# Instanton Counting and Donaldson–Thomas Theory
on Toric Calabi–Yau Four-Orbifolds

Richard J. Szabo<EMAIL_ADDRESS>Department of Mathematics, Heriot–Watt
University
Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK
Maxwell Institute for Mathematical Sciences, Edinburgh, UK
Higgs Centre for Theoretical Physics, Edinburgh, UK Michelangelo Tirelli
<EMAIL_ADDRESS>Department of Mathematics, Heriot–Watt University
Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK
Maxwell Institute for Mathematical Sciences, Edinburgh, UK
Higgs Centre for Theoretical Physics, Edinburgh, UK

###### Abstract

We study rank $r$ cohomological Donaldson–Thomas theory on a toric Calabi–Yau
orbifold of $\mathbbm{C}^{4}$ by a finite abelian subgroup $\mathsf{\Gamma}$
of $\mathsf{SU}(4)$, from the perspective of instanton counting in
cohomological gauge theory on a noncommutative crepant resolution of the
quotient singularity. We describe the moduli space of noncommutative
instantons on $\mathbbm{C}^{4}/\mathsf{\Gamma}$ and its generalized ADHM
parametrization. Using toric localization, we compute the orbifold instanton
partition function as a combinatorial series over $r$-vectors of
$\mathsf{\Gamma}$-coloured solid partitions. When the $\mathsf{\Gamma}$-action
fixes an affine line in $\mathbbm{C}^{4}$, we exhibit the dimensional
reduction to rank $r$ Donaldson–Thomas theory on the toric Kähler three-
orbifold $\mathbbm{C}^{3}/\mathsf{\Gamma}$. Based on this reduction and
explicit calculations, we conjecture closed infinite product formulas, in
terms of generalized MacMahon functions, for the instanton partition functions
on the orbifolds $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$ and
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$,
finding perfect agreement with new mathematical results of Cao, Kool and
Monavari.

EMPG–23–01

###### Contents

  1. 1 Introduction
  2. 2 Eight-Dimensional Cohomological Gauge Theory
    1. 2.1 $\mathsf{Spin}(7)$-Instanton Equations
    2. 2.2 Cohomological Gauge Theory
    3. 2.3 Equivariant Gauge Theory
    4. 2.4 Noncommutative Field Theory
    5. 2.5 Quiver Matrix Model
    6. 2.6 Instanton Partition Function
    7. 2.7 Pure $\mathcal{N}_{\textrm{\tiny T}}=2$ Gauge Theory
  3. 3 Orbifolds of the Eight-Dimensional Theory
    1. 3.1 Noncommutative Gauge Theory on $\mathbbm{C}^{4}/\mathsf{\Gamma}$
    2. 3.2 Crepant Resolutions
    3. 3.3 Noncommutative Crepant Resolutions and their ADHM Representations
    4. 3.4 Cohomological Field Theory on Noncommutative Crepant Resolutions
    5. 3.5 Orbifold Instanton Partition Functions
    6. 3.6 Pure $\mathcal{N}_{\textrm{\tiny T}}=2$ Gauge Theory on $[\mathbbm{C}^{4}/\mathsf{\Gamma}]$
  4. 4 The (2,0) Orbifolds $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$
    1. 4.1 Noncommutative $\mathsf{U}(1)$ Instantons on $\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$
    2. 4.2 Higher Rank Generalizations
    3. 4.3 Pure $\mathcal{N}_{\textrm{\tiny T}}=2$ Gauge Theory on $[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}$
  5. 5 The (3,0) Orbifold $\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$
    1. 5.1 Noncommutative $\mathsf{U}(1)$ Instantons on $\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$
    2. 5.2 Higher Rank Generalization
    3. 5.3 Pure $\mathcal{N}_{\textrm{\tiny T}}=2$ Gauge Theory on $[\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}]\times\mathbbm{C}$
  6. A Generalized ADHM Construction
  7. B Infinite Product Formulas for Instanton Partition Functions
    1. B.1 Evidence for Conjecture 2.105
    2. B.2 Evidence for Conjecture 4.7

##  1 Introduction

The counting of BPS states in string theory and quantum field theory often
leads to deep mathematical insights into counting problems of enumerative
geometry. Of particular interest are vacuum moduli spaces of D-branes and
instantons, which often provide alternative contructions of the relevant
moduli spaces that are otherwise technically difficult to define rigorously in
geometrical approaches. Looking in the other direction, mathematical
constructions of enumerative geometry can shed light on structural properties
of the BPS spectrum of particles in string theory and quantum field theory.

In this paper we are concerned with the connections between BPS state counting
problems and Donaldson–Thomas theory [1, 2]. Donaldson–Thomas invariants are
virtual numbers counting sheaves on a complex variety. They are defined as
integrals of cohomology classes over virtual cycles of moduli spaces of
sheaves. Their best understood physics connection is to type II string theory
compactified on a Kähler threefold $M$, where BPS states preserve half of the
$\mathcal{N}=2$ supersymmetry and correspond to bound states of D-branes. The
vacuum degeneracies are computed by the Witten index and the corresponding
partition function reproduces the generating function for the Donaldson–Thomas
invariants of $M$ (see e.g. [3, 4] for reviews).

The enumeration of D-brane bound states on $M$ can be equivalently
reformulated as an instanton counting problem in a six-dimensional
$\mathcal{N}_{\textrm{\tiny T}}=2$ cohomological gauge theory on $M$. The
computation of Donaldson–Thomas partition functions from this perspective has
been studied in great detail by [5, 6, 7, 8] in the case where $M$ is a toric
Calabi–Yau threefold. For this, the moduli space of $\mathsf{U}(r)$ instantons
is compactified in two ways: by deforming the first order partial differential
equations defining BPS states to operator algebraic equations for
noncommutative instantons, and by a local $\varOmega$-deformation of $M$ which
preserves its $\mathsf{SU}(3)$-holonomy.

For the simplest example $M=\mathbbm{C}^{3}$, the Donaldson–Thomas partition
function enumerates BPS bound states of D$p$-branes inside $r$
D$(p{+}6)$-branes. A six-dimensional version of the ADHM construction in four
dimensions [9] establishes that, from a geometric point of view, the
compactified instanton moduli space is isomorphic to a moduli space of
torsion-free sheaves on complex projective space $\mathbbm{P}^{3}$ with
suitable characteristic classes and framing conditions [6, 7]. The Coulomb
branch instanton partition function can be written in a simple compact form as
[5, 6]

$\displaystyle
Z^{r}_{\mathbbm{C}^{3}}(\mathtt{q})=M\big{(}(-1)^{r}\,\mathtt{q}\big{)}^{r}\
,$ (1.1)

where $M(q)=\prod_{n\geq 1}\,(1-q^{n})^{-n}$ is the MacMahon function and
$\mathtt{q}$ is the Boltzmann weight parameter for instantons. See [10, 11]
for reviews of various physical and geometrical aspects of this
correspondence.

General D$p$–D$p^{\prime}$-brane systems can be considered after turning on
fluxes for the Neveu–Schwarz (NS) $B$-field, in such a way that supersymmetry
is restored in the vacuum [12, 13]. In this paper we are interested in the
codimension $p^{\prime}-p=8$. Many new features appear in eight dimensions.
The corresponding eight-dimensional cohomological gauge theories were
constructed and studied in the late 1990s [14, 15, 16]. They are in many
respects similar to their four-dimensional counterparts, namely
Donaldson–Witten theory [17] and Vafa–Witten theory [18] on complex surfaces.
The equivariant instanton partition functions on $\mathbbm{C}^{4}$ were
recently studied by Nekrasov and Piazzalunga in [19, 20].

The renewed interest in these eight-dimensional quantum field theories has
been sparked by recent mathematical advances in the Donaldson–Thomas theory of
Calabi–Yau fourfolds, starting with the seminal work of Cao and Leung [21]
which constructed a virtual fundamental class in special cases. Subsequently,
virtual cycles of the Donaldson–Thomas moduli spaces for fourfolds were
developed more generally by Borisov and Joyce [22] in the setting of derived
differential geometry, as well as by Oh and Thomas [23] in the setting of
algebraic geometry. The virtual cycle is defined via a suitable choice of a
local orientation at each point of the moduli space. This corresponds to a
choice of _signs_ , a well-known phenomenon in eight dimensions that does not
arise in lower dimensions. These signs are the most important and non-trivial
aspects of the theory, while at the same time presenting one of the major
difficulties. The computations of Cao and Kool [24] show that the choice of
signs seems unique (up to overall orientation), and that they always give the
simplest answer for the Donaldson–Thomas partition functions.

In the following we shall elaborate on several aspects of the construction of
noncommutative instantons in eight dimensions and the evaluation of their
partition functions. We give a new derivation of the equivariant instanton
partition function on $\mathbbm{C}^{4}$ that carefully incorporates the
correct sign choices. Our sign choice differs from other choices that have
appeared so far in the literature.

The central achievement of this paper is a detailed systematic study and
computations of the rank $r$ degree zero cohomological Donaldson–Thomas
invariants of the toric Calabi–Yau four-orbifolds
$\mathbbm{C}^{4}/\mathsf{\Gamma}$, extending the flat space treatment on
$\mathbbm{C}^{4}$. These have so far received only limited attention in both
the physics and mathematics literature. The analogous instanton counting
problem in six dimensions has been studied in detail in [7, 8]; see [4] for a
review with comparisons to the instanton counting problems in the
Donaldson–Witten and Vafa–Witten theories. In eight dimensions, only the
instanton partition functions for some simple cyclic orbifolds have been
briefly discussed in [25, 26]. A thorough mathematical treatment of the rank
one K-theoretic Donaldson–Thomas theory of Calabi–Yau four-orbifolds appears
in parallel to our work in [27], complementing the results of the present
paper, as we discuss further below.

In this paper we construct and study rank $r$ cohomological gauge theory on
quotient stacks $\big{[}\mathbbm{C}^{4}/\mathsf{\Gamma}\big{]}$, where
$\mathsf{\Gamma}$ is a finite abelian subgroup of
$\mathsf{SL}(4,\mathbbm{C})$. This is equivalent to the gauge theory on a
noncommutative crepant resolution of the quotient singularity
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ provided by the path algebra ${\sf A}$ of a
generalization of the McKay quiver determined by representation theory data of
$\mathsf{\Gamma}$, with relations given by a generalized ADHM parametrization
of the orbifold noncommutative instanton equations. The topological gauge
theory localizes by construction on $\mathsf{\Gamma}$-equivariant instanton
configurations. Using toric localization, the orbifold instanton partition
function can be reduced to the fixed points of the moduli space under the
action of the maximal torus of $\mathsf{SU}(4)$ which are also
$\mathsf{\Gamma}$-invariant. These are classified by $r$-vectors of
$\mathsf{\Gamma}$-coloured solid partitions whose Boltzmann weights depend on
the representations of $\mathsf{\Gamma}$; this was also mentioned by [25] and
is analogous to the combinatorial description of orbifold instantons in six
dimensions in terms of plane partitions [7]. We assert that the BPS partition
function on $[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ conjecturally provides the
corresponding orbifold Donaldson–Thomas invariants.

Noncommutative Donaldson–Thomas invariants of four-Calabi–Yau algebras already
appear in the seminal work of [21]; see also [28] for the example of the local
resolved conifold. Our path algebras ${\sf A}$ are always four-Calabi–Yau, and
in fact they are Koszul since
$\mathsf{\Gamma}\subset\mathsf{SL}(4,\mathbbm{C})$. In the three-orbifold
case, this fact was used repeatedly in [7] to establish the relation between
equivariant sheaves on $\mathbbm{C}^{3}$ and BPS states in the noncommutative
resolution chamber of resolved Calabi–Yau singularities. The wall-and-chamber
structure of the Kähler moduli space is discussed in [28] for the example of
the local resolved conifold, building on the standard threefold case [29].

If there exists a toric crepant resolution $X$ of the quotient singularity
$\mathbbm{C}^{4}/\mathsf{\Gamma}$, we can expect an analogous relation, as
well as a version of the McKay correspondence which would establish an
equivalence between the derived category of ${\sf A}$-modules and the derived
category of $X$, when the former admits a tilting object. This should then
relate the instanton partition functions in the ‘orbifold’ and ‘large radius’
phases by changes of variables and wall-crossing formulas. As such an analysis
is out of reach with our current techniques, we defer it to future
investigations.

#### Summary of Results

Before starting a systematic study of our main topic, we first review in some
detail the analogous problem on the flat space $\mathbbm{C}^{4}$, in order to
set the stage as the extension to orbifolds is then relatively
straightforward. We elaborate on the analysis of noncommutative instantons in
$\mathsf{U}(r)$ cohomological gauge theory on $\mathbbm{C}^{4}$ with
$\mathsf{SU}(4)$-holonomy and their generalized ADHM parametrization. They are
realized in a dimensional reduction of $\mathcal{N}=1$ supersymmetric
Yang–Mills theory from ten dimensions, and also in type II string theory where
the noncommutative deformation corresponds to turning on a non-zero constant
background $B$-field in the flat ten-dimensional target spacetime [30]. The
instanton partition function on an $\varOmega$-background is then regarded as
an equivariant integral over the instanton moduli space [31] and can be
evaluated using toric localization techniques as a combinatorial expansion in
random solid partitions.

We evaluate the equivariant instanton partition function of the
$\mathsf{U}(r)$ cohomological gauge theory on $\mathbbm{C}^{4}$ with $r$
massive fundamental matter fields. It can be expressed in an exact closed form
as (Conjecture 2.105)

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{\epsilon},m)=M(-\mathtt{q})^{-\frac{r\,m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}}$
(1.2)

with

$\displaystyle m=\frac{1}{r}\,\sum_{l=1}^{r}\,(m_{l}-a_{l})\ ,$ (1.3)

where $a_{l}$ and $m_{l}$ are the Coulomb and mass parameters associated to
the maximal tori of the global $\mathsf{U}(r)$ colour and flavour symmetry
groups, respectively, while $\epsilon_{a}$ are coordinates on the maximal
torus of $\mathsf{SU}(4)$ satisfying the Calabi–Yau constraint
$\epsilon_{1}+\cdots+\epsilon_{4}=0$; we use the notation
$\epsilon_{ab}=\epsilon_{a}+\epsilon_{b}$.

The formula (1.2) for the equivariant Donaldson–Thomas partition function on
$\mathbbm{C}^{4}$ is well-known and has appeared many times before in the
literature. We sketch a possible alternative analytic proof which, while
incomplete and lacking conceptual insight, highlights the symmetries of the
theory which are not evident in other approaches and can be potentially
extended to theories on more complicated spaces, like some of our orbifolds.
It is based on the fact that in six dimensions the Coulomb branch instanton
partition function is known for generic $\varOmega$-deformation of
$\mathbbm{C}^{3}$ [32, 33, 34] and is recovered from the eight-dimensional
theory through the mass specialization $m_{l}=\epsilon_{4}+a_{l}$ (Proposition
2.71). The idea then is to show that (1.2) is the unique expression determined
by the instanton deformation complex which correctly reduces in the six-
dimensional limit and preserves all symmetries of the cohomological matrix
model representation of the instanton partition function.

Moving on to our orbifold theories, we prove an orbifold version of
Proposition 2.71 which relates the equivariant instanton partition function on
an orbifold of the form $\mathbbm{C}^{3}/\mathsf{\Gamma}\times\mathbbm{C}$,
where $\mathsf{\Gamma}\subset\mathsf{SL}(3,\mathbbm{C})$, to the
noncommutative Donaldson–Thomas partition function for the toric Kähler
orbifold $\mathbbm{C}^{3}/\mathsf{\Gamma}$ through mass specialisation
(Proposition 3.63). In the rank one case $r=1$, the noncommutative
Donaldson–Thomas partition function on $\mathbbm{C}^{3}/\mathsf{\Gamma}$ is
expressed by a closed formula precisely for the the Kleinian group
$\mathsf{\Gamma}=\mathbbm{Z}_{n}$ in $\mathsf{SL}(2,\mathbbm{C})$ and the
orbifold group $\mathsf{\Gamma}=\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ in
$\mathsf{SL}(3,\mathbbm{C})$ at the Calabi–Yau specialization
$\epsilon_{1}+\epsilon_{2}+\epsilon_{3}=0$ on $\mathbbm{C}^{3}$ [35, 7].

For $\mathsf{\Gamma}=\mathbbm{Z}_{n}$ the infinite product formula is extended
to generic triples $(\epsilon_{1},\epsilon_{2},\epsilon_{3})$ in [36]. From
this we assert that our proposed proof of Conjecture 2.105 should be possible
to adapt to show that 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 (Conjecture
4.7)

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{n}]\times\mathbbm{C}^{2}}(\vec{\mathtt{q}};\vec{\epsilon},m)&=M\big{(}(-1)^{n}\,\mathtt{Q}\big{)}^{-n\,\frac{m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}-\frac{n^{2}-1}{n}\,\frac{m\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}}\\\
&\quad\,\times\prod_{0<p\leq
s<n}\,\widetilde{M}\big{(}(-1)^{p-s+1}\,\mathtt{q}_{[p,s]},(-1)^{n}\,\mathtt{Q}\big{)}^{-\frac{m\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\
,\end{split}$ (1.4)

where $\mathtt{Q}=\mathtt{q}_{0}\,\mathtt{q}_{1}\cdots\mathtt{q}_{n-1}$,
$\mathtt{q}_{[p,s]}=\mathtt{q}_{p}\,\mathtt{q}_{p+1}\cdots\mathtt{q}_{s-1}\,\mathtt{q}_{s}$,
and we weigh the fractional instanton contributions with fugacities
$\mathtt{q}_{s}$ indexed by the irreducible representations of
$\mathsf{\Gamma}=\mathbbm{Z}_{n}$. The infinite product $M(x,q)=\prod_{k\geq
1}\,(1-x\,q^{k})^{-k}$ is the generalized MacMahon function and we have set
$\widetilde{M}(x,q)=M(x,q)\,M(x^{-1},q)$.

For $\mathsf{\Gamma}=\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ we do not yet have
available results for a generic $\varOmega$-deformation of $\mathbbm{C}^{3}$,
but we provide strong evidence in favour of the closed formula of Conjecture
5.10. It seems unlikely that there are any other orbifolds
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ for which exact infinite product expressions
for the instanton partition functions are even conjecturally possible; see
[27] for a geometric explanation of this.

For higher rank theories we formulate various conjectural closed formulas for
orbifold instanton partition functions with particular framing decompositions
$\vec{r}$ of the rank $r$ according to the irreducible representations of
$\mathsf{\Gamma}$, for the orbifolds
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}^{2}$ and
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$
(Conjectures 4.11, 4.21 and 5.15). These conjectures are developed from the
$\mathsf{\Gamma}$-equivariant instanton deformation complex associated to the
ADHM-type parametrization of the orbifold instanton moduli space. From this we
obtain a combinatorial expression for the orbifold instanton partition
function as a sum over $r$-vectors of $\mathsf{\Gamma}$-coloured solid
partitions, from which we deduce its symmetries and properties that are
enforced in our conjectures. Then we require that they correctly reduce to the
known expressions on $\mathbbm{C}^{3}/\mathsf{\Gamma}$.

One of the novelties in eight dimensions is that, contrary to the standard
noncommutative Donaldson–Thomas invariants [29], our invariants depend
explicitly on the framing vector $\vec{r}$, and contrary to the invariants of
[7, 8], the $\vec{r}$ dependence here is genuine and cannot simply be encoded
in a multiplicative factor. In the case of the orbifold
$\mathbbm{C}^{2}/\mathbbm{Z}_{2}\times\mathbbm{C}^{2}$ we obtain a conjectural
formula for the equivariant instanton partition function of type
$\vec{r}=(r_{0},r_{1})$ for the rank $r$ cohomological gauge theory with $r$
massive fundamental matter fields on
$[\mathbbm{C}^{2}/\mathbbm{Z}_{2}]\times\mathbbm{C}^{2}$. It is given by
(Conjecture 4.21)

$\displaystyle\begin{split}Z_{[\mathbbm{C}^{2}/\mathbbm{Z}_{2}]\times\mathbbm{C}^{2}}^{\vec{r}}(\vec{\mathtt{q}};\vec{\epsilon},\vec{m})&=M(\mathtt{Q})^{-2\,\frac{m\,r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{3}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}-\frac{3}{2}\,\frac{m\,r\,\epsilon_{12}}{\epsilon_{1}\,\epsilon_{2}}}\\\
&\quad\,\times\widetilde{M}(-\mathtt{q}_{1},\mathtt{q}_{0}\,\mathtt{q}_{1})^{-\frac{m^{0}\,r_{0}\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\
\widetilde{M}(-\mathtt{q}_{0},\mathtt{q}_{0}\,\mathtt{q}_{1})^{-\frac{m^{1}\,r_{1}\,\epsilon_{12}}{\epsilon_{3}\,\epsilon_{4}}}\
,\end{split}$ (1.5)

where $\vec{m}=(m^{0},m^{1})$ is the $\mathsf{\Gamma}$-module decomposition of
the center of mass parameter (1.3).

As offsprings of our conjectures, we obtain predictions for the rank $r$
Donaldson–Thomas partition functions on the Kähler three-orbifolds
$\mathbbm{C}^{2}/\mathbbm{Z}_{n}\times\mathbbm{C}$ and
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with generic
$\mathsf{U}(3)$-holonomy, in the cases where they are not yet known. See
Propositions 4.16, 4.24, 5.12 and 5.18.

The $\varOmega$-deformation also enables the definition of the “pure”
cohomological gauge theory on $\mathbbm{C}^{4}$ without the massive
fundamental matter fields. The instanton partition function for the pure gauge
theory is related to (1.2) by the infinite mass limit of Proposition 2.112
which decouples the fundamental hypermultiplet. For rank $r=1$ this gives
(Corollary 2.115)

$\displaystyle Z_{\mathbbm{C}^{4}}(\Lambda;\vec{\epsilon}\,)^{\rm
pure}=\exp\Big{(}-\Lambda\
\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\,\Big{)}\
,$ (1.6)

where $\Lambda$ is the ultraviolet scale of the quantum field theory, whereas
for higher rank $r>1$ the instanton contributions all vanish.

Analogously, we define the “pure” gauge theory on orbifolds. The orbifold
instanton partition function for the pure gauge theory is related to the
orbifold instanton partition function of the cohomological gauge theory with
massive fundamental matter by the infinite mass limit of Proposition 3.99. For
rank $r>1$ the orbifold instanton contributions also all vanish (Propositions
4.27 and 5.20).

######  Remark 1.7.

As the writing of this paper was nearing completion, a related work by Kimura
appeared [26], which writes down the same matrix integral expression (3.62)
from a different perspective for the instanton partition function on the
orbifold $\mathbbm{C}^{4}/\mathsf{\Gamma}$, in the special case where
$\mathsf{\Gamma}=\mathbbm{Z}_{n}$ is a generic cyclic subgroup of
$\mathsf{SU}(4)$. Kimura also gives a free field representation of the
orbifold instanton partition function in that case, in the spirit of the
BPS/CFT correspondence [37], but does not discuss the explicit evaluation of
the contour integrals nor the geometrical applications to Donaldson–Thomas
theory.

#### Relation to the Work of Cao, Kool and Monavari

Our work is complementary to new independent mathematical work by Cao, Kool
and Monavari [27] who consider Donaldson–Thomas invariants of general
Calabi–Yau four-orbifolds using an algebro-geometric approach. In the local
toric model, they compute rank one Donaldson–Thomas invariants using
equivariant localization and a vertex formalism on quotient singularities
$\mathbbm{C}^{4}/\mathsf{\Gamma}$, where $\mathsf{\Gamma}$ is a finite abelian
subgroup of $\mathsf{SL}(4,\mathbbm{C})$. When $\mathsf{\Gamma}$ is the
Kleinian subgroup $\mathbbm{Z}_{n}\subset\mathsf{SL}(2,\mathbbm{C}$) or the
subgroup
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}\subset\mathsf{SL}(3,\mathbbm{C})$, they
also conjecture closed formulas for the equivariant K-theoretic partition
functions, generalizing the Nekrasov–Piazzalunga partition function [20]. They
recover the cohomological invariants as well as three-orbifold invariants via
the dimensional reduction of [38].

They also consider the Pandharipande–Thomas invariants of the crepant
resolutions of $\mathbbm{C}^{4}/\mathsf{\Gamma}$ (when they exist), including
the cases of non-abelian orbifold groups $\mathsf{\Gamma}$. They again
conjecture closed formulas for the equivariant K-theoretic partition
functions, as well as relate the Donaldson–Thomas invariants for orbifold
groups $\mathbbm{Z}_{n}$ and $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with the
Pandharipande–Thomas invariants of the corresponding crepant resolutions by
certain changes of variables.

Their work complements ours by developing the rank one K-theory version of the
story from a rigorous purely algebro-geometric perspective, whereas we also
develop the higher rank cohomological Donaldson–Thomas invariants of toric
Calabi–Yau four-orbifolds from the viewpoint of quantum field theory. Up to
slightly different conventions, our results perfectly match. More precisely,
we can summarise the main agreements as follows:

  * •

Conjecture 4.7 agrees with the cohomological limit of [27, Conjecture 5.13]
(see [27, Corollary 6.6]).

  * •

The rank one case of Proposition 4.27 agrees with [27, Corollary 6.10].

  * •

Conjecture 5.10 agrees with the cohomological limit of [27, Conjecture 5.14]
(see [27, Corollary 6.6]).

  * •

The rank one case of Proposition 5.20 agrees with [27, Corollary 6.10].

#### Outline

Throughout our presentation we gloss over numerous technical subtleties
without comment. The organisation of the remainder of this paper can be
briefly summarised as follows:

  * •

In Section 2 we provide an elaborate review of the construction of an eight-
dimensional cohomological gauge theory for the holonomy group
$\mathsf{SU}(4)$. We study its instanton solutions, discuss properties of the
instanton moduli space and define an ADHM-type parametrization. We evaluate
the equivariant instanton partition function using a quiver matrix model for
the ADHM data and the tangent-obstruction complex of the instanton moduli
space. We discuss the relations to Donaldson–Thomas invariants of
$\mathbbm{C}^{4}$ and the analogous enumerative theory in six dimensions.

  * •

In Section 3 we study instantons on orbifolds
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ with $\mathsf{\Gamma}$ a finite abelian
subgroup of $\mathsf{SL}(4,\mathbbm{C})$. We extend the results of Section 2
to the cohomological gauge theories on quotient stacks
$[\mathbbm{C}^{4}/\mathsf{\Gamma}]$ for generic $\mathsf{\Gamma}$-actions,
providing numerous explicit examples. We also give a supplementary overview of
crepant resolutions of $\mathbbm{C}^{4}/\mathsf{\Gamma}$ as well as their
relevance to the BPS state counting problems for D-brane bound states and
instantons.

  * •

In the final two Sections 4 and 5 we focus on two examples of orbifolds where
it appears possible to explicitly sum the instanton series, namely
$\mathbbm{C}^{2}/{\mathbbm{Z}_{n}}\times\mathbbm{C}^{2}$ and
$\mathbbm{C}^{3}/(\mathbbm{Z}_{2}\times\mathbbm{Z}_{2})\times\mathbbm{C}$. We
conjecture closed infinite product formulas for the orbifold instanton
partition functions in terms of combinations of generalized MacMahon
functions.

  * •

Appendix A presents some details of the generalized ADHM construction of the
instanton moduli space, while in Appendix B we sketch possible analytic proofs
of the closed expressions for the instanton partition functions on
$\mathbbm{C}^{4}$ for general rank $r$, as well as on the orbifold
$\mathbbm{C}^{2}/{\mathbbm{Z}_{n}}\times\mathbbm{C}^{2}$ in the rank one case.

#### Acknowledgements

We thank Yalong Cao, Martijn Kool and Sergej Monavari for helpful discussions
and correspondence, and for sharing a draft of their manuscript [27] with us
prior to submission. M.T. is particularly grateful to Sergej Monavari for
valuable comments on our manuscript and enlightening conversations. We thank
an anonymous referee whose comments prompted us to clarify various aspects of
our presentation. R.J.S. is grateful to the Mainz Institute for Theoretical
Physics (MITP) of the DFG Cluster of Excellence PRISMA+ (Project ID 39083149)
for its hospitality and support during the completion of this work. The work
of R.J.S. was supported in part by the STFC Consolidated Grant ST/P000363/1.
The work of M.T. is supported by an EPSRC Doctoral Training Partnership grant.

##  2 Eight-Dimensional Cohomological Gauge Theory

In this section we review and elaborate on several well-known aspects of
generalized instantons and their equivariant partition functions in
cohomological gauge theory on $\mathbbm{C}^{4}$. We do this in some detail, as
the treatment is needed and is parallel to the set-up used in discussing the
orbifold theories. In particular, we discuss in detail
$\mathsf{Spin}(7)$-instantons in noncommutative gauge theory and their ADHM
parametrizations, which also figure in the generalized gauge theories of [39,
40], where some explicit solutions are also discussed. It is instructive for
the reader to keep in mind the analogous treatments in lower dimensions; see
[41] for a review and comparison of instanton counting theories in four, six
and eight dimensions, and [42] for the analogue discussion of ADHM-type quiver
matrix models.

###  2.1 $\mathsf{Spin}(7)$-Instanton Equations

Let $M\simeq\mathbbm{R}^{8}$ be an oriented eight-dimensional real vector
space endowed with the standard flat Euclidean metric. Instantons on $M$ are
solutions to first order self-duality equations for gauge fields, which are
preserved by a holonomy group $\mathsf{H}\subset\mathsf{SO}(8)$ [15, 16]. The
largest possible holonomy group is $\mathsf{Spin}(7)\subset\mathsf{SO}(8)$
[43], where the $\mathsf{Spin}(7)$-structure is specified by the closed non-
degenerate Cayley four-form $\Phi$. It can be expressed in an oriented
orthonormal coframe $\\{e^{\mu}\\}_{\mu=1}^{8}$ of $M\simeq\mathbbm{R}^{8}$,
where the metric is $\delta_{\mu\nu}\,e^{\mu}\otimes e^{\nu}$, as [44]

$\begin{split}\Phi&=e^{1256}+e^{1278}+e^{3456}+e^{3478}+e^{1257}-e^{1368}-e^{2456}\\\
&\quad\,+e^{2468}-e^{1458}-e^{1467}-e^{2358}-e^{2367}+e^{1234}+e^{5678}\
,\end{split}$ (2.1)

where we generally use the shorthand notation
$e^{\mu_{1}\cdots\mu_{n}}:=e^{\mu_{1}}\wedge\cdots\wedge e^{\mu_{n}}$. The
four-form $\Phi$ is self-dual, $\ast\,\Phi=\Phi$, where
$\ast:\text{\Large$\wedge$}^{k}\,\mathbbm{R}^{8}\longrightarrow\text{\Large$\wedge$}^{8-k}\,\mathbbm{R}^{8}$
is the Hodge duality operator on the space of $k$-forms associated to the
metric on $M$.

The generalized instanton equations can be written as

$\displaystyle\lambda\,F=\ast\,(\Phi\wedge F)\ ,$ (2.2)

where $F=\mathrm{d}A+A\wedge A$ is the curvature two-form of a $\mathsf{U}(r)$
gauge connection $A$ on $M$. The eigenvalues $\lambda=-3$ and $\lambda=1$
correspond respectively to the two eigenspaces of the self-adjoint operator
$\ast\,(\Phi\wedge-)$ in the decomposition of the $28$-dimensional space of
two-forms $\text{\Large$\wedge$}^{2}\,\mathbbm{R}^{8}$ into two irreducible
representations $\mathbf{28}|_{\mathsf{Spin}(7)}=\mathbf{7}\oplus\mathbf{21}$
of $\mathsf{Spin}(7)$. In this paper we focus on solutions of (2.2) with
$\lambda=1$, which are called $\mathsf{Spin}(7)$-instantons and form the basis
of Donaldson–Thomas theory in eight dimensions [1, 2].

The Bianchi identity $DF=\mathrm{d}F+A\wedge F=0$ together with (2.2) for
$\lambda=1$ imply

$\displaystyle D\ast F=D(\Phi\wedge F)=\Phi\wedge DF=0\ ,$ (2.3)

where we used $\mathrm{d}\Phi=0$. Thus every solution of the
$\mathsf{Spin}(7)$-instanton equations is a solution of the second order
Yang–Mills equations. A typical solution has moduli parametrizing the center
of the instanton in $\mathbbm{R}^{8}$ and its scale $\ell\in\mathbbm{R}_{>0}$
(see e.g. [45]). Thus the instanton moduli space $\mathfrak{M}_{r}$, i.e. the
space of smooth solutions to (2.2) for $\lambda=1$ modulo gauge
transformations, is not compact because instantons can be shrunk to zero size
($\ell\rightarrow 0^{+}$) where they become singular. Deforming the equations
(2.2) to noncommutative instanton equations provides an ultraviolet
regularization which resolves these small instanton singularities.

The noncommutative deformation amounts to quantizing a constant Poisson
structure on $\mathbbm{R}^{8}$ defined by a bivector $\theta$ of maximal rank.
After using an $\mathsf{SO}(8)$ rotation to transform $\theta$ into its Jordan
canonical form and a suitable rescaling of coordinates, it defines a complex
structure on $\mathbbm{R}^{8}$. Conversely, we can choose one of the seven
complex structures on $\mathbbm{R}^{8}$, consider the associated Kähler form
$\omega$ corresponding to the metric $\delta_{\mu\nu}\,e^{\mu}\otimes
e^{\nu}$, and set $\theta=\zeta\,\omega^{-1}$, where $\zeta$ is a positive
real parameter. The noncommutative deformation introduces an additional length
scale $\zeta$ which serves as a minimum size for the instantons, so it
resolves the moduli space singularity as instantons can no longer reach the
singularity where their size vanishes. The introduction of a complex structure
$J$ breaks the holonomy group to
$\mathsf{SU}(4)\simeq\mathsf{Spin}(6)\subset\mathsf{Spin}(7)$, so that the
instanton equations are now defined on the Calabi–Yau fourfold
$\mathbbm{C}^{4}$.

Let $\Theta^{a}=e^{2a-1}+\mathrm{i}\,e^{2a}$ and
$\Theta^{\bar{a}}=\bar{\Theta}^{a}$ be a coframe on $M=\mathbbm{C}^{4}$ which
forms a basis of $(1,0)$-forms and $(0,1)$-forms with respect to $J$, that is

$\displaystyle J\,\Theta^{a}=\mathrm{i}\,\Theta^{a}\qquad\mbox{for}\quad a\
\in\ \underline{\sf 4}:=\\{1,2,3,4\\}\ .$ (2.4)

Then there is a non-degenerate $(4,0)$-form $\Omega$ and a $(1,1)$-form
$\omega$ defined by [46]

$\displaystyle\Omega=\Theta^{1}\wedge\Theta^{2}\wedge\Theta^{3}\wedge\Theta^{4}\qquad\mbox{and}\qquad\omega=\tfrac{\mathrm{i}}{2}\,\big{(}\Theta^{1}\wedge\Theta^{\bar{1}}+\Theta^{2}\wedge\Theta^{\bar{2}}+\Theta^{3}\wedge\Theta^{\bar{3}}+\Theta^{4}\wedge\Theta^{\bar{4}}\big{)}\
,$ (2.5)

which obey

$\displaystyle\Omega\wedge\omega=0\qquad\mbox{and}\qquad\Omega\wedge\bar{\Omega}=\tfrac{2}{3}\,\omega^{4}=16\,{\tiny\tt
vol}\ .$ (2.6)

The pair $(\omega,\Omega)$ defines an $\mathsf{SU}(4)$-structure on $M$ and
there is a compatible $\mathsf{Spin}(7)$-structure determined by

$\displaystyle\Phi=\tfrac{1}{2}\,\omega\wedge\omega-{\rm Re}\,\Omega\ ,$ (2.7)

which coincides with the Cayley four-form (2.1). The corresponding Kähler
metric is $\omega\circ
J=\delta_{a\bar{a}}\,\Theta^{a}\otimes\Theta^{\bar{a}}$.

We can write the field strength $F$ in the basis of $(1,0)$-forms and
$(0,1)$-forms as

$\displaystyle
F=F^{2,0}+F^{1,1}+F^{0,2}=\tfrac{1}{2}\,F_{ab}\,\Theta^{a}\wedge\Theta^{b}+F_{a\bar{b}}\,\Theta^{a}\wedge\Theta^{\bar{b}}+\tfrac{1}{2}\,F_{\bar{a}\bar{b}}\,\Theta^{\bar{a}}\wedge\Theta^{\bar{b}}\
.$ (2.8)

When the $\mathsf{Spin}(7)$-structure is determined by an
$\mathsf{SU}(4)$-structure via (2.7), the $\mathsf{Spin}(7)$-instanton
equations (2.2) can be reduced via the inclusion
$\mathsf{SU}(4)\subset\mathsf{Spin}(7)$ to the equations [1, 2]

$\displaystyle
F_{ab}=\tfrac{1}{2}\,\varepsilon_{ab\bar{c}\bar{d}}\,F_{\bar{c}\bar{d}}\qquad\mbox{and}\qquad\delta^{a\bar{a}}\,F_{a\bar{a}}=0\
,$ (2.9)

for $a,b\in\underline{\sf 4}$, where $\varepsilon_{ab\bar{c}\bar{d}}$ is the
Levi–Civita tensor in four dimensions.

The $\mathsf{Spin}(7)$-instanton equations (2.9) are weaker than the Hermitian
Yang–Mills equations that appear in the analogue instanton equations in four
and six dimensions, which amount to replacing the first equation of (2.9) with
$F_{ab}=0$ and describe semistable holomorphic vector bundles on $M$. Any
solution of the Hermitian Yang–Mills equations is automatically a solution of
the $\mathsf{Spin}(7)$-instanton equations, but not conversely in general.

###  2.2 Cohomological Gauge Theory

The cohomological Yang–Mills theory which describes the instanton moduli
problem in eight dimensions corresponding to the $\mathsf{SU}(4)$-structure
defined by (2.5) was constructed in [47, 14] using the BRST formalism to gauge
fix the topological action functional

$\displaystyle S_{0}:=\int_{M}\,\Omega\wedge\mathrm{Tr}\big{(}F^{0,2}\wedge
F^{0,2}\big{)}\ ,$ (2.10)

where $\mathrm{Tr}$ denotes the trace in the fundamental representation of the
(complexified) gauge group. It can be equivalently obtained by dimensional
reduction from ten-dimensional $\mathcal{N}=1$ supersymmetric Yang–Mills
theory with gauge group $\mathsf{U}(r)$ [48], whose field content consists of
the gauge field $A$ and a Majorana–Weyl spinor in the adjoint representation
of $\mathsf{U}(r)$. This also identifies it as the low-energy effective field
theory on D-branes in type II string theory.

One starts by compactifying the time dimension and the ninth spatial dimension
of $\mathbbm{R}^{1,9}$ on a square torus
$\mathbbm{T}^{1,1}=\mathbbm{R}^{1,1}\,\big{/}\,2\pi\,R\,\mathbbm{Z}^{2}$ of
radius $R$. Accordingly, the field theory is defined on the spacetime
$\mathbbm{R}^{8}\times\mathbbm{T}^{1,1}$, and in the limit $R\rightarrow 0$
the unbroken global symmetry group is

$\displaystyle\mathsf{SO}(1,9)\ \supset\ \mathsf{SO}(8)\times\mathsf{SO}(1,1)\
.$ (2.11)

The components of the gauge field $A_{0}$ and $A_{9}$ become a pair of adjoint
scalar fields on $M=\mathbbm{R}^{8}$ after dimensional reduction. The reduced
field theory has a non-compact R-symmetry group $\mathsf{SO}(1,1)$ which in
the bosonic sector acts only on these scalars. The Majorana–Weyl spinor field
reduces according to the decomposition of the positive chirality real spinor
representation of $\mathsf{SO}(1,9)$ into irreducible representations of the
subgroup (2.11) given by

$\displaystyle{{\bm{1}}6}_{+}\big{|}_{\mathsf{SO}(8)\times\mathsf{SO}(1,1)}={\bm{(8}}{}_{\rm
s}{\bm{,+1)}}\,\oplus\,{\bm{(8}}{}_{\rm c}{\bm{,-1)}}\ ,$ (2.12)

where ${{\bm{8}}}_{\rm s}$ and ${{\bm{8}}}_{\rm c}$ are respectively the
chiral and antichiral real spinor representations of $\mathsf{SO}(8)$.

With the reduced holonomy group $\mathsf{SU}(4)\subset\mathsf{Spin}(7)$, the
ambient space becomes the Calabi–Yau fourfold $\mathbbm{C}^{4}$, characterized
by a complex structure $J$ and a compatible $\mathsf{SU}(4)$-structure
$(\omega,\Omega)$ from (2.4)–(2.6). The doublet of real scalars are naturally
grouped into a complex Higgs field $\varphi$. Under $\mathsf{SU}(4)$ the
curvature two-form $F$ decomposes as in (2.8).

The $\mathsf{SU}(4)$ holonomy preserves two supercharges. Under
$\mathsf{SU}(4)$ the spinor representations $\mathbf{8}_{\rm s}$ and
$\mathbf{8}_{\rm c}$ branch into irreducible representations as

$\displaystyle\begin{split}\mathbf{8}_{\rm
s}\big{|}_{\mathsf{SU}(4)}&={\bm{1}}\oplus{\bm{6}}\oplus{\bm{1}}\simeq\text{\Large$\wedge$}^{0}\,\mathbbm{C}^{4}\,\oplus\,\text{\Large$\wedge$}^{2,0}\,\mathbbm{C}^{4}\,\oplus\,\text{\Large$\wedge$}^{4,0}\,\mathbbm{C}^{4}\
,\\\\[4.0pt] \mathbf{8}_{\rm
c}\big{|}_{\mathsf{SU}(4)}&={\bm{4}}\oplus\bar{{\bm{4}}}\simeq\text{\Large$\wedge$}^{1,0}\,\mathbbm{C}^{4}\,\oplus\,\text{\Large$\wedge$}^{3,0}\,\mathbbm{C}^{4}\
.\end{split}$ (2.13)

Accordingly, in the fermionic sector the spectrum consists of a complex scalar
and a $(2,0)$-form, along with $(1,0)$-forms and $(0,1)$-forms. This matches
the field content of the bosonic spectrum, which consists of the complex gauge
field

$\displaystyle
A=A^{1,0}+A^{0,1}=A_{a}\,\Theta^{a}+A_{\bar{a}}\,\Theta^{\bar{a}}$ (2.14)

and Higgs field $\varphi$ along with their complex conjugates. This recovers
the field content of the holomorphic $\mathcal{N}_{\textrm{\tiny T}}=2$
topological Yang–Mills theory in eight dimensions [47, 14].

Here we shall focus on the purely bosonic sector of the dimensionally reduced
field theory on $\mathbbm{C}^{4}$. For this, let
$\ast:\text{\Large$\wedge$}^{k,l}\,\mathbbm{C}^{4}\longrightarrow\text{\Large$\wedge$}^{4-l,4-k}\,\mathbbm{C}^{4}$
be the map induced by the Hodge duality operator. Together with the
holomorphic four-form $\Omega$, it defines an anti-linear operator
$\star_{\Omega}:\text{\Large$\wedge$}^{0,k}\,\mathbbm{C}^{4}\longrightarrow\text{\Large$\wedge$}^{0,4-k}\,\mathbbm{C}^{4}$
by $\star_{\Omega}\,\beta:=\overline{\ast(\beta\wedge\Omega)}$ for
$\beta\in\text{\Large$\wedge$}^{0,k}\,\mathbbm{C}^{4}$. With
$\mathfrak{g}:=\mathfrak{gl}(r,\mathbbm{C})$, this operation has the property
that the standard inner product of
$\alpha,\beta\in\text{\Large$\wedge$}^{0,k}\,T^{*}\mathbbm{C}^{4}\otimes\mathfrak{g}$
can be expressed as

$\displaystyle\langle\alpha,\beta\rangle:=\int_{M}\,\mathrm{Tr}\big{(}\alpha\wedge\ast\,\beta^{\dagger}\big{)}=\int_{M}\,\Omega\wedge\mathrm{Tr}\big{(}\alpha\wedge\star_{\Omega}\,\beta\big{)}\
.$ (2.15)

There is a related gauge-invariant complex quadratic form defined for
$\alpha\in\text{\Large$\wedge$}^{0,2}\,T^{*}\mathbbm{C}^{4}\otimes\mathfrak{g}$
by

$\displaystyle(\alpha,\alpha)_{\Omega}:=\int_{M}\,\Omega\wedge\mathrm{Tr}(\alpha\wedge\alpha)\
.$ (2.16)

The Yang–Mills Lagrangian density on $M$ can be written as ${\tiny\tt
vol}\,\mathrm{Tr}\big{(}-\frac{1}{4}\,F_{\mu\nu}^{2}\big{)}=\frac{1}{2}\,\mathrm{Tr}(F\wedge\ast
F)$, and so the Yang–Mills action functional is expressed in terms of the norm
induced by the inner product (2.15) as $\frac{1}{2}\,\|F\|^{2}$. This may be
rewritten for gauge fields with suitable boundary conditions at infinity and
vanishing first Chern form (i.e.
$F\in\text{\Large$\wedge$}^{2}\,T^{*}\mathbbm{R}^{8}\otimes\mathfrak{su}(r)$)
using the useful identity [49]

$\displaystyle\tfrac{1}{2}\,\|F\|^{2}=2\,\big{\|}F^{0,2}\big{\|}^{2}+\tfrac{1}{2}\,\|\omega\,\lrcorner\,F\|^{2}+\mathcal{K}\
,$ (2.17)

where $\omega\,\lrcorner\,F=\ast\,(\omega\wedge\ast\,F)$ is the contraction of
the two-form $F$ with the $(1,1)$-form $\omega$, while
$\mathcal{K}=6\,\int_{M}\,\omega^{2}\wedge\mathrm{Tr}(F^{2})$ is a topological
term. This enables us to write the purely bosonic part of the dimensionally
reduced action functional in the form

$\displaystyle
S=2\,\big{\|}F^{0,2}\big{\|}^{2}+\tfrac{1}{2}\,\|\omega\,\lrcorner\,F\|^{2}+\|D\varphi\|^{2}+V(\varphi,\bar{\varphi})+\mathcal{K}\
,$ (2.18)

where $V(\varphi,\bar{\varphi})=\frac{1}{2}\,\int_{M}\,{\tiny\tt vol}\
\mathrm{Tr}\big{(}[\varphi,\bar{\varphi}]^{2}\big{)}$ is the Higgs potential.

The involution
$\star_{\Omega}:\text{\Large$\wedge$}^{0,2}\,\mathbbm{C}^{4}\longrightarrow\text{\Large$\wedge$}^{0,2}\,\mathbbm{C}^{4}$
enables an orthogonal decomposition of the field strength
$F^{0,2}=\bar{\partial}A^{0,1}+A^{0,1}\wedge A^{0,1}=F_{+}^{0,2}+F_{-}^{0,2}$
into eigencurvatures as

$\displaystyle
F_{\pm}^{0,2}=\tfrac{1}{2}\,\big{(}F^{0,2}\pm\star_{\Omega}\,F^{0,2}\big{)}\qquad\mbox{with}\quad\star_{\Omega}F_{\pm}^{0,2}=\pm\,F_{\pm}^{0,2}\
.$ (2.19)

This self-duality condition arises from the reality of the representation $\bf
6$ in the branching of ${\bf 8}_{\rm s}$ from (2.13), which yields a splitting

$\displaystyle\text{\Large$\wedge$}^{0,2}\,\mathbbm{C}^{4}=\text{\Large$\wedge$}_{+}^{0,2}\,\mathbbm{C}^{4}\oplus\text{\Large$\wedge$}_{-}^{0,2}\,\mathbbm{C}^{4}$
(2.20)

into _real_ $\pm 1$-eigenspaces
$\text{\Large$\wedge$}^{0,2}_{\pm}\,\mathbbm{C}^{4}$ of $\star_{\Omega}$,
since $(\star_{\Omega})^{2}=\mathbbm{1}$.

The quadratic form (2.16) is maximally positive/negative definite on
$\text{\Large$\wedge$}_{\pm}^{0,2}\,T^{*}\mathbbm{C}^{4}\otimes\mathfrak{g}$.
This enables us to write the topological action functional (2.10) as

$\displaystyle
S_{0}=\big{(}F^{0,2},F^{0,2}\big{)}_{\Omega}=\big{\|}F_{+}^{0,2}\big{\|}^{2}-\big{\|}F_{-}^{0,2}\big{\|}^{2}\
.$ (2.21)

From
$\big{\|}F^{0,2}\big{\|}^{2}=\big{\|}F_{+}^{0,2}\big{\|}^{2}+\big{\|}F_{-}^{0,2}\big{\|}^{2}=2\,\big{\|}F_{-}^{0,2}\big{\|}^{2}+S_{0}$
we can then bring (2.18) to the form

$\displaystyle
S=4\,\big{\|}F_{-}^{0,2}\big{\|}^{2}+\tfrac{1}{2}\,\|\omega\,\lrcorner\,F\|^{2}+\|D\varphi\|^{2}+V(\varphi,\bar{\varphi})+2\,S_{0}+\mathcal{K}\
.$ (2.22)

In particular, it follows that $S\geq 2\,S_{0}+\mathcal{K}$.

Since $\mathsf{SU}(4)\subset\mathsf{Spin}(7)$, and dimensional reduction of
ten-dimensional $\mathcal{N}=1$ supersymmetric Yang–Mills theory on a manifold
of $\mathsf{Spin}(7)$-holonomy is equivalent to a topological twist of the
resulting eight-dimensional supersymmetric gauge theory [14], this field
theory is cohomological and has a BRST symmetry. Hence in the topological
sector with $2\,S_{0}+\mathcal{K}$ fixed, it localizes onto the moduli space
of solutions of the BRST fixed point equations given by

$\displaystyle F_{-}^{0,2}=0\
,\quad\omega\,\lrcorner\,F=0\qquad\mbox{and}\qquad D\varphi=0\ .$ (2.23)

The first equation implies
$F_{ab}=\frac{1}{2}\,\varepsilon_{ab\bar{c}\bar{d}}\,F_{\bar{c}\bar{{d}}}$,
while the second equation gives $\delta^{a\bar{a}}\,F_{a\bar{a}}=0$. In other
words, the gauge theory localizes on the instanton equations (2.9) on the
Calabi–Yau fourfold $\mathbbm{C}^{4}$. The solutions of these equations
minimize the action functional (2.22). When $S_{0}=0$, the
$\mathsf{Spin}(7)$-instanton equations are equivalent to the Hermitian
Yang–Mills equations.

###  2.3 Equivariant Gauge Theory

We shall now describe the moduli space $\mathfrak{M}_{r}$ of solutions to the
$\mathsf{Spin}(7)$-instanton equations (2.9), and use this to compute the
instanton partition function of the eight-dimensional cohomological gauge
theory. We begin with an informal discussion of ‘integration’ over instanton
moduli spaces, and then quantify how to compute these formal expressions
practically in the remainder of this section through a generalized ADHM
parametrization of $\mathfrak{M}_{r}$.

Generally, the BRST symmetry of a rank $r$ cohomological gauge theory
localizes its path integral (in a fixed topological sector) to an integral
over a virtual fundamental class $[\mathfrak{M}_{r}]^{\rm vir}$, which is
represented by a coarse moduli space $\mathfrak{M}_{r}$ and a vector bundle
${\sf Ob}_{r}\longrightarrow\mathfrak{M}_{r}$ called the obstruction bundle,
whose fibres are spanned by the antighost zero modes; it is non-trivial when
there is a quadratic Kuranishi map obstructing the extension of first order
deformations, parametrized by the fibres of the tangent bundle
$T\mathfrak{M}_{r}\longrightarrow\mathfrak{M}_{r}$, to second order. If
$\mathfrak{M}_{r}$ is smooth and oriented (which is seldom the case), then
$[\mathfrak{M}_{r}]^{\rm vir}$ is the Poincaré dual of the Euler class $e({\sf
Ob}_{r})$. Partition functions of the topological field theory are then
defined geometrically by integrating the Euler class $e(\mathscr{K})$ of a
‘matter bundle’ $\mathscr{K}\longrightarrow\mathfrak{M}_{r}$ whose rank equals
the virtual dimension of the moduli space, i.e. the rank of the virtual
tangent bundle $T^{\rm vir}\mathfrak{M}_{r}=T\mathfrak{M}_{r}\ominus{\sf
Ob}_{r}$.

In our eight-dimensional theory, we let
$P_{\Omega}^{-}=\frac{1}{2}\,(\mathbbm{1}-\star_{\Omega}):\text{\Large$\wedge$}^{0,2}\,\mathbbm{C}^{4}\longrightarrow\text{\Large$\wedge$}_{-}^{0,2}\,\mathbbm{C}^{4}$
be the projection to the $-1$-eigenspace of the involution $\star_{\Omega}$
induced by the Kähler metric and the holomorphic $(4,0)$-form $\Omega$ on the
Calabi–Yau fourfold $M=\mathbbm{C}^{4}$. The local geometry of the instanton
moduli space $\mathfrak{M}_{r}$ is captured by the instanton deformation
complex [14]

$\displaystyle
0\longrightarrow\text{\Large$\wedge$}^{0}\,T^{*}\mathbbm{C}^{4}\otimes\mathfrak{g}\xrightarrow{\
\bar{\partial}_{A}\
}\text{\Large$\wedge$}^{0,1}\,T^{*}\mathbbm{C}^{4}\otimes\mathfrak{g}\xrightarrow{\
\bar{\partial}_{A}^{-}\
}\text{\Large$\wedge$}_{-}^{0,2}\,T^{*}\mathbbm{C}^{4}\otimes\mathfrak{g}\longrightarrow
0\ ,$ (2.24)

where the first arrow is an infinitesimal complex gauge transformation, while
the second arrow with
$\bar{\partial}_{A}^{-}:=P_{\Omega}^{-}\circ\bar{\partial}_{A}$ is the
linearization of the first equation in (2.23). Associated to this complex is a
local cyclic $L_{\infty}$-algebra which describes the full cohomological field
theory in the BV formalism and $\mathfrak{M}_{r}$ as the corresponding
Maurer–Cartan moduli space.

The cochain complex (2.24) is elliptic and its degree one cohomology
represents the (complex) tangent space to the moduli space $\mathfrak{M}_{r}$
at a fixed holomorphic self-dual connection $A$. We assume that the degree
zero cohomology vanishes, i.e. $\ker\big{(}\bar{\partial}_{A}\big{)}=0$, which
amounts to restricting to irreducible connections with only trivial
automorphisms. There is also a non-trivial degree two cohomology which defines
the real self-dual obstruction bundle ${\sf
Ob}_{r}^{-}\longrightarrow\mathfrak{M}_{r}$, whose fibre over $A$ is given by
${\rm
coker}\big{(}\bar{\partial}_{A}^{-}\big{)}\subset\text{\Large$\wedge$}_{-}^{0,2}\,T^{*}\mathbbm{C}^{4}\otimes\mathfrak{g}$.

This is the starting point for the construction of a corresponding (real)
virtual fundamental class $[\mathfrak{M}_{r}]^{\rm vir}$ in [21, 50, 22, 51,
52, 23, 53]. We consider the stratification of the moduli space
$\mathfrak{M}_{r}$ into its connected components $\mathfrak{M}_{r,k}$ which
are labelled by the instanton number (fourth Chern class)

$\displaystyle k=\frac{1}{384\pi^{4}}\,\int_{M}\,\mathrm{Tr}(F\wedge F\wedge
F\wedge F)\ .$ (2.25)

The obstruction bundle correspondingly restricts to real vector bundles ${\sf
Ob}_{r,k}^{-}\longrightarrow\mathfrak{M}_{r,k}$, which are orientable. The
real virtual dimension is $2\,r\,k$ and the virtual fundamental class
$[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm vir}$ depends on the choice of an
orientation $\mathtt{o}$ of ${\sf Ob}_{r,k}^{-}$.

The Euler class of the self-dual obstruction bundle, which is naturally
induced by integration over the BRST antighost fields in the cohomological
gauge theory, can be thought of as defining a square root Euler class for the
complexification ${\sf Ob}_{r,k}:={\sf
Ob}_{r,k}^{-}\otimes_{\mathbbm{R}}\mathbbm{C}$ through

$\displaystyle\sqrt{e}({\sf Ob}_{r,k}):=e({\sf Ob}^{-}_{r,k})\ .$ (2.26)

In [23] an alternative definition of the square root Euler class is given as
follows. The complex vector bundle ${\sf Ob}_{r,k}$ carries a natural
quadratic form induced by (2.16), which restricts to a metric on ${\sf
Ob}_{r,k}^{-}$. Then $\sqrt{e}({\sf Ob}_{r,k})$ can be defined using a
maximally isotropic holomorphic sub-bundle of ${\sf Ob}_{r,k}$ with respect to
this quadratic form, which is isomorphic to ${\sf Ob}_{r,k}^{-}$ as a real
bundle. This comes with a sign determined by the choice of orientation. It has
the advantage of yielding a class that lifts to Chow cohomology (with
$\mathbbm{Z}[\frac{1}{2}]$-coefficients) and provides a construction of
$[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm vir}$ as an algebraic cycle.

A natural choice of ‘matter bundle’ $\mathscr{K}$ in this case is provided by
the bundle $\mathscr{V}_{r,k}\longrightarrow\mathfrak{M}_{r,k}$ whose fibre
over a gauge orbit $[A]$ is the complex vector space $V$ entering the
generalized ADHM parametrization of $\mathfrak{M}_{r,k}$ in Section 2.4 below;
this is a complex vector bundle of rank $k$, and taking $r$ tensor powers of
it gives a vector bundle $\mathscr{K}$ of the desired real rank $2\,r\,k$. To
incorporate masses $\vec{m}=(m_{1},\dots,m_{r})$ for the matter fields we use
the Chern polynomial of the vector bundle $\mathscr{V}_{r,k}$ which is defined
by the formula

$\displaystyle
c(\mathscr{V}_{r,k};m)=\sum_{i=0}^{k}\,m^{i}\,c_{k-i}(\mathscr{V}_{r,k})\ ,$
(2.27)

where $c_{i}(\mathscr{V}_{r,k})$ is the $i$-th Chern class of
$\mathscr{V}_{r,k}$. The Euler class is the top Chern class
$e(\mathscr{V}_{r,k})=c_{k}(\mathscr{V}_{r,k})$.

The instanton partition function of the eight-dimensional cohomological gauge
theory then schematically has the form

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{m})=\sum_{k=0}^{\infty}\,\mathtt{q}^{k}\
\int_{[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm vir}}\
\prod_{l=1}^{r}\,c(\mathscr{V}_{r,k};m_{l})\ ,$ (2.28)

where the counting parameter $\mathtt{q}$ weighs the instanton number and is
determined by the couplings of the gauge theory.

To make sense of the symbolic expression (2.28), we will work in the setting
of equivariant gauge theory and _define_ it via an equivariant integral over
the instanton moduli space. The global symmetry group of the cohomological
field theory with matter is

$\displaystyle\mathsf{G}=\mathsf{U}(r)_{\rm
col}\,\times\,\mathsf{SU}(4)\,\times\,\mathsf{U}(r)_{\rm fla}\ ,$ (2.29)

where the colour symmetry $\mathsf{U}(r)_{\rm col}$ acts by rotating the
framing of the gauge bundle at infinity in $\mathbbm{C}^{4}$, and the flavour
symmetry $\mathsf{U}(r)_{\rm fla}$ acts on its vector representation
${\bm{r}}$. This group can be rotated into its maximal torus

$\displaystyle\mathsf{T}=\mathsf{T}_{\vec{a}}\,\times\,\mathsf{T}_{\vec{\epsilon}}\,\times\,\mathsf{T}_{\vec{m}}\
,$ (2.30)

where we label the torus factors by the corresponding complexified Cartan
subalgebra elements, which we assume to consist of generic complex parameters
in the following.

The framing rotation parameters $\vec{a}=(a_{1},\dots,a_{r})$ are vacuum
expectation values of the complex Higgs field $\varphi$, while
$\vec{\epsilon}=(\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4})$ satisfy

$\displaystyle\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4}=0$ (2.31)

and correspond to the natural complex scaling action of the three-torus
$\mathsf{T}_{\vec{\epsilon}}$ on $\mathbbm{C}^{4}$. This couples the gauge
theory to Nekrasov’s $\varOmega$-background [31] through an
$\mathsf{SU}(4)$-invariant deformation of the BRST supercharges. The
$\varOmega$-deformation does not change the instanton equations (2.9), but it
confines their solutions to the fixed point $0\in\mathbbm{C}^{4}$ and so
provides an infrared regularization of the singularities of
$\mathfrak{M}_{r,k}$ due to instantons that escape to infinity. The flavour
rotation parameters $\vec{m}=(m_{1},\dots,m_{r})$ are masses of $r$
fundamental matter fields.

The torus $\mathsf{T}$ acts on the moduli space $\mathfrak{M}_{r,k}$ and its
obstruction bundle ${\sf Ob}_{r,k}$ (with $\mathsf{T}_{\vec{m}}$ acting
trivially), as well as on the vector bundle $\mathscr{V}_{r,k}\otimes{\bm{r}}$
(with $\mathsf{T}_{\vec{m}}$ acting trivially on $\mathscr{V}_{r,k}$ and
$\mathsf{T}_{\vec{a},\vec{\epsilon}}:=\mathsf{T}_{\vec{a}}\times\mathsf{T}_{\vec{\epsilon}}$
acting trivially on ${\bm{r}}$). The product of Chern polynomials in (2.28)
can be regarded as the equivariant Euler class
$e_{\mathsf{T}}(\mathscr{V}_{r,k}\otimes{\bm{r}})$. The integrations are then
interpreted as the pushforwards $\int_{[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm
vir}}^{\mathsf{T}}\,e_{\mathsf{T}}(\mathscr{V}_{r,k}\otimes{\bm{r}})$ to a
point in the $\mathsf{T}$-equivariant Chow cohomology of $\mathfrak{M}_{r,k}$,
whose coefficient ring is
$\mathbbm{C}[\vec{a},\vec{\epsilon},\vec{m}]/\langle\epsilon_{1}+\cdots+\epsilon_{4}\rangle$.
The $\varOmega$-deformation localizes the moduli space $\mathfrak{M}_{r,k}$
onto its isolated $\mathsf{T}$-fixed points
$\vec{\sigma}\in\mathfrak{M}_{r,k}^{\mathsf{T}}$. The virtual localization
formula of [54, 23] computes the integrals in (2.28) as a sum over these fixed
points, giving the instanton partition function as a combinatorial expansion
in $\mathtt{q}$ whose terms are rational functions of the equivariant
parameters $(\vec{a},\vec{\epsilon},\vec{m})$. It reads as

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m})=\sum_{k=0}^{\infty}\,\mathtt{q}^{k}\
\sum_{\vec{\sigma}\in\mathfrak{M}_{r,k}^{\mathsf{T}}}\,\frac{\sqrt{e_{\mathsf{T}}}\big{(}({\sf
Ob}_{r,k})_{\vec{\sigma}}\big{)}\
e_{\mathsf{T}}\big{(}(\mathscr{V}_{r,k})_{\vec{\sigma}}\otimes{\bm{r}}\big{)}}{e_{\mathsf{T}}\big{(}T_{\vec{\sigma}}\mathfrak{M}_{r,k}\big{)}}\
.$ (2.32)

The fixed points of the instanton moduli spaces have a combinatorial
significance that we discuss below. The square root Euler class
$\sqrt{e_{\mathsf{T}}}\big{(}({\sf Ob}_{r,k})_{\vec{\sigma}}\big{)}$ in (2.32)
is defined up to a sign which depends explicitly on the orientation of ${\sf
Ob}_{r,k}^{-}$ at the fixed point $\vec{\sigma}$. The formula (2.32) assumes
that the spaces $T_{\vec{\sigma}}\mathfrak{M}_{r,k}$ and $({\sf
Ob}_{r,k})_{\vec{\sigma}}$ carry only non-zero weights for the action of the
torus $\mathsf{T}_{\vec{a},\vec{\epsilon}}\,$, so that the equivariant Chow
cohomology classes $e_{\mathsf{T}}(T_{\vec{\sigma}}\mathfrak{M}_{r,k})$ and
$\sqrt{e_{\mathsf{T}}}\big{(}({\sf Ob}_{r,k})_{\vec{\sigma}}\big{)}$ are
invertible. We shall elaborate further on these points later on in this
section.

The equivariant theory also allows for the definition of the instanton
partition function of the cohomological gauge theory without matter [19], that
is, the “pure” holomorphic $\mathcal{N}_{\textrm{\tiny T}}=2$ topological
Yang–Mills theory in eight dimensions defined by integrating $1$ over
$[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm vir}$. This is analogous to the
definition of the Nekrasov partition function for the pure $\mathcal{N}=2$
supersymmetric Yang–Mills theory in four dimensions [31]. The field theory is
now equivariant with respect to the torus
$\mathsf{T}_{\vec{a},\vec{\epsilon}}\subset\mathsf{T}$, and the localization
theorem gives

$\displaystyle Z_{\mathbbm{C}^{4}}^{r}(\Lambda;\vec{a},\vec{\epsilon}\,)^{\rm
pure}:=\sum_{k=0}^{\infty}\,\Lambda^{k}\
\int_{[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm
vir}}^{\mathsf{T}_{\vec{a},\vec{\epsilon}}}\,1=\sum_{k=0}^{\infty}\,\Lambda^{k}\
\sum_{\vec{\sigma}\in\mathfrak{M}_{r,k}^{\mathsf{T}}}\,\frac{\sqrt{e_{\mathsf{T}}}\big{(}({\sf
Ob}_{r,k})_{\vec{\sigma}}\big{)}}{e_{\mathsf{T}}\big{(}T_{\vec{\sigma}}\mathfrak{M}_{r,k}\big{)}}\
.$ (2.33)

###  2.4 Noncommutative Field Theory

As we discussed in Section 2.1, the noncommutative deformation is equivalent
to the choice of a complex structure on $\mathbbm{R}^{8}$, which induces a
Poisson bivector $\theta=\zeta\,\omega^{-1}$ that we wish to quantize, where
$\zeta\in\mathbbm{R}_{>0}$ and $\omega$ is the corresponding Kähler form of
$\mathbbm{R}^{8}$. The quantization map sends the local complex coordinates
$z_{a},\bar{z}_{\bar{a}}$ of $M=\mathbbm{C}^{4}$ to operators with the
commutation relations

$\displaystyle[z_{a},z_{b}]=0=[\bar{z}_{\bar{a}},\bar{z}_{\bar{b}}]\qquad\mbox{and}\qquad[z_{a},\bar{z}_{\bar{b}}]=\zeta\,\delta_{a\bar{b}}\
,$ (2.34)

for $a,\bar{a},b,\bar{b}\in\underline{\sf 4}$. We write $\mathcal{A}$ for the
noncommutative algebra generated by these operators over $\mathbbm{C}$. The
unique irreducible representation of this algebra is given by the Fock module

$\displaystyle\mathcal{H}=\mathbbm{C}[\bar{z}_{1},\bar{z}_{2},\bar{z}_{3},\bar{z}_{4}]|\vec{0}\,\rangle=\bigoplus_{\vec{n}\in\mathbbm{Z}_{\geq
0}^{4}}\,\mathbbm{C}|\vec{n}\,\rangle\ ,$ (2.35)

with vacuum vector $|\vec{0}\,\rangle$.

Let $W\simeq\mathbbm{C}^{r}$ be the fundamental representation of the colour
symmetry $\mathsf{U}(r)_{\rm col}$, regarded as a Hermitian vector space.
Using the quantization map, we send all fields of the eight-dimensional
cohomological gauge theory to operators acting on the separable Hilbert space
$\mathcal{H}^{r}:=W\otimes\mathcal{H}$, i.e. to elements of the algebra ${\sf
Mat}_{r{\times}r}(\mathcal{A}):=\mathsf{End}_{\mathbbm{C}}(W)\otimes\mathcal{A}$
of $r{\times}r$ matrices valued in $\mathcal{A}$. This turns the gauge theory
into an infinite-dimensional matrix model. The fixed point equations (2.23)
become operator algebraic equations for the noncommutative fields given by

$\displaystyle[Z_{a},Z_{b}]=\tfrac{1}{2}\,\varepsilon_{ab\bar{c}\bar{d}}\,\big{[}Z^{\dagger}_{\bar{c}},Z^{\dagger}_{\bar{d}}\big{]}\
,\quad\sum_{a=1}^{4}\,\big{[}Z_{\bar{a}}^{\dagger},Z_{a}\big{]}=\zeta\,\mathbbm{1}_{\mathcal{H}^{r}}\qquad\mbox{and}\qquad[Z_{a},\varphi]=0\,$
(2.36)

for $a,b\in\underline{\sf 4}$, where the operators
$Z_{a}:=\frac{1}{2}\,(z_{a}+\mathrm{i}\,A_{\bar{a}})$ are called _covariant
coordinates_. The $\varOmega$-deformation leaves unchanged the instanton
equations but alters the last equation of (2.36) to

$\displaystyle[Z_{a},\varphi]=\epsilon_{a}\,Z_{a}\ .$ (2.37)

It follows that charge $k$ noncommutative $\mathsf{U}(r)$ instantons in eight
dimensions are described as elements of the algebra ${\sf
Mat}_{r{\times}r}(\mathcal{A})$ acting on the free $\mathcal{A}$-module
$\mathcal{A}^{r}:=W\otimes\mathcal{A}$. Through the natural isomorphism of
$\mathcal{A}$-modules $\mathcal{E}^{r,k}\simeq\mathcal{A}^{r}$, they can be
related to connections
$\nabla:\mathcal{E}^{r,k}\longrightarrow\mathcal{E}^{r,k}\otimes_{\mathcal{A}}\Omega_{\mathcal{A}}^{1}$
on the projective modules
$\mathcal{E}^{r,k}:=\mathcal{H}^{k}\oplus\mathcal{A}^{r}$ over $\mathcal{A}$,
where $\mathcal{H}^{k}:=V\otimes\mathcal{H}$ for a Hermitian vector space
$V\simeq\mathbbm{C}^{k}$ and $\Omega_{\mathcal{A}}^{1}$ is the bimodule of
one-forms over the algebra $\mathcal{A}$. This induces a decomposition of the
covariant coordinates $Z_{a}\in\mathsf{End}_{\mathcal{A}}(\mathcal{E}^{r,k})$
as

$\displaystyle Z_{1}={\small\bigg{(}\begin{matrix}B_{1}&\hat{I}_{1}\\\
\hat{I}^{\prime}_{1}&R_{1}\end{matrix}\bigg{)}}\normalsize\ ,\quad
Z_{2}={\small\bigg{(}\begin{matrix}B_{2}&\hat{I}_{2}^{\prime}\\\
\hat{I}_{2}&R_{2}\end{matrix}\bigg{)}}\normalsize\ ,\quad
Z_{3}={\small\bigg{(}\begin{matrix}B_{3}&\hat{I}_{3}^{\prime}\\\
\hat{I}_{3}&R_{3}\end{matrix}\bigg{)}}\normalsize\qquad\mbox{and}\qquad
Z_{4}={\small\bigg{(}\begin{matrix}B_{4}&\hat{I}_{4}^{\prime}\\\
\hat{I}_{4}&R_{4}\end{matrix}\bigg{)}}\normalsize\ ,$ (2.38)

where the diagonal blocks consist of linear maps
$B_{a}\in\mathsf{End}_{\mathbbm{C}}(V)$ and operators
$R_{a}\in\mathsf{End}_{\mathcal{A}}(W\otimes\mathcal{A})$, while the off-
diagonal blocks are operators

$\displaystyle\hat{I}_{1},\hat{I}_{2}^{\prime},\hat{I}_{3}^{\prime},\hat{I}_{4}^{\prime}\
\in\
\mathsf{Hom}_{\mathcal{A}}(W\otimes\mathcal{A},V)\qquad\mbox{and}\qquad\hat{I}_{1}^{\prime},\hat{I}_{2},\hat{I}_{3},\hat{I}_{4}\
\in\ \mathsf{Hom}_{\mathcal{A}}(V,W\otimes\mathcal{A})\ .$ (2.39)

We can set $\hat{I}_{a}^{\prime}=0$ for $a\in\underline{\sf 4}$ thanks to the
freedom of gauge choice.

Using the isomorphisms

$\displaystyle\begin{split}\mathsf{Hom}_{\mathcal{A}}(W\otimes\mathcal{A},V)&\simeq\mathsf{Hom}_{\mathbbm{C}}(W,V)\otimes\mathsf{End}_{\mathcal{A}}(\mathcal{A}^{r})\end{split}$
(2.40)

and

$\displaystyle\begin{split}\mathsf{Hom}_{\mathcal{A}}(V,W\otimes\mathcal{A})&\simeq\mathsf{Hom}_{\mathbbm{C}}(V,W)\otimes\mathsf{End}_{\mathcal{A}}(\mathcal{A}^{r})\end{split}$
(2.41)

we decompose the operators from (2.39) as $\hat{I}_{a}=I_{a}\otimes\psi_{a}$,
where $I_{1}\in\mathsf{Hom}_{\mathbbm{C}}(W,V)$,
$I_{\alpha}\in\mathsf{Hom}_{\mathbbm{C}}(V,W)$ for $\alpha\in\\{2,3,4\\}$ and
the operators
$\psi_{a}\in\mathsf{End}_{\mathcal{A}}(\mathcal{A}^{r})\simeq{\sf
Mat}_{r{\times}r}(\mathcal{A})$ satisfy
$\psi_{1}\,\psi_{\alpha}=\mathbbm{1}_{\mathcal{A}^{r}}$. Relabelling
$I:=I_{1}$ and $J_{\alpha}:=I_{\alpha+1}$, it is then straightforward to see
that the first and second equations of (2.36) yield the matrix equations

$\displaystyle\begin{split}[B_{1},B_{\alpha}]+I\,J_{\alpha}-\tfrac{1}{2}\,\varepsilon_{1\alpha\bar{\beta}\bar{\gamma}}\,\big{[}B_{\bar{\beta}}^{\dagger},B_{\bar{\gamma}}^{\dagger}\big{]}&=0\
,\\\\[4.0pt]
[B_{\alpha},B_{\beta}]-\tfrac{1}{2}\,\varepsilon_{1\alpha\bar{\beta}\bar{\gamma}}\,\big{(}\big{[}B_{\bar{1}}^{\dagger},B_{\bar{\gamma}}^{\dagger}\big{]}-J_{\bar{\gamma}}^{\dagger}\,I^{\dagger}\big{)}&=0\
,\\\\[4.0pt]
\sum_{a=1}^{4}\,\big{[}B_{a},B_{\bar{a}}^{\dagger}\big{]}+I\,I^{\dagger}-\sum_{\alpha=1}^{3}\,J_{\bar{\alpha}}^{\dagger}\,J_{\alpha}&=\zeta\,\mathbbm{1}_{k}\
,\end{split}$ (2.42)

for $\alpha,\beta,\gamma\in\\{1,2,3\\}$, along with other relations which are
not relevant here.

We observe that the space of solutions to the system of equations (2.42) is
independent of the value of $\zeta>0$ as a consequence of the scaling symmetry
$B_{a}\to\kappa\,B_{a}$, $I\to\kappa\,I$, $J_{\alpha}\to\kappa\,J_{\alpha}$
and $\zeta\to\kappa^{2}\,\zeta$ for $\kappa\in\mathbbm{R}$. If the
$\mathsf{Spin}(7)$-instanton equations reduce to the Hermitian Yang–Mills
equations (i.e. $S_{0}=0$), then the first instanton equation of (2.36) would
reduce to the holomorphic equations $[Z_{a},Z_{b}]=0$ and the non-holomorphic
terms in the first two matrix equations of (2.42) would drop out. In this
instance one can repeat the stability argument of [6] to infer that, in the
equivariant gauge theory of Section 2.3, one may set $J_{\alpha}=0$ for
$\alpha\in\\{1,2,3\\}$. In fact, it is the equations $[B_{a},B_{b}]=0$ that
arise for the vacua in the string theory setting discussed below, which we
shall see are equivalent to the non-holomorphic equations with $J_{\alpha}=0$
that retain the information of the original $\mathsf{Spin}(7)$-instantons.
With this in mind, we restrict the solutions of (2.42) to the closed
subvariety defined by the condition

$\displaystyle J_{\alpha}=0\qquad\mbox{for}\quad\alpha\in\\{1,2,3\\}\ .$
(2.43)

Altogether we have shown that the noncommutative instanton equations (2.36)
can be reduced to the ADHM-type equations

$\displaystyle\mu^{\mathbbm{C}}_{ab}:=[B_{a},B_{b}]-\tfrac{1}{2}\,\varepsilon_{ab\bar{c}\bar{d}}\,\big{[}B_{\bar{c}}^{\dagger},B_{\bar{d}}^{\dagger}\big{]}=0\qquad\mbox{and}\qquad\mu^{\mathbbm{R}}:=\sum_{a=1}^{4}\,\big{[}B_{a},B_{\bar{a}}^{\dagger}\big{]}+I\,I^{\dagger}=\zeta\,\mathbbm{1}_{k}\
.$ (2.44)

In Appendix A we describe how to explicitly construct an
$\mathsf{SU}(4)$-instanton connection $A$ from the generalized ADHM data. For
later considerations, it is convenient to encode the ADHM data in a framed
representation of the four-loop quiver $\mathsf{L}_{4}$:

${{\boxed{W}}}$${{\boxed{V}}}$$\scriptstyle{B_{1}}$$\scriptstyle{B_{2}}$$\scriptstyle{B_{3}}$$\scriptstyle{B_{4}}$$\scriptstyle{I}$
(2.45)

#### Physical Interpretation

The ADHM equations (2.44) describe supersymmetric bound states of $k$
coincident D$1$-branes along $\mathbbm{R}^{1,1}$ with $r$ D$9$-branes filling
$\mathbbm{R}^{1,9}\simeq\mathbbm{R}^{1,1}\times\mathbbm{C}^{4}$ in the low
energy limit of type IIB string theory (or the T-dual D$0$–D$8$ and
D$(-1)$–D$7$ systems), with a large constant Kalb–Ramond field preserving
$\frac{1}{8}$ of the supercharges [55, 39, 19, 20, 56, 57, 40]. The massless
spectrum of D9–D9 strings yields ten-dimensional $\mathcal{N}=1$ Yang–Mills
theory with gauge group $\mathsf{U}(r)$, and the noncommutative instanton
equations (2.36) describe D$1$–D$9$-brane bound states from the perspective of
the D$9$-branes. Our choice of vanishing first Chern form in Section 2.2
excludes D$7$–D$9$ bound states, while the restriction of vanishing second
Chern form, i.e. $S_{0}=0$, also excludes D$5$–D$9$ bound states.

From the point of view of the D$1$-branes, the D$9$-branes are heavy and the
degrees of freedom supported on them are frozen to their vacuum expectation
values. Then the Chan–Paton gauge symmetry becomes a global
$\mathsf{U}(r)_{\rm col}$ colour symmetry on the D$9$-branes. The parameter
$\zeta$ is determined by the constant background $B$-field along
$\mathbbm{C}^{4}$ and it plays the role of the coupling of a Fayet–Iliopoulos
term in the low energy $\mathcal{N}=(0,2)$ field theory with gauge group
$\mathsf{U}(k)$ on the D$1$-branes. The arrows $B_{a}$ of the ADHM quiver
representation (2.45) for $a\in\underline{\sf 4}$ are the lowest components of
chiral superfields on $\mathbbm{R}^{1,1}$ that arise from quantizing the
D$1$–D$1$ strings which generate four complex scalar fields in the adjoint
representation of $\mathsf{U}(k)$, while the arrow $I$ is the lowest component
of a chiral superfield on $\mathbbm{R}^{1,1}$ associated to the massless
spectrum of D$1$–D$9$ open strings in the Neveu–Schwarz sector which give rise
to a complex scalar field transforming in the bifundamental representation of
$\mathsf{U}(k)\times\mathsf{U}(r)_{\rm col}$.

The ADHM equations (2.44) are the equations for the classical Higgs branch of
the D1-brane theory [39, 40]. Since

$\displaystyle\mathrm{Tr}\big{(}\mu^{\mathbbm{R}}\big{)}=\mathrm{Tr}\big{(}I\,I^{\dagger}\big{)}=\zeta\,k\
,$ (2.46)

and the term in the middle is non-negative, a solution to
$\mu^{\mathbbm{R}}=\zeta\,\mathbbm{1}_{k}$ exists only if $\zeta\geq 0$. If
$\zeta=0$ then $I=0$, which corresponds to the two-dimensional
$\mathcal{N}=(8,8)$ supersymmetric Yang–Mills theory with gauge group
$\mathsf{U}(k)$ on $\mathbbm{R}^{1,1}$ associated to the massless spectrum of
D1–D1 strings alone. It can be obtained as the dimensional reduction of ten-
dimensional $\mathcal{N}=1$ supersymmetric Yang–Mills theory with gauge group
$\mathsf{U}(k)$.

Thus the interesting case for us is $\zeta>0$; in this case the D1–D9-brane
system is tachyonic and unstable [13], so it decays to a supersymmetric vacuum
via tachyon condensation. The presence of the Fayet–Iliopoulos term with
coupling $\zeta$ in string theory corresponds to a noncommutative deformation
of the $\mathsf{U}(r)$ gauge theory on the D9-branes [58, 30], and this is why
one has to work with noncommutative instantons. Thus the situation here is
somewhat different from the four-dimensional case: for instantons on
$\mathbbm{R}^{4}\simeq\mathbbm{C}^{2}$ a similar analysis, now done for a
D1–D5-brane system, leads to a theory where $\zeta=0$ is a permitted and non-
trivial choice, though $\zeta\neq 0$ prevents the system from entering the
Coulomb branch through the small instanton singularity [58].

Finally, as discussed in Section 2.3, the partition function of the
topological field theory is defined by adding $r$ fundamental fermions whose
functional integration brings down the Euler class of the matter bundle
$\mathscr{V}_{r,k}\otimes{\bm{r}}\longrightarrow\mathfrak{M}_{r,k}$. In string
theory language this is conjecturally equivalent to adding $r$ anti-D9-branes
to the interacting system of D-branes [19, 20], with the masses
$\vec{m}=(m_{1},\dots,m_{r})$ corresponding to the equivariant Chern roots of
the Chan–Paton bundle on the $\overline{\rm D9}$-branes. Similarly to the
D9–D9 strings, the degrees of freedom supported on the $\overline{\rm
D9}$–$\overline{\rm D9}$ strings are frozen to their vacuum expectation
values, leaving a global $\mathsf{U}(r)_{\rm fla}$ flavour symmetry on the
$\overline{\rm D9}$-branes. Although the D9-branes and anti-D9-branes need not
annihilate completely to a supersymmetric vacuum in the background of a
$B$-field [59], any remaining open string ground states will be invisible to
the D1-branes.

Altogether the spectrum of stable states (free from tachyonic modes) in the
presence of $r$ anti-D9-branes involves reversed GSO projections and changes
only the Fermi field content in the $\mathcal{N}=(0,2)$ field theory on the
D$1$-branes, adding to the ADHM data an antichiral spinor on
$\mathbbm{R}^{1,1}$ in the bifundamental representation of
$\mathsf{U}(k)\times\mathsf{U}(r)_{\rm fla}$ associated to the
D1–$\overline{\rm D9}$-strings from the Ramond sector, whose lowest component
we denote by $\bar{I}$. It does not modify the form of the ADHM equations
(2.44) nor the non-trivial values of the Fayet–Iliopoulos coupling $\zeta$.

#### Geometrical Interpretation

The moduli space $\mathfrak{M}_{r,k}$ of $\mathsf{U}(r)$ noncommutative
$k$-instantons can be described as the ADHM moduli space, that is, the space
of quintuples $(B_{a},I)_{a\in\underline{\sf 4}}$ satisfying the ADHM
equations (2.44) modulo the action of the unitary group
$\mathsf{G}^{\textrm{\tiny$V$}}=\mathsf{U}(k)$ of the vector space $V$ given
by

$\displaystyle g\cdot(B_{a},I)_{a\in\underline{\sf
4}}=(g\,B_{a}\,g^{-1},g\,I)_{a\in\underline{\sf 4}}\qquad\mbox{with}\quad
g\in\mathsf{U}(k)\ ,$ (2.47)

which leaves invariant the ADHM equations (2.44). In this parametrization, the
vector space $V$ descends to $\mathfrak{M}_{r,k}$ as the vector bundle
$\mathscr{V}_{r,k}$ discussed in Section 2.3.

Let us consider the complex moment map equations in (2.44), given by
$\mu^{\mathbbm{C}}_{ab}=0$ for $1\leq a<b\leq 4$. As noted by [19], there is
an identity

$\displaystyle\sum_{1\leq a<b\leq
4}\,\big{\|}\mu^{\mathbbm{C}}_{ab}\big{\|}_{\textrm{\tiny F}}^{2}$
$\displaystyle=\sum_{1\leq a<b\leq
4}\,\big{\|}[B_{a},B_{b}]\big{\|}_{\textrm{\tiny F}}^{2}\ ,$ (2.48)

where $\|-\|_{\textrm{\tiny F}}$ is the Frobenius norm on
$\mathsf{End}_{\mathbbm{C}}(V)$. From (2.48) it follows that the equations
$\mu^{\mathbbm{C}}_{ab}=0$ are equivalent to the equations $[B_{a},B_{b}]=0$.
Instead, the real moment map equation
$\mu^{\mathbbm{R}}=\zeta\,\mathbbm{1}_{k}$ in (2.44) for $\zeta>0$ is
equivalent to a _stability_ condition: There is no proper subspace
$S\subset\mathbbm{C}^{k}$ such that $B_{a}(S)\subset S$ for
$a\in\underline{\sf 4}$ and ${\rm im}(I)\subset S$. The stability condition is
equivalent to the condition that the operators $B_{a}$ for $a\in\underline{\sf
4}$ acting on ${\rm im}(I)$ generate the vector space
$V=\mathbbm{C}[B_{1},B_{2},B_{3},B_{4}]\,I(W)$; it can be interpreted
physically as the requirement that all D1-branes are bound to D9-branes.

Writing $\mathfrak{B}^{\rm st}$ for the space of stable quintuples, it follows
that the instanton moduli space has an explicit holomorphic parametrization as
the geometric invariant theory (GIT) quotient

$\displaystyle\mathfrak{M}_{r,k}=\big{\\{}(B_{a},I)_{a\in\underline{\sf
4}}\in\mathfrak{B}^{\rm st}\ \big{|}\ [B_{a},B_{b}]=0\ ,\ 1\leq a<b\leq
4\big{\\}}\,\big{/}\,\mathsf{GL}(k,\mathbbm{C})\ .$ (2.49)

From this one easily sees that the real virtual dimension of
$\mathfrak{M}_{r,k}$ is $(8\,k^{2}+2\,r\,k)-6\,k^{2}-2\,k^{2}=2\,r\,k$.

As a consequence, by results of [60, 61] the instanton moduli space
$\mathfrak{M}_{r,k}$ is isomorphic to the Quot scheme ${\sf
Quot}^{k}_{r}(\mathbbm{C}^{4})$ of zero-dimensional quotients of the free
sheaf $\mathcal{O}_{\mathbbm{C}^{4}}^{\oplus r}$ with length $k$,

$\displaystyle\mathfrak{M}_{r,k}\,\simeq\,{\sf Quot}^{k}_{r}(\mathbbm{C}^{4})\
,$ (2.50)

which parametrizes framed torsion free sheaves $\mathcal{E}$ on complex
projective space ${\mathbbm{P}}^{4}$ of rank $r$ and
$\mathrm{ch}_{4}(\mathcal{E})=k$. Any such sheaf sits in a short exact
sequence

$\displaystyle
0\longrightarrow\mathcal{E}\longrightarrow\mathcal{O}_{{\mathbbm{P}}^{4}}^{\oplus
r}\longrightarrow\mathcal{S}_{Z}\longrightarrow 0\ ,$ (2.51)

where $\mathcal{S}_{Z}$ is a pure torsion sheaf of length $k$ supported on a
zero-dimensional subscheme $Z\subset{\mathbbm{P}}^{4}$.

######  Remark 2.52.

In the rank one case $r=1$, it follows from (2.50) that the instanton moduli
space is isomorphic to the Hilbert scheme of $k$ points on $\mathbbm{C}^{4}$,

$\displaystyle\mathfrak{M}_{1,k}\,\simeq\,{\sf Hilb}^{k}(\mathbbm{C}^{4})\ ,$
(2.53)

which is parametrized by ideals $J$ of codimension $k$ in the polynomial ring
$\mathbbm{C}[z_{1},z_{2},z_{3},z_{4}]$. The correspondence follows from
defining the $k$-dimensional vector space
$V=\mathbbm{C}[z_{1},z_{2},z_{3},z_{4}]/J$ for an ideal $J\in{\sf
Hilb}^{k}(\mathbbm{C}^{4})$. We then take
$B_{a}\in\mathsf{End}_{\mathbbm{C}}(V)$ to be given as multiplication by
$z_{a}$ mod $J$ and $I\in\mathsf{Hom}_{\mathbbm{C}}(\mathbbm{C},V)$ as
$I(1)=1$ mod $J$.

###  2.5 Quiver Matrix Model

From the perspective of D$9$-branes, the noncommutative field theory of
Section 2.4 computes the instanton partition function (2.28) for the
cohomological field theory of Section 2.2 [41]. Here we shall study the theory
from the perspective of the D$1$-branes. Following [6, 7], this uses the
generalized ADHM parametrization of the instanton moduli space
$\mathfrak{M}_{r,k}$ to construct a matrix model representation of the
integral in (2.28), which is based on the fields of the quiver (2.45). The
construction of the cohomological matrix model proceeds analogously to the
construction of the eight-dimensional cohomological gauge theory from [47,
14].

The symmetry group of the quiver matrix model is
$\mathsf{U}(k)\times\mathsf{U}(r)_{\rm col}\times\mathsf{SU}(4)$. The gauge
and global framing symmetries act on the ADHM variables
$(B_{a},I)_{a\in\underline{\sf 4}}$ as

$\displaystyle(g,h)\cdot(B_{a},I)_{a\in\underline{\sf
4}}=(g\,B_{a}\,g^{-1},g\,I\,h^{-1})_{a\in\underline{\sf
4}}\qquad\mbox{for}\quad(g,h)\in\mathsf{U}(k)\times\mathsf{U}(r)_{\rm col}\ .$
(2.54)

We work equivariantly with respect to the (complex) maximal torus
$\mathsf{T}_{\vec{\epsilon}}$ of $\mathsf{SU}(4)$, which acts on the ADHM data
as

$\displaystyle(t_{a})_{a\in\underline{\sf
4}}\cdot(B_{a},I)_{a\in\underline{\sf
4}}=(t_{a}^{-1}\,B_{a},I)_{a\in\underline{\sf
4}}\qquad\mbox{for}\quad(t_{a})_{a\in\underline{\sf
4}}=\big{(}\mathrm{e}^{\,\mathrm{i}\,\epsilon_{a}}\big{)}_{a\in\underline{\sf
4}}\in\mathsf{T}_{\vec{\epsilon}}\ .$ (2.55)

The BRST multiplets are $(B_{a},\psi_{a})_{a\in\underline{\sf 4}}$ and
$(I,\varrho)$ with the BRST transformations

$\displaystyle\begin{split}\mathcal{Q}B_{a}=\psi_{a}\qquad,&\qquad\mathcal{Q}\psi_{a}=[\phi,B_{a}]-\epsilon_{a}\,B_{a}\
,\\\\[4.0pt]
\mathcal{Q}I=\varrho\qquad,&\qquad\mathcal{Q}\varrho=\phi\,I-I\,{\bm{a}}\
,\end{split}$ (2.56)

for $a\in\underline{\sf 4}$, where $\phi$ is the generator of $\mathsf{U}(k)$
gauge transformations and ${\bm{a}}={\rm diag}(a_{1},\dots,a_{r})$ is a
background field which parametrizes an element of the (complex) Cartan
subalgebra of $\mathsf{U}(r)_{\rm col}$.

The matrix fields $(B_{a},I)_{a\in\underline{\sf 4}}$ are required to satisfy
the seven constraints (2.44). Note that the complex moment map equations
$\mu^{\mathbbm{C}}_{ab}=0$, for $a,b\in\underline{\sf 4}$ with $a<b$, only
contribute three independent constraints, which we choose to be

$\displaystyle\mu^{\mathbbm{C}}_{\alpha\beta}=0\qquad\mbox{with}\quad(\alpha,\beta)\
\in\ \underline{\sf 3}^{\perp}:=\big{\\{}(1,2)\,,\,(1,3)\,,\,(2,3)\big{\\}}\
.$ (2.57)

To implement the equations (2.57) and
$\mu^{\mathbbm{R}}=\zeta\,\mathbbm{1}_{k}$ in (2.44), we add the Fermi
multiplets $(\vec{\chi},\vec{H})$, where
$\vec{\chi}=(\chi^{\mathbbm{C}}_{\alpha\beta},\chi^{\mathbbm{R}})_{(\alpha,\beta)\in\underline{\sf
3}^{\perp}}$ are the antighost fields in $\mathsf{End}_{\mathbbm{C}}(V)$ and
$\vec{H}=(H_{\alpha\beta}^{\mathbbm{C}},H^{\mathbbm{R}})_{(\alpha,\beta)\in\underline{\sf
3}^{\perp}}$ are the auxiliary fields. Their BRST transformations are

$\displaystyle\begin{split}\mathcal{Q}\chi^{\mathbbm{C}}_{\alpha\beta}=H^{\mathbbm{C}}_{\alpha\beta}\qquad,&\qquad\mathcal{Q}H^{\mathbbm{C}}_{\alpha\beta}=[\phi,\chi^{\mathbbm{C}}_{\alpha\beta}]-\epsilon_{\alpha\beta}\,\chi^{\mathbbm{C}}_{\alpha\beta}\
,\\\\[4.0pt]
\mathcal{Q}\chi^{\mathbbm{R}}=H^{\mathbbm{R}}\qquad,&\qquad\mathcal{Q}H^{\mathbbm{R}}=[\phi,\chi^{\mathbbm{R}}]\
,\end{split}$ (2.58)

for $(\alpha,\beta)\in\underline{\sf 3}^{\perp}$, where generally we
abbreviate $\epsilon_{ab\cdots}:=\epsilon_{a}+\epsilon_{b}+\cdots$ throughout.
Finally, we add the scalar gauge multiplet $(\phi,\bar{\phi},\eta)$ to close
the algebra of BRST transformations as

$\displaystyle\mathcal{Q}\phi=0\
,\quad\mathcal{Q}\bar{\phi}=\eta\qquad\mbox{and}\qquad\mathcal{Q}\eta=[\bar{\phi},\phi]\
.$ (2.59)

We also add matter fields to the theory, associated to the matter bundle
$\mathscr{V}_{r,k}\otimes{\bm{r}}$, which transform under a
$\mathsf{U}(r)_{\rm fla}$ flavour symmetry. Corresponding to $r$
anti-D$9$-branes, in the cohomological matrix model we include the Fermi
multiplet $(\bar{I},\bar{\varrho})$ of fields in
$\mathsf{Hom}_{\mathbbm{C}}({\bm{r}},V)$ with the BRST transformations

$\displaystyle\mathcal{Q}\bar{I}=\bar{\varrho}\qquad\mbox{and}\qquad\mathcal{Q}\bar{\varrho}=\phi\,\bar{I}-\bar{I}\,{\bm{m}}\
,$ (2.60)

where the masses ${\bm{m}}={\rm diag}(m_{1},\dots,m_{r})$ parametrize an
element of the (complex) Cartan subalgebra of $\mathsf{U}(r)_{\rm fla}$.

The action functional which corresponds to this system of fields and equations
is given by

$\displaystyle\begin{split}S&=\mathcal{Q}\,\mathrm{Tr}\Big{(}\sum_{(\alpha,\beta)\in\underline{\sf
3}^{\perp}}\,{\chi}_{\alpha\beta}^{\mathbbm{C}\,\dagger}\,\big{(}g_{\mathbbm{C}}\,{H}^{\mathbbm{C}}_{\alpha\beta}-\mu^{\mathbbm{C}}_{\alpha\beta}\big{)}+\chi^{\mathbbm{R}}\,\big{(}g_{\mathbbm{R}}\,H^{\mathbbm{R}}-\mu^{\mathbbm{R}}-\zeta\,\mathbbm{1}_{k}\big{)}\\\
&\hskip 56.9055pt+\sum_{a\in\underline{\sf
4}}\,\psi_{a}\,\big{[}\bar{\phi},B^{\dagger}_{a}\big{]}+I^{\dagger}\,\bar{\phi}\,\varrho+\bar{I}^{\dagger}\,\bar{\phi}\,\bar{\varrho}+\eta\,\big{[}\phi,\bar{\phi}\,\big{]}+{\rm
c.c.}\Big{)}\\\
&\quad\,+g_{1}\,\mathcal{Q}\,\mathrm{Tr}\big{(}\chi^{\mathbbm{R}}\,\bar{\phi}\,\big{)}+g_{2}\,\mathcal{Q}\,\mathrm{Tr}\Big{(}\sum_{a\in\underline{\sf
4}}\,\big{(}B_{a}\,\psi_{a}^{\dagger}-\psi_{a}\,B_{a}^{\dagger}\big{)}+I\,\varrho^{\dagger}-\varrho\,I^{\dagger}+\bar{I}\,\bar{\varrho}^{\dagger}-\bar{\varrho}\,\bar{I}^{\dagger}\Big{)}\
,\end{split}$ (2.61)

where ${\rm c.c.}$ means complex conjugate and we introduced appropriate
coupling constants for convenience. The last two terms have been added by hand
to respectively give a nondegenerate mass matrix for $\chi^{\mathbbm{R}}$ as
well as kinetic terms for $\psi_{a}$, $\varrho$ and $\bar{\varrho}$. After
dividing out by the volume of the gauge group, the path integral form of the
equivariant integral over the ADHM variables can then be written symbolically
as

$\displaystyle\int\,\frac{\mathscr{D}[{\rm fields}]}{{\tiny\tt
vol}\big{(}\mathsf{U}(k)\big{)}}\ \mathrm{e}^{-S}\ .$ (2.62)

To evaluate the integral (2.62), we follow the technique employed in [6]. The
first step is to use the $\mathsf{U}(k)$ gauge invariance to diagonalize the
gauge generator $\phi$. This produces a measure on $\mathbbm{R}^{k}$ with
Vandermonde determinant

$\displaystyle\frac{1}{k!}\,\prod_{i=1}^{k}\,\frac{\mathrm{d}\phi_{i}}{2\pi\,\mathrm{i}}\
\prod^{k}_{\stackrel{{\scriptstyle\scriptstyle i,j=1}}{{\scriptstyle i\neq
j}}}\,(\phi_{i}-\phi_{j})\ ,$ (2.63)

where $k!$ is the order of the Weyl group of $\mathsf{U}(k)$.

By construction of the cohomological theory, the path integral (2.62) is
independent of the coupling constants $g_{\mathbbm{C}}$, $g_{\mathbbm{R}}$,
$g_{1}$ and $g_{2}$. Taking the limit $g_{1}\rightarrow\infty$, the relevant
part of the action functional is
$\mathrm{Tr}(H^{\mathbbm{R}}\,\bar{\phi}+\chi^{\mathbbm{R}}\,\eta)$, which
shows that the fields $(H^{\mathbbm{R}},\bar{\phi},\chi^{\mathbbm{R}},\eta)$
can simply be integrated out and do not contribute non-trivially to the path
integral.

Sending $g_{\mathbbm{C}}\rightarrow\infty$, the relevant part of the action
functional is

$\displaystyle\sum_{(\alpha,\beta)\in\underline{\sf
3}^{\perp}}\,\mathrm{Tr}\Big{(}H_{\alpha\beta}^{\mathbbm{C}\,{\dagger}}\,H^{\mathbbm{C}}_{\alpha\beta}+\chi_{\alpha\beta}^{\mathbbm{C}\,{\dagger}}\,\big{(}[\phi,\chi^{\mathbbm{C}}_{\alpha\beta}]-\epsilon_{\alpha\beta}\,\chi^{\mathbbm{C}}_{\alpha\beta}\big{)}\Big{)}\
.$ (2.64)

This shows that the auxiliary fields $H^{\mathbbm{C}}_{\alpha\beta}$ do not
contribute non-trivially. On the other hand, integrating the antighost fields
$\chi^{\mathbbm{C}}_{\alpha\beta}$ produces

$\displaystyle\prod_{i,j=1}^{k}\ \prod_{(\alpha,\beta)\in\underline{\sf
3}^{\perp}}\,(\phi_{i}-\phi_{j}-\epsilon_{\alpha\beta})\ ,$ (2.65)

which contributes to the equivariant Euler class of the self-dual obstruction
bundle ${\sf Ob}_{r,k}^{-}$, with global orientation corresponding to the
choice (2.57) and our choice of ordering of antighosts in the Grassmann
integration measure.

Finally, for $g_{2}\rightarrow\infty$ the contribution of the ADHM fields
$(B_{a},I)_{a\in\underline{\sf 4}}$ to the path integral (2.62) is

$\displaystyle\prod_{i,j=1}^{k}\ \prod_{a\in\underline{\sf
4}}\,\frac{1}{\phi_{i}-\phi_{j}-\epsilon_{a}}\ \prod_{i=1}^{k}\
\prod_{l=1}^{r}\,\frac{1}{\phi_{i}-a_{l}}\ ,$ (2.66)

while the contribution of the matter fields $\bar{I}$ is

$\displaystyle\prod_{i=1}^{k}\ \prod_{l=1}^{r}\,(\phi_{i}-m_{l})\ .$ (2.67)

Putting everything together we arrive at the matrix model representation of
the equivariant integral in (2.32) given by

$\begin{split}Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon},\vec{m}):=\int^{\mathsf{T}}_{[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm
vir}}\,e_{\mathsf{T}}(\mathscr{V}_{r,k}\otimes{\bm{r}})&=\frac{(-1)^{k}}{k!}\,\Big{(}\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\Big{)}^{k}\\\
&\quad\,\times\ \oint_{\varGamma_{r,k}}\
\prod_{i=1}^{k}\,\frac{\mathrm{d}\phi_{i}}{2\pi\,\mathrm{i}}\,\frac{\mathcal{P}_{r}(\phi_{i}|\vec{m})}{\mathcal{P}_{r}(\phi_{i}|\vec{a})}\
\prod_{\stackrel{{\scriptstyle\scriptstyle i,j=1}}{{\scriptstyle i\neq
j}}}^{k}\,\mathcal{R}(\phi_{i}-\phi_{j}|\vec{\epsilon}\,)\end{split}$ (2.68)

at a generic point $\vec{\epsilon}$ of the $\varOmega$-deformation, where

$\displaystyle\mathcal{P}_{r}(x|\vec{w})=\prod_{l=1}^{r}\,(x-w_{l})\qquad\mbox{and}\qquad\mathcal{R}(x|\vec{\epsilon}\,)=\frac{x\,(x-\epsilon_{12})\,(x-\epsilon_{13})\,(x-\epsilon_{23})}{(x-\epsilon_{1})\,(x-\epsilon_{2})\,(x-\epsilon_{3})\,(x-\epsilon_{4})}\
.$ (2.69)

The instanton partition function (2.32) is then the generating function for
the integrals (2.68), obtained from a weighted sum over all the instanton
sectors:

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m})=1+\sum_{k=1}^{\infty}\,\mathtt{q}^{k}\
Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon},\vec{m})\ .$ (2.70)

The polynomial $\mathcal{P}_{r}(x|\vec{w})$ encodes the $S_{r}$-invariants for
the action of the Weyl group $S_{r}$ of $\mathsf{U}(r)$ by permutations of the
entries of $\vec{w}=(w_{1},\dots,w_{r})$. There is also a symmetry under the
group $\mathbbm{Z}_{2}$ which acts on the $\varOmega$-deformation parameters
$\vec{\epsilon}$ by permuting
$\epsilon_{1}\leftrightarrow\epsilon_{4}=-\epsilon_{123}$. Note that the
action of the center $\mathsf{U}(1)\subset\mathsf{U}(r)_{\rm col}$ on the
partition function (2.68) can be absorbed into an action of the center
$\mathsf{U}(1)\subset\mathsf{U}(r)_{\rm fla}$: an overall shift $a_{l}\to
a_{l}+a$ of the Coulomb parameters can be absorbed by a simultaneous overall
shift $\phi_{i}\to\phi_{i}+a$ of the integration variables and redefinition
$m_{l}\to m_{l}+a$ of the masses. Hence there is a natural action of ${\sf
P}\big{(}\mathsf{U}(r)_{\rm col}\times\mathsf{U}(r)_{\rm fla}\big{)}$ on
$Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon},\vec{m})$.

The integral (2.68) also appears in [20, 56, 57, 26] from different
perspectives. It is ill-defined as a Lebesgue integral over $\mathbbm{R}^{k}$
because its integrand does not converge to zero at infinity. Instead, as
suggested by the notation, it should be interpreted as a contour integral
where the sum of residues reproduces the sum over $\mathsf{T}$-fixed points in
(2.32). However, assigning a choice of contour
$\varGamma_{r,k}\subset\mathbbm{C}^{k}$ to the integral (2.68) directly is not
straightforward. We shall discuss this point further below.

#### Dimensional Reduction

From (2.68) we may immediately deduce

######  Proposition 2.71.

The equivariant instanton partition function of the cohomological gauge theory
with a massive fundamental hypermultiplet on $\mathbbm{C}^{4}$ is related to
the Coulomb branch partition function
$Z_{\mathbbm{C}^{3}}^{r}(\mathtt{q};\vec{a},\epsilon_{1},\epsilon_{2},\epsilon_{3})$
for Donaldson–Thomas theory on the toric Kähler threefold $\mathbbm{C}^{3}$
through the mass specialisations

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},m_{l}=a_{l}+\epsilon_{4})=Z_{\mathbbm{C}^{3}}^{r}\big{(}(-1)^{r+1}\,\mathtt{q};\epsilon_{1},\epsilon_{2},\epsilon_{3}\big{)}={M}(-\mathtt{q})^{-\frac{r\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}}}\
,$ (2.72)

where

$\displaystyle{M}(q)=\prod_{n=1}^{\infty}\,\frac{1}{(1-q^{n})^{n}}$ (2.73)

is the MacMahon function.

###### Proof.

Using the Calabi–Yau condition $\epsilon_{4}=-\epsilon_{123}$ on
$\mathbbm{C}^{4}$ one finds that the matrix integral (2.68) gives

$\displaystyle\begin{split}&Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon},m_{l}=a_{l}-\epsilon_{123})\\\\[4.0pt]
&\qquad=\frac{1}{k!}\,\Big{(}\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{123}}\Big{)}^{k}\,\oint_{\varGamma_{r,k}}\
\prod_{i=1}^{k}\,\frac{\mathrm{d}\phi_{i}}{2\pi\,\mathrm{i}}\,\frac{\mathcal{P}_{r}(\phi_{i}+\epsilon_{123}|\vec{a})}{\mathcal{P}_{r}(\phi_{i}|\vec{a})}\
\prod_{\stackrel{{\scriptstyle\scriptstyle i,j=1}}{{\scriptstyle i\neq
j}}}^{k}\,\mathcal{R}(\phi_{i}-\phi_{j}|\epsilon_{1},\epsilon_{2},\epsilon_{3},-\epsilon_{123})\
.\end{split}$ (2.74)

Up to an overall sign $(-1)^{(r+1)\,k}$, this is exactly the charge $k$
integral contribution
$Z_{\mathbbm{C}^{3}}^{r,k}(\vec{a},\epsilon_{1},\epsilon_{2},\epsilon_{3})$
from [6, eq. (5.23)] to the rank $r$ instanton partition function on
$\mathbbm{C}^{3}$ at a generic point
$(\epsilon_{1},\epsilon_{2},\epsilon_{3})$ of the $\varOmega$-deformation.
This proves the first equality in (2.72).

In the rank one case $r=1$, the second equality in (2.72) was originally
proven in [32, Theorem 1], where
$Z_{\mathbbm{C}^{3}}^{1,k}(\epsilon_{1},\epsilon_{2},\epsilon_{3})$ computes
the equivariant volume of the Hilbert scheme of $k$ points on
$\mathbbm{C}^{3}$:

$\displaystyle
Z_{\mathbbm{C}^{4}}^{1,k}(\vec{\epsilon},m=a-\epsilon_{123})=Z_{\mathbbm{C}^{3}}^{1,k}(\epsilon_{1},\epsilon_{2},\epsilon_{3})=\int^{\mathsf{T}_{\vec{\epsilon}}}_{[{\sf
Hilb}^{k}(\mathbbm{C}^{3})]^{\rm vir}}\,1\ .$ (2.75)

The general higher rank result for $r>1$ was conjectured in [62, 33] and
proven in [34], where
$Z_{\mathbbm{C}^{3}}^{r,k}(\vec{a},\epsilon_{1},\epsilon_{2},\epsilon_{3})$
computes the equivariant volume of the Quot scheme ${\sf
Quot}^{k}_{r}(\mathbbm{C}^{3})$ of zero-dimensional quotients of
$\mathcal{O}_{\mathbbm{C}^{3}}^{\oplus r}$ of length $k$. ∎

######  Remark 2.76.

From the string theory perspective, Proposition 2.71 superficially supports
Sen’s conjecture [59]: it implies that the specification
$m_{l}=a_{l}+\epsilon_{4}$, which corresponds to the diagonal coordinates for
the maximal torus $\mathsf{U}(1)^{r}_{\rm fla}\times\mathsf{U}(1)^{r}_{\rm
col}$ of the global symmetry group $\mathsf{U}(r)_{\rm
fla}\times\mathsf{U}(r)_{\rm col}$, can be interpreted as a particular
configuration of D9-branes and anti-D9-branes that decay into D7-branes, whose
bound states with D1-branes correspond to instantons on $\mathbbm{C}^{3}$.
This was noted in the K-theory version of the theory for $r=1$ by Nekrasov
[19]. The rank one version of the dimensional reduction of Proposition 2.71
was also studied rigorously by [24] and its K-theory counterpart by [38].

#### Fixed Points and Solid Partitions

Starting from the integral (2.68) directly, we can identify the combinatorial
enumeration problem computed by the sum over residues. The integrand has poles
along the hyperplanes

$\displaystyle\phi_{i}-\phi_{j}-\epsilon_{a}=0\qquad\mbox{and}\qquad\phi_{i}-a_{l}=0\
.$ (2.77)

These are exactly the fixed points of the equivariant action of
$\mathsf{T}_{\vec{\epsilon}}$ on the ADHM variables, which are given by

$\displaystyle t_{a}\,B_{a}=g\,B_{a}\,g^{-1}\qquad\mbox{and}\qquad
I=g\,I\,h^{-1}\ ,$ (2.78)

for
$t_{a}=\mathrm{e}^{\,\mathrm{i}\,\epsilon_{a}}\in\mathsf{T}_{\vec{\epsilon}}$,
$g=\exp(\mathrm{i}\,\phi)\in\mathsf{U}(k)$ and
$h=\exp(\mathrm{i}\,{\bm{a}})\in\mathsf{U}(r)_{\rm col}$. These equations
define a group representation
$\mathsf{T}_{\vec{a}}\times\mathsf{T}_{\vec{\epsilon}}\longrightarrow\mathsf{U}(k)$,
and they are equivalent to

$\displaystyle(B_{a})_{ij}\,(\phi_{i}-\phi_{j}-\epsilon_{a})=0\qquad\mbox{and}\qquad
I_{il}\,(\phi_{i}-a_{l})=0\ .$ (2.79)

Together with $\bar{\varrho}_{il}=0$, these are the equations for the fixed
points of the BRST charge $\mathcal{Q}$, onto which the path integral of the
topological field theory localizes.

The fixed points (2.77) have a standard combinatorial description. A subtlety
here is that the fixed point equations (2.78) are given for the equivariant
action of the (complex) maximal torus $\mathsf{T}_{\vec{\epsilon}}$ of
$\mathsf{SU}(4)$, rather than the maximal torus
$\mathsf{T}_{\vec{\epsilon}\,^{\prime}}$ of $\mathsf{U}(4)$. Following
Nekrasov’s approach from [19], one can argue that the two sets of fixed point
equations are equivalent: The gauge transformations generated by $\phi$ can be
replaced with $\mathsf{U}(k)$ gauge transformations generated by
$\phi^{\prime}$ such that

$\displaystyle[B_{a},\phi^{\prime}]=\epsilon^{\prime}_{a}\,B_{a}\ ,$ (2.80)

for $a\in\underline{\sf 4}$ and _generic_
$\vec{\epsilon}\,^{\prime}=(\epsilon_{1}^{\prime},\epsilon_{2}^{\prime},\epsilon_{3}^{\prime},\epsilon^{\prime}_{4})$.
Clearly if (2.80) holds generically, then it holds in particular at the
Calabi–Yau specialisation with $\epsilon_{4}=-\epsilon_{123}$.

Conversely, if $[B_{a},\phi]=\epsilon_{a}\,B_{a}$ with $\epsilon_{1234}=0$,
then $X:=B_{1}\,B_{2}\,B_{3}\,B_{4}$ is nilpotent, that is, $[X,\phi]=0$. By
the Jacobson–Morozov theorem, it follows that $X$ can be extended to an
$\mathfrak{sl}(2,\mathbbm{C})$ triple $(X,Y,H)$ in
$\mathsf{End}_{\mathbbm{C}}(V)$:

$\displaystyle[X,H]=2\,X\ ,\quad[H,Y]=-2\,Y\qquad\mbox{and}\qquad[X,Y]=H\ ,$
(2.81)

where $Y$ is nilpotent and $H$ is the generator of the associated Cartan
subalgebra. Given arbitrary $\vec{\epsilon}\,^{\prime}$, we define
$\phi^{\prime}=\frac{1}{2}\,\epsilon^{\prime}_{1234}\,H$ with
$\epsilon^{\prime}_{1234}\neq 0$. Then
$[X,\phi^{\prime}]=\epsilon^{\prime}_{1234}\,X$, and in particular

$\displaystyle\begin{split}&[B_{1},\phi^{\prime}]\,B_{2}\,B_{3}\,B_{4}+B_{1}\,[B_{2},\phi^{\prime}]\,B_{3}\,B_{4}+B_{1}\,B_{2}\,[B_{3},\phi^{\prime}]\,B_{4}+B_{1}\,B_{2}\,B_{3}\,[B_{4},\phi^{\prime}]=\epsilon^{\prime}_{1234}\,X\
.\end{split}$ (2.82)

It follows that $[B_{a},\phi^{\prime}]=\epsilon^{\prime}_{a}\,B_{a}$ (up to
relabelling of the entries of $\vec{\epsilon}\,^{\prime}$), as required. A
geometric proof of this fact (at the level of fixed points of Hilbert schemes
of points on $\mathbbm{C}^{4}$) is found in [24].

With this equivalence in mind, we write a generic fixed point (2.77) by
setting the eigenvalues of $\phi$ to

$\displaystyle\phi_{(a_{l};\vec{p}\,)}=a_{l}+\vec{p}\cdot\vec{\epsilon}\ ,$
(2.83)

where $l\in\\{1,\dots,r\\}$ and
$\vec{p}=(p_{1},p_{2},p_{3},p_{4})\in\mathbbm{Z}^{4}_{>0}$ is a quadruple of
positive integers; here and in the following we denote
$\vec{p}\cdot\vec{\epsilon}:=\sum_{a\in\underline{\sf
4}}\,p_{a}\,\epsilon_{a}$ and we used the Calabi–Yau condition (2.31). In
other words, a fixed point is parametrized by a set of tuples
$(a_{l};\vec{p}\,)$. There are $k$ different tuples, because there are $k$
eigenvalues. Given an instanton number $k$, we can partition it into integers
$k=k_{1}+\cdots+k_{r}$ such that for fixed $l$, and hence fixed $a_{l}$,
$k_{l}\geq 0$ is the total number of quadruples $\vec{p}$ of integers.

Since the quadruple $(1,1,1,1)$ is always present, the collection of tuples
$(a_{l};\vec{p}\,)$ for fixed $l$ may be represented by a _solid partition_
$\sigma_{l}=(\sigma_{i,j,n})_{i,j,n\geq 1}$, that is, a sequence of non-
negative integers $\sigma_{i,j,n}\in\mathbbm{Z}_{\geq 0}$ satisfying

$\displaystyle\sigma_{i,j,n}\geq\sigma_{i+1,j,n}\
,\quad\sigma_{i,j,n}\geq\sigma_{i,j+1,n}\qquad\mbox{and}\qquad\sigma_{i,j,n}\geq\sigma_{i,j,n+1}\
,$ (2.84)

for all $i,j,n\geq 1$. The _size_ of the solid partition $\sigma_{l}$ is

$\displaystyle|\sigma_{l}|:=\sum_{i,j,n\geq 1}\,\sigma_{i,j,n}=k_{l}\ .$
(2.85)

It follows that there is a one-to-one correspondence between fixed points
$(a_{l};\vec{p}\,)$ and solid partitions
$\sigma_{l}=\\{\vec{p}\in\mathbbm{Z}^{4}_{>0}\leavevmode\nobreak\
|\leavevmode\nobreak\ 1\leq p_{4}\leq\sigma_{p_{1},p_{2},p_{3}}\\}$, and
therefore one may associate to any fixed point an array of solid partitions
$\vec{\sigma}=(\sigma_{1},\dots,\sigma_{r})$ whose total size is the instanton
number:

$\displaystyle|\vec{\sigma}|=\sum_{l=1}^{r}\,|\sigma_{l}|=\sum_{l=1}^{r}\,k_{l}=k\
.$ (2.86)

Note that here the only role of the ADHM field $I$, which arises from D1–D9
strings, is to assign a ‘colouring’ of the solid partitions into the $r$-tuple
$\vec{\sigma}$.

######  Remark 2.87.

Recall that a plane partition is a collection of triples of positive integers
$(p_{1},p_{2},p_{3})$ where the pairs $(p_{1},p_{2})$ define a Young diagram
$\lambda$ and $p_{3}\leq\pi_{p_{1},p_{2}}$, with $\pi$ a
$\mathbbm{Z}_{>0}$-valued function of $(i,j)\in\lambda$ such that
$\pi_{i,j}\geq\pi_{i+1,j}$ and $\pi_{i,j}\geq\pi_{i,j+1}$. The rank one
instanton partition function for the $\mathcal{N}_{\textrm{\tiny T}}=2$
cohomological gauge theory on $\mathbbm{C}^{3}$ at the Calabi–Yau
specialisation of the $\varOmega$-deformation, or equivalently the MacMahon
function (2.73), is the generating function for the number of plane partitions
$\pi$ of fixed size $|\pi|=\sum_{i,j\geq 1}\,\pi_{i,j}$:

$\displaystyle
Z_{\mathbbm{C}^{3}}^{r=1}\big{(}\mathtt{q};\epsilon_{1},\epsilon_{2},\epsilon_{3}\big{)}\big{|}_{\epsilon_{123}=0}=M(-\mathtt{q})=\sum_{\pi}\,(-\mathtt{q})^{|\pi|}\
.$ (2.88)

More generally,
$Z_{\mathbbm{C}^{3}}^{r}\big{(}\mathtt{q};\epsilon_{1},\epsilon_{2},\epsilon_{3}\big{)}\big{|}_{\epsilon_{123}=0}=M\big{(}(-1)^{r}\,\mathtt{q}\big{)}$
is the generating function for $r$-tuples $\vec{\pi}=(\pi_{1},\dots,\pi_{r})$
of plane partitions.

Given a _solid_ partition $\sigma=(\sigma_{i,j,n})_{i,j,n\geq 1}$, we can view
it as a stack of _plane_ partitions, with cubes piled in the positive
$16$-multiant $(x_{1},x_{2},x_{3},x_{4})\in\mathbbm{Z}_{\geq 0}^{4}$, where
$\sigma_{i,j,n}$ is the height function of the stack of hypercubes defined on
the $(x_{1},x_{2},x_{3})$ hyperplane.

###  2.6 Instanton Partition Function

We shall now discuss the explicit evaluation of the equivariant partition
function (2.32) as a combinatorial expansion based on the generalized ADHM
parametrization of the instanton moduli spaces $\mathfrak{M}_{r,k}$. As
discussed in Section 2.5, the partition function (2.68) receives contributions
from the fixed points $[(B_{a},I)_{a\in\underline{\sf
4}}]\in\mathfrak{M}^{\mathsf{T}}_{r,k}$ of the ADHM moduli space. Thus we need
to describe the local geometry of the instanton moduli space
$\mathfrak{M}_{r,k}$ around these fixed points, and suitably incorporate the
moduli of the matter fields.

Let $\vec{\sigma}$ be a fixed point, corresponding to an $r$-tuple of solid
partitions, and consider the ADHM deformation complex

$\displaystyle
0\longrightarrow\mathsf{End}_{\mathbbm{C}}(V_{\vec{\sigma}})\xrightarrow{\
\mathrm{d}_{1}\
}\begin{matrix}\mathsf{End}_{\mathbbm{C}}(V_{\vec{\sigma}})\otimes
Q\\\\[4.30554pt] \oplus\\\\[4.30554pt]
\mathsf{Hom}_{\mathbbm{C}}(W_{\vec{\sigma}},V_{\vec{\sigma}})\end{matrix}\xrightarrow{\
\mathrm{d}_{2}\
}\mathsf{End}_{\mathbbm{C}}(V_{\vec{\sigma}})\otimes\text{\Large$\wedge$}_{-}^{0,2}\,Q\longrightarrow
0\ ,$ (2.89)

where $Q\simeq\mathbbm{C}^{4}$ is the four-dimensional fundamental
$\mathsf{T}_{\vec{\epsilon}}\,$-module with weight decomposition

$\displaystyle Q=t_{1}^{-1}+t_{2}^{-1}+t_{3}^{-1}+t_{4}^{-1}\ ,$ (2.90)

and the weights $t_{a}=\mathrm{e}^{\,\mathrm{i}\,\epsilon_{a}}$ satisfy the
Calabi–Yau condition

$\displaystyle t_{1}\,t_{2}\,t_{3}\,t_{4}=1\ .$ (2.91)

The map $\mathrm{d}_{1}$ is an infinitesimal $\mathsf{GL}(k,\mathbbm{C})$
gauge transformation, while $\mathrm{d}_{2}$ is the linearization of the
holomorphic ADHM equations $[B_{\alpha},B_{\beta}]=0$; explicitly

$\displaystyle\mathrm{d}_{1}\phi=\bigg{(}\begin{matrix}\big{(}[\phi,B_{a}]\big{)}_{a\in\underline{\sf
4}}\\\
\phi\,I\end{matrix}\bigg{)}\qquad\mbox{and}\qquad\mathrm{d}_{2}\bigg{(}\begin{matrix}(b_{a})_{a\in\underline{\sf
4}}\\\
i\end{matrix}\bigg{)}=\big{(}[b_{\alpha},B_{\beta}]+[B_{\alpha},b_{\beta}]\big{)}_{(\alpha,\beta)\in\underline{\sf
3}^{\perp}}\ .$ (2.92)

By construction, the first cohomology of the cochain complex (2.89) is a local
model for the tangent space $T_{\vec{\sigma}}\mathfrak{M}_{r,k}$ at the fixed
point $\vec{\sigma}$, while the second cohomology parametrizes the local
obstruction space $({\sf Ob}_{r,k}^{-})_{\vec{\sigma}}$. We assume that
$\ker(\mathrm{d}_{1})=0$. Then the equivariant index of this complex computes
the virtual sum

$\displaystyle\sqrt{\mathrm{ch}_{\mathsf{T}}}\big{(}T_{[\mathcal{E}]}^{\rm
vir}\,\mathfrak{M}_{r,k}\big{)}={\sf
Ext}_{\mathcal{O}_{{\mathbbm{P}}^{4}}}^{1}(\mathcal{E},\mathcal{E})\,\ominus\,{\sf
Ext}_{\mathcal{O}_{{\mathbbm{P}}^{4}}}^{2\,-}(\mathcal{E},\mathcal{E})$ (2.93)

of cohomology groups, where $[\mathcal{E}]$ is the isomorphism class of a
framed torsion free sheaf on ${\mathbbm{P}}^{4}$ corresponding to the fixed
point $\vec{\sigma}$. This gives

$\displaystyle\begin{split}\sqrt{\mathrm{ch}_{\mathsf{T}}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\,\mathfrak{M}_{r,k}\big{)}&=V_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}\,\big{(}t^{-1}_{1}+t^{-1}_{2}+t^{-1}_{3}+t^{-1}_{4}\big{)}+W_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}\\\ &\quad\,-V_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}\,\big{(}1+t^{-1}_{1}\,t^{-1}_{2}+t^{-1}_{1}\,t^{-1}_{3}+t^{-1}_{2}\,t^{-1}_{3}\big{)}\
.\end{split}$ (2.94)

Recalling that the fibre of the matter bundle
$(\mathscr{V}_{r,k})_{\vec{\sigma}}\otimes\,{\bm{r}}$ at the fixed point
$\vec{\sigma}$ is $\mathsf{Hom}_{\mathbbm{C}}({\bm{r}},V_{\vec{\sigma}})$, the
total index we wish to compute is

$\displaystyle\begin{split}\chi_{\vec{\sigma}}:\\!&=\sqrt{\mathrm{ch}_{\mathsf{T}}}\big{(}T^{\rm
vir}_{\vec{\sigma}}\,\mathfrak{M}_{r,k}\big{)}-\mathrm{ch}_{\mathsf{T}}\big{(}(\mathscr{V}_{r,k})_{\vec{\sigma}}\otimes{\bm{r}}\big{)}\\\\[4.0pt]
&=-V_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}\,\big{(}1-t^{-1}_{1}-t^{-1}_{2}-t^{-1}_{3}-t^{-1}_{4}+t^{-1}_{1}\,t^{-1}_{2}+t^{-1}_{1}\,t^{-1}_{3}+t^{-1}_{2}\,t^{-1}_{3}\big{)}\\\
&\quad\,+\big{(}W_{\vec{\sigma}}^{*}-{\bm{r}}^{*}\big{)}\otimes
V_{\vec{\sigma}}\ .\end{split}$ (2.95)

In these expressions we regard the various vector spaces in (2.94) and (2.95)
as elements of the representation ring of the torus
$\mathsf{T}=\mathsf{T}_{\vec{a}}\times\mathsf{T}_{\vec{\epsilon}}\times\mathsf{T}_{\vec{m}}$,
i.e. as polynomials in $t_{a}$ for $a\in\underline{\sf 4}$ as well as in
$e_{l}:=\mathrm{e}^{\,\mathrm{i}\,a_{l}}$ and
$f_{l}:=\mathrm{e}^{\,\mathrm{i}\,m_{l}}$ for $l=1,\dots,r$. The dual
involution acts on the weights as $t_{a}^{*}=t_{a}^{-1}$,
$e_{l}^{*}=e_{l}^{-1}$ and $f_{l}^{*}=f_{l}^{-1}$.

As the notation suggests, the character (2.94) can be identified as a square
root of the character of the virtual tangent bundle at the fixed point
$\vec{\sigma}$:

$\displaystyle\begin{split}\mathrm{ch}_{\mathsf{T}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{r,k}\big{)}&=-V_{\vec{\sigma}}^{*}\otimes V_{\vec{\sigma}}\
\prod_{a\in\underline{\sf
4}}\,\big{(}1-t^{-1}_{a}\big{)}+W_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}+V_{\vec{\sigma}}^{*}\otimes W_{\vec{\sigma}}\\\\[4.0pt]
&=\sqrt{\mathrm{ch}_{\mathsf{T}}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{r,k}\big{)}+\sqrt{\mathrm{ch}_{\mathsf{T}}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{r,k}\big{)}^{*}\ .\end{split}$ (2.96)

The choice of square root is not unique and different choices will produce
contributions to the $k$-instanton partition functions
$Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon},\vec{m})$ below which
coincide up to a sign $\pm 1$ which we fix by hand, see e.g. [24, 20, 38, 63,
27]. Every sign choice is equivalent to a choice of local orientation at each
$\mathsf{T}_{\vec{a},\vec{\epsilon}}\,$-fixed point $\vec{\sigma}$ of the
instanton moduli space [23], and it produces a sign factor
$(-1)^{\mathtt{O}_{\vec{\sigma}}}$.

With suitable choices of bases we can decompose the $\mathsf{U}(r)_{\rm
col}$-module $W$ at a fixed point $\vec{\sigma}$ and the $\mathsf{U}(r)_{\rm
fla}$-module ${\bm{r}}$ into one-dimensional vector spaces for the
$\mathsf{T}_{\vec{a}}\,$-action and the $\mathsf{T}_{\vec{m}}$-action,
respectively, with the weight decompositions

$\displaystyle
W_{\vec{\sigma}}=\sum_{l=1}^{r}\,e_{l}\qquad\mbox{and}\qquad{\bm{r}}=\sum_{l=1}^{r}\,f_{l}\
.$ (2.97)

On the other hand, from the fixed point equations (2.78) and the stability
condition (cf. Section 2.4) it follows that the
$\mathsf{T}_{\vec{a},\vec{\epsilon}}\,$-module decomposition of the vector
space $V$ at the fixed point $\vec{\sigma}$ is given by

$\displaystyle V_{\vec{\sigma}}=\sum_{l=1}^{r}\,e_{l}\
\sum_{\vec{p}\,\in\sigma_{l}}\,t_{1}^{p_{1}}\,t_{2}^{p_{2}}\,t_{3}^{p_{3}}\,t_{4}^{p_{4}}\
.$ (2.98)

Hence each term in the weight decomposition of the vector space
$V_{\vec{\sigma}}$ corresponds to an element in the collection of solid
partitions $\vec{\sigma}=(\sigma_{1},\dots,\sigma_{r})$.

After inserting the decompositions (2.97) and (2.98) into (2.95), we can use
the equivariant characters $\chi_{\vec{\sigma}}$ to evaluate the instanton
partition function (2.32) by using the operation

$\displaystyle\widehat{\tt e}\Big{[}\sum_{I}\,n_{I}\,\mathrm{e}^{\,{\sf
w}_{I}}\Big{]}=\prod_{{\sf w}_{I}\neq 0}\,{\sf w}_{I}^{n_{I}}$ (2.99)

that converts the additive Chern characters to multiplicative (top form) Euler
classes. Then ${\sf w}_{I}(\vec{a},\vec{\epsilon},\vec{m})$ gives the weights
of the $\mathsf{T}$-action on the virtual tangent bundle and on the matter
bundle.

Altogether this leads to the general combinatorial formula

$\displaystyle\begin{split}Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon},\vec{m})&=\sum_{|\vec{\sigma}|=k}\,(-1)^{{\tt
O}_{\vec{\sigma}}}\ \widehat{\tt e}[-\chi_{\vec{\sigma}}]\\\\[4.0pt]
&=\sum_{|\vec{\sigma}|=k}\,(-1)^{{\tt O}_{\vec{\sigma}}}\ \prod_{l=1}^{r}\
\prod_{\vec{p}_{l}\in\sigma_{l}}^{\neq
0}\,\frac{\mathcal{P}_{r}(a_{l}+\vec{p}_{l}\cdot\vec{\epsilon}\,|\vec{m})}{\mathcal{P}_{r}(a_{l}+\vec{p}_{l}\cdot\vec{\epsilon}\,|\vec{a})}\\\
&\hskip 142.26378pt\times\prod_{l^{\prime}=1}^{r}\
\prod_{\vec{p}^{\,\prime}_{l^{\prime}}\in\sigma_{l^{\prime}}}^{\neq
0}\,\mathcal{R}\big{(}a_{l}-a_{l^{\prime}}+(\vec{p}_{l}-\vec{p}^{\,\prime}_{l^{\prime}})\cdot\vec{\epsilon}\,\big{|}\vec{\epsilon}\,\big{)}\end{split}$
(2.100)

for the partition function (2.68), where the polynomial $\mathcal{P}_{r}$ and
the rational function $\mathcal{R}$ are defined in (2.69). The superscripts ≠0
on the products indicate the omission of terms with zero numerator or
denominator from the formula (2.100), according to the prescription in (2.99).
The sign factors $(-1)^{\mathtt{O}_{\vec{\sigma}}}$ are given by sums of
cardinalities

$\displaystyle{\tt
O}_{\vec{\sigma}}=\sum_{l=1}^{r}\,\Big{|}\Big{\\{}(\vec{p}_{l},\vec{p}_{l}^{\,\prime})\in\sigma_{l}\times\sigma_{l}\
\Big{|}\ {\small\begin{matrix}(p_{l})_{\alpha}\neq(p_{l}^{\prime})_{\alpha}\\\
(p_{l})_{\alpha}-(p_{l})_{4}+1=(p_{l}^{\prime})_{\alpha}-(p_{l}^{\prime})_{4}\end{matrix}}\normalsize\
,\ \alpha\in\\{1,2,3\\}\Big{\\}}\Big{|}\ .$ (2.101)

The full partition function (2.70), including all instanton sectors, is given
by

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m})=\sum_{\vec{\sigma}}\,(-1)^{{\tt
O}_{\vec{\sigma}}}\ \mathtt{q}^{|\vec{\sigma}|}\ \widehat{\tt
e}[-\chi_{\vec{\sigma}}]\ ,$ (2.102)

where the sum runs over all $r$-tuples
$\vec{\sigma}=(\sigma_{1},\dots,\sigma_{r})$ of solid partitions $\sigma_{l}$
(including the empty solid partitions of sizes $|\sigma_{l}|=0$).

######  Remark 2.103.

The expression for ${\mathtt{O}_{\vec{\sigma}}}$ in (2.101) differs from the
sign factor conjectured by Nekrasov and Piazzalunga in [20], which is proved
by Kool and Rennemo in [64], and also from those of [38, 23, 27], because our
choice of square root (2.94) is different. It was found by comparing the
character $\sqrt{\mathrm{ch}_{\mathsf{T}}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{r,k}\big{)}$ in (2.94) with the character of [27, Section
5.7] and [64]: they differ in the sign factor $(-1)^{k\,(r-1)+{\rm rank}\,{\sf
A}_{\vec{\sigma}}}$ given by

$\displaystyle{\sf A}_{\vec{\sigma}}=\big{(}V_{\vec{\sigma}}^{*}\otimes
V_{\vec{\sigma}}\ t_{4}^{-1}\big{)}^{\rm fix}\ ,$ (2.104)

where the superscript fix stands for the
$\mathsf{T}_{\vec{a},\vec{\epsilon}}\,$-fixed part. This is tantamount to
counting the zeroes of
$a_{l}-a_{l^{\prime}}+(\vec{p}_{l}-\vec{p}^{\,\prime}_{l^{\prime}})\cdot\vec{\epsilon}-\epsilon_{4}$,
for $\vec{p}_{l}\in\sigma_{l}$ and
$\vec{p}^{\,\prime}_{l^{\prime}}\in\sigma_{l^{\prime}}$. Then the sign factor
(2.101) is calculated as the difference of (2.104) and the sign factor from
[27].

Since our choice of square root differs from the square root of [27] by
conjugation of a finite number of terms, which is shown to be movable in [20],
our square root is movable as well.111We thank an anonymous referee for
pointing this out to us. That is, each term contains weights $t_{a}$ and
$e_{l}$.

With our choice of square root
$\sqrt{\mathrm{ch}_{\mathsf{T}}}\big{(}T_{\vec{\sigma}}^{\rm
vir}\mathfrak{M}_{r,k}\big{)}$ in (2.94), and subsequently of the index
$\chi_{\vec{\sigma}}$, the first term affected by the sign
$(-1)^{\mathtt{O}_{\vec{\sigma}}}$ appears at instanton number $k=16$.

The formula (2.100) makes evident the evaluation of (2.68) as a sum over
residues; the signs $(-1)^{\mathtt{O}_{\vec{\sigma}}}$ should then come from a
careful residue calculation. From this perspective the dimensional reduction
of Proposition 2.71 can be seen to arise through the introduction of extra
fixed parts to the index $\chi_{\vec{\sigma}}$ from terms involving
$f^{-1}_{l}\,e_{l}\,t_{4}$, which keeps only contributions to (2.102) from
solid partitions $\vec{\sigma}$ with $(p_{l})_{4}=1$ for all
$\vec{p}_{l}\in\sigma_{l}$, $l=1,\dots,r$; these correspond to arrays
$\vec{\pi}$ of plane partitions (cf. Remark 2.87) and the sign factors (2.101)
become $\mathtt{O}_{\vec{\pi}}=|\vec{\pi}|$. In fact, it is possible to
“uplift” the result of Proposition 2.71 to infer

######  Conjecture 2.105.

The equivariant instanton partition function of the cohomological gauge theory
with a massive fundamental hypermultiplet on $\mathbbm{C}^{4}$ is given by

$\displaystyle
Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m})=Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{\epsilon},m)=M(-\mathtt{q})^{-\frac{r\,m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}}\qquad\mbox{with}\quad
m=\frac{1}{r}\,\sum_{l=1}^{r}\,(m_{l}-a_{l})\ .$ (2.106)

######  Remark 2.107.

We checked explicitly that the order $\mathtt{q}$ and $\mathtt{q}^{2}$ terms
of the series (2.102) agree with (2.106) in the rank one case $r=1$, as well
as with an elementary Cauchy residue evaluation of the contour integrals in
(2.68) if one considers contours $\varGamma_{r,k}\subset\mathbbm{C}^{k}$ which
enclose all singularities of the integrand. The rank one case of Conjecture
2.105 was conjectured by Cao and Kool in [24]. The $k$-instanton contributions
with $k\geq 1$ give the rank $r$ equivariant Donaldson–Thomas invariants of
$\mathbbm{C}^{4}$:

$\displaystyle\int^{\mathsf{T}}_{[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm
vir}}\,e_{\mathsf{T}}(\mathscr{V}_{r,k}\otimes{\bm{r}})=\sum_{n=1}^{k}\,\frac{1}{n!}\,\Big{(}\frac{r\,m\,\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\Big{)}^{n}\
\sum_{\begin{subarray}{c}k_{1},\dots,k_{n}\geq 1\\\
k_{1}+\cdots+k_{n}=k\end{subarray}}\ \prod_{i=1}^{n}\
\sum_{d|k_{i}}\,\frac{k_{1}\cdots k_{n}}{d^{2}}\ .$ (2.108)

Following [65, 66], one can write a well-defined general integration
prescription starting from (2.68) which calculates the partition function
_$Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon},\vec{m})$_ by uplifting the
theory to an $8{+}1$-dimensional theory on
$\mathbbm{C}^{4}\times\mathbb{S}_{\beta}^{1}$, where $\mathbb{S}_{\beta}^{1}$
is the circle of circumference $\beta$. This defines the K-theory version of
the gauge theory, wherein the eight-dimensional instantons are viewed as
constant loops on $\mathbbm{C}^{4}\times\mathbb{S}_{\beta}^{1}$; the original
theory is recovered in the cohomological limit $\beta\rightarrow 0$ where the
circle shrinks to a point. We choose as integration contour for the uplifted
integral any contour large enough to enclose all singularities of the
integrand, and finally we apply a residue theorem to evaluate the contour
integrals.

Nekrasov and Piazzalunga propose a plethystic exponential formula in [20] for
the uplifted instanton partition function on
$\mathbbm{C}^{4}\times\mathbb{S}_{\beta}^{1}$, generalizing the rank one
conjecture of [19]; in their approach, the sign factors arise from evaluation
of the Jeffrey–Kirwan residue formula. Their conjectured formula is proven by
Kool and Rennemo in [64]. It is straightforward to show that this formula
reproduces Conjecture 2.105 in the cohomological limit (see [19, 20, 38] for
the rank one case).

In Appendix B.1 we sketch the steps of a direct but less conceptual proof of
the formula (2.106) using the combinatorial evaluation (2.100) of the quiver
matrix model (2.68), which is based on the dimensional reduction of
Proposition 2.71. We believe that such an argument, while presently
incomplete, provides useful insights into the symmetries of the theory, which
are not evident through other approaches. These arguments can also be extended
to more general settings, such as some of our orbifold theories below, for
which rigorous results are not yet available.

###  2.7 Pure $\mathcal{N}_{\textrm{\tiny T}}=2$ Gauge Theory

As we discussed in Section 2.3, Nekrasov’s $\varOmega$-deformation permits the
definition of the equivariant partition function (2.33) for the eight-
dimensional cohomological gauge theory even in the absence of fundamental
matter fields. It should come as no surprise that the only difference from our
previous calculations is the absence of the matter bundle
$\mathscr{V}_{r,k}\otimes{\bm{r}}$, i.e. the theory is based entirely on the
generalized ADHM data. The same BRST construction from Section 2.5 applies by
dropping the matter field contribution to the path integral, giving the
equivariant integral

$\displaystyle\begin{split}Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon}\,)^{\rm
pure}:\\!&=\int^{\mathsf{T}_{\vec{a},\vec{\epsilon}}}_{[\mathfrak{M}_{r,k}]_{\mathtt{o}}^{\rm
vir}}\,1\\\\[4.0pt]
&=\frac{(-1)^{k}}{k!}\,\Big{(}\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\Big{)}^{k}\,\oint_{\varGamma_{r,k}}\
\prod_{i=1}^{k}\,\frac{\mathrm{d}\phi_{i}}{2\pi\,\mathrm{i}}\,\frac{1}{\mathcal{P}_{r}(\phi_{i}|\vec{a})}\
\prod_{\stackrel{{\scriptstyle\scriptstyle i,j=1}}{{\scriptstyle i\neq
j}}}^{k}\,\mathcal{R}(\phi_{i}-\phi_{j}|\vec{\epsilon}\,)\end{split}$ (2.109)

from the field theory perspective.

The contour integral (2.109) can be computed from the Chern character (2.94)
alone, giving the combinatorial expansion into solid partitions

$\begin{split}Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon}\,)^{\rm
pure}&=\sum_{|\vec{\sigma}|=k}\,(-1)^{{\tt O}_{\vec{\sigma}}}\
\prod_{l=1}^{r}\ \prod_{\vec{p}_{l}\in\sigma_{l}}^{\neq
0}\,\frac{1}{\mathcal{P}_{r}(a_{l}+\vec{p}_{l}\cdot\vec{\epsilon}\,|\vec{a})}\\\
&\hskip 142.26378pt\times\prod_{l^{\prime}=1}^{r}\
\prod_{\vec{p}^{\,\prime}_{l^{\prime}}\in\sigma_{l^{\prime}}}^{\neq
0}\,\mathcal{R}\big{(}a_{l}-a_{l^{\prime}}+(\vec{p}_{l}-\vec{p}^{\,\prime}_{l^{\prime}})\cdot\vec{\epsilon}\,\big{|}\vec{\epsilon}\,\big{)}\
.\end{split}$ (2.110)

The instanton partition function (2.33) is then given by

$\displaystyle Z_{\mathbbm{C}^{4}}^{r}(\Lambda;\vec{a},\vec{\epsilon}\,)^{\rm
pure}=1+\sum_{k=1}^{\infty}\,\Lambda^{k}\
Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon}\,)^{\rm pure}\ .$ (2.111)

From the field theory perspective, we expect that (2.32) reduces to (2.33) in
a suitable infinite mass limit which decouples the fundamental matter
hypermultiplets [19]. This physical expectation is confirmed by

######  Proposition 2.112.

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

$\displaystyle Z_{\mathbbm{C}^{4}}^{r}(\Lambda;\vec{a},\vec{\epsilon}\,)^{\rm
pure}=\lim_{m_{1},\dots,m_{r}\to\infty}\ \lim_{\mathtt{q}\to
0}\,Z_{\mathbbm{C}^{4}}^{r}(\mathtt{q};\vec{a},\vec{\epsilon},\vec{m})\,\Big{|}_{\Lambda=(-1)^{r}\,m_{1}\cdots
m_{r}\,\mathtt{q}}\ .$ (2.113)

###### Proof.

From (2.100) the relevant terms of the combinatorial expansion (2.102) in the
limit of large masses are given by

$\displaystyle\begin{split}\mathtt{q}^{|\vec{\sigma}|}\,\prod_{l=1}^{r}\
\prod_{\vec{p}_{l}\in\sigma_{l}}^{\neq
0}\,\mathcal{P}_{r}(a_{l}+\vec{p}_{l}\cdot\vec{\epsilon}\,|\vec{m})\xrightarrow{m_{1},\dots,m_{r}\gg
1}\mathtt{q}^{|\vec{\sigma}|}\,\prod_{l=1}^{r}\
\prod_{\vec{p}_{l}\in\sigma_{l}}\,(-1)^{r}\,&m_{1}\cdots m_{r}\\\\[4.0pt]
&=\big{(}(-1)^{r}\,m_{1}\cdots m_{r}\,\mathtt{q}\big{)}^{|\vec{\sigma}|}\
,\end{split}$ (2.114)

and the result now follows from (2.110) and (2.111). ∎

######  Corollary 2.115.

Assume Conjecture 2.105 is true. Then the equivariant instanton partition
function of the pure cohomological gauge theory on $\mathbbm{C}^{4}$ is given
by

$\displaystyle Z_{\mathbbm{C}^{4}}^{r}(\Lambda;\vec{\epsilon}\,)^{\rm
pure}=\left\\{\begin{matrix}\displaystyle\exp\Big{(}-\Lambda\
\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\,\Big{)}\qquad\mbox{for}\quad
r=1\ ,\\\\[12.91663pt] 1\qquad\mbox{for}\quad r>1\ .\end{matrix}\right.$
(2.116)

###### Proof.

We combine Propositions 2.112 and Conjecture 2.105 using the series
representation

$\displaystyle\log
M(q)=\sum_{n=1}^{\infty}\,\frac{1}{n}\,\frac{q^{n}}{(1-q^{n})^{2}}$ (2.117)

for the logarithm of the MacMahon function (2.73), which converges for any
$q\neq 1$. For $\mathtt{q}\ll 1$ this yields the behaviour $\log
M(-\mathtt{q})=-\mathtt{q}+O(\mathtt{q}^{2})$. For $r=1$ the result follows by
fixing $\Lambda=-m\,\mathtt{q}$ in the limits $m\to\infty$ and $\mathtt{q}\to
0$. For $r>1$, the partition function is identically equal to one in the
double scaling limit $m_{l}\to\infty$ and $\mathtt{q}\to 0$ with
$(-1)^{r}\,m_{1}\cdots m_{r}\,\mathtt{q}=\Lambda$ fixed. ∎

######  Remark 2.118.

We checked explicitly that the order $\Lambda$ and $\Lambda^{2}$ terms of the
series (2.111) agree with (2.116) in the rank one case $r=1$, and that the
contributions for $r=2$ with $k=1,2$ and for $r=3$ with $k=1$ are trivial:
$Z_{\mathbbm{C}^{4}}^{r,k}(\vec{a},\vec{\epsilon}\,)^{\rm pure}=0$.

Geometrically, the large mass limit of Proposition 2.112 defines the
insertion-free equivariant Donaldson–Thomas invariants of $\mathbbm{C}^{4}$
[38]. The rank one formula of Corollary 2.115 was originally discussed in [24,
38]. It implies that the equivariant volume of the Hilbert scheme of $k$
points on $\mathbbm{C}^{4}$ is given by

$\displaystyle\int_{[{\sf Hilb}^{k}(\mathbbm{C}^{4})]_{\mathtt{o}}^{\rm
vir}}^{\mathsf{T}_{\vec{\epsilon}}}\,1=\frac{(-1)^{k}}{k!}\,\Big{(}\frac{\epsilon_{12}\,\epsilon_{13}\,\epsilon_{23}}{\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\epsilon_{4}}\Big{)}^{k}\
,$ (2.119)

whereas the equivariant volume of the Quot scheme ${\sf
Quot}_{r}^{k}(\mathbbm{C}^{4})$ of zero-dimensional quotients of
$\mathcal{O}_{\mathbbm{C}^{4}}^{\oplus r}$ with length $k$ vanishes for all
$r>1$. This is in marked contrast to the equivariant volumes of the Quot
schemes of zero-dimensional quotients of
$\mathcal{O}_{\mathbbm{C}^{2}}^{\oplus r}$ and
$\mathcal{O}_{\mathbbm{C}^{3}}^{\oplus r}$, which are all non-zero.

##  3 Orbifolds of the Eight-Dimensional Theory

In Section 2 we have studied eight-dimensional noncommutative instantons on
flat space. As a natural non-trivial generalization beyond flat space
$\mathbbm{C}^{4}$, on the way towards local Calabi–Yau fourfolds, in this
section we will extend these considerations to quotients
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ by a finite group $\mathsf{\Gamma}$ acting
linearly on $\mathbbm{C}^{4}$. In order to preserve the holonomy group
$\mathsf{SU}(4)$ of the cohomological gauge theory, the group
$\mathsf{\Gamma}$ has to be a subgroup of $\mathsf{SU}(4)$. To define the
equivariant instanton partition functions, we will further restrict to toric
Calabi–Yau orbifolds $\mathbbm{C}^{4}/\mathsf{\Gamma}$, which requires that
$\mathsf{\Gamma}$ is abelian and that the $\mathsf{\Gamma}$-action commutes
with the action of the maximal torus
$\mathsf{T}_{\vec{\epsilon}}\subset\mathsf{SU}(4)$ on $\mathbbm{C}^{4}$. This
restricts to orbifold groups of the form
$\mathsf{\Gamma}=\mathbbm{Z}_{n_{1}}\times\mathbbm{Z}_{n_{2}}\times\mathbbm{Z}_{n_{3}}$,
with order $n=n_{1}\,n_{2}\,n_{3}$. Such an orbifold theory was studied in
[25, 26] for a specific example; here we will vastly extend and generalize
their considerations using the framework of Section 2.

###  3.1 Noncommutative Gauge Theory on $\mathbbm{C}^{4}/\mathsf{\Gamma}$

A conventional quantum field theory on an orbifold
$\mathbbm{C}^{4}/\mathsf{\Gamma}$ is singular. However, an orbifold field
theory can be constructed by allowing fields to be equivariant under the
action of $\mathsf{\Gamma}$ and gauging the $\mathsf{\Gamma}$-action, together
with a subsequent projection to $\mathsf{\Gamma}$-invariant states of the
cohomological gauge theory of Section 2.2. Such a construction incorporates
“twisted sectors” into the theory, analogously to string orbifolds, and can be
thought of as a cohomological gauge theory on the quotient stack
$[\mathbbm{C}^{4}/\mathsf{\Gamma}]$. It is naturally realised in
noncommutative field theory.

Suppose that the generators of the orbifold group $\mathsf{\Gamma}$ act on the
coordinates of $\mathbbm{C}^{4}$ as

$\displaystyle(z_{1},z_{2},z_{3},z_{4})\longmapsto(\mathrm{e}^{\,2\pi\,\mathrm{i}\,s_{1}/n}\,z_{1},\mathrm{e}^{\,2\pi\,\mathrm{i}\,s_{2}/n}\,z_{2},\mathrm{e}^{\,2\pi\,\mathrm{i}\,s_{3}/n}\,z_{3},\mathrm{e}^{\,2\pi\,\mathrm{i}\,s_{4}/n}\,z_{4})\
,$ (3.1)

where $n$ is the order of $\mathsf{\Gamma}$. This induces a decomposition of
the fundamental representation $Q\simeq\mathbbm{C}^{4}$ of
$\mathsf{SL}(4,\mathbbm{C})$ as

$\displaystyle
Q=\rho_{s_{1}}\oplus\rho_{s_{2}}\oplus\rho_{s_{3}}\oplus\rho_{s_{4}}\ ,$ (3.2)

where $\rho_{s_{a}}$ denotes the irreducible representation of
$\mathsf{\Gamma}$ with weight $s_{a}$. The Calabi–Yau constraint implies
$\rho_{s_{1}}\otimes\cdots\otimes\rho_{s_{4}}\simeq\rho_{0}$, where $\rho_{0}$
is the trivial representation of weight zero.

Let $\widehat{\mathsf{\Gamma}}$ denote the finite abelian group of irreducible
representations of $\mathsf{\Gamma}$; since each representation $\rho_{s}$ for
$s\in\widehat{\mathsf{\Gamma}}$ is one-dimensional, we can regard elements of
$\widehat{\mathsf{\Gamma}}$ as characters
$\chi_{s}:\mathsf{\Gamma}\longrightarrow\mathbbm{C}^{\times}$ of
$\mathsf{\Gamma}$. Then the decomposition of $Q$ also defines a
$\widehat{\mathsf{\Gamma}}$-colouring $\mathbbm{Z}_{\geq 0}^{\oplus
4}\longrightarrow\mathsf{\Gamma}$ through the identification
$\widehat{\mathsf{\Gamma}}\simeq\mathsf{\Gamma}$ by

$\displaystyle(n_{1},n_{2},n_{3},n_{4})\longmapsto\rho_{s_{1}}^{\otimes
n_{1}}\otimes\rho_{s_{2}}^{\otimes n_{2}}\otimes\rho_{s_{3}}^{\otimes
n_{3}}\otimes\rho_{s_{4}}^{\otimes n_{4}}\ .$ (3.3)

Via this colouring, the $\mathsf{\Gamma}$-action on $\mathbbm{C}^{4}$ induces
an isotopical decomposition of the Fock space (2.35) into irreducible
representations as

$\displaystyle\mathcal{H}=\bigoplus_{s\in\widehat{\mathsf{\Gamma}}}\,\mathcal{H}_{s}\qquad\mbox{with}\quad\mathcal{H}_{s}={\sf
Span}_{\mathbbm{C}}\big{\\{}|\vec{n}\,\rangle\ \big{|}\ \rho_{s_{1}}^{\otimes
n_{1}}\otimes\cdots\otimes\rho_{s_{4}}^{\otimes n_{4}}\simeq\rho_{s}\big{\\}}\
.$ (3.4)

Consequently, in the rank one case $r=1$ the covariant coordinates $Z_{a}$
decompose into maps
$Z=\bigoplus_{s\in\widehat{\mathsf{\Gamma}}}\,(Z_{a}^{s})_{a\in\underline{\sf
4}}:\mathcal{H}\longrightarrow Q\otimes\mathcal{H}$ with
$Z_{a}^{s}:\mathcal{H}_{s}\longrightarrow\mathcal{H}_{s+s_{a}}$ for
$a\in\underline{\sf 4}\,$, and the Higgs field $\varphi$ into maps
$\varphi=\bigoplus_{s\in\widehat{\mathsf{\Gamma}}}\,\varphi^{s}:\mathcal{H}\longrightarrow\mathcal{H}$
with $\varphi^{s}:\mathcal{H}_{s}\longrightarrow\mathcal{H}_{s}$. The operator
algebraic equations (2.36) now read as

$\displaystyle\begin{split}Z_{a}^{s+s_{b}}\,Z_{b}^{s}-Z_{b}^{s+s_{a}}\,Z_{a}^{s}&=\tfrac{1}{2}\,\varepsilon_{ab\bar{c}\bar{d}}\,\big{(}Z_{\bar{c}}^{s-s_{d}-s_{c}\dagger}\,Z_{\bar{d}}^{s-s_{d}\dagger}-Z_{\bar{d}}^{s-s_{d}-s_{c}\dagger}\,Z_{\bar{c}}^{s-s_{c}\dagger}\big{)}\
,\\\\[4.0pt]
\sum_{a=1}^{4}\,\big{(}Z_{\bar{a}}^{s\dagger}\,Z_{a}^{s}-Z_{a}^{s-s_{a}}\,Z_{\bar{a}}^{s-s_{a}\dagger}\big{)}&=\zeta_{s}\,\mathbbm{1}_{\mathcal{H}_{s}}\
,\\\\[4.0pt]
Z_{a}^{s}\,\varphi^{s}-\varphi^{s+s_{a}}\,Z_{a}^{s}&=\epsilon_{a}\,Z_{a}^{s}\
,\end{split}$ (3.5)

where the Fayet–Iliopoulos parameters $\zeta_{s}>0$ for
$s\in{\widehat{\mathsf{\Gamma}}}$ are determined by the decomposition of the
$B$-field into NS–NS twisted sectors of type II string theory on
$\mathbbm{C}^{4}/\mathsf{\Gamma}$. Note that the first equation of (3.5) makes
sense because of the Calabi–Yau condition $\rho_{s_{1234}}\simeq\rho_{0}$.

The generalization of the set of equations (3.5) to higher rank $r>1$ involves
an action of the orbifold group $\mathsf{\Gamma}$ on the Chan–Paton space
$W\simeq\mathbbm{C}^{r}$ that we describe below, which is defined by a
homomorphism $\mathsf{\Gamma}\longrightarrow\mathsf{U}(r)_{\rm col}$. This
breaks the colour symmetry to the centralizer of the image of
$\mathsf{\Gamma}$. Solutions of these equations describe noncommutative
$\mathsf{U}(r)$ instantons on the orbifold $\mathbbm{C}^{4}/\mathsf{\Gamma}$.

The splitting of the covariant coordinates induces an equivariant
decomposition of the ADHM data $({B}_{a},I)_{a\in\underline{\sf 4}}$ from
(2.38). With respect to the decomposition of the orbifold group action into
irreducible representations, the vector spaces on which the ADHM variables are
defined decompose as

$\displaystyle
V=\bigoplus_{s\in\widehat{\mathsf{\Gamma}}}\,V_{s}\otimes\rho_{s}^{*}\qquad\mbox{and}\qquad
W=\bigoplus_{s\in\widehat{\mathsf{\Gamma}}}\,W_{s}\otimes\rho_{s}^{*}\ .$
(3.6)

The multiplicity spaces
$V_{s}=\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(\rho_{s},V)$ and
$W_{s}=\mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(\rho_{s},W)$ consist of
$\mathsf{\Gamma}$-equivariant homomorphisms, where
$\mathbbm{C}[\mathsf{\Gamma}]$ is the group ring of $\mathsf{\Gamma}$ over
$\mathbbm{C}$. The dimensions $k=\dim_{\mathbbm{C}}V$ and
$r=\dim_{\mathbbm{C}}W$ correspondingly decompose as

$\displaystyle
k=\sum_{s\in\widehat{\mathsf{\Gamma}}}\,k_{s}=\sum_{s\in\widehat{\mathsf{\Gamma}}}\,\dim_{\mathbbm{C}}V_{s}\qquad\mbox{and}\qquad
r=\sum_{s\in\widehat{\mathsf{\Gamma}}}\,r_{s}=\sum_{s\in\widehat{\mathsf{\Gamma}}}\,\dim_{\mathbbm{C}}W_{s}\
,$ (3.7)

which define arrays of fractional instanton charges
$\vec{k}=(k_{s})_{s\in\widehat{\mathsf{\Gamma}}}$ with ranks
$\vec{r}=(r_{s})_{s\in{\widehat{\mathsf{\Gamma}}}}$ in $\mathbbm{Z}_{\geq
0}^{|\mathsf{\Gamma}|}$, whose _size_ is
$|\vec{k}\,|:=\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\,k_{s}=k$ and
$|\vec{r}\,|:=\sum_{s\in{\widehat{\mathsf{\Gamma}}}}\,r_{s}:=r$.

By Schur’s lemma this implies the decompositions

$\displaystyle
B=\bigoplus_{s\in\widehat{\mathsf{\Gamma}}}\,(B_{a}^{s})_{a\in\underline{\sf
4}}\ \in\ \mathsf{Hom}_{\mathbbm{C}[\mathsf{\Gamma}]}(V,V\otimes
Q)\qquad\mbox{with}$ $\displaystyle\quad B^{s}_{a}:V_{s}\longrightarrow