yau manifold which is a
resolution of the original orbifold, for generic choices of Fayet–Iliopoulos
parameters. In marked contrast, for $d=4$ the Higgs moduli space is not
necessarily a smooth Calabi–Yau fourfold even if the orbifold admits a smooth
resolution: $\mathfrak{M}_{\vec{r},\vec{k}}^{\mathsf{\Gamma}}$ is a Calabi–Yau
fourfold only if $\mathbbm{C}^{4}/\mathsf{\Gamma}$ admits a crepant blow-up to
it [67].

Now suppose that $\mathsf{\Gamma}$ is abelian, and that

$\displaystyle\pi:X\longrightarrow\mathbbm{C}^{d}/\mathsf{\Gamma}$ (3.20)

is a crepant resolution. Under certain technical conditions which are spelled
out in [82, Theorem 5.2], the crepant resolution $X$ can be obtained as a fine
moduli space of stable $\mathsf{\Gamma}$-equivariant coherent sheaves
$\mathcal{E}$ on $\mathbbm{C}^{d}$ such that $H^{0}(\mathcal{E})\simeq R$ as a
$\mathsf{\Gamma}$-module, which are called _$\mathsf{\Gamma}$
-constellations_; it can be parametrized by $\mathsf{\Gamma}$-equivariant
modules over the coordinate ring $\mathbbm{C}[z_{1},\dots,z_{d}]$ which are
isomorphic to $R$ as $\mathsf{\Gamma}$-modules. This includes the cases where
$X$ is the $\mathsf{\Gamma}$-Hilbert scheme: the structure sheaf
$\mathcal{O}_{Z}$ of a $\mathsf{\Gamma}$-cluster $Z$ in $\mathbbm{C}^{d}$ is a
$\mathsf{\Gamma}$-constellation. Physically $X$ can be interpreted as an
orbifold resolution by regular D-branes [81], with Fayet–Iliopoulos parameters
controlling the size of the blow-ups.

Specialising to $d=4$, if $\mathbbm{C}^{4}/\mathsf{\Gamma}$ admits a suitable
crepant resolution $X$, then $X$ can be realised as a moduli space of
$\mathsf{\Gamma}$-constellations, which in turn should be isomorphic to a
moduli space of noncommutative instantons in the Coulomb branch. This is the
starting point for a generalization of the $d=3$ construction of [7], which is
based on the $\mathsf{\Gamma}$-Hilbert scheme and uses Beilinson’s theorem
along with some homological algebra to interpret orbifold noncommutative
instantons geometrically in terms of equivariant sheaves. This would lend a
geometrical parametrization of the instanton moduli space
$\mathfrak{M}_{r,k}^{\mathsf{\Gamma}}$ in terms of
$\mathsf{\Gamma}$-equivariant coherent sheaves on $\mathbbm{C}^{4}$. Such a
construction is briefly sketched in [25] for a special case where $X$ is the
$\mathsf{\Gamma}$-Hilbert scheme by directly applying the techniques of [7].
We do not pursue these technical considerations in this paper, and instead
proceed to look at these blow-ups in more detail for the types of theories we
are interested in; the orbifold example of [25] is contained in our general
formalism as a special case.

#### Toric Orbifold Resolutions

Let $\mathsf{\Gamma}$ be a finite abelian subgroup of
$\mathsf{SL}(d,\mathbbm{C})$ acting linearly on $\mathbbm{C}^{d}$, which we
require to be a subgroup of the maximal torus of the holonomy group
$\mathsf{SU}(d)$. The space $\mathbbm{C}^{d}$ is a toric Calabi–Yau manifold
and the coarse moduli space of the toric Calabi–Yau orbifold
$\mathbbm{C}^{d}/\mathsf{\Gamma}$ has quotient singularities. These can be
resolved through a blow-up process [72] which defines a crepant resolution. To
discuss the blow-up we will use the language of toric geometry [83, 84].

We start by recalling the relevant notions from toric geometry. Let
$N\simeq\mathbbm{Z}^{d}$ be a $\mathbbm{Z}$-lattice and let
$\\{v_{1},\dots,v_{p}\\}$ be a set of linearly independent vectors in the
vector space
$N_{\mathbbm{R}}=N\otimes_{\mathbbm{Z}}\mathbbm{R}\simeq\mathbbm{R}^{d}$. The
set

$\displaystyle\sigma:=\mathbbm{R}_{\geq 0}\,v_{1}+\cdots+\mathbbm{R}_{\geq
0}\,v_{p}\ \subset\ N_{\mathbbm{R}}$ (3.21)

is called a polyhedral cone if $\sigma\cap(-\sigma)=\\{0\\}$. We say that
$\sigma$ is rational if $v_{i}\in N$, and that the cone is smooth if its
generators $v_{1},\dots,v_{p}$ form part of an integral basis of $N$. In the
following we use the notation $\langle v_{1},\dots,v_{p}\rangle$ to indicate
the rational cone generated by the vectors $v_{1},\dots,v_{p}$. For a lattice
point $\vec{\alpha}=(\alpha_{1},\dots\alpha_{d})$ of $N\cap\sigma,$ the age of
$\vec{\alpha}$ is defined by ${\rm
age}(\vec{\alpha})=\sum_{a=1}^{d}\,\alpha_{a}$.

A fan $\Delta\subseteq N_{\mathbbm{R}}$ is a finite collection of polyhedral
cones in $N_{\mathbbm{R}}$ such that every face of any cone $\sigma\in\Delta$
is also a cone of $\Delta$, and the intersection of any two cones of $\Delta$
is also a cone of $\Delta$; it is smooth if all of its cones are smooth. For
our purposes it suffices to say that a toric variety $X_{\Delta}$ is a
$d$-dimensional complex variety defined combinatorially by a fan $\Delta$,
which has an open covering by invariant subsets $X_{\sigma}\subset X_{\Delta}$
with $\sigma\in\Delta$ for the natural action of the algebraic torus
$N_{\mathbbm{C}^{\times}}=N\otimes_{\mathbbm{Z}}\mathbbm{C}^{\times}\simeq(\mathbbm{C}^{\times})^{d}$;
it is smooth if $\Delta$ is smooth.

Our main example is the quotient variety
$\mathbbm{C}^{d}/\mathsf{\Gamma}:={\rm
Spec}\big{(}\mathbbm{C}[z_{1},\dots,z_{d}]^{\mathsf{\Gamma}}\big{)}$, the
spectrum of the $\mathsf{\Gamma}$-invariant subring of
$\mathbbm{C}[z_{1},\dots,z_{d}]$, which can be represented by a fan containing
a single rational polyhedral cone $\sigma$. Indeed, if $\mathsf{\Gamma}$ is a
finite abelian group of order $n$ whose generators act on the coordinates of
$\mathbbm{C}^{d}$ as

$\displaystyle(z_{1},\dots,z_{d})\longmapsto(\mathrm{e}^{\,2\pi\,\mathrm{i}\,s_{1}/n}\,z_{1},\dots,\mathrm{e}^{\,2\pi\,\mathrm{i}\,s_{d}/n}\,z_{d})\
,$ (3.22)

then the local coordinates of $X_{\sigma}$ are
$U^{a}=(z_{1})^{(v_{1})_{a}}\cdots(z_{d})^{(v_{d})_{a}}$ for $a=1,\dots,d$,
where $v_{1},\dots,v_{d}$ are the generators of the cone $\sigma$ [85]. We
require the coordinates $U^{a}$ to be invariant under the action of
$\mathsf{\Gamma}$, thus the cone is given by the solution of the system of
equations

$\displaystyle s_{1}\,(v_{1})_{a}+\cdots+s_{d}\,(v_{d})_{a}=0\quad\mbox{mod}\
n\ ,$ (3.23)

for $a=1,\dots,d$.

Let $\Sigma$ be a subdivision of $\sigma$ through lattice points
$\vec{\alpha}$ with ${\rm age}(\vec{\alpha})=1$, which defines a hyperplane
$\Pi$ in $N_{\mathbbm{R}}$. If $\Sigma$ is smooth, then the toric variety
$X_{\Sigma}$ determined by $\Sigma$ is a crepant resolution of
$\mathbbm{C}^{d}/\mathsf{\Gamma}$; it is Calabi–Yau because its canonical
bundle $K_{X_{\Sigma}}$ is trivial, and so is acted on by a
$d{-}1$-dimensional Calabi–Yau subtorus in $N_{\mathbbm{C}^{\times}}$. The
McKay correspondence associates the prime exceptional divisors of $X_{\Sigma}$
to the elements of $\mathsf{\Gamma}$ with age
$\frac{1}{n}\,(s_{1}+\cdots+s_{d})=1$ [82].

In what follows we will consider two particular classes of orbifolds of
$\mathbbm{C}^{4}$ explicitly for which crepant resolutions are always
guaranteed. They both involve non-generic subgroups
$\mathsf{\Gamma}\subset\mathsf{SL}(4,\mathbbm{C})$ which also preserve a
smaller holonomy subgroup of $\mathsf{SU}(4)$.

####  ${\bm{(2,0)}}$ Orbifolds
${\bm{\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}}}$

Consider the quotient singularity
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$, where
$\mathsf{\Gamma}=\mathbbm{Z}_{n}$ is the cyclic group of order $n$ in
$\mathsf{SL}(2,\mathbbm{C})$ whose generator acts on $\mathbbm{C}^{4}$ as

$\displaystyle(z_{1},z_{2},z_{3},z_{4})\longmapsto(\omega\,z_{1},\omega^{-1}\,z_{2},z_{3},z_{4})\
,$ (3.24)

with $\omega=\mathrm{e}^{\,2\pi\,\mathrm{i}/n}$ a primitive $n$-th root of
unity. We call these _(2,0) orbifolds_. They have $\mathsf{SU}(2)$ holonomy,
leading to an enhanced $\mathcal{N}=(0,4)$ supersymmetry in the two-
dimensional D1-brane theory, which can be obtained by dimensional reduction of
the six-dimensional $\mathcal{N}=2$ worldvolume theories of D5-branes at ALE
singularities $\mathbbm{C}^{2}/\mathbbm{Z}_{n}$.

The conditions (3.23) leave an arbitrariness in choosing the generators
$v_{3}$ and $v_{4}$. Instead, the generators $v_{1}$ and $v_{2}$ define a
plane which, with an appropriate choice of an orthonormal basis
$\\{e_{a}\\}_{a\in\underline{\sf 4}}$ of $\mathbbm{R}^{4}$, is spanned by
$e_{1}$ and $e_{2}$. Consider now the rational cone $\sigma_{12}=\langle
v_{1},v_{2}\rangle$. Since $\mathbbm{C}^{2}/\mathbbm{Z}_{n}$ admits a crepant
resolution (by ${\sf Hilb}^{\mathbbm{Z}_{n}}(\mathbbm{C}^{2})$), there is a
subdivision $\Sigma_{12}$ of $\sigma_{12}$ which is smooth. By arbitrariness
of $v_{3}$ and $v_{4}$ we can pick them such that they define a smooth
rational cone; for instance, $v_{3}=e_{3}$ and $v_{4}=e_{4}$ with respect to
the basis $\\{e_{a}\\}_{a\in\underline{\sf 4}}$.

The fan

$\displaystyle\Sigma=\big{\\{}\langle\beta_{12},v_{3},v_{4}\rangle\ \big{|}\
\beta_{12}\in\Sigma_{12}\big{\\}}$ (3.25)

is a smooth subdivision of $\sigma=\langle v_{1},v_{2},v_{3},v_{4}\rangle$.
Therefore the orbifold
$\mathbbm{C}^{4}/\mathbbm{Z}_{n}\simeq\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$,
with the action of
$\mathbbm{Z}_{n}\subset\mathsf{SL}(2,\mathbbm{C})\subset\mathsf{SL}(4,\mathbbm{C})$
given by (3.24), admits a crepant resolution $X_{\Sigma}$, which is a
(trivial) fibration of $A_{n-1}$ ALE spaces over the affine plane, or
equivalently the total space of the rank three bundle
$\mathcal{O}_{{\mathbbm{P}}^{1}}(-n+1)\oplus\mathcal{O}_{{\mathbbm{P}}^{1}}\oplus\mathcal{O}_{{\mathbbm{P}}^{1}}$.

######  Example 3.26.

Take the orbifold $\mathbbm{C}^{2}/\mathbbm{Z}_{2}\times\mathbbm{C}^{2}$
considered in [25]. By the conditions (3.23) we can choose the generators of
the rational cone $\sigma$ to be

$\displaystyle v_{1}=(2,-1,0,0)\quad,\quad v_{2}=(0,1,0,0)\quad,\quad
v_{3}=(0,0,1,0)\quad,\quad v_{4}=(0,0,0,1)\ .$ (3.27)

Its subdivision $\Sigma$ using the hyperplane $\Pi$ is

$\displaystyle\begin{split}\Sigma=\big{\\{}\langle
v_{1},v_{3},v_{4},v_{5}\rangle\,,\,\langle
v_{2},v_{3},v_{4},v_{5}\rangle\,,\,\langle
v_{1},v_{3},v_{4}\rangle\,,\,\langle v_{2},v_{3},v_{4}\rangle\,,\,\langle
v_{3},v_{4},v_{5}\rangle\,,\,\langle v_{3},v_{4}\rangle\big{\\}}\
,\end{split}$ (3.28)

where

$\displaystyle v_{5}=(1,0,0,0)\ .$ (3.29)

Each tuple of vectors $v_{i}$ forms part of a basis of the lattice
$\mathbbm{Z}^{4}$, so $\Sigma$ is smooth and $X_{\Sigma}$ is a crepant
resolution of $\mathbbm{C}^{2}/\mathbbm{Z}_{2}\times\mathbbm{C}^{2}$.

####  ${\bm{(3,0)}}$ Orbifolds
${\bm{\mathbbm{C}^{3}/\mathsf{\Gamma}\times\mathbbm{C}}}$

Another tractable quotient singularity we will consider explicitly is
$\mathbbm{C}^{4}/\mathsf{\Gamma}\simeq\mathbbm{C}^{3}/\mathsf{\Gamma}\times\mathbbm{C}$
where $\mathsf{\Gamma}$ is a finite abelian subgroup of
$\mathsf{SL}(3,\mathbbm{C})\subset\mathsf{SL}(4,\mathbbm{C})$. We call these
_(3,0) orbifolds_. They have $\mathsf{SU}(3)$ holonomy and again lead to an
enhancement of the supersymmetry in the two-dimensional D1-brane theory, which
can be obtained from dimensional reduction of the four-dimensional
$\mathcal{N}=2$ worldvolume theories of D3-branes at toric Calabi–Yau
threefold singularities $\mathbbm{C}^{3}/\mathsf{\Gamma}$.

The construction for $(2,0)$ orbifolds above can be generalized directly here,
with obvious modifications, starting from the key fact that
$\mathbbm{C}^{3}/\mathsf{\Gamma}$ always admits a crepant resolution for any
finite abelian subgroup $\mathsf{\Gamma}$ of $\mathsf{SL}(3,\mathbbm{C}$) (for
example by ${\sf Hilb}^{\mathsf{\Gamma}}(\mathbbm{C}^{3})$). Note that any
$(2,0)$ orbifold is also a $(3,0)$ orbifold.

###  3.3 Noncommutative Crepant Resolutions and their ADHM Representations

As in the example studied in [25], all information about the moduli space of
orbifold instantons can be encoded in the data of a quiver ${\sf Q}$ which
generalizes the McKay quiver for surface singularities and is analogous to the
four-loop quiver ${\sf L}_{4}$ of (2.45). It is determined by the
representation theory data of the $\mathsf{\Gamma}$-action, with ${\sf L}_{4}$
corresponding to the trivial group $\mathsf{\Gamma}=\\{\mathbbm{1}\\}$, and
the generalized McKay correspondence asserts that it captures the geometry of
the orbifold $\mathbbm{C}^{4}/\mathsf{\Gamma}$. The quiver ${\sf Q}$ encodes
the isotopical decomposition of the usual ADHM data according to the
$\mathsf{\Gamma}$-action and it has the orbifold ADHM equations (3.13) as
relations. For background, see [86] for a concise overview of the theory of
quivers.

One starts from the irreducible representations $\widehat{\mathsf{\Gamma}}$ of
$\mathsf{\Gamma}\subset\mathsf{SL}(4,\mathbbm{C})$. To each representation
$\rho_{s}$ in $\widehat{\mathsf{\Gamma}}$, including the trivial
representation $\rho_{0}$ of weight zero, we associate a node of a quiver
${\sf Q}$. A node $s$ is connected to a node $s^{\prime}$ by a number of
arrows $a_{ss^{\prime}}$ from $s$ to $s^{\prime}$ determined by the adjacency
matrix $A=(a_{ss^{\prime}})$ in the decomposition of $\mathsf{\Gamma}$-modules

$\displaystyle
Q\otimes\rho_{s}=\bigoplus_{s^{\prime}\in\widehat{\mathsf{\Gamma}}}\,a_{ss^{\prime}}\,\rho_{s^{\prime}}\
,$ (3.30)

where the multiplicities are given by

$\displaystyle
a_{ss^{\prime}}=\dim_{\mathbbm{C}}\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(\rho_{s^{\prime}},Q\otimes\rho_{s})=\frac{1}{|\mathsf{\Gamma}|}\,\sum_{g\in\mathsf{\Gamma}}\,\chi_{Q}(g)\,\chi_{s}(g)\,\overline{\chi_{s^{\prime}}(g)}\
.$ (3.31)

In this graphical representation the sum over the irreducible representations
$\widehat{\mathsf{\Gamma}}$ of $\mathsf{\Gamma}$ in (3.8) becomes a sum over
the nodes ${\sf Q}_{0}$ of the quiver and the matrices $B_{a}^{s}$ are
associated to the arrows ${\sf Q}_{1}$. The resulting quiver is known as the
(generalized) bounded McKay quiver $({\sf Q},{\sf R})$ and it is associated
with a two-sided ideal of relations $\langle{\sf R}\rangle$ in the
corresponding path algebra $\mathbbm{C}\,{\sf Q}$ generated by the arrows
${\sf Q}_{1}$; recall that the product in $\mathbbm{C}\,{\sf Q}$ is the
concatenation of paths in ${\sf Q}$ whenever this makes sense and $0$
otherwise. The remaining data in (3.9) define a framing of the quiver.

Representations of the McKay quiver ${\sf Q}=({\sf Q}_{0},{\sf Q}_{1})$ with
relations (3.13) are functorially equivalent to finitely-generated left
modules over its path algebra ${\sf A}:=\mathbbm{C}\,{\sf Q}/\langle{\sf
R}\rangle$. This can be identified as the standard _noncommutative_ crepant
resolution of the abelian quotient singularity
$\mathbbm{C}^{4}/\mathsf{\Gamma}$, in the sense that ${\sf A}$ is Morita
equivalent to the skew group ring
$\mathbbm{C}[z_{1},z_{2},z_{3},z_{4}]\rtimes\mathbbm{C}[\mathsf{\Gamma}]$ of
the quotient stack $[\mathbbm{C}^{4}/\mathsf{\Gamma}]$, whereas the centre
${\sf Z}({\sf A})$ of ${\sf A}$ is isomorphic to the
$\mathsf{\Gamma}$-invariant subring
$\mathbbm{C}[z_{1},z_{2},z_{3},z_{4}]^{\mathsf{\Gamma}}$. Thus while
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ may not have a geometric crepant resolution,
it always possesses a noncommutative crepant resolution by ${\sf A}$. This can
be seen as a consequence of the fact that the derived category of coherent
sheaves of ${\sf A}$-modules on $\mathbbm{C}^{4}/\mathsf{\Gamma}$ is a
_categorical_ crepant resolution of $\mathbbm{C}^{4}/\mathsf{\Gamma}$ [87], in
the sense that it mimicks the functorial behaviour one expects from the
derived category of a (geometric) crepant resolution; see e.g. [88, 89] for
introductions.

The moduli space $\mathfrak{M}_{r,k}^{\mathsf{\Gamma}}$ of instantons on
$\mathbbm{C}^{4}/\mathsf{\Gamma}$, viewed as the moduli space of
$\mathsf{\Gamma}$-equivariant instantons on $\mathbbm{C}^{4}$, is then
identified as the moduli space of stable framed representations of the bounded
quiver $({\sf Q},{\sf R})$. The connected component
$\mathfrak{M}_{\vec{r},\vec{k}}$ is the moduli space of stable framed quiver
representations with fixed dimension vector $(\vec{r},\vec{k}\,)$, that is,
the (stacky) quotient by the action of the group (3.16) of the subvariety of
the framed quiver representation space

$\displaystyle{\mathfrak{Rep}}_{\vec{r},\vec{k}}=\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(V,V\otimes
Q)\,\oplus\,\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(W,V)$ (3.32)

cut out by the ADHM equations (3.13). In particular, regular instantons with
$V\simeq R$ the regular representation of $\mathsf{\Gamma}$ correspond to
orbits in $\mathfrak{M}_{\vec{r},\vec{k}}$ for $r=1$ and
$\vec{k}=(1,\dots,1)$, which is a moduli space parametrizing isomorphism
classes of $\mathsf{\Gamma}$-constellations by [90, 82].

From this perspective, the (complex) obstruction bundle over the instanton
moduli space has the explicit description [21] as the pullback along
$\mathfrak{M}_{\vec{r},\vec{k}}\lhook\joinrel\longrightarrow{\mathfrak{Rep}}_{\vec{r},\vec{k}}$
of the bundle

$\displaystyle{\sf
Ob}_{\vec{r},\vec{k}}={\mathfrak{Rep}}_{\vec{r},\vec{k}}\,\times_{\mathsf{G}_{\vec{k}}}\,{\sf
Ext}_{\sf
A}^{2}\Big{(}\textstyle\bigoplus\limits_{s\in{\widehat{\mathsf{\Gamma}}}}\,D_{s}\otimes
V_{s}\,,\,\bigoplus\limits_{s\in{\widehat{\mathsf{\Gamma}}}}\,D_{s}\otimes
V_{s}\Big{)}\ ,$ (3.33)

whose fibre computes the relations ${\sf R}$ of the quiver ${\sf Q}$. Here
$D_{s}$ is the one-dimensional simple ${\sf A}$-module defined by
$(D_{s})_{s^{\prime}}=\delta_{s,s^{\prime}}\,\mathbbm{C}$ for
$s,s^{\prime}\in{\widehat{\mathsf{\Gamma}}}$.

Recall from Section 2.4 that the introduction of a flat $B$-field on
$\mathbbm{C}^{4}$ leads to a representation of D-branes by complexes of
modules over the noncommutative algebra ${\sf Mat}_{r{\times}r}(\mathcal{A})$.
This description is valid when the $B$-field is large compared to the metric
on $\mathbbm{C}^{4}$. For generic orbifold groups
$\mathsf{\Gamma}\subset\mathsf{SL}(4,\mathbbm{C})$, the theory on
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ is regarded as a limit where the volume of a
subspace containing the orbifold fixed point shrinks to zero but the $B$-field
is non-vanishing, which regularises the string worldsheet theory and leads to
a noncommutative crepant resolution of the quotient singularity; this is a
highly non-geometric limit of the worldvolume gauge theory. The vacua of the
D1-brane theory of Section 3.1, and therefore the stable D-brane
configurations, then correspond to stable representations of the
noncommutative algebra ${\sf A}$ associated to $({\sf Q},{\sf R})$. In
particular, a regular D-brane corresponds to a module over ${\sf A}$ of
dimension vector $\vec{k}=(1,\dots,1)$ which is the ${\sf A}$-module of global
sections $H^{0}(\mathcal{E})$ of a $\mathsf{\Gamma}$-constellation
$\mathcal{E}$.

######  Example 3.34.

Let $\mathsf{\Gamma}=\mathbbm{Z}_{4}$ acting on $\mathbbm{C}^{4}$ with
generator

$\displaystyle(z_{1},z_{2},z_{3},z_{4})\longmapsto(\mathrm{i}\,z_{1},-\mathrm{i}\,z_{2},z_{3},z_{4})\
.$ (3.35)

This $\mathbbm{Z}_{4}$-action defines a $(2,0)$ quotient singularity
$\mathbbm{C}^{4}/\mathbbm{Z}_{4}\simeq\mathbbm{C}^{2}/\mathbbm{Z}_{4}\times\mathbbm{C}^{2}$.
From the general discussion of Section 3.2, it admits a geometric crepant
resolution by a fibration of $A_{3}$ ALE spaces over $\mathbbm{C}^{2}$.

With respect to this $\mathbbm{Z}_{4}$-action, the fundamental representation
$Q\simeq\mathbbm{C}^{4}$ decomposes into irreducible $\mathbbm{Z}_{4}$-modules
as $Q=\rho_{1}\oplus\rho_{3}\oplus\rho_{0}\oplus\rho_{0}$, which implies

$\displaystyle\begin{split}Q\otimes\rho_{0}=\rho_{1}\oplus\rho_{3}\oplus\rho_{0}\oplus\rho_{0}\qquad,\qquad
Q\otimes\rho_{2}=\rho_{3}\oplus\rho_{1}\oplus\rho_{2}\oplus\rho_{2}\
,\\\\[4.0pt]
Q\otimes\rho_{1}=\rho_{2}\oplus\rho_{0}\oplus\rho_{1}\oplus\rho_{1}\qquad,\qquad
Q\otimes\rho_{3}=\rho_{0}\oplus\rho_{2}\oplus\rho_{3}\oplus\rho_{3}\
.\end{split}$ (3.36)

The finite abelian group ${\widehat{\mathsf{\Gamma}}}$ is defined by the
character table of $\mathbbm{Z}_{4}$:

| $\rho_{0}$ | $\rho_{1}$ | $\rho_{2}$ | $\rho_{3}$ |
---|---|---|---|---|---
$\chi_{0}$ | $1$ | $1$ | $1$ | $1$ |
$\chi_{1}$ | $1$ | $\mathrm{i}$ | $-1$ | $-\mathrm{i}$ |
$\chi_{2}$ | $1$ | $-1$ | $1$ | $-1$ |
$\chi_{3}$ | $1$ | $-\mathrm{i}$ | $-1$ | $\mathrm{i}$ |
(3.37)

The generalized McKay quiver ${\sf Q}$ is

${0}$${1}$${3}$${2}$ (3.38)

The generalized ADHM equations, which determine the relations ${\sf R}$ of the
quiver (3.38), assume the form

$\displaystyle
B^{s+s_{a}}_{b}\,B_{a}^{s}=B^{s+s_{b}}_{a}\,B_{b}^{s}\qquad\text{with}\quad\begin{cases}\
B_{1}^{s}:V_{s}\longrightarrow V_{s+1}\\\ \ B_{2}^{s}:V_{s}\longrightarrow
V_{s+3}\\\ \ B_{3,4}^{s}:V_{s}\longrightarrow V_{s}\end{cases}\ .$ (3.39)

Explicitly

$B_{2}^{1}\,B_{1}^{0}=B_{1}^{3}\,B_{2}^{0}$ , |  $B_{3}^{1}\,B_{1}^{0}=B_{1}^{0}\,B_{3}^{0}$ , |  $B_{4}^{1}\,B_{1}^{0}=B_{1}^{0}\,B_{4}^{0}$ , |  $B_{3}^{0}\,B_{2}^{0}=B_{2}^{0}\,B_{3}^{0}$ ,
---|---|---|---
$B_{4}^{3}\,B_{2}^{0}=B_{2}^{0}\,B_{4}^{0}$ , |  $B_{4}^{0}\,B_{3}^{0}=B_{3}^{0}\,B_{4}^{0}$ , |  $B_{2}^{2}\,B_{1}^{1}=B_{1}^{0}\,B_{2}^{1}$ , |  $B_{3}^{2}\,B_{1}^{1}=B_{1}^{1}\,B_{3}^{1}$ ,
$B_{4}^{2}\,B_{1}^{1}=B_{1}^{1}\,B_{4}^{1}$ , |  $B_{3}^{0}\,B_{2}^{1}=B_{2}^{1}\,B_{3}^{1}$ , |  $B_{4}^{0}\,B_{2}^{1}=B_{2}^{1}\,B_{4}^{1}$ , |  $B_{4}^{1}\,B_{3}^{1}=B_{3}^{1}\,B_{4}^{1}$ ,
$B_{2}^{3}\,B_{1}^{2}=B_{1}^{1}\,B_{2}^{2}$ , |  $B_{3}^{3}\,B_{1}^{2}=B_{1}^{2}\,B_{3}^{2}$ , |  $B_{4}^{3}\,B_{1}^{2}=B_{1}^{2}\,B_{4}^{2}$ , |  $B_{3}^{1}\,B_{2}^{2}=B_{2}^{2}\,B_{3}^{2}$ ,
$B_{4}^{1}\,B_{2}^{2}=B_{2}^{2}\,B_{4}^{2}$ , |  $B_{4}^{2}\,B_{3}^{2}=B_{3}^{2}\,B_{4}^{2}$ , |  $B_{2}^{0}\,B_{1}^{3}=B_{1}^{2}\,B_{2}^{3}$ , |  $B_{3}^{0}\,B_{1}^{3}=B_{1}^{3}\,B_{3}^{3}$ ,
$B_{4}^{0}\,B_{1}^{3}=B_{1}^{3}\,B_{4}^{3}$ , |  $B_{3}^{0}\,B_{2}^{1}=B_{2}^{1}\,B_{3}^{1}$ , |  $B_{4}^{2}\,B_{2}^{3}=B_{2}^{3}\,B_{4}^{3}$ , | $B_{4}^{3}\,B_{3}^{3}=B_{3}^{3}\,B_{4}^{3}\ \ .$
(3.40)

The center ${\sf Z}({\sf A})$ of the path algebra ${\sf A}$ of $({\sf Q},{\sf
R})$ is generated as a ring by elements

$\displaystyle\begin{split}{\tt
X}_{11}=B^{3}_{1}\,B_{1}^{2}\,B_{1}^{1}\,B_{1}^{0}\quad,\quad&{\tt
X}_{22}=B^{1}_{2}\,B^{2}_{2}\,B^{3}_{2}\,B^{0}_{2}\quad,\quad{\tt
X}_{12}=B^{3}_{1}\,B^{0}_{2}\ ,\\\\[4.0pt] {\tt
Y}_{3}&=B^{0}_{3}\quad,\quad{\tt Y}_{4}=B^{0}_{4}\ .\end{split}$ (3.41)

Then given the $\mathbbm{Z}_{4}$-action (3.35), we can identify these
generators with the $\mathbbm{Z}_{4}$-invariant elements in the ring
$\mathbbm{C}[z_{1},z_{2},z_{3},z_{4}]$ through

$\displaystyle\begin{split}{\tt X}_{11}\leadsto z_{1}^{4}\quad,\quad&{\tt
X}_{22}\leadsto z_{2}^{4}\quad,\quad{\tt X}_{12}\leadsto z_{1}\,z_{2}\
,\\\\[4.0pt] &{\tt Y}_{3}\leadsto z_{3}\quad,\quad{\tt Y}_{4}\leadsto z_{4}\
.\end{split}$ (3.42)

It follows that ${\rm Spec}({\sf Z}({\sf
A}))\simeq\mathbbm{C}^{2}/\mathbbm{Z}_{4}\times\mathbbm{C}^{2}$ and the path
algebra ${\sf A}$ is a noncommutative crepant resolution of the quotient
singularity $\mathbbm{C}^{4}/\mathbbm{Z}_{4}$.

Finally, the generalized ADHM data $(B_{a},I)_{a\in\underline{\sf 4}}$ define
a framed representation of the quiver (3.38), which we depict by the oriented
graph

${{\boxed{W_{0}}}}$${{\boxed{V_{0}}}}$${{\boxed{W_{1}}}}$${{\boxed{V_{1}}}}$${{\boxed{V_{3}}}}$${{\boxed{W_{3}}}}$${{\boxed{V_{2}}}}$${{\boxed{W_{2}}}}$
(3.43)

######  Example 3.44.

Let $\mathsf{\Gamma}=\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ be the subgroup of
order four in $\mathsf{SL}(4,\mathbbm{C})$ acting non-trivially on all four
coordinates of $\mathbbm{C}^{4}$ with weights $s_{1}=(1,1,0,0)$,
$s_{2}=(0,0,1,1)$ and $s_{3}=s_{1}+s_{2}=(1,1,1,1)$. Although this does not
define a $(3,0)$ orbifold, the existence of a geometric crepant resolution of
$\mathbbm{C}^{4}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ is guaranteed by [84,
Proposition 3.1], which may be taken to be ${\sf
Hilb}^{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}(\mathbbm{C}^{4})$ by [84,
Proposition 3.2].

The generators of $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with respect to this
action are

$\displaystyle g_{1}={\small\begin{pmatrix}-1&&&\\\ &-1&&\\\ &&1&\\\
&&&1\end{pmatrix}}\normalsize\qquad\mbox{and}\qquad
g_{2}={\small\begin{pmatrix}1&&&\\\ &1&&\\\ &&-1&\\\
&&&-1\end{pmatrix}}\normalsize\ .$ (3.45)

The group action has four irreducible representations $\rho_{s}$ where
$\rho_{0}$ is the trivial representation, $\rho_{1}$ and $\rho_{2}$ have
weights $s_{1}$ and $s_{2}$, respectively, while
$\rho_{3}=\rho_{1}\otimes\rho_{2}$ has weight $s_{3}$. The character table of
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ is

| $\rho_{0}$ | $\rho_{1}$ | $\rho_{2}$ | $\rho_{3}$ |
---|---|---|---|---|---
$\chi_{0}$ | $1$ | $1$ | $1$ | $1$ |
$\chi_{1}$ | $1$ | $1$ | $-1$ | $-1$ |
$\chi_{2}$ | $1$ | $-1$ | $1$ | $-1$ |
$\chi_{3}$ | $1$ | $-1$ | $-1$ | $1$ |
(3.46)

The ADHM equations are

$B_{2}^{1}\,B_{1}^{0}=B_{1}^{1}\,B_{2}^{0}$ , |  $B_{3}^{1}\,B_{1}^{0}=B_{1}^{2}\,B_{3}^{0}$ , |  $B_{4}^{1}\,B_{1}^{0}=B_{1}^{2}\,B_{4}^{0}$ , |  $B_{3}^{1}\,B_{2}^{0}=B_{2}^{2}\,B_{3}^{0}$ ,
---|---|---|---
$B_{4}^{1}\,B_{2}^{0}=B_{2}^{2}\,B_{4}^{0}$ , |  $B_{4}^{2}\,B_{3}^{0}=B_{3}^{2}\,B_{4}^{0}$ , |  $B_{2}^{0}\,B_{1}^{1}=B_{1}^{0}\,B_{2}^{1}$ , |  $B_{3}^{0}\,B_{1}^{1}=B_{1}^{3}\,B_{3}^{1}$ ,
$B_{4}^{0}\,B_{1}^{1}=B_{1}^{3}\,B_{4}^{1}$ , |  $B_{3}^{0}\,B_{2}^{1}=B_{2}^{3}\,B_{3}^{1}$ , |  $B_{4}^{0}\,B_{2}^{1}=B_{2}^{3}\,B_{4}^{1}$ , |  $B_{4}^{3}\,B_{3}^{1}=B_{3}^{3}\,B_{4}^{1}$ ,
$B_{2}^{3}\,B_{1}^{1}=B_{1}^{3}\,B_{2}^{2}$ , |  $B_{3}^{3}\,B_{1}^{2}=B_{1}^{0}\,B_{3}^{2}$ , |  $B_{4}^{3}\,B_{1}^{2}=B_{1}^{0}\,B_{4}^{2}$ , |  $B_{3}^{3}\,B_{2}^{2}=B_{2}^{0}\,B_{3}^{2}$ ,
$B_{4}^{3}\,B_{2}^{2}=B_{2}^{0}\,B_{4}^{2}$ , |  $B_{4}^{0}\,B_{3}^{2}=B_{3}^{0}\,B_{4}^{2}$ , |  $B_{2}^{2}\,B_{1}^{3}=B_{1}^{2}\,B_{2}^{3}$ , |  $B_{3}^{2}\,B_{1}^{3}=B_{1}^{1}\,B_{3}^{3}$ ,
$B_{4}^{2}\,B_{1}^{3}=B_{1}^{1}\,B_{4}^{3}$ , |  $B_{3}^{2}\,B_{2}^{3}=B_{2}^{1}\,B_{3}^{3}$ , |  $B_{4}^{2}\,B_{2}^{3}=B_{2}^{1}\,B_{4}^{3}$ , |  $B_{4}^{1}\,B_{3}^{3}=B_{3}^{1}\,B_{4}^{3}$ .
(3.47)

These are relations ${\sf R}$ for the corresponding McKay quiver ${\sf Q}$
given by

${0}$${1}$${3}$${2}$ (3.48)

The centre ${\sf Z}({\sf A})$ of the path algebra ${\sf A}$ of the quiver
(3.48) with relations (3.47) is generated as a ring by the elements

$\displaystyle\begin{split}{\tt
X}_{abcd}&=B^{2}_{a}\,B_{b}^{3}\,B_{c}^{1}\,B_{d}^{0}\qquad\mbox{with}\quad
b\leq d<3\leq a\leq c\leq 4\ ,\\\\[4.0pt] {\tt
X}_{ab}&=B^{2}_{a}\,B^{0}_{b}\qquad\mbox{with}\quad b\leq 2<a\leq 4\
,\end{split}$ (3.49)

which are again identified with the
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$-invariant elements in
$\mathbbm{C}[z_{1},z_{2},z_{3},z_{4}]$. It follows that ${\rm Spec}({\sf
Z}({\sf A}))\simeq\mathbbm{C}^{4}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$.

Similarly to Example 3.34, the ADHM data define a framed quiver representation

${{\boxed{W_{0}}}}$${{\boxed{V_{0}}}}$${{\boxed{W_{1}}}}$${{\boxed{V_{1}}}}$${{\boxed{V_{2}}}}$${{\boxed{W_{2}}}}$${{\boxed{V_{3}}}}$${{\boxed{W_{3}}}}$
(3.50)

of the bounded McKay quiver (3.48,3.47).

###  3.4 Cohomological Field Theory on Noncommutative Crepant Resolutions

We shall now modify the construction of Section 2.5 to calculate the instanton
partition function of the cohomological field theory on orbifolds
$\mathbbm{C}^{4}/\mathsf{\Gamma}$, which we regard as a gauge theory on the
quotient stack $[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ following [7]. We begin
with a framed representation of the generalized McKay quiver associated with
the abelian quotient singularity, which is uniquely determined by the
decomposition of the fundamental representation
$Q=\rho_{s_{1}}\oplus\cdots\oplus\rho_{s_{4}}$ of the orbifold group into
irreducible $\mathsf{\Gamma}$-modules. The topological field theory on
$[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ is invariant under a set of
$\mathsf{\Gamma}$-equivariant BRST transformations and it localises onto the
relations ${{\sf R}}$ of the generalized McKay quiver ${\sf Q}$. The resulting
orbifold instanton partition functions encode noncommutative Donaldson–Thomas
invariants associated to $({\sf Q},{\sf R})$, which count semistable
representations of the noncommutative path algebra ${\sf A}=\mathbbm{C}\,{\sf
Q}/\langle{\sf R}\rangle$.

If the quotient singularity $\mathbbm{C}^{4}/\mathsf{\Gamma}$ admits a
geometric crepant resolution
$\pi:X\longrightarrow\mathbbm{C}^{4}/\mathsf{\Gamma}$, it is natural to
conjecture that the noncommutative crepant resolution ${\sf A}$ of
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ is realised in the stringy Kähler moduli
space of $X$. In particular, the orbifold instanton partition function should
be related to the large radius partition functions which compute the
Donaldson–Thomas and Pandharipande–Thomas invariants of $X$ via a version of
the McKay correspondence, through suitable changes of variables and wall-
crossing formulas in the derived category of $X$ using the methods of [7]. As
they presently stand, our techniques do not immediately extend to these
resolutions and so we defer further discussion to future work. A conjectural
mathematical treatment of the rank one partition functions in this setting is
discussed in [27].

We begin from the BRST transformations with $\mathsf{\Gamma}$-module
structure, where now the matrix fields decompose according to the irreducible
representations of $\mathsf{\Gamma}$. The action of the symmetry group
$\big{(}\operatorname*{\scalerel*{\times}{\textstyle\sum}}_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathsf{U}(k_{s})\times\mathsf{U}(r_{s})_{\rm
col}\big{)}\times\mathsf{SU}(4)$ is described in Section 3.1, and the BRST
transformations of the ADHM variables are given by

$\begin{split}\mathcal{Q}_{\mathsf{\Gamma}}B_{a}^{s}=\psi_{a}^{s}\qquad&,\qquad\mathcal{Q}_{\mathsf{\Gamma}}\psi_{a}^{s}=\phi^{s+s_{a}}\,B_{a}^{s}-B_{a}^{s}\,\phi^{s}-\epsilon_{a}\,B_{a}^{s}\
,\\\\[4.0pt]
\mathcal{Q}_{\mathsf{\Gamma}}I^{s}=\varrho^{s}\qquad&,\qquad\mathcal{Q}_{\mathsf{\Gamma}}\varrho^{s}=\phi^{s}\,I^{s}-I^{s}\,{\bm{a}}^{s}\
,\end{split}$ (3.51)

for $a\in\underline{\sf 4}$ and $s\in{\widehat{\mathsf{\Gamma}}}$. Here
$\phi^{s}$ parametrizes $\mathsf{U}(k_{s})$ gauge transformations, while the
vector ${\bm{a}}^{s}$ collects the $r_{s}$ Higgs field eigenvalues $a_{l}$
associated with the irreducible representation $\rho_{s}$. The latter defines
a map $l\longmapsto s(l)\in{\widehat{\mathsf{\Gamma}}}$ for
$l\in\\{1,\dots,r\\}$.

The $\mathsf{\Gamma}$-module structure of the antighost fields in
$\mathsf{End}_{\mathbbm{C}[\mathsf{\Gamma}]}(V)$ are dictated by the
generalized ADHM equations (3.10), where again we choose
$\mu_{\alpha\beta}^{\mathbbm{C}s}=0$ for $(\alpha,\beta)\in\underline{\sf
3}^{\perp}:=\big{\\{}(1,2)\,,\,(1,3)\,,\,(2,3)\big{\\}}$ as the independent
complex moment map equations. Their decomposition into equivariant maps is
given by

$\displaystyle\chi_{\alpha\beta}^{\mathbbm{C}s}:V_{s}\longrightarrow
V_{s+s_{\alpha\beta}}\qquad\mbox{and}\qquad\chi^{\mathbbm{R}s}:V_{s}\longrightarrow
V_{s}\ .$ (3.52)

Together with the auxiliary fields their BRST transformations are

$\begin{split}\mathcal{Q}_{\mathsf{\Gamma}}\chi_{\alpha\beta}^{\mathbbm{C}s}=H_{\alpha\beta}^{\mathbbm{C}s}\qquad&,\qquad\mathcal{Q}_{\mathsf{\Gamma}}H_{\alpha\beta}^{\mathbbm{C}s}=\phi^{s+s_{\alpha\beta}}\,\chi_{\alpha\beta}^{\mathbbm{C}s}-\chi_{\alpha\beta}^{\mathbbm{C}s}\,\phi^{s}-\epsilon_{\alpha\beta}\,\chi^{\mathbbm{C}s}_{\alpha\beta}\
,\\\\[4.0pt]
\mathcal{Q}_{\mathsf{\Gamma}}\chi^{\mathbbm{R}s}=H^{\mathbbm{R}s}\qquad&,\qquad\mathcal{Q}_{\mathsf{\Gamma}}H^{\mathbbm{R}s}=[\phi^{s},\chi^{\mathbbm{R}s}]\
,\end{split}$ (3.53)

while the gauge multiplet closes the BRST algebra

$\displaystyle\mathcal{Q}_{\mathsf{\Gamma}}\phi^{s}=0\
,\quad\mathcal{Q}_{\mathsf{\Gamma}}\bar{\phi}^{s}=\eta^{s}\qquad\mbox{and}\qquad\mathcal{Q}_{\mathsf{\Gamma}}\eta^{s}=[\phi^{s},\bar{\phi}^{s}]\
.$ (3.54)

For the fundamental matter fields, we treat the global flavour symmetry
identically to the global colour symmetry, which is thereby broken to the
subgroup
$\operatorname*{\scalerel*{\times}{\textstyle\sum}}_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathsf{U}(r_{s})_{\rm
fla}$ by the $\mathsf{\Gamma}$-action; its maximal torus is

$\displaystyle\mathsf{T}_{\vec{m}}=\operatorname*{\scalerel*{\times}{\displaystyle\sum}}_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathsf{T}_{\vec{m}^{s}}\
,$ (3.55)

where $\mathsf{T}_{\vec{m}^{s}}$ is the maximal torus of
$\mathsf{U}(r_{s})_{\rm fla}$. From the string theory perspective of Section
2.4, this aligns the Chan–Paton factors of D9-branes and $\overline{\rm
D9}$-branes on the orbifold such that they still annihilate. Since both the
framing of the gauge bundle and the fundamental matter fields transform in the
fundamental representation ${\bm{r}}$ of $\mathsf{U}(r)$, it decomposes into
irreducible representations of the orbifold group $\mathsf{\Gamma}$ as

$\displaystyle{\bm{r}}=\bigoplus_{s\in{\widehat{\mathsf{\Gamma}}}}\,{\bm{r}}_{s}\otimes\rho_{s}^{*}\
,$ (3.56)

with multiplicity spaces
${\bm{r}}_{s}=\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(\rho_{s},{\bm{r}})$
and

$\displaystyle
r=\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\,r_{s}=\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\,\dim_{\mathbbm{C}}{\bm{r}}_{s}\
.$ (3.57)

We use Schur’s lemma to decompose the Fermi multiplet in
$\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}({\bm{r}},V)$ into isotopical
components

$\displaystyle\bar{I}^{s}:{\bm{r}}_{s}\longrightarrow V_{s}\ .$ (3.58)

Then the BRST transformations are given by

$\begin{split}\mathcal{Q}_{\mathsf{\Gamma}}\bar{I}^{s}=\bar{\varrho}^{s}\qquad\mbox{and}\qquad\mathcal{Q}_{\mathsf{\Gamma}}\bar{\varrho}^{s}=\phi^{s}\,\bar{I}^{s}-\bar{I}^{s}\,{\bm{m}}^{s}\
,\end{split}$ (3.59)

where the vector ${\bm{m}}^{s}$ collects the $r_{s}$ masses $m_{l}$ associated
with the irreducible representation $\rho_{s}$.

Following the same steps as in Section 2.5 we can now write down the instanton
partition function. We introduce a set of fugacities
$\vec{\mathtt{q}}=(\mathtt{q}_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}$ for the
corresponding fractional instanton sectors
$\vec{k}=(k_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}\in\mathbbm{Z}_{\geq
0}^{|\mathsf{\Gamma}|}$ and define

$\displaystyle
Z_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}^{\vec{r}}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},\vec{m})=\sum_{\vec{k}\in\mathbbm{Z}_{\geq
0}^{|\mathsf{\Gamma}|}}\,\vec{\mathtt{q}}^{\,\vec{k}}\
Z_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}^{\vec{r},\vec{k}}(\vec{a},\vec{\epsilon},\vec{m})\
,$ (3.60)

where

$\displaystyle\vec{\mathtt{q}}^{\,\vec{k}}:=\prod_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathtt{q}_{s}^{k_{s}}$
(3.61)

and the quiver matrix model is defined by the integral

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}^{\vec{r},\vec{k}}(\vec{a},\vec{\epsilon},\vec{m})=\oint_{\varGamma_{\vec{r},\vec{k}}}\
\prod_{s\in{\widehat{\mathsf{\Gamma}}}}\,\frac{1}{k_{s}!}\
&\prod_{i=1}^{k_{s}}\,\frac{\mathrm{d}\phi_{i}^{s}}{2\pi\,\mathrm{i}}\,\frac{\mathcal{P}_{r_{s}}(\phi_{i}^{s}|\vec{m}^{s})}{\mathcal{P}_{r_{s}}(\phi_{i}^{s}|\vec{a}^{s})}\
\prod_{\stackrel{{\scriptstyle\scriptstyle i,j=1}}{{\scriptstyle i\neq
j}}}^{k_{s}}\,\big{(}\phi^{s}_{i}-\phi^{s}_{j}\big{)}\\\ &\hskip
85.35826pt\times\
\prod_{i,j=1}^{k_{s}}\,\frac{\displaystyle\prod_{(\alpha,\beta)\in\underline{\sf
3}^{\perp}}\,\big{(}\phi_{i}^{s+s_{\alpha\beta}}-\phi_{j}^{s}-\epsilon_{\alpha\beta}\big{)}}{\displaystyle\prod_{a\in\underline{\sf
4}}\,\big{(}\phi_{i}^{s+s_{a}}-\phi_{j}^{s}-\epsilon_{a}\big{)}}\
,\end{split}$ (3.62)

which again we make sense of via a suitable contour integral prescription.

#### Dimensional Reduction

From the matrix model representation (3.62) we immediately obtain the orbifold
version of Proposition 2.71, which reads

######  Proposition 3.63.

The equivariant instanton partition function on a $(3,0)$ orbifold
$\mathbbm{C}^{3}/\mathsf{\Gamma}\times\mathbbm{C}$ is related to the partition
function
$Z_{[\mathbbm{C}^{3}/\mathsf{\Gamma}]}^{\vec{r}}(\vec{\mathtt{q}};\vec{a},\epsilon_{1},\epsilon_{2},\epsilon_{3})$
for noncommutative Donaldson–Thomas invariants of type $\vec{r}$ for the toric
Kähler orbifold $\mathbbm{C}^{3}/\mathsf{\Gamma}$ through the mass
specialisation

$\displaystyle
Z_{[\mathbbm{C}^{3}/\mathsf{\Gamma}]\times\mathbbm{C}}^{\vec{r}}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},m_{l}^{s}=a_{l}^{s}+\epsilon_{4})=Z_{[\mathbbm{C}^{3}/\mathsf{\Gamma}]}^{\vec{r}}(\vec{\mathtt{q}}\,^{\prime};\vec{a},\epsilon_{1},\epsilon_{2},\epsilon_{3})\
,$ (3.64)

where
$\vec{\mathtt{q}}\,^{\prime}=\big{(}(-1)^{r_{s}+1}\,\mathtt{q}_{s}\big{)}_{s\in{\widehat{\mathsf{\Gamma}}}}$.

###### Proof.

Since any $(3,0)$ orbifold has weight $s_{4}=0$, and $s_{123}=0$ since
$\mathsf{\Gamma}\subset\mathsf{SL}(3,\mathbbm{C})$, using the Calabi–Yau
condition $\epsilon_{4}=-\epsilon_{123}$ on
$\mathbbm{C}^{3}/\mathsf{\Gamma}\times\mathbbm{C}$ one finds that the matrix
integral (3.62) gives

$\displaystyle\begin{split}&Z_{[\mathbbm{C}^{3}/\mathsf{\Gamma}]\times\mathbbm{C}}^{\vec{r},\vec{k}}(\vec{a},\vec{\epsilon},m_{l}^{s}=a_{l}^{s}-\epsilon_{123})\\\\[4.0pt]
&\quad=\oint_{\varGamma_{\vec{r},\vec{k}}}\
\prod_{s\in{\widehat{\mathsf{\Gamma}}}}\,\frac{1}{k_{s}!}\
\prod_{i=1}^{k_{s}}\,\frac{\mathrm{d}\phi_{i}^{s}}{2\pi\,\mathrm{i}}\,\frac{\mathcal{P}_{r_{s}}(\phi_{i}^{s}+\epsilon_{123}|\vec{a}^{s})}{\mathcal{P}_{r_{s}}(\phi_{i}^{s}|\vec{a}^{s})}\
\prod_{\stackrel{{\scriptstyle\scriptstyle i,j=1}}{{\scriptstyle i\neq
j}}}^{k_{s}}\,\big{(}\phi^{s}_{i}-\phi^{s}_{j}\big{)}\\\ &\hskip
48.36958pt\times\
\prod_{i,j=1}^{k_{s}}\,\frac{\displaystyle\big{(}\phi_{i}^{s+s_{12}}-\phi_{j}^{s}-\epsilon_{12}\big{)}\,\big{(}\phi_{i}^{s+s_{13}}-\phi_{j}^{s}-\epsilon_{13}\big{)}\,\big{(}\phi_{i}^{s+s_{23}}-\phi_{j}^{s}-\epsilon_{23}\big{)}}{\displaystyle\big{(}\phi_{i}^{s}-\phi_{j}^{s}-\epsilon_{123}\big{)}\,\big{(}\phi_{i}^{s+s_{1}}-\phi_{j}^{s}-\epsilon_{1}\big{)}\,\big{(}\phi_{i}^{s+s_{2}}-\phi_{j}^{s}-\epsilon_{2}\big{)}\,\big{(}\phi_{i}^{s+s_{3}}-\phi_{j}^{s}-\epsilon_{3}\big{)}}\
.\end{split}$ (3.65)

Up to an overall sign
$\prod_{s\in{\widehat{\mathsf{\Gamma}}}}\,(-1)^{(r_{s}+1)\,k_{s}}$, this is
the same as the straightforward generalization of the matrix integral
representation from [6, eq. (5.23)] for the instanton partition function on
$\mathbbm{C}^{3}/\mathsf{\Gamma}$ using the orbifold quiver matrix model from
[7, Section 5.5], slightly modified to include a generic point of the
$\varOmega$-deformation on $\mathbbm{C}^{3}$ with $\epsilon_{123}\neq 0$. ∎

######  Remark 3.66.

The noncommutative crepant resolution of $\mathbbm{C}^{3}/\mathsf{\Gamma}$ in
this dimensional reduction is obtained by erasing a single loop from each node
of the McKay quiver for the quotient singularity
$\mathbbm{C}^{3}/\mathsf{\Gamma}\times\mathbbm{C}$.

#### Fixed Points and Coloured Solid Partitions

The integrand of (3.62) has poles along the hyperplanes

$\displaystyle\phi^{s+s_{a}}_{i}-\phi^{s}_{j}-\epsilon_{a}=0\qquad\mbox{and}\qquad\phi^{s}_{i}-a^{s}_{l}=0$
(3.67)

in $\mathbbm{R}^{k}$, for $a\in\underline{\sf 4}$ and
$s\in\widehat{\mathsf{\Gamma}}$. In completely analogy with the matrix model
of Section 2.5, these are the fixed points of the orbifold ADHM data
$(B_{a}^{s},I^{s})_{a\in\underline{\sf 4},\,s\in{\widehat{\mathsf{\Gamma}}}}$
under the equivariant action of the symmetry group
$\big{(}\operatorname*{\scalerel*{\times}{\textstyle\sum}}_{s\in{\widehat{\mathsf{\Gamma}}}}\,\mathsf{U}(k_{s})\times\mathsf{U}(r_{s})_{\rm
col}\big{)}\times\mathsf{SU}(4)$. They reside on the locus of fixed points of
the BRST charge $\mathcal{Q}_{\mathsf{\Gamma}}$ of the cohomological gauge
theory on $[\mathbbm{C}^{4}/\mathsf{\Gamma}]$.

Since the actions of $\mathsf{\Gamma}$ and $\mathsf{T}$ commute, we can argue
as in Section 2.5 that these are parametrized by arrays of solid partitions
$\vec{\sigma}=(\sigma_{1},\dots,\sigma_{r})$, where the splitting of the ADHM
data into irreducible representations of $\mathsf{\Gamma}$ induces a
${\widehat{\mathsf{\Gamma}}}$-colouring of the solid partitions according to
(3.3). A coloured solid partition $\sigma_{l}$ is in one-to-one correspondence
with the fixed points

$\displaystyle\phi^{s}_{(a_{l}^{s(l)},\vec{p}\,)}=a^{s(l)}_{l}+\vec{p}\cdot\vec{\epsilon}\
,$ (3.68)

for $\vec{p}=(p_{1},p_{2},p_{3},p_{4})\in\mathbbm{Z}_{>0}^{4}$, which carry an
irreducible representation $\rho_{s}$ of the orbifold group $\mathsf{\Gamma}$
given by

$\displaystyle\rho_{s}=\rho_{s(l)}\otimes\rho_{\vec{p}}\qquad\mbox{with}\quad\rho_{\vec{p}}:=\rho_{s_{1}}^{\otimes
p_{1}}\otimes\rho_{s_{2}}^{\otimes p_{2}}\otimes\rho_{s_{3}}^{\otimes
p_{3}}\otimes\rho_{s_{4}}^{\otimes p_{4}}\ .$ (3.69)

The ${\widehat{\mathsf{\Gamma}}}$-colouring of the array of solid partitions
defines the total number of boxes of colour $\rho_{s}$ in $\vec{\sigma}$ for
each $s\in{\widehat{\mathsf{\Gamma}}}$ as the fractional instanton number
$|\vec{\sigma}|_{s}=k_{s}$; we write this condition as
$|\vec{\sigma}|_{{\widehat{\mathsf{\Gamma}}}}=\vec{k}$.

###  3.5 Orbifold Instanton Partition Functions

The partition function of the cohomological theory on
$[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ can be computed explicitly by considering
the $\mathsf{\Gamma}$-equivariant version of the instanton deformation complex
(2.89), which reads

$\displaystyle
0\longrightarrow\mathsf{End}_{\mathbbm{C}[\mathsf{\Gamma}]}(V_{\vec{\sigma}})\xrightarrow{\
\mathrm{d}_{1}^{\mathsf{\Gamma}}\
}\begin{matrix}\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(V_{\vec{\sigma}},V_{\vec{\sigma}}\otimes
Q)\\\\[4.30554pt] \oplus\\\\[4.30554pt]
\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(W_{\vec{\sigma}},V_{\vec{\sigma}})\end{matrix}\xrightarrow{\
\mathrm{d}_{2}^{\mathsf{\Gamma}}\
}\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(V_{\vec{\sigma}},V_{\vec{\sigma}}\otimes\text{\Large$\wedge$}_{-}^{0,2}\,Q)\longrightarrow
0\ ,$ (3.70)

where the map $\mathrm{d}_{1}^{\mathsf{\Gamma}}$ is an infinitesimal
$\mathsf{G}_{\vec{k}}$ gauge transformation, while
$\mathrm{d}_{2}^{\mathsf{\Gamma}}$ is the linearization of the holomorphic
ADHM equations
$B_{\alpha}^{s+s_{\beta}}\,B_{\beta}^{s}=B_{\beta}^{s+s_{\alpha}}\,B_{\alpha}^{s}$
for $(\alpha,\beta)\in\underline{\sf 3}^{\perp}$.

Since the actions of the groups $\mathsf{\Gamma}$ and $\mathsf{T}$ commute,
the character we wish to calculate is now the $\mathsf{\Gamma}$-invariant part
of the character of the complex (2.89), that is

$\displaystyle\begin{split}\chi^{\mathsf{\Gamma}}_{\vec{\sigma}}:\\!&=\sqrt{\mathrm{ch}^{\mathsf{\Gamma}}_{\mathsf{T}}}\big{(}T^{\rm
vir}_{\vec{\sigma}}\mathfrak{M}_{\vec{r},\vec{k}}\big{)}-\mathrm{ch}^{\mathsf{\Gamma}}_{\mathsf{T}}\big{(}(\mathscr{V}_{r,k})_{\vec{\sigma}}\otimes{\bm{r}}\big{)}=\sqrt{\mathrm{ch}^{\mathsf{\Gamma}}_{\mathsf{T}}}\big{(}T^{\rm
vir}_{\vec{\sigma}}\mathfrak{M}_{\vec{r},\vec{k}}\big{)}-\big{[}{\bm{r}}^{*}\otimes
V_{\vec{\sigma}}\big{]}^{\mathsf{\Gamma}}\ ,\end{split}$ (3.71)

with

$\displaystyle\begin{split}\sqrt{\mathrm{ch}^{\mathsf{\Gamma}}_{\mathsf{T}}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{\vec{r},\vec{k}}\big{)}&=\big{[}V_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}\,\big{(}t^{-1}_{1}+t^{-1}_{2}+t^{-1}_{3}+t^{-1}_{4}\big{)}+W_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}\\\ &\qquad\,-V_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}\,\big{(}1+t^{-1}_{1}\,t^{-1}_{2}+t^{-1}_{1}\,t^{-1}_{3}+t^{-1}_{2}\,t^{-1}_{3}\big{)}\big{]}^{\mathsf{\Gamma}}\
.\end{split}$ (3.72)

Since the dual involution commutes with the $\mathsf{\Gamma}$-action, by
taking the $\mathsf{\Gamma}$-invariant part of (2.96) it follows that

$\displaystyle\mathrm{ch}^{\mathsf{\Gamma}}_{\mathsf{T}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{\vec{r},\vec{k}}\big{)}=\sqrt{\mathrm{ch}^{\mathsf{\Gamma}}_{\mathsf{T}}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{\vec{r},\vec{k}}\big{)}+\sqrt{\mathrm{ch}^{\mathsf{\Gamma}}_{\mathsf{T}}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{\vec{r},\vec{k}}\big{)}^{*}\ .$ (3.73)

The subgroup inclusion
$\mathsf{\Gamma}\lhook\joinrel\longrightarrow\mathsf{T}_{\vec{\epsilon}}$
defines the irreducible representations of $\mathsf{\Gamma}$ associated to the
toric generators $t_{a}$ for $a\in\underline{\sf 4}$. Consequently the vector
spaces $V$ and $W$ at a fixed point $\vec{\sigma}$ decompose into

$\displaystyle\begin{split}V_{\vec{\sigma}}=\sum_{l=1}^{r}\,e_{l}\
\sum_{\vec{p}\,\in{\sigma}_{l}}\,t_{1}^{p_{1}}\,t_{2}^{p_{2}}\,t_{3}^{p_{3}}\,t_{4}^{p_{4}}\otimes\rho^{*}_{\vec{p}}\otimes\rho^{*}_{s(l)}\qquad\mbox{and}\qquad
W_{\vec{\sigma}}=\sum_{l=1}^{r}\,e_{l}\otimes\rho^{*}_{s(l)}\ ,\end{split}$
(3.74)

and the $\mathsf{U}(r)_{\rm fla}$-module ${\bm{r}}$ into

$\displaystyle{\bm{r}}=\sum_{l=1}^{r}\,f_{l}\otimes\rho^{*}_{s(l)}\ ,$ (3.75)

as elements of the representation ring of the group
$\mathsf{T}\times\mathsf{\Gamma}$. The index (3.71) is calculated by
projecting the character (2.95) onto those terms which carry the trivial
representation $\rho_{0}$, giving an element in the representation ring of
$\mathsf{T}$.

Then the orbifold instanton partition function is evaluated by using the top
form operation (2.99) to get the combinatorial formula

$\displaystyle
Z^{\vec{r}}_{[\mathbb{C}^{4}/\mathsf{\Gamma}]}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},\vec{m})=\sum_{\vec{k}\in\mathbbm{Z}_{\geq
0}^{|\mathsf{\Gamma}|}}\,\vec{\mathtt{q}}^{\,\vec{k}}\
\sum_{|\vec{\sigma}|_{\widehat{\mathsf{\Gamma}}}=\vec{k}}\,(-1)^{{\tt
O}^{\mathsf{\Gamma}}_{\vec{\sigma}}}\ \widehat{\tt
e}\big{[}-\chi^{\mathsf{\Gamma}}_{\vec{\sigma}}\,\big{]}\ ,$ (3.76)

where

$\displaystyle\begin{split}\widehat{\tt
e}\big{[}-\chi_{\vec{\sigma}}^{\mathsf{\Gamma}}\,\big{]}=\prod_{l=1}^{r}\
\prod_{\vec{p}_{l}\in{\sigma}_{l}}^{\neq
0}\,\frac{\mathcal{P}_{r}\circ\delta^{\mathsf{\Gamma}}_{0}(a_{l}+\vec{p}_{l}\cdot\vec{\epsilon}\,|\vec{m})}{\mathcal{P}_{r}\circ\delta^{\mathsf{\Gamma}}_{0}(a_{l}+\vec{p}_{l}\cdot\vec{\epsilon}\,|\vec{a})}\
\prod_{l^{\prime}=1}^{r}\
\prod_{\vec{p}^{\,\prime}_{l^{\prime}}\in{\sigma}_{l^{\prime}}}^{\neq
0}\,\mathcal{R}\circ\delta^{\mathsf{\Gamma}}_{0}\big{(}a_{l}-a_{l^{\prime}}+(\vec{p}_{l}-\vec{p}^{\,\prime}_{l^{\prime}})\cdot\vec{\epsilon}\,\big{|}\vec{\epsilon}\,\big{)}\
.\end{split}$ (3.77)

Here the operation $\delta_{0}^{\mathsf{\Gamma}}$ acts on a combination of
equivariant parameters $x$ as the identity if $x$ is associated to the trivial
representation $\rho_{0}$ and returns $1$ otherwise; for example

$\displaystyle\delta_{0}^{\mathsf{\Gamma}}\big{(}a_{l}-a_{l^{\prime}}+(\vec{p}_{l}-\vec{p}^{\,\prime}_{l^{\prime}})\cdot\vec{\epsilon}\,\big{)}=\begin{cases}a_{l}-a_{l^{\prime}}+(\vec{p}_{l}-\vec{p}^{\,\prime}_{l^{\prime}})\cdot\vec{\epsilon}\quad\text{if}\
\
\rho_{\vec{p}}\otimes\rho_{s(l)}\otimes\rho^{*}_{\vec{p}^{\,\prime}}\otimes\rho_{s(l^{\prime})}^{*}\simeq\rho_{0}\
,\\\\[4.0pt] 1\quad\text{otherwise}\ .\end{cases}$ (3.78)

As previously, we understand (3.77) as a residue contribution to the matrix
integral (3.62), regarded as a contour integral over
$\varGamma_{\vec{r},\vec{k}}\subset\mathbbm{C}^{k}$ where $k=|\vec{k}\,|$ is
the size of $\vec{k}\in\mathbbm{Z}_{\geq 0}^{|\mathsf{\Gamma}|}$.

######  Remark 3.79.

Since the character (3.72) is obtained by projection to the
$\mathsf{\Gamma}$-invariant part of the character (2.94), we believe that the
sign factors $\mathtt{O}_{\vec{\sigma}}^{\mathsf{\Gamma}}$ in (3.76), which
should come from a careful residue calculation of (3.62), do not depend on the
$\mathsf{\Gamma}$-colourings of the solid partitions and coincide with
(2.101). This same assertion is made in [27].

####  ${\bm{\mathsf{U}(1)}}$ Gauge Theories

Let us consider the rank one case $r=1$. Then there are only single
equivariant parameters $a^{s}$ and $m^{s}$ for the gauge and flavour symmetry,
which are both associated to the same irreducible representation $\rho_{s}$ of
$\mathsf{\Gamma}$. Since $\rho_{s}\otimes\rho_{s}^{*}\simeq\rho_{0}$, the
partition function (3.76) does not depend on the choice of framing vector
$\vec{r}=(r_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}$, where we use the
convention that $r_{0}$ is the first entry of $\vec{r}$. In other words, all
framing vectors of the form $\vec{r}=(0,\dots,0,1,0,\dots,0)$, with zeroes in
all but one entry, give the same partition function. Moreover, the partition
function (3.76) depends only on the combination $m:=m^{s}-a^{s}$ and hence is
effectively independent of the Coulomb parameter. The rank one partition
functions are therefore simply denoted as

$\displaystyle
Z_{[\mathbb{C}^{4}/\mathsf{\Gamma}]}(\vec{\mathtt{q}};\vec{\epsilon},m)=\sum_{\sigma}\,(-1)^{{\tt
O}^{\mathsf{\Gamma}}_{{\sigma}}}\
\vec{\mathtt{q}}^{\,|{\sigma}|_{\widehat{\mathsf{\Gamma}}}}\
\prod_{\vec{p}\,\in{\sigma}}^{\neq
0}\,\frac{\delta^{\mathsf{\Gamma}}_{0}(\vec{p}\cdot\vec{\epsilon}-m)}{\delta^{\mathsf{\Gamma}}_{0}(\vec{p}\cdot\vec{\epsilon}\,)}\
\prod_{\vec{p}^{\,\prime}\in{\sigma}}^{\neq
0}\,\mathcal{R}\circ\delta^{\mathsf{\Gamma}}_{0}\big{(}(\vec{p}-\vec{p}^{\,\prime})\cdot\vec{\epsilon}\,\big{|}\vec{\epsilon}\,\big{)}\
.$ (3.80)

######  Example 3.81.

Let $\mathsf{\Gamma}=\mathbbm{Z}_{3}$ act on $\mathbbm{C}^{4}$ with generator

$\displaystyle(z_{1},z_{2},z_{3},z_{4})\longmapsto(\xi\,z_{1},\xi\,z_{2},\xi\,z_{3},z_{4})\
,$ (3.82)

where $\xi=\mathrm{e}^{\,2\pi\,\mathrm{i}/3}$ is a primitive third root of
unity. This defines a $(3,0)$ orbifold
$\mathbbm{C}^{3}/\mathbbm{Z}_{3}\times\mathbbm{C}$, whose natural geometric
crepant resolution induced by the $\mathbbm{Z}_{3}$-Hilbert scheme ${\sf
Hilb}^{\mathbbm{Z}_{3}}(\mathbbm{C}^{3})$ is a fibration of local del Pezzo
surfaces of degree zero over the affine line, or equivalently the total space
of the rank two bundle
$\mathcal{O}_{{\mathbbm{P}}^{2}}(-3)\oplus\mathcal{O}_{{\mathbbm{P}}^{2}}$.
Its McKay quiver is

${0}$${1}$${2}$ (3.83)

which is obtained from the Beilinson quiver $\sf B$ by adding a loop at each
node. The fractional instanton contributions to the corresponding
$\mathsf{U}(1)$ partition function are given by

$\displaystyle\begin{split}&Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{3}]\times\mathbbm{C}}^{\vec{k}}(\vec{\epsilon},m)\\\\[4.0pt]
&\quad=\sum_{|{\sigma}|_{\mathbbm{Z}_{3}}=\vec{k}}\,(-1)^{{\tt
O}^{\mathbbm{Z}_{3}}_{{\sigma}}}\
\prod_{\begin{subarray}{c}\vec{p}\,\in\sigma\\\
p_{1}+p_{2}+p_{3}\,\equiv_{3}\,0\end{subarray}}^{\neq
0}\,\frac{\vec{p}\cdot\vec{\epsilon}-m}{\vec{p}\cdot\vec{\epsilon}}\
\prod_{\begin{subarray}{c}\vec{p},\vec{p}\,^{\prime}\in\sigma\\\
p_{1}+p_{2}+p_{3}\,\equiv_{3}\,p_{1}^{\prime}+p_{2}^{\prime}+p_{3}^{\prime}\end{subarray}}^{\neq
0}\,\frac{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}}{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{4}}\\\
&\hskip
28.45274pt\times\prod_{\begin{subarray}{c}\vec{p},\vec{p}\,^{\prime}\in\sigma\\\
p_{1}+p_{2}+p_{3}+1\,\equiv_{3}\,p_{1}^{\prime}+p_{2}^{\prime}+p_{3}^{\prime}\end{subarray}}^{\neq
0}\,\frac{\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{12}\big{)}\,\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{13}\big{)}\,\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{23}\big{)}}{\big{(}(\vec{p}\,^{\prime}-\vec{p}\,)\cdot\vec{\epsilon}-\epsilon_{1}\big{)}\,\big{(}(\vec{p}\,^{\prime}-\vec{p}\,)\cdot\vec{\epsilon}-\epsilon_{2}\big{)}\,\big{(}(\vec{p}\,^{\prime}-\vec{p}\,)\cdot\vec{\epsilon}-\epsilon_{3}\big{)}}\
,\end{split}$ (3.84)

where $\equiv_{3}$ denotes congruence modulo $3$.

#### Higher Rank Gauge Theories

In contrast to the rank one case, the partition function for higher rank $r>1$
depends explicitly on the choice of decomposition
$\vec{r}=(r_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}$ of the rank $r$ according
to the irreducible representations of $\mathsf{\Gamma}$, and different framing
$\mathsf{\Gamma}$-modules $W\simeq\mathbbm{C}^{r}$ generally lead to
inequivalent theories.

######  Example 3.85.

Consider the $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$-action of Example 3.44.
Three $\vec{k}=(2,0,0,0)$ contributions to rank two partition functions
$Z_{[\mathbbm{C}^{4}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]}^{\vec{r}}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},\vec{m})$
are

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{4}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]}^{\vec{r}=(2,0,0,0),\vec{k}=(2,0,0,0)}(\vec{a},\vec{\epsilon},\vec{m})&=\frac{\epsilon_{12}^{2}\,m_{1}\,m_{2}}{(a_{1}-a_{2})^{2}}\,(a_{1}-a_{2}+m_{1})\,(a_{1}-a_{2}-m_{2})\
,\\\\[4.0pt]
Z_{[\mathbbm{C}^{4}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]}^{\vec{r}=(1,1,0,0),\vec{k}=(2,0,0,0)}(\vec{a},\vec{\epsilon},\vec{m})&=\epsilon_{12}^{2}\,m_{1}\,m_{2}=Z_{[\mathbbm{C}^{4}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]}^{\vec{r}=(1,0,1,0),\vec{k}=(2,0,00)}(\vec{a},\vec{\epsilon},\vec{m})\
.\end{split}$ (3.86)

However, some higher rank theories are equivalent. Looking at the contribution
(3.77) from the index $\chi_{\vec{\sigma}}^{\mathsf{\Gamma}}$, we see that all
framing vectors $\vec{r}$ of the form $(0,\dots,0,r,0,\dots,0)$ yield
equivalent partition functions, irrespecitve of the location of the entry $r$.
Indeed, in these cases the equivariant parameters $\vec{a}$ and $\vec{m}$ are
associated the same irreducible $\mathsf{\Gamma}$-representation. Thus the
differences $a_{l}-a_{l^{\prime}}$ and $a_{l}-m_{l^{\prime}}$ are associated
to the trivial representation $\rho_{0}$ and one always counts the same
contributions
$Z^{\vec{r},\vec{k}}_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}(\vec{a},\vec{\epsilon},\vec{m})$.

######  Example 3.87.

Consider $\mathsf{\Gamma}=\mathbbm{Z}_{4}$ acting on $\mathbbm{C}^{4}$ with
generator

$\displaystyle(z_{1},z_{2},z_{3},z_{4})\longmapsto(\mathrm{i}\,z_{1},\mathrm{i}\,z_{2},\mathrm{i}\,z_{3},\mathrm{i}\,z_{4})\
.$ (3.88)

The quotient singularity $\mathbbm{C}^{4}/\mathbbm{Z}_{4}$ admits a geometric
crepant resolution by [67, Claim 2] and [84, Proposition 3.1]. Its McKay
quiver is

${0}$${1}$${3}$${2}$ (3.89)

and the noncommutative crepant resolution can be associated with the total
space of the canonical bundle $\mathcal{O}_{{\mathbbm{P}}^{3}}(-4)$ [21]. The
leading contributions for $\vec{r}=(2,0,0,0)$ and $\vec{r}=(0,2,0,0)$ to the
rank two partition functions
$Z_{[\mathbbm{C}^{4}/\mathbbm{Z}_{4}]}^{\vec{r}}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},\vec{m})$
coincide and are given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{4}/\mathbbm{Z}_{4}]}^{\vec{r}=(2,0,0,0)}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},\vec{m})&=Z_{[\mathbbm{C}^{4}/\mathbbm{Z}_{4}]}^{\vec{r}=(0,2,0,0)}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},\vec{m})\\\\[4.0pt]
&=-(m_{1}+m_{2})\,\mathtt{q}_{0}\\\ &\hskip
28.45274pt\quad\,+\frac{\epsilon_{12}^{2}\,m_{1}\,m_{2}}{(a_{1}-a_{2})^{2}}\,(a_{1}-a_{2}+m_{1})\,(a_{1}-a_{2}-m_{2})\,\mathtt{q}_{0}^{2}+\cdots\
.\end{split}$ (3.90)

From (3.62) it follows that
$Z^{\vec{r},\vec{k}}_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}(\vec{a},\vec{\epsilon},\vec{m})$
is invariant under the action of the Weyl group
$\operatorname*{\scalerel*{\times}{\textstyle\sum}}_{s\in{\widehat{\mathsf{\Gamma}}}}\,S_{r_{s}}$
of the colour and flavour symmetries by permuting the entries of the
parameters $\vec{a}$ and $\vec{m}$. It is natural to ask if the permutation
symmetry of the type $\vec{r}=(0,\dots,0,r,0,\dots,0)$ instanton partition
functions observed above persists in more generality. That is, whether the
theories with rank vectors $\vec{r}=(r_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}$
and
${}^{\varsigma}\vec{r}:=(r_{\varsigma(s)})_{s\in{\widehat{\mathsf{\Gamma}}}}$
are equivalent, for a permutation $\varsigma\in S_{|\mathsf{\Gamma}|}$ of
degree $|\mathsf{\Gamma}|$. Despite the appearance of the symmetric case in
Example 3.85, it is easy to see that this is not true in general.

######  Example 3.91.

For the $\mathbbm{Z}_{4}$-action of Example 3.34, consider the rank two
partition functions with framing vectors $\vec{r}=(1,1,0,0)$ and
$\vec{r}=(1,0,1,0)$ for $\vec{k}=(1,1,0,0)$. In this case there are only two
pairs of $\mathbbm{Z}_{4}$-coloured solid partitions that contribute to (3.76)
which are given by

$\displaystyle\vec{\sigma}_{1}=(\sigma,\emptyset)\qquad\mbox{and}\qquad\vec{\sigma}_{2}=(\emptyset,\sigma)\
,$ (3.92)

where

$\displaystyle\sigma=\big{\\{}(1,1,1,1)\,,\,(2,1,1,1)\big{\\}}\ .$ (3.93)

The corresponding charge two contributions are given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{4}]\times\mathbbm{C}^{2}}^{\vec{r}=(1,1,0,0),\vec{k}=(1,1,0,0)}(\vec{a},\vec{\epsilon},\vec{m})&=-\frac{\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\,\Big{(}m_{1}+m_{2}-\frac{m_{1}\,m_{2}}{a_{1}-a_{2}+\epsilon_{1}}\Big{)}\
,\\\\[4.0pt]
Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{4}]\times\mathbbm{C}^{2}}^{\vec{r}=(1,0,1,0),\vec{k}=(1,1,0,0)}(\vec{a},\vec{\epsilon},\vec{m})&=-\frac{\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\,(m_{1}+m_{2})\
.\end{split}$ (3.94)

###  3.6 Pure $\mathcal{N}_{\textrm{\tiny T}}=2$ Gauge Theory on
$[\mathbbm{C}^{4}/\mathsf{\Gamma}]$

By dropping the matter bundle contribution to the matrix integral (3.62), we
obtain the quiver matrix model for the pure gauge theory on
$[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ given by

$\displaystyle\begin{split}&Z_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}^{\vec{r},\vec{k}}(\vec{a},\vec{\epsilon}\,)^{\rm
pure}=\oint_{\varGamma_{\vec{r},\vec{k}}}\
\prod_{s\in{\widehat{\mathsf{\Gamma}}}}\,\frac{1}{k_{s}!}\
\prod_{i=1}^{k_{s}}\,\frac{\mathrm{d}\phi_{i}^{s}}{2\pi\,\mathrm{i}}\
\frac{\displaystyle\prod_{\stackrel{{\scriptstyle\scriptstyle
j=1}}{{\scriptstyle j\neq
i}}}^{k_{s}}\,\big{(}\phi^{s}_{i}-\phi^{s}_{j}\big{)}}{\mathcal{P}_{r_{s}}(\phi_{i}^{s}|\vec{a}^{s})}\\\
&\hskip
227.62204pt\times\prod_{i,j=1}^{k_{s}}\,\frac{\displaystyle\prod_{(\alpha,\beta)\in\underline{\sf
3}^{\perp}}\,\big{(}\phi_{i}^{s+s_{\alpha\beta}}-\phi_{j}^{s}-\epsilon_{\alpha\beta}\big{)}}{\displaystyle\prod_{a\in\underline{\sf
4}}\,\big{(}\phi_{i}^{s+s_{a}}-\phi_{j}^{s}-\epsilon_{a}\big{)}}\
.\end{split}$ (3.95)

The contour integral (LABEL:Orb_Zpure) can be evaluated by using the index
(3.72) to get

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}^{\vec{r},\vec{k}}(\vec{a},\vec{\epsilon}\,)^{\rm
pure}&=\sum_{|\vec{\sigma}|_{\widehat{\mathsf{\Gamma}}}=\vec{k}}\,(-1)^{{\tt
O}^{\mathsf{\Gamma}}_{\vec{\sigma}}}\ \prod_{l=1}^{r}\
\prod_{\vec{p}_{l}\in{\sigma}_{l}}^{\neq
0}\,\frac{1}{\mathcal{P}_{r}\circ\delta^{\mathsf{\Gamma}}_{0}(a_{l}+\vec{p}_{l}\cdot\vec{\epsilon}\,|\vec{a})}\\\
&\hskip 113.81102pt\times\prod_{l^{\prime}=1}^{r}\
\prod_{\vec{p}^{\,\prime}_{l^{\prime}}\in{\sigma}_{l^{\prime}}}^{\neq
0}\,\mathcal{R}\circ\delta^{\mathsf{\Gamma}}_{0}\big{(}a_{l}-a_{l^{\prime}}+(\vec{p}_{l}-\vec{p}^{\,\prime}_{l^{\prime}})\cdot\vec{\epsilon}\,\big{|}\vec{\epsilon}\,\big{)}\
.\end{split}$ (3.96)

The corresponding orbifold instanton partition function is

$\displaystyle
Z_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}^{\vec{r}}(\vec{\Lambda};\vec{a},\vec{\epsilon}\,)^{\rm
pure}=\sum_{\vec{k}\in\mathbbm{Z}_{\geq
0}^{|\mathsf{\Gamma}|}}\,\vec{\Lambda}^{\,\vec{k}}\
Z_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}^{\vec{r},\vec{k}}(\vec{a},\vec{\epsilon}\,)^{\rm
pure}\ ,$ (3.97)

where $\vec{\Lambda}=(\Lambda_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}$ and

$\displaystyle\vec{\Lambda}^{\,\vec{k}}=\prod_{s\in{\widehat{\mathsf{\Gamma}}}}\,\Lambda_{s}^{k_{s}}\
.$ (3.98)

Similarly to the pure gauge theory on flat space $\mathbbm{C}^{4}$ from
Section 2.7, the pure gauge theory on the quotient stack
$[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ can be computed as an appropriate limit of
the cohomological gauge theory on $[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ with
massive matter. The orbifold version of Proposition 2.112 reads

######  Proposition 3.99.

The instanton partition function for the pure cohomological gauge theory is
related to the partition function with a massive fundamental hypermultiplet on
$[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ through the double scaling limit

$\displaystyle
Z^{\vec{r}}_{[\mathbbm{C}^{4}/\mathsf{\Gamma}]}(\vec{\Lambda};\vec{a},\vec{\epsilon}\,)^{\rm
pure}=\lim_{m_{1},\dots,m_{r}\to\infty}\ \lim_{\mathtt{q}_{0}\to
0}\,Z^{\vec{r}}_{[\mathbb{C}^{4}/\mathsf{\Gamma}]}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},\vec{m})\,\Big{|}_{\Lambda_{0}=(-1)^{r}\,m_{1}\cdots
m_{r}\,\mathtt{q}_{0}}\ ,$ (3.100)

with $\Lambda_{s}=\mathtt{q}_{s}$ for $s\neq 0$.

###### Proof.

This follows immediately from (3.77) and (3.96). ∎

##  4 The (2,0) Orbifolds
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$

Proposition 3.63 suggests that the equivariant partition functions on $(2,0)$
orbifolds $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$ can be
expressed as a generalization of equivariant partition functions on the toric
three-orbifolds $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}$, when the
latter are known explicitly, similarly to the uplifting from Proposition 2.71
to Conjecture 2.105. In this section we will propose uplifts of the rank one
instanton partition functions, building on the known generating function for
$\mathbbm{Z}_{n}$-coloured plane partitions from [35] and its extension to the
toric three-orbifolds $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}$ with
$\mathsf{U}(3)$ holonomy from [36]. We shall also propose conjectural closed
formulas for the higher rank instanton partition functions, as well as
consider the corresponding pure gauge theories.

###  4.1 Noncommutative $\mathsf{U}(1)$ Instantons on
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$

Generalizing Example 3.34, consider noncommutive $\mathsf{U}(1)$ instantons on
the toric Calabi–Yau four-orbifold
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$. The
$\mathbbm{Z}_{n}$-action is generated by (3.24). The cyclic group
$\mathbbm{Z}_{n}$ has $n$ irreducible representations $\rho_{s}$, with
$s=0,1,\dots,n-1$ and $\rho_{s}=\rho_{1}^{\otimes s}$. The fundamental
representation of $\mathsf{SU}(4)$ restricts to $\mathbbm{Z}_{n}$ as
$Q=\rho_{1}\oplus\rho_{n-1}\oplus\rho_{0}\oplus\rho_{0}$, and the McKay quiver
assumes the form

${0}$${1}$${{n-1}}$${2}$${3}$ (4.1)

This is obtained as the double of the cyclic quiver ${\sf C}_{n}$ with arrows
$s\to s+1$, whose underlying graph is the extended Dynkin diagram of type
$\hat{A}_{n-1}$, and adding a pair of loops at each node.

The fractional instanton contributions to the corresponding $\mathsf{U}(1)$
partition function are given by

$\displaystyle\begin{split}&Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}^{\vec{k}}(\vec{\epsilon},m)\\\\[4.0pt]
&\hskip
28.45274pt=\sum_{|{\sigma}|_{\mathbbm{Z}_{n}}=\vec{k}}\,(-1)^{\mathtt{O}_{\sigma}^{\mathbbm{Z}_{n}}}\
\prod_{\begin{subarray}{c}\vec{p}\,\in\sigma\\\
p_{1}\,\equiv_{n}\,p_{2}\end{subarray}}^{\neq
0}\,\frac{\vec{p}\cdot\vec{\epsilon}-m}{\vec{p}\cdot\vec{\epsilon}}\\\ &\hskip
85.35826pt\times\prod_{\begin{subarray}{c}\vec{p},\vec{p}\,^{\prime}\in\sigma\\\
p_{1}+p_{2}+p_{3}\,\equiv_{n}\,p_{1}^{\prime}+p_{2}^{\prime}+p_{3}^{\prime}\end{subarray}}^{\neq
0}\,\frac{\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}\,\big{)}\,\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{12}\big{)}}{\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{3}\big{)}\,\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{4}\big{)}}\\\
&\hskip
142.26378pt\times\prod_{\begin{subarray}{c}\vec{p},\vec{p}\,^{\prime}\in\sigma\\\
p_{1}-p_{2}+1\,\equiv_{n}\,p_{1}^{\prime}-p_{2}^{\prime}\end{subarray}}^{\neq
0}\,\frac{\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{13}\big{)}\,\big{(}(\vec{p}\,^{\prime}-\vec{p}\,)\cdot\vec{\epsilon}-\epsilon_{23}\big{)}}{\big{(}(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{1}\big{)}\,\big{(}(\vec{p}\,^{\prime}-\vec{p}\,)\cdot\vec{\epsilon}-\epsilon_{2}\big{)}}\
,\end{split}$ (4.2)

where $\equiv_{n}$ denotes congruence modulo $n$.

The equivariant partition function for rank one noncommutative
Donaldson–Thomas invariants of the toric three-orbifold
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}$ with $\mathsf{U}(3)$
holonomy can be written in the closed form [36]

$\displaystyle
Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}}(\vec{\mathtt{q}};\epsilon_{1},\epsilon_{2},\epsilon_{3})=M(-\mathtt{Q})^{-n\,\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}+\frac{n^{2}-1}{n}\,\frac{\epsilon_{12}\,\epsilon_{123}}{\epsilon_{1}\,\epsilon_{2}}}\
\prod_{0<p\leq
s<n}\,\widetilde{M}(\mathtt{q}_{[p,s]},-\mathtt{Q})^{-\frac{\epsilon_{12}}{\epsilon_{3}}}\
,$ (4.3)

where

$\displaystyle\mathtt{Q}=\mathtt{q}_{0}\,\mathtt{q}_{1}\cdots\mathtt{q}_{n-1}\qquad\mbox{and}\qquad\mathtt{q}_{[p,s]}=\mathtt{q}_{p}\,\mathtt{q}_{p+1}\cdots\mathtt{q}_{s-1}\,\mathtt{q}_{s}\
,$ (4.4)

while

$\displaystyle
M(x,q)=\prod_{k=1}^{\infty}\,\frac{1}{(1-x\,q^{k})^{k}}\qquad\mbox{and}\qquad\widetilde{M}(x,q)=M(x,q)\,M(x^{-1},q)\
.$ (4.5)

The generalized MacMahon function $M(x,q)$ counts weighted plane partitions
(cf. Remark 2.87) and specialises to (2.73) at $x=1$: $M(1,q)=M(q)$.

At the Calabi–Yau specialization $\epsilon_{1}+\epsilon_{2}+\epsilon_{3}=0$ of
the $\varOmega$-deformation, the formula (4.3) reduces to the instanton
partition function on $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}$ with
$\mathsf{SU}(3)$ holonomy and agrees with the closed formula found in [35, 7]:

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}}(\vec{\mathtt{q}};\epsilon_{1},\epsilon_{2},\epsilon_{3})\big{|}_{\epsilon_{123}=0}&=M(-\mathtt{Q})^{n}\
\prod_{0<p\leq s<n}\,\widetilde{M}(\mathtt{q}_{[p,s]},-\mathtt{Q})\\\\[4.0pt]
&=\sum_{\pi}\,(-\mathtt{q}_{0})^{|\pi|_{0}}\,\mathtt{q}_{1}^{|\pi|_{1}}\,\cdots\mathtt{q}_{n-1}^{|\pi|_{n-1}}\
.\end{split}$ (4.6)

In the second equality the sum runs through plane partitions $\pi$ which are
$\mathbbm{Z}_{n}$-coloured with respect to (3.3).

Together with Proposition 3.63, the forms of the partition functions (4.3) and
(4.6) suggest that the partition function for $\mathsf{U}(1)$ noncommuative
instantons on $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$ assumes a
closed form as a combination of generalized MacMahon functions $M(x,q)$. This
is encapsulated in

######  Conjecture 4.7.

The equivariant instanton partition function of the cohomological
$\mathsf{U}(1)$ gauge theory with massive fundamental matter on
$[\mathbb{C}^{2}/\mathbb{Z}_{n}]\times\mathbb{C}^{2}$ is given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\mathtt{q}};\vec{\epsilon},m)&=M\big{(}(-1)^{n}\,\mathtt{Q}\big{)}^{-n\,\frac{m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}-\frac{n^{2}-1}{n}\,\frac{m\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}}\\\
&\quad\,\times\prod_{0<p\leq
s<n}\,\widetilde{M}\big{(}(-1)^{p-s+1}\,\mathtt{q}_{[p,s]},(-1)^{n}\,\mathtt{Q}\big{)}^{-\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\
.\end{split}$ (4.8)

Using the combinatorial expression (4.2) we explicitly checked

######  Proposition 4.9.

Conjecture 4.7 is true for $n=2,3,4$ and $k=|\vec{k}\,|=1,2,3$.

######  Remark 4.10.

In Appendix B.2 we sketch the steps of a possible direct proof of Conjecture
4.7, analogous to the sketch of the analytic proof of Conjecture 2.105 .

###  4.2 Higher Rank Generalizations

Conjecture 2.105 shows that the equivariant instanton partition function of
the rank $r>1$ cohomological gauge theory with a massive fundamental
hypermultiplet on $\mathbbm{C}^{4}$ is given by the simple closed formula
(2.106), expressing it as the $r$-th power of the rank one partition function
with the mass parameter $m$ given by coordinates on the center
$\mathsf{U}(1|1)$ of the global symmetry
$\mathsf{U}(r|r)\supset\mathsf{U}(r)_{\rm col}\times\mathsf{U}(r)_{\rm fla}$.
We would now like to generalize Conjecture 4.7 to the case of higher rank
framing vectors $\vec{r}$. Unfortunately, we do not have available any higher
rank noncommutative Donaldson–Thomas partition functions on the orbifold
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}$ with generic
$\mathsf{U}(3)$ holonomy to uplift, so our results will be limited to a set of
well-substantiated conjectures.

#### Instantons on
${\bm{\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}}}$ of Type
$\vec{r}=(0,\dots,0,r,0,\dots,0)$

For the framing vectors of the form $\vec{r}=(r,0,\dots,0)$, we propose

######  Conjecture 4.11.

Set

$\displaystyle m=\frac{1}{r}\,\sum_{l=1}^{r}\,(m_{l}-a_{l})\ .$ (4.12)

Then the equivariant instanton partition function of type
$\vec{r}=(r,0,\dots,0)$ for the cohomological $\mathsf{U}(r)$ gauge theory
with massive fundamental matter on
$[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}$ is given by

$\displaystyle\begin{split}Z^{\vec{r}=(r,0,\dots,0)}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\mathtt{q}};\vec{\epsilon},m)&=M\big{(}(-1)^{n}\,\mathtt{Q}\big{)}^{-n\,\frac{m\,r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}-\frac{n^{2}-1}{n}\,\frac{m\,r\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}}\\\
&\quad\,\times\prod_{0<p\leq
s<n}\,\widetilde{M}\big{(}(-1)^{p-s+1}\,\mathtt{q}_{[p,s]},(-1)^{n}\,\mathtt{Q}\big{)}^{-\frac{m\,r\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\
.\end{split}$ (4.13)

Using the combinatorial expression (3.76) we explicitly checked

######  Proposition 4.14.

Conjecture 4.11 is true for $r=2,3$, $k=|\vec{k}\,|=1,2$ and $n=2,3$.

######  Remark 4.15.

The partition function (4.13) is invariant under permutation of the location
of the entry $r$ in the array $\vec{r}=(r,0,\dots,0)$ (cf. Section 3.5).

######  Proposition 4.16.

Assume Conjecture 4.11 is true. Then the partition function for rank $r$
noncommutative Donaldson–Thomas invariants of type $\vec{r}=(r,0,\dots,0)$ for
the orbifold $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}$ with
$\mathsf{U}(3)$ holonomy is given by

$\displaystyle\begin{split}Z^{\vec{r}=(r,0,\dots,0)}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}}(\vec{\mathtt{q}};\epsilon_{1},\epsilon_{2},\epsilon_{3})&=M\big{(}(-1)^{r}\,\mathtt{Q}\big{)}^{-n\,\frac{r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}+\frac{n^{2}-1}{n}\,\frac{r\,\epsilon_{12}\,\epsilon_{123}}{\epsilon_{1}\,\epsilon_{2}}}\\\
&\quad\,\times\prod_{0<p\leq
s<n}\,\widetilde{M}\big{(}\mathtt{q}_{[p,s]},(-1)^{r}\,\mathtt{Q}\big{)}^{-\frac{r\,\epsilon_{12}}{\epsilon_{3}}}\
.\end{split}$ (4.17)

###### Proof.

This follows immediately from (4.13) by using Proposition 3.63. ∎

#### Instantons on
${\bm{\mathbbm{C}^{2}/\mathbbm{Z}_{2}\times\mathbbm{C}^{2}}}$ of Type
$\vec{r}=(r_{0},r_{1})$

It appears to be difficult to make a concrete conjecture for the closed form
of the instanton partition function on
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$ for generic framing
vectors $\vec{r}=(r_{0},r_{1},\dots,r_{n-1})$, as the calculations become much
more cumbersome in general. Here we consider the simplest case $n=2$, and
propose a closed formula which holds in full generality.

######  Example 4.18.

The leading terms in the expansions of the rank two partition functions for
the orbifold $\mathbbm{C}^{2}/\mathbbm{Z}_{2}\times\mathbbm{C}^{2}$ are

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{2}]\times\mathbbm{C}^{2}}^{\vec{r}=(2,0)}(\mathtt{q}_{0},\mathtt{q}_{1};\vec{a},\vec{\epsilon},\vec{m})&=2\,\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\,\mathtt{q}_{0}+\frac{m\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}\,\Big{(}4\,\frac{\epsilon_{13}\,\epsilon_{3}}{\epsilon_{3}\,\epsilon_{4}}+3\Big{)}\,\mathtt{q}_{0}\,\mathtt{q}_{1}\\\
&\quad\,+\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\,\Big{(}2\,\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}-1\Big{)}\,\mathtt{q}^{2}_{0}+\cdots\
,\\\\[4.0pt]
Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{2}]\times\mathbbm{C}^{2}}^{\vec{r}=(1,1)}(\mathtt{q}_{0},\mathtt{q}_{1};\vec{a},\vec{\epsilon},\vec{m})&=2\,\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\,\mathtt{q}_{0}+\Big{(}4\,\frac{m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{3}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}+3\,\frac{m\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}\\\
&\hskip
142.26378pt+\frac{(m_{1}-a_{1})\,(m_{2}-a_{2})\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\Big{)}\,\mathtt{q}_{0}\,\mathtt{q}_{1}\\\
&\quad\,+\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\Big{(}2\,\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}-1\Big{)}\,\mathtt{q}^{2}_{0}+\cdots\
,\end{split}$ (4.19)

where $m=\frac{1}{2}\,(m_{1}-a_{1}+m_{2}-a_{2})$.

Since we are working with the $\mathbbm{Z}_{2}$-action of Section 4.1 in
evaluating the contribution (3.77) from the index
$\chi^{\mathbbm{Z}_{2}}_{\vec{\sigma}}$, we can use

$\displaystyle\rho_{s}\otimes\rho_{s^{\prime}}=\begin{cases}\rho_{0}\quad\text{if}\
s=s^{\prime}\ ,\\\ \rho_{1}\quad\mbox{otherwise}\ ,\end{cases}$ (4.20)

for $s,s^{\prime}\in\\{0,1\\}$. Then the partition function (3.76) is
symmetric under permutation of $r_{0}$ and $r_{1}$, and it must reduce to
(4.13) with $n=2$ for $\vec{r}=(r,0)$ and $\vec{r}=(0,r)$. These properties
together with Example 4.18 prompt us to formulate

######  Conjecture 4.21.

The equivariant instanton partition function of type $\vec{r}=(r_{0},r_{1})$
for the cohomological gauge theory with massive fundamental matter on
$[\mathbbm{C}^{2}/\mathbbm{Z}_{2}]\times\mathbbm{C}^{2}$ is given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{2}]\times\mathbbm{C}^{2}}^{\vec{r}=(r_{0},r_{1})}(\vec{\mathtt{q}};\vec{a},\vec{\epsilon},\vec{m})&=M(\mathtt{Q})^{-2\,\frac{m\,r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{3}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}-\frac{3}{2}\,\frac{m\,r\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}}\\\
&\quad\,\times\widetilde{M}(-\mathtt{q}_{1},\mathtt{q}_{0}\,\mathtt{q}_{1})^{-\frac{m^{0}\,r_{0}\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\,\widetilde{M}(-\mathtt{q}_{0},\mathtt{q}_{0}\,\mathtt{q}_{1})^{-\frac{m^{1}\,r_{1}\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\
,\end{split}$ (4.22)

where $r=r_{0}+r_{1}$ and

$\displaystyle m^{s}=\frac{1}{r_{s}}\,\sum_{s(l)=s}\,(m_{l}-a_{l})$ (4.23)

are coordinates on the center of the global symmetry supergroup
$\mathsf{U}(r_{s}|r_{s})$ associated to the irreducible representation
$\rho_{s}$ of $\mathbbm{Z}_{2}$, for $s\in\\{0,1\\}$.

######  Proposition 4.24.

Assume Conjecture 4.21 is true. Then the partition function for noncommutative
Donaldson–Thomas invariants of type $\vec{r}=(r_{0},r_{1})$ for the orbifold
$\mathbbm{C}^{2}/\mathbbm{Z}_{2}\times\mathbbm{C}$ with $\mathsf{U}(3)$
holonomy is given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{2}]\times\mathbbm{C}}^{\vec{r}=(r_{0},r_{1})}(\vec{\mathtt{q}};\epsilon_{1},\epsilon_{2},\epsilon_{3})&=M\big{(}(-1)^{r}\,\mathtt{Q}\big{)}^{-2\,\frac{r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}+\frac{3}{2}\,\frac{r\,\epsilon_{12}\,\epsilon_{123}}{\epsilon_{1}\,\epsilon_{2}}}\\\
&\hskip
28.45274pt\times\widetilde{M}\big{(}(-1)^{r_{1}}\,\mathtt{q}_{1},(-1)^{r}\,\mathtt{q}_{0}\,\mathtt{q}_{1}\big{)}^{-\frac{r_{0}\,\epsilon_{12}}{\epsilon_{3}}}\\\
&\hskip
56.9055pt\times\widetilde{M}\big{(}(-1)^{r_{0}}\,\mathtt{q}_{0},(-1)^{r}\,\mathtt{q}_{0}\,\mathtt{q}_{1}\big{)}^{-\frac{r_{1}\,\epsilon_{12}}{\epsilon_{3}}}\
.\end{split}$ (4.25)

###### Proof.

This follows immediately from (4.22) by using Proposition 3.63. ∎

######  Remark 4.26.

At the Calabi–Yau specialization $\epsilon_{123}=0$, both partition functions
(4.17) and (4.25) agree with the generating functions for Coulomb branch
invariants found in [7, Section 7].

###  4.3 Pure $\mathcal{N}_{\textrm{\tiny T}}=2$ Gauge Theory on
$[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}$

Using Proposition 3.99 together with the results of Sections 4.1 and 4.2 we
can immediately infer corresponding closed formulas for the partition
functions of the pure gauge theories on the quotient stack
$[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}$. They read as

######  Proposition 4.27.

If Conjecture 4.7 is true, then the equivariant instanton partition function
of the pure cohomological $\mathsf{U}(1)$ gauge theory on
$[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}$ is given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\Lambda};\vec{\epsilon}\,)^{\rm
pure}&=\exp\,(-1)^{n}\,\varLambda\,\Big{(}n\,\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}+\frac{n^{2}-1}{n}\,\frac{\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\\\
&\hskip
85.35826pt-\frac{\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\,\sum_{0<p\leq
s<n}\,(-1)^{p-s}\,\big{(}\Lambda_{[p,s]}+\Lambda_{[p,s]}^{-1}\big{)}\Big{)}\
,\end{split}$ (4.28)

where

$\displaystyle\varLambda=\Lambda_{0}\,\Lambda_{1}\cdots\Lambda_{n}\qquad\mbox{and}\qquad\Lambda_{[p,s]}=\Lambda_{p}\,\Lambda_{p+1}\cdots\Lambda_{s-1}\,\Lambda_{s}\
.$ (4.29)

If in addition Conjecture 4.11 is true, then the higher rank partition
functions of type $\vec{r}=(r,0,\dots,0)$ are all trivial:

$\displaystyle
Z^{\vec{r}=(r,0,\dots,0)}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\Lambda};\vec{\epsilon}\,)^{\rm
pure}=1\qquad\mbox{for}\quad r>1\ .$ (4.30)

###### Proof.

The proof is completely analogous to the proof of Corollary 2.115 using the
series representation

$\displaystyle\log
M(x,q)=\sum_{k=1}^{\infty}\,\frac{x^{k}}{k}\,\frac{q^{k}}{(1-q^{k})^{2}}$
(4.31)

for the logarithm of the generalized MacMahon function. ∎

##  5 The (3,0) Orbifold
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$

As another explicit example, we repeat our treatment from Section 4 in the
case of the $(3,0)$ orbifold
$\mathbbm{C}^{3}/\mathsf{\Gamma}\times\mathbbm{C}$ for the action of the group
$\mathsf{\Gamma}=\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ in
$\mathsf{SL}(3,\mathbbm{C})$ defined below. By the results of [91] it admits
four geometric projective toric crepant resolutions related to each other by
flops, which can each be constructed as fine moduli spaces of stable
$\mathsf{\Gamma}$-constellations; for the symmetric resolution induced by
${\sf Hilb}^{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}(\mathbbm{C}^{3})$, the
geometric crepant resolution of
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$ is a
fibration of closed topological vertex geometries over $\mathbbm{C}$. Whereas
the dimensionally reduced partition functions are well understood for the
Calabi–Yau three-orbifolds
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ by the results of [35,
7], we are currently lacking closed expressions for generic $\mathsf{U}(3)$
holonomy, even in the rank one case. Hence in this section our considerations
will again be limited to conjectural but well-substantiated closed formulas.

###  5.1 Noncommutative $\mathsf{U}(1)$ Instantons on
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$

Consider the toric Calabi–Yau four-orbifold
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$,
where the action of the group
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}=\\{\mathbbm{1},g_{1},g_{2},g_{3}\\}$ on
$\mathbbm{C}^{4}$ is given by the $\mathsf{SU}(4)$ matrices

$\displaystyle g_{1}={\small\begin{pmatrix}-1&&&\\\ &-1&&\\\ &&1&\\\
&&&1\end{pmatrix}}\normalsize\qquad\mbox{and}\qquad
g_{2}={\small\begin{pmatrix}-1&&&\\\ &1&&\\\ &&-1&\\\
&&&1\end{pmatrix}}\normalsize\ ,$ (5.1)

together with $g_{3}=g_{1}\,g_{2}$. The four irreducible representations
${\widehat{\mathsf{\Gamma}}}=\\{\rho_{0},\rho_{1},\rho_{2},\rho_{3}\\}$ have
weights $s_{0}=(0,0,0,0)$, $s_{1}=(1,1,0,0)$, $s_{2}=(1,0,1,0)$ and
$s_{3}=s_{1}+s_{2}=(0,1,1,0)$, respectively.

The tensor product decomposition of the fundamental representation $Q$ gives
an adjacency matrix

$\displaystyle A=(a_{ss^{\prime}})={\small\begin{pmatrix}1&1&1&1\\\ 1&1&1&1\\\
1&1&1&1\\\ 1&1&1&1\end{pmatrix}}\normalsize\ .$ (5.2)

The generalized McKay quiver constructed from the representation theory of
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ is

${1}$${3}$${0}$${2}$ (5.3)

To write the corresponding $\mathsf{U}(1)$ instanton partition function
explicitly, it is convenient to use the imaginary unit quaternions
$\mathrm{i}$, $\mathrm{j}$ and $\mathrm{k}$ which satisfy the relations

$\displaystyle\mathrm{i}^{2}=\mathrm{j}^{2}=\mathrm{k}^{2}=\mathrm{i}\,\mathrm{j}\,\mathrm{k}=-1\
.$ (5.4)

Then the fractional instanton contributions are given by

$\displaystyle\begin{split}&Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}}^{\vec{k}}(\vec{\epsilon},m)\\\\[4.0pt]
&\quad=\sum_{|{\sigma}|_{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}=\vec{k}}\,(-1)^{\mathtt{O}_{\sigma}^{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}}\
\prod_{\begin{subarray}{c}\vec{p}\,\in\sigma\\\
\mathrm{i}^{p_{1}}\,\mathrm{j}^{p_{2}}\,\mathrm{k}^{p_{3}}=\pm
1\end{subarray}}^{\neq
0}\,\frac{\vec{p}\cdot\vec{\epsilon}-m}{\vec{p}\cdot\vec{\epsilon}}\
\prod_{\begin{subarray}{c}\vec{p},\vec{p}\,^{\prime}\in\sigma\\\
\mathrm{i}^{p_{1}-p_{1}^{\prime}}\,\mathrm{j}^{p_{2}-p_{2}^{\prime}}\,\mathrm{k}^{p_{3}-p_{3}^{\prime}}=\pm
1\end{subarray}}^{\neq
0}\,\frac{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}}{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{4}}\\\
&\hskip
56.9055pt\times\prod_{\begin{subarray}{c}\vec{p},\vec{p}\,^{\prime}\in\sigma\\\
\mathrm{i}^{p_{1}-p_{1}^{\prime}}\,\mathrm{j}^{p_{2}-p_{2}^{\prime}}\,\mathrm{k}^{p_{3}-p_{3}^{\prime}}=\pm\mathrm{i}\end{subarray}}^{\neq
0}\,\frac{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{23}}{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{1}}\
\prod_{\begin{subarray}{c}\vec{p},\vec{p}\,^{\prime}\in\sigma\\\
\mathrm{i}^{p_{1}-p_{1}^{\prime}}\,\mathrm{j}^{p_{2}-p_{2}^{\prime}}\,\mathrm{k}^{p_{3}-p_{3}^{\prime}}=\pm\mathrm{j}\end{subarray}}^{\neq
0}\,\frac{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{13}}{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{2}}\\\
&\hskip
113.81102pt\times\prod_{\begin{subarray}{c}\vec{p},\vec{p}\,^{\prime}\in\sigma\\\
\mathrm{i}^{p_{1}-p_{1}^{\prime}}\,\mathrm{j}^{p_{2}-p_{2}^{\prime}}\,\mathrm{k}^{p_{3}-p_{3}^{\prime}}=\pm\mathrm{k}\end{subarray}}^{\neq
0}\,\frac{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{12}}{(\vec{p}-\vec{p}\,^{\prime})\cdot\vec{\epsilon}-\epsilon_{3}}\
.\end{split}$ (5.5)

The rank one Donaldson–Thomas partition function for the three-orbifold
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with holonomy group
$\mathsf{SU}(3)$ is discussed in detail in [35, 7]. It can be expressed as the
closed formula

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]}(\vec{\mathtt{q}};\epsilon_{1},\epsilon_{2},\epsilon_{3})\big{|}_{\epsilon_{123}=0}&=\frac{M(-\mathtt{Q})^{4}}{L(\mathtt{q}_{1},\mathtt{q}_{2},\mathtt{q}_{3},-\mathtt{Q})}\
\prod_{1\leq p<s\leq
3}\,\widetilde{M}(\mathtt{q}_{p}\,\mathtt{q}_{s},-\mathtt{Q})\\\\[4.0pt]
&=\sum_{\pi}\,(-1)^{|\pi|_{1}+|\pi|_{2}+|\pi|_{3}}\,\mathtt{q}_{0}^{|\pi|_{0}}\,\mathtt{q}_{1}^{|\pi|_{1}}\,\mathtt{q}_{2}^{|\pi|_{2}}\,\mathtt{q}_{3}^{|\pi|_{3}}\
,\end{split}$ (5.6)

where the sum runs through $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$-coloured
plane partitions $\pi$. We have set
$\mathtt{Q}=\mathtt{q}_{0}\,\mathtt{q}_{1}\,\mathtt{q}_{2}\,\mathtt{q}_{3}$
and

$\displaystyle
L(x_{1},x_{2},x_{3},q)=\widetilde{M}(x_{1},q)\,\widetilde{M}(x_{2},q)\,\widetilde{M}(x_{3},q)\,\widetilde{M}(x_{1}\,x_{2}\,x_{3},q)\
.$ (5.7)

With an argument analogous to that of Section 4.1, we expect that the rank one
instanton partition function for
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$
assumes a closed form in terms of a generalization of the formula (5.6). Using
(3.76) we evaluate the leading terms in the expansion of
$Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}}(\vec{\mathtt{q}};\vec{\epsilon},m)$
to be

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}}(\vec{\mathtt{q}};\vec{\epsilon},m)&=\frac{m}{\epsilon_{4}}\,\mathtt{q}_{0}+\frac{(m-\epsilon_{4})\,m}{2\,\epsilon_{4}^{2}}\,\mathtt{q}_{0}^{2}+\frac{(\epsilon_{1}-\epsilon_{23})\,m}{2\,\epsilon_{1}\,\epsilon_{4}}\,\mathtt{q}_{0}\,\mathtt{q}_{1}+\frac{(\epsilon_{2}-\epsilon_{13})\,m}{2\,\epsilon_{2}\,\epsilon_{4}}\,\mathtt{q}_{0}\,\mathtt{q}_{2}\\\
&\quad\,+\frac{(\epsilon_{3}-\epsilon_{12})\,m}{2\,\epsilon_{2}\,\epsilon_{4}}\,\mathtt{q}_{0}\,\mathtt{q}_{3}+\frac{m}{\epsilon_{4}}\,\mathtt{q}_{0}\,\mathtt{q}_{1}\,\mathtt{q}_{2}+\cdots\
.\end{split}$ (5.8)

Dimensionally reducing this result according to Proposition 3.63 at the
Calabi–Yau specialization of the $\varOmega$-deformation on $\mathbbm{C}^{3}$
yields the leading contributions predicted by (5.6):

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}}(\vec{\mathtt{q}};\vec{\epsilon},m=\epsilon_{4})\big{|}_{\epsilon_{123}=0}&=\mathtt{q}^{\prime}_{0}-\mathtt{q}^{\prime}_{0}\,\mathtt{q}^{\prime}_{1}+\mathtt{q}^{\prime}_{0}\,\mathtt{q}^{\prime}_{2}+\mathtt{q}^{\prime}_{0}\,\mathtt{q}^{\prime}_{3}+\mathtt{q}^{\prime}_{0}\,\mathtt{q}^{\prime}_{1}\,\mathtt{q}^{\prime}_{2}+\cdots\\\\[4.0pt]
&=Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]}(\vec{\mathtt{q}}\,^{\prime};\epsilon_{1},\epsilon_{2},\epsilon_{3})\big{|}_{\epsilon_{123}=0}\
,\end{split}$ (5.9)

where
$\vec{\mathtt{q}}\,^{\prime}=(\mathtt{q}_{0},-\mathtt{q}_{1},-\mathtt{q}_{2},-\mathtt{q}_{3})$.

This prompts us to formulate

######  Conjecture 5.10.

The equivariant instanton partition function of the cohomological
$\mathsf{U}(1)$ gauge theory with massive fundamental matter on
$[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}$ is
given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}}(\vec{\mathtt{q}};\vec{\epsilon},m)&=\frac{M(\mathtt{Q})^{m\,\frac{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}-\epsilon_{1}^{2}\,\epsilon_{2}-\epsilon^{2}_{1}\,\epsilon_{3}-\epsilon^{2}_{2}\,\epsilon_{3}-\epsilon_{1}\,\epsilon_{2}^{2}-\epsilon_{1}\,\epsilon_{3}^{2}-\epsilon_{2}\,\epsilon^{2}_{3}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}}}{L(-\mathtt{q}_{1},-\mathtt{q}_{2},-\mathtt{q}_{3},\mathtt{Q})^{\frac{m}{\epsilon_{4}}}}\\\
&\quad\,\times\prod_{1\leq p<s\leq
3}\,\widetilde{M}(\mathtt{q}_{p}\,\mathtt{q}_{s},\mathtt{Q})^{m\,\frac{\epsilon_{(ps)^{-}}-\epsilon_{ps}}{2\,\epsilon_{4}\,\epsilon_{(ps)^{-}}}}\
,\end{split}$ (5.11)

where $(ps)^{-}\in\\{1,2,3\\}\setminus\\{p,s\\}$.

######  Proposition 5.12.

Assume Conjecture 5.10 is true. Then the partition function for rank one
noncommutative Donaldson–Thomas invariants of the orbifold
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with holonomy group
$\mathsf{U}(3)$ is given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]}(\vec{\mathtt{q}};\epsilon_{1},\epsilon_{2},\epsilon_{3})&=\frac{M(-\mathtt{Q})^{\frac{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}-\epsilon_{1}^{2}\,\epsilon_{2}-\epsilon^{2}_{1}\,\epsilon_{3}-\epsilon^{2}_{2}\,\epsilon_{3}-\epsilon_{1}\,\epsilon_{2}^{2}-\epsilon_{1}\,\epsilon_{3}^{2}-\epsilon_{2}\,\epsilon^{2}_{3}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}}}{L(\mathtt{q}_{1},\mathtt{q}_{2},\mathtt{q}_{3},-\mathtt{Q})}\\\
&\quad\,\times\prod_{1\leq p<s\leq
3}\,\widetilde{M}(\mathtt{q}_{p}\,\mathtt{q}_{s},-\mathtt{Q})^{\frac{\epsilon_{(ps)^{-}}-\epsilon_{ps}}{2\,\epsilon_{(ps)^{-}}}}\
.\end{split}$ (5.13)

###### Proof.

This follows immediately from (5.11) by using Proposition 3.63. ∎

######  Remark 5.14.

For holonomy group $\mathsf{SU}(3)$, the partition function (5.13) reduces to
(5.6). As noted by [27], it is possible to adapt the techniques of [36] to the
orbifold $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ and hence
provide a direct proof of (5.13).

###  5.2 Higher Rank Generalization

For higher rank noncommutative instantons on
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$ of
type $\vec{r}=(0,\dots,0,r,0,\dots,0)$, the analogue of Conjecture 4.11 reads
as

######  Conjecture 5.15.

The equivariant instanton partition function of type $\vec{r}=(r,0,\dots,0)$
for the cohomological $\mathsf{U}(r)$ gauge theory with massive fundamental
matter on
$[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}$ is
given by

$\displaystyle\begin{split}Z^{\vec{r}=(r,0\dots,0)}_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}}(\vec{\mathtt{q}};\vec{\epsilon},m)&=\frac{M(\mathtt{Q})^{m\,r\,\frac{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}-\epsilon_{1}^{2}\,\epsilon_{2}-\epsilon^{2}_{1}\,\epsilon_{3}-\epsilon^{2}_{2}\,\epsilon_{3}-\epsilon_{1}\,\epsilon_{2}^{2}-\epsilon_{1}\,\epsilon_{3}^{2}-\epsilon_{2}\,\epsilon^{2}_{3}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}}}{L(-\mathtt{q}_{1},-\mathtt{q}_{2},-\mathtt{q}_{3},\mathtt{Q})^{\frac{m\,r}{\epsilon_{4}}}}\\\
&\quad\,\times\prod_{1\leq p<s\leq
3}\,\widetilde{M}(\mathtt{q}_{p}\,\mathtt{q}_{s},\mathtt{Q})^{m\,r\,\frac{\epsilon_{(ps)^{-}}-\epsilon_{ps}}{2\,\epsilon_{4}\,\epsilon_{(ps)^{-}}}}\
.\end{split}$ (5.16)

Using the combinatorial expansion (3.76) we explicitly checked

######  Proposition 5.17.

Conjecture 5.15 is true for $r=2,3$ and $k=|\vec{k}\,|=1,2$.

######  Proposition 5.18.

Assume Conjecture 5.15 is true. Then the partition function for rank $r$
noncommutative Donaldson–Thomas invariants of type $\vec{r}=(r,0,\dots,0)$ for
the orbifold $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with
$\mathsf{U}(3)$ holonomy is given by

$\displaystyle\begin{split}Z^{\vec{r}=(r,0\dots,0)}_{[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]}(\vec{\mathtt{q}};\epsilon_{1},\epsilon_{2},\epsilon_{3})&=\frac{M\big{(}(-1)^{r}\,\mathtt{Q}\big{)}^{r\,\frac{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}-\epsilon_{1}^{2}\,\epsilon_{2}-\epsilon^{2}_{1}\,\epsilon_{3}-\epsilon^{2}_{2}\,\epsilon_{3}-\epsilon_{1}\,\epsilon_{2}^{2}-\epsilon_{1}\,\epsilon_{3}^{2}-\epsilon_{2}\,\epsilon^{2}_{3}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}}}{L\big{(}\mathtt{q}_{1},\mathtt{q}_{2},\mathtt{q}_{3},(-1)^{r}\,\mathtt{Q}\big{)}^{r}}\\\
&\quad\,\times\prod_{1\leq p<s\leq
3}\,\widetilde{M}\big{(}\mathtt{q}_{p}\,\mathtt{q}_{s},(-1)^{r}\,\mathtt{Q}\big{)}^{r\,\frac{\epsilon_{(ps)^{-}}-\epsilon_{ps}}{2\,\epsilon_{(ps)^{-}}}}\
.\end{split}$ (5.19)

###### Proof.

This follows immediately from (5.16) by using Proposition 3.63. ∎

###  5.3 Pure $\mathcal{N}_{\textrm{\tiny T}}=2$ Gauge Theory on
$[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}$

For the pure gauge theories on the quotient stack
$[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}$,
obtained via the decoupling limit of Proposition 3.99, the analogue of
Proposition 4.27 reads as

######  Proposition 5.20.

Assume Conjectures 5.10 and 5.15 are true. Then the equivariant instanton
partition function of the pure cohomological $\mathsf{U}(1)$ gauge theory on
$[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}$ is
given by

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}]}(\vec{\Lambda};\vec{\epsilon}\,)^{\rm
pure}&=\exp-\varLambda\,\Big{(}\frac{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}-\epsilon_{1}^{2}\,\epsilon_{2}-\epsilon^{2}_{1}\,\epsilon_{3}-\epsilon^{2}_{2}\,\epsilon_{3}-\epsilon_{1}\,\epsilon_{2}^{2}-\epsilon_{1}\,\epsilon_{3}^{2}-\epsilon_{2}\,\epsilon^{2}_{3}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\\\
&\hskip 71.13188pt+\sum_{1\leq p<s\leq
3}\,\frac{\big{(}\epsilon_{(ps)^{-}}-\epsilon_{ps}\big{)}\,\big{(}\Lambda_{p}\,\Lambda_{s}+\Lambda_{p}^{-1}\,\Lambda_{s}^{-1}\big{)}}{2\,\epsilon_{4}\,\epsilon_{(ps)^{-}}}\\\
&\hskip
71.13188pt+\frac{1}{\epsilon_{4}}\,\sum_{s=1}^{3}\,\big{(}\Lambda_{s}+\Lambda_{s}^{-1}\big{)}-\frac{1}{\epsilon_{4}}\,\big{(}\Lambda_{123}+\Lambda_{123}^{-1}\big{)}\Big{)}\
,\end{split}$ (5.21)

where $\varLambda=\Lambda_{0}\,\Lambda_{1}\,\Lambda_{2}\,\Lambda_{3}$ and
$\Lambda_{123}=\Lambda_{1}\,\Lambda_{2}\,\Lambda_{3}$, while the higher rank
partition functions of type $\vec{r}=(r,0,\dots,0)$ are all trivial:

$\displaystyle
Z^{\vec{r}=(r,0,\dots,0)}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}}(\vec{\Lambda};\vec{\epsilon}\,)^{\rm
pure}=1\qquad\mbox{for}\quad r>1\ .$ (5.22)

###### Proof.

The proof is completely analogous to the proof of Proposition 4.27. ∎

##  Appendix A Generalized ADHM Construction

In this appendix we construct the ADHM type finite-dimensional matrix model of
the moduli space of finite action solutions to the noncommutative instanton
equations (2.36). To write these generalized ADHM equations, we first
introduce two Hermitian vector spaces $V$ and $W$ of complex dimensions $k$
and $r$, respectively. Let $U$ be an $(8\,k{+}r){\times}r$ matrix which solves
the Weyl equation

$\displaystyle\Delta^{\dagger}\,U=0\ ,$ (A.1)

where $\Delta$ is the $(8\,k+r){\times}8\,k$ matrix

$\displaystyle\Delta={\small\begin{pmatrix}b_{1}^{\dagger}&b_{2}&b_{3}&b_{4}&0&0&0&0\\\
b_{2}^{\dagger}&-b_{1}&0&0&b_{3}&-b_{4}&0&0\\\
b_{3}^{\dagger}&0&-b_{1}&0&b_{2}&0&b_{4}&0\\\
b_{4}^{\dagger}&0&0&b_{1}^{\dagger}&0&b_{2}&b_{3}&0\\\
0&0&0&0&b_{4}^{\dagger}&b_{3}^{\dagger}&b_{2}^{\dagger}&b_{1}\\\
0&0&-b_{4}^{\dagger}&b_{3}^{\dagger}&0&0&-b_{1}^{\dagger}&b_{2}\\\
0&b_{4}^{\dagger}&0&-b_{2}^{\dagger}&0&-b_{2}^{\dagger}&-b_{1}^{\dagger}&b_{3}\\\
0&-b_{3}^{\dagger}&-b_{2}^{\dagger}&0&-b_{1}^{\dagger}&0&0&b_{4}\\\
I^{\dagger}&0&0&0&0&0&0&0\end{pmatrix}}\ .\normalsize$ (A.2)

Here $b_{a}=B_{a}-z_{a}\,\mathbbm{1}_{k}$, with
$B_{a}\in\mathsf{End}_{\mathbbm{C}}(V)$ for $a\in\underline{\sf 4}$ and
$I\in\mathsf{Hom}_{\mathbbm{C}}(W,V)$.

The auxiliary matrix $\Delta$ is required to satisfy the equation

$\displaystyle\Delta^{\dagger}\,\Delta=\mathbbm{1}_{8}\otimes f_{k}^{-1}\ ,$
(A.3)

where $f_{k}$ is an invertible $k{\times}k$ matrix. This leads to the
equations for the ADHM data $(B_{a},I)_{a\in\underline{\sf 4}}$ given by
(2.44):

$\displaystyle[B_{a},B_{b}]-\tfrac{1}{2}\,\epsilon_{ab\bar{c}\bar{d}}\,\big{[}B_{\bar{c}}^{\dagger},B_{\bar{d}}^{\dagger}\big{]}=0\qquad\mbox{and}\qquad\sum_{a=1}^{4}\,\big{[}B_{a},B_{\bar{a}}^{\dagger}\big{]}+I\,I^{\dagger}=\xi\,\mathbbm{1}_{k}\
.$ (A.4)

One asks that the matrix $U$ be normalized: $U^{\dagger}\,U=\mathbbm{1}_{r}$.
Then the columns of $U$ together with $\Delta$ form a complete basis in
$\mathbbm{C}^{8\,k+r}$, and therefore

$\displaystyle\mathbbm{1}_{8\,k+r}-U\,U^{\dagger}=\Delta\,(\mathbbm{1}_{8}\otimes
f_{k})\,\Delta^{\dagger}\ .$ (A.5)

The $\mathsf{Spin}(7)$-instanton connection can now be written as
$A=U^{\dagger}\,\mathrm{d}U$. Indeed, using (A.5) we compute the components of
its curvature two-form to get

$\displaystyle\begin{split}F_{\mu\nu}&=\partial_{\mu}U^{\dagger}\,\partial_{\nu}U-\partial_{\nu}U^{\dagger}\,\partial_{\mu}U+\big{[}U^{\dagger}\,\partial_{\mu}U,U^{\dagger}\,\partial_{\nu}U\big{]}\\\\[4.0pt]
&=\partial_{[\mu}U^{\dagger}\,\big{(}\mathbbm{1}_{8\,k+r}-U\,U^{\dagger}\big{)}\,\partial_{\nu]}U\\\\[4.0pt]
&=U^{\dagger}\,\partial_{[\mu}\Delta\,(\mathbbm{1}_{8}\otimes
f_{k})\,\partial_{\nu]}\Delta^{\dagger}\,U=U^{\dagger}\,\big{(}\Sigma_{\mu\nu}^{\textrm{\tiny$(+)$}}\otimes
f_{k}\big{)}\,U\ ,\end{split}$ (A.6)

which satisfy the self-duality equations (2.2) $(\lambda=1)$ for
$\mathsf{Spin}(7)$-holonomy. Here we have introduced the eight-dimensional
counterparts of the ’t Hooft symbols

$\displaystyle\Sigma^{\textrm{\tiny$(+)$}}_{\mu\nu}=\bar{\Sigma}_{\mu}\,\Sigma_{\nu}-\bar{\Sigma}_{\nu}\,\Sigma_{\mu}\qquad\mbox{and}\qquad\Sigma^{\textrm{\tiny$(-)$}}_{\mu\nu}=\Sigma_{\mu}\,\bar{\Sigma}_{\nu}-\Sigma_{\nu}\,\bar{\Sigma}_{\mu}\
.$ (A.7)

The $8{\times}8$ spin matrices $\Sigma_{\mu}$ and $\bar{\Sigma}_{\mu}$ for
$\mu=1,\dots,8$ are generators of the Clifford algebra $\mathsf{C\ell}(8)$,
where ${\Sigma}_{8}=\bar{\Sigma}_{8}=\mathbbm{1}_{8}$ and the matrices
${\Sigma}_{a}=-\bar{\Sigma}_{a}$ for $a=1,\dots,7$ satisfy the anticommutation
relations $\\{\Sigma_{a},\Sigma_{b}\\}=-2\,\delta_{ab}\,\mathbbm{1}_{8}$.

Consequently, the complex connection

$\displaystyle A_{a}=\tfrac{1}{\sqrt{2\,\xi}}\,U^{\dagger}\,\partial_{a}U$
(A.8)

for $a\in\underline{\sf 4}$ satisfies the instanton equations (2.9).

##  Appendix B Infinite Product Formulas for Instanton Partition Functions

In this appendix we outline a possible alternative proof of Conjecture 2.105
based on the quiver matrix model and its combinatorial evaluation, and then
proceed to sketch how this can be extended to provide a potential similar
proof of Conjecture 4.7.

###  B.1 Evidence for Conjecture 2.105

We start by explicitly computing the $k=1$ contribution to the instanton
partition function (2.102), resulting in

######  Lemma B.1.

For any rank $r\geq 1$, the one-instanton contribution to
$Z^{r}_{\mathbbm{C}^{4}}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m})$ is given
by

$\displaystyle Z^{r,1}_{\mathbbm{C}^{4}}(\vec{a},\vec{\epsilon},\vec{m})$
$\displaystyle=\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\,r\,m\qquad\mbox{with}\quad
m=\frac{1}{r}\,\sum_{l=1}^{r}\,(m_{l}-a_{l})\ .$ (B.2)

###### Proof.

Using the formula (2.100) we immediately see the result for $r=1$. So we
assume $r>1$, and shifting the masses $m_{l}$ to $m_{l}^{\prime}:=m_{l}-a_{l}$
for $l=1,\dots,r$ we get

$\displaystyle Z^{r,1}_{\mathbbm{C}^{4}}(\vec{a},\vec{\epsilon},\vec{m})$
$\displaystyle=\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\,\sum^{r}_{l=1}\,m^{\prime}_{l}\
\prod^{r}_{\stackrel{{\scriptstyle\scriptstyle p=1}}{{\scriptstyle p\neq
l}}}\,\Big{(}1-\frac{m_{p}^{\prime}}{a_{lp}}\Big{)}\ ,$ (B.3)

where $a_{lp}=a_{l}-a_{p}=-a_{pl}$. Using (B.3) the result is easy to check
for $r=2$, so henceforth we restrict to ranks $r>2$. The sum in (B.3) can be
rewritten as

$\displaystyle\begin{split}\sum^{r}_{l=1}\,m^{\prime}_{l}\
\prod^{r}_{\stackrel{{\scriptstyle\scriptstyle p=1}}{{\scriptstyle p\neq
l}}}\,\Big{(}1-\frac{m^{\prime}_{p}}{a_{lp}}\Big{)}&=\sum_{l=1}^{r}\,m_{l}^{\prime}\,\Big{(}1+\sum_{p=1}^{r-1}\
\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{p}\leq r\\\ i_{j}\neq
l\end{subarray}}\,\frac{m^{\prime}_{i_{1}}\cdots
m^{\prime}_{i_{p}}}{a_{i_{1}l}\cdots a_{i_{p}l}}\Big{)}\\\\[4.0pt]
&=r\,m+\sum_{p=1}^{r-1}\ \sum_{l=1}^{r}\,m_{l}^{\prime}\
\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{p}\leq r\\\ i_{j}\neq
l\end{subarray}}\,\frac{m^{\prime}_{i_{1}}\cdots
m^{\prime}_{i_{p}}}{a_{i_{1}l}\cdots a_{i_{p}l}}\ .\end{split}$ (B.4)

For each $1\leq p\leq r-1$, the second sum of (B.4) can be expressed in the
form

$\displaystyle\begin{split}&\sum_{l=1}^{r}\,m_{l}^{\prime}\
\sum_{\begin{subarray}{c}1\leq i_{1}<\cdots<i_{p}\leq r\\\ i_{j}\neq
l\end{subarray}}\,\frac{m^{\prime}_{i_{1}}\cdots
m^{\prime}_{i_{p}}}{a_{i_{1}l}\cdots a_{i_{p}l}}\\\\[4.0pt] &\hskip
56.9055pt=\Big{(}\sum_{1\leq l<i_{1}<\cdots<i_{p}\leq r}+\sum_{1\leq
i_{1}<l<i_{2}<\cdots<i_{p}\leq r}+\cdots+\sum_{1\leq i_{1}<\cdots<i_{p}<l\leq
r}\Big{)}\,\frac{m^{\prime}_{i_{1}}\cdots
m^{\prime}_{i_{p}}\,m^{\prime}_{l}}{a_{i_{1}l}\cdots a_{i_{p}l}}\\\\[4.0pt]
&\hskip 56.9055pt=\sum_{1\leq l<i_{1}<\cdots<i_{p}\leq
r}\,m^{\prime}_{i_{1}}\cdots
m^{\prime}_{i_{p}}\,m^{\prime}_{l}\,\Big{(}(a_{i_{1}l}\cdots
a_{i_{p}l})^{-1}+\sum_{j=1}^{p}\,a_{li_{j}}^{-1}\
\prod_{\begin{subarray}{c}n=1\\\ n\neq
j\end{subarray}}^{p}\,a_{i_{n}i_{j}}^{-1}\Big{)}\\\\[4.0pt] &\hskip
56.9055pt=\sum_{1\leq l<i_{1}<\cdots<i_{p}\leq
r}\,\frac{m^{\prime}_{i_{1}}\cdots
m^{\prime}_{i_{p}}\,m^{\prime}_{l}}{a_{i_{1}l}\cdots a_{i_{p}l}\
\displaystyle\prod_{1\leq j<n\leq p}\,a_{i_{j}i_{n}}}\
\mathcal{A}_{i_{1}\cdots i_{p}l}\ ,\end{split}$ (B.5)

where, for each increasing sequence $1\leq l<i_{1}<\cdots<i_{p}\leq r$, we set

$\displaystyle\begin{split}\mathcal{A}_{i_{1}\cdots i_{p}l}:\\!&=\prod_{1\leq
j<n\leq p}\,a_{i_{j}i_{n}}-\sum_{j=1}^{p}\,(-1)^{p-j}\
\prod_{\begin{subarray}{c}j^{\prime}=1\\\ j^{\prime}\neq
j\end{subarray}}^{p}\,a_{i_{j^{\prime}}l}\ \prod_{\begin{subarray}{c}1\leq
q<n\leq p\\\ n,q\neq j\end{subarray}}\,a_{i_{q}i_{n}}\\\\[4.0pt]
&=\sum_{t=1}^{p+1}\,(-1)^{p-t-1}\ \prod_{\begin{subarray}{c}1\leq j<n\leq
p+1\\\ j,n\neq t\end{subarray}}a_{i_{j}i_{n}}\ ,\end{split}$ (B.6)

with the convention $i_{p+1}:=l$.

Although it should be possible to show directly that (B.6) vanishes (indeed we
have checked this explicitly up to $p=4$), a more straightforward proof uses
Proposition 2.71 to assert that (B.2) holds at the mass specialisations
$m_{l}^{\prime}=\epsilon_{4}$ for $l=1,\dots,r$. Since (B.6) is independent of
the shifted mass parameters, it follows that

$\displaystyle\mathcal{A}_{i_{1}\cdots i_{p}l}=0$ (B.7)

as required. ∎

We are now ready to sketch an argument that may prove Conjecture 2.105.
Consider the instanton partition function

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m}^{\prime})=1+\sum_{k=1}^{\infty}\,\mathtt{q}^{k}\,Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})$
(B.8)

with the explicit combinatorial expansion (2.100), where
$\vec{m}^{\prime}:=\vec{m}-\vec{a}$. By Proposition 2.71 we know that

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},m^{\prime}_{l}=-\epsilon_{123})=Z_{\mathbbm{C}^{3}}^{r}\big{(}(-1)^{r+1}\,\mathtt{q};\epsilon_{1},\epsilon_{2},\epsilon_{3}\big{)}={M}(-\mathtt{q})^{-\frac{r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}}\
.$ (B.9)

Armed with this information, we can assume that the partition function takes a
form given by

$\displaystyle\log
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m}^{\prime})=f_{r}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})\log
M(-\mathtt{q})+\log G_{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m}^{\prime})\
,$ (B.10)

where $f_{r}$ is a rational function of the equivariant parameters with

$\displaystyle
f_{r}(\vec{a},\vec{\epsilon},m_{l}^{\prime}=-\epsilon_{123})=-\frac{r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}\
,$ (B.11)

and the function $G_{r}$ has a power series expansion

$\displaystyle
G_{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m}^{\prime})=1+\sum_{k=1}^{\infty}\,\mathtt{q}^{k}\,G_{r}^{(k)}(\vec{a},\vec{\epsilon},{\vec{m}^{\prime}})$
(B.12)

whose coefficients $G_{r}^{(k)}$ are rational functions of the equivariant
parameters with

$\displaystyle
G^{(k)}_{r}(\vec{a},\vec{\epsilon},m_{l}^{\prime}=-\epsilon_{123})=0\ .$
(B.13)

From Lemma B.1 it follows that

$\displaystyle
f_{r}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})-G_{r}^{(1)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=-\frac{r\,m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\
.$ (B.14)

Hence, by redefining the functions $f_{r}$ and $G_{r}^{(1)}$ if necessary, we
can assume that

$\displaystyle
f_{r}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=-\frac{r\,m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\qquad\mbox{and}\qquad
G_{r}^{(1)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=0\ .$ (B.15)

The idea now is to prove that
$G_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=0$ by induction on $k$.
We know this for $k=1$, so we suppose
$G_{r}^{(n)}(\vec{a},\vec{\epsilon},\vec{m},r)=0$ for $1\leq n\leq k-1$ with
$k>1$. Then

$\displaystyle
Z^{r,k}_{\mathbbm{C}^{4}}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=F_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})+G_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})\
,$ (B.16)

where $F_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})$ is the
coefficient of $\mathtt{q}^{k}$ in the series expansion of
$\exp(f_{r}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})\log M(-\mathtt{q}))$.
Recalling the symmetries of the matrix integral (2.68), we know that
$Z^{r,k}_{\mathbbm{C}^{4}}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})$ is
invariant under permutation of $\epsilon_{1}$ and
$\epsilon_{4}=-\epsilon_{123}$, as well as under permutation of the entries of
$\vec{m}^{\prime}=(m_{1}^{\prime},\dots,m_{r}^{\prime})$. Since
$F_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})$ is invariant under
these permutations, so is
$G_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})$.

Looking at (2.100), we can decompose the $k$-instanton contributions for $k>1$
into

$\displaystyle
Z^{r,k}_{\mathbbm{C}^{4}}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=\sum_{\imath_{1},\dots,\imath_{r}=1}^{k}\,\mathfrak{Z}_{\mathbbm{C}^{4}}^{r,k;\vec{\imath}}(\vec{a},\vec{\epsilon}\,)\
\prod_{l=1}^{r}\,(m^{\prime}_{l}+\epsilon_{123})^{\imath_{l}}+(-1)^{(r+1)\,k}\,Z^{r,k}_{\mathbbm{C}^{3}}(\epsilon_{1},\epsilon_{2},\epsilon_{3})$
(B.17)

for some functions
$\mathfrak{Z}_{\mathbbm{C}^{4}}^{r,k;\vec{\imath}}(\vec{a},\vec{\epsilon}\,)$.
We now write

$\displaystyle
f_{r}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=-\frac{r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}-\sum_{l=1}^{r}\,\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}\,(m^{\prime}_{l}+\epsilon_{123})}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}$
(B.18)

and use this to separate out the polynomial mass dependence in
$F_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})$, similarly to (B.17),
as

$\displaystyle
F_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=\sum_{\imath_{1},\dots,\imath_{r}=1}^{k}\,\mathfrak{F}_{r}^{(k);\vec{\imath}}(\vec{a},\vec{\epsilon}\,)\
\prod_{l=1}^{r}\,(m^{\prime}_{l}+\epsilon_{123})^{\imath_{l}}+(-1)^{(r+1)\,k}\,Z^{r,k}_{\mathbbm{C}^{3}}(\epsilon_{1},\epsilon_{2},\epsilon_{3})$
(B.19)

with some functions
$\mathfrak{F}_{r}^{(k);\vec{\imath}}(\vec{a},\vec{\epsilon}\,)$. The
coefficient functions
$\mathfrak{Z}_{\mathbbm{C}^{4}}^{r,k;\vec{\imath}}(\vec{a},\vec{\epsilon}\,)$
and $\mathfrak{F}_{r}^{(k);\vec{\imath}}(\vec{a},\vec{\epsilon}\,)$ are
independent of the masses $\vec{m}^{\prime}$ and symmetric in the entries of
$\vec{\imath}=(\imath_{1},\dots,\imath_{r})$.

Then (B.16) determines $G_{r}^{(k)}$ as the Taylor expansion

$\displaystyle
G_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})=\sum_{\imath_{1},\dots,\imath_{r}=1}^{k}\,\big{(}\mathfrak{Z}_{\mathbbm{C}^{4}}^{r,k;\vec{\imath}}(\vec{a},\vec{\epsilon}\,)-\mathfrak{F}_{r}^{(k);\vec{\imath}}(\vec{a},\vec{\epsilon}\,)\big{)}\
\prod_{l=1}^{r}\,(m^{\prime}_{l}+\epsilon_{123})^{\imath_{l}}\ .$ (B.20)

At this stage one should be able to exploit the $\mathbbm{Z}_{2}$-symmetry
$\epsilon_{1}\leftrightarrow-\epsilon_{123}$ of
$G_{r}^{(k)}(\vec{a},\vec{\epsilon},\vec{m}^{\prime})$, together with the
analytic behaviour of (2.100) in $\vec{\epsilon}\,$, to infer that
$\mathfrak{Z}_{\mathbbm{C}^{4}}^{r,k;\vec{\imath}}(\vec{a},\vec{\epsilon}\,)=\mathfrak{F}_{r}^{(k);\vec{\imath}}(\vec{a},\vec{\epsilon}\,)$
for each $\vec{\imath}$. It would be very interesting to understand this
further and complete the proof of Conjecture 2.105 along these lines.

###  B.2 Evidence for Conjecture 4.7

A possible proof of Conjecture 4.7 follows the same line of reasoning as in
Appendix B.1, starting with the dimensional reduction (4.3) according to
Proposition 3.63. We choose $\vec{r}=(1,0,\dots,0)$ without loss of
generality. We can assume that

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\mathtt{q}};\vec{\epsilon},m)&=M\big{(}(-1)^{n}\,\mathtt{Q}\big{)}^{-n\,\frac{m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}-\frac{n^{2}-1}{n}\,\frac{m\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}}\\\
&\quad\,\times\prod_{0<p\leq
s<n}\,\widetilde{M}\big{(}(-1)^{p-s+1}\,\mathtt{q}_{[p,s]},(-1)^{n}\,\mathtt{Q}\big{)}^{-\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\
G(\vec{\mathtt{q}};\vec{\epsilon},m)\end{split}\ ,$ (B.21)

where the function $G$ has a power series expansion

$\displaystyle
G(\vec{\mathtt{q}};\vec{\epsilon},m)=1+\sum_{\vec{k}\in\mathbbm{Z}_{\geq
0}^{n}\setminus\,\vec{0}}\,\vec{\mathtt{q}}^{\,\vec{k}}\
G^{(\vec{k}\,)}(\vec{\epsilon},m)$ (B.22)

whose coefficients are rational functions of the equivariant parameters with
$G^{(\vec{k}\,)}(\vec{\epsilon},m=-\epsilon_{123})=0$.

The idea is to proceed by induction on the size of
$\vec{k}\in\mathbbm{Z}_{\geq 0}^{n}\setminus\vec{0}$ to show that
$G^{(\vec{k}\,)}(\vec{\epsilon},m)=0$. For $|\vec{k}\,|=1$ the only
contribution to the instanton partition function is

$\displaystyle\mathtt{q}_{0}\,\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}\
.$ (B.23)

Thus $G^{(\vec{k}\,)}(\vec{\epsilon},m)=0$ for all $\vec{k}$ of size one. Now
suppose $G^{(\vec{k}\,)}(\vec{\epsilon},m)=0$ for all $\vec{k}$ of sizes
$1\leq|\vec{k}\,|\leq k-1$ with $k>1$. Then for $\vec{k}$ of size
$|\vec{k}\,|=k$ we can write

$\displaystyle
Z^{\vec{k}}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\epsilon},m)=F^{(\vec{k}\,)}(\vec{\epsilon},m)+G^{(\vec{k}\,)}(\vec{\epsilon},m)\
,$ (B.24)

where $F^{(\vec{k}\,)}(\vec{\epsilon},m)$ is the coefficient of
$\vec{\mathtt{q}}^{\,\vec{k}}$ in the power series expansion of

$\displaystyle
M\big{(}(-1)^{n}\,\mathtt{Q}\big{)}^{-n\,\frac{m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}-\frac{n^{2}-1}{n}\,\frac{m\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}}\
\prod_{0<p\leq
s<n}\,\widetilde{M}\big{(}(-1)^{p-s+1}\,\mathtt{q}_{[p,s]},(-1)^{n}\,\mathtt{Q}\big{)}^{-\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\
.$ (B.25)

From the matrix integral (3.62) with $\vec{r}=(1,0,\dots,0)$ for the
$\mathbbm{Z}_{n}$-action of Section 4.1, with weights $s_{1}=1$, $s_{2}=n-1$
and $s_{3}=s_{4}=0$, it follows that the fractional instanton contribution
$Z^{\vec{k}}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\epsilon},m)$
is invariant under the permutation of $\epsilon_{3}$ and
$\epsilon_{4}=-\epsilon_{123}$. From the combinatorial expansion (4.2) it
follows that it can be decomposed into

$\displaystyle
Z^{\vec{k}}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\epsilon},m)=\sum_{\imath=1}^{|\vec{k}\,|}\,(m+\epsilon_{123})^{\imath}\
\mathfrak{Z}^{\vec{k};\imath}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\epsilon}\,)+(-1)^{|\vec{k}\,|+k_{0}}\,Z^{\vec{k}}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}}(\epsilon_{1},\epsilon_{2},\epsilon_{3})\
,$ (B.26)

with some functions
$\mathfrak{Z}^{\vec{k};\imath}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\epsilon}\,)$
for $\imath=1,\dots,|\vec{k}\,|$ which are independent of the mass $m$.

The powers of the generalized MacMahon functions in (B.25) can be rewritten
respectively as

$\displaystyle-n\,\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}+\frac{n^{2}-1}{n}\,\frac{\epsilon_{12}\,\epsilon_{123}}{\epsilon_{1}\,\epsilon_{2}}-\Big{(}n\,\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}-\frac{n^{2}-1}{n}\,\frac{\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}\Big{)}\,(m+\epsilon_{123})$
(B.27)

and

$\displaystyle-\frac{\epsilon_{12}}{\epsilon_{3}}-\frac{\epsilon_{12}\,(m+\epsilon_{123})}{\epsilon_{3}\,\epsilon_{4}}\
.$ (B.28)

Since $F^{(\vec{k}\,)}(\vec{\epsilon},m)$ is symmetric under
$\epsilon_{3}\leftrightarrow-\epsilon_{123}$, it follows from (B.24) that so
is $G^{(\vec{k}\,)}(\vec{\epsilon},m)$, and that it is given as the Taylor
expansion

$\displaystyle
G^{(\vec{k}\,)}(\vec{\epsilon},m)=\sum_{\imath=1}^{|\vec{k}\,|}\,\big{(}\mathfrak{Z}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}^{\vec{k};\imath}(\vec{\epsilon}\,)-\mathfrak{F}^{(\vec{k}\,);\imath}(\vec{\epsilon}\,)\big{)}\,(m+\epsilon_{123})^{\imath}\
,$ (B.29)

with some functions $\mathfrak{F}^{(\vec{k}\,);\imath}(\vec{\epsilon}\,)$ for
$\imath=1,\dots,|\vec{k}\,|$ which are independent of the mass parameter $m$.
At this stage one should be able to exploit invariance under the
$\mathbbm{Z}_{2}$-action $\epsilon_{3}\leftrightarrow-\epsilon_{123}$,
together with the analytic behaviour of (4.2) in $\vec{\epsilon}\,$, to show
that
$\mathfrak{F}^{(\vec{k}\,);\imath}(\vec{\epsilon}\,)=\mathfrak{Z}_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}^{\vec{k};\imath}(\vec{\epsilon}\,)$
for all $\imath\in\\{1,\dots,|\vec{k}\,|\\}$.

## References

  * [1] S. K. Donaldson and R. P. Thomas, _Gauge theory in higher dimensions_ , in _Conference on Geometric Issues in Foundations of Science in honor of Sir Roger Penrose’s 65th Birthday_ , pp. 31–47, 1996.
  * [2] S. K. Donaldson and E. Segal, _Gauge theory in higher dimensions II_ , _Surv. Diff. Geom._ XVI (2011) 1 [0902.3239].
  * [3] M. Yamazaki, _Crystal Melting and Wall Crossing Phenomena_ , _Int. J. Mod. Phys. A_ 26 (2011) 1097 [1002.1709].
  * [4] M. Cirafici and R. J. Szabo, _Curve counting, instantons and McKay correspondences_ , _J. Geom. Phys._ 72 (2013) 54 [1209.1486].
  * [5] A. Iqbal, N. Nekrasov, A. Okounkov and C. Vafa, _Quantum foam and topological strings_ , _JHEP_ 04 (2008) 011 [hep-th/0312022].
  * [6] M. Cirafici, A. Sinkovics and R. J. Szabo, _Cohomological gauge theory, quiver matrix models and Donaldson–Thomas theory_ , _Nucl. Phys. B_ 809 (2009) 452 [0803.4188].
  * [7] M. Cirafici, A. Sinkovics and R. J. Szabo, _Instantons, Quivers and Noncommutative Donaldson–Thomas Theory_ , _Nucl. Phys. B_ 853 (2011) 508 [1012.2725].
  * [8] M. Cirafici, A. Sinkovics and R. J. Szabo, _Instanton counting and wall-crossing for orbifold quivers_ , _Ann. Henri Poincaré_ 14 (2013) 1001 [1108.3922].
  * [9] M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Y. I. Manin, _Construction of Instantons_ , _Phys. Lett. A_ 65 (1978) 185.
  * [10] R. J. Szabo, _Instantons, Topological Strings and Enumerative Geometry_ , _Adv. Math. Phys._ 2010 (2010) 107857 [0912.1509].
  * [11] R. J. Szabo, _Crystals, instantons and quantum toric geometry_ , _Acta Phys. Polon. Supp._ 4 (2011) 461 [1102.3861].
  * [12] E. Witten, _Bound states of strings and $p$-branes_, _Nucl. Phys. B_ 460 (1996) 335 [hep-th/9510135].
  * [13] E. Witten, _BPS Bound states of D0–D6 and D0–D8 systems in a $B$-field_, _JHEP_ 04 (2002) 012 [hep-th/0012054].
  * [14] L. Baulieu, H. Kanno and I. M. Singer, _Special quantum field theories in eight-dimensions and other dimensions_ , _Commun. Math. Phys._ 194 (1998) 149 [hep-th/9704167].
  * [15] L. Baulieu, H. Kanno and I. M. Singer, _Cohomological Yang–Mills theory in eight-dimensions_ , in _APCTP Winter School on Dualities of Gauge and String Theories_ , pp. 365–373, 1997, [hep-th/9705127].
  * [16] B. S. Acharya, M. O’Loughlin and B. J. Spence, _Higher dimensional analogs of Donaldson–Witten theory_ , _Nucl. Phys. B_ 503 (1997) 657 [hep-th/9705138].
  * [17] E. Witten, _Topological Quantum Field Theory_ , _Commun. Math. Phys._ 117 (1988) 353.
  * [18] C. Vafa and E. Witten, _A strong coupling test of S-duality_ , _Nucl. Phys. B_ 431 (1994) 3 [hep-th/9408074].
  * [19] N. Nekrasov, _Magnificent four_ , _Adv. Theor. Math. Phys._ 24 (2020) 1171 [1712.08128].
  * [20] N. Nekrasov and N. Piazzalunga, _Magnificent Four with Colors_ , _Commun. Math. Phys._ 372 (2019) 573 [1808.05206].
  * [21] Y. Cao and N. C. Leung, _Donaldson–Thomas theory for Calabi–Yau 4-folds_ , [1407.7659].
  * [22] D. Borisov and D. Joyce, _Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds_ , _Geom. Topol._ 21 (2017) 3231 [1504.00690].
  * [23] J. Oh and R. P. Thomas, _Counting sheaves on Calabi–Yau 4-folds, I_ , [2009.05542].
  * [24] Y. Cao and M. Kool, _Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds_ , _Adv. Math._ 338 (2018) 601 [1712.07347].
  * [25] G. Bonelli, N. Fasola, A. Tanzini and Y. Zenkevich, _ADHM in 8d, coloured solid partitions and Donaldson–Thomas invariants on orbifolds_ , [2011.02366].
  * [26] T. Kimura, _Double Quiver Gauge Theory and BPS/CFT Correspondence_ , [2212.03870].