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# Tetrahedron instantons in Donaldson-Thomas theory

Nadir Fasola, Sergej Monavari  School of Mathematics and Statistics,
University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, United Kingdom
<EMAIL_ADDRESS>Ecole Polytechnique Fédérale de Lausanne (EPFL),
CH-1015 Lausanne, Switzerland<EMAIL_ADDRESS>
###### Abstract.

Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the
moduli space of tetrahedron instantons as a Quot scheme and describe it as a
moduli space of quiver representations. We construct a virtual fundamental
class and virtual structure sheaf à la Oh-Thomas, by which we define
$K$-theoretic invariants. We show that the partition function of such
invariants reproduces the one studied by Pomoni-Yan-Zhang, and explicitly
determine it, as a product of shifted partition functions of rank one
Donaldson-Thomas invariants of the three-dimensional affine space. Our
geometric construction answers a series of questions of Pomoni-Yan-Zhang on
the geometry of the moduli space of tetrahedron instantons and the behaviour
of its partition function, and provides a new application of the recent work
of Oh-Thomas.

##  1\. Introduction

###  1.1. Tetrahedron instantons

Pomoni-Yan-Zhang [62] recently introduced the _tetrahedron instantons_ , which
can be realized in type IIB string theory as D1-branes probing a configuration
of intersecting D7-branes in flat spacetime with constant $B$-field turned on.
This is equivalently interpreted as describing instanton dynamics on
${\mathbb{C}}^{3}$ in the presence of real-codimension-two supersymmetric
defects.

Their construction of the moduli space of tetrahedron instantons is realised
via gauge-theoretic methods, so that supersymmetric localisation techniques
apply to define and study the associated instanton partition function non-
perturbatively. In this article, we propose a rigorous mathematical
perspective on the problem. We realise the moduli space of tetrahedron
instantons as a Quot scheme and compute explicitly the generating series of
the resulting virtual invariants, thus shedding light on the geometry of the
constructions in [62] from the point of view of Donaldson-Thomas theory.

####  1.1.1. Mathematical formulation

The idea of mathematically studying instanton partition functions — defined in
_topological string theory_ — via moduli spaces of sheaves (in our case, Quot
schemes) is not new in the literature, and follows the circle of ideas for
which _systems of D-branes_ should be identified with (complexes of) sheaves
with prescribed Chern character and suitable stability condition. In this
setting, the instanton partition function is usually realised as a generating
series of _virtual_ invariants on the corresponding moduli space and exactly
computed via (virtual) localisation techniques. See [60, 1, 33, 18, 46, 19,
42, 47] for some recent examples where this circle of ideas has been applied,
providing an interpretation in Donaldson-Thomas theory of some results
initially introduced in topological string theory (for instance [38, 53, 3, 7,
54, 57]). A novel feature of our proposal is that the relevant moduli space we
study — _geometrising_ the work of [62] — is the Quot scheme of a _singular_
variety, parametrising quotients of a _torsion_ sheaf.

####  1.1.2. The moduli space

For $i=1,\dots,4$ denote the $i$-th coordinate hyperplane in
${\mathbb{C}}^{4}$ by ${\mathbb{C}}^{3}_{i}:=Z(x_{i})$ and set
$\Delta=\bigcup_{i=1}^{4}{\mathbb{C}}^{3}_{i}\subset{\mathbb{C}}^{4}$ to be
the union of the four coordinate hyperplanes. For a tuple
$\overline{r}=(r_{1},r_{2},r_{3},r_{4})$ of non-negative integers, define the
torsion sheaf on $\Delta$

${\mathcal{E}}_{\overline{r}}=\bigoplus\iota_{i,*}\operatorname{\mathcal{O}}^{\oplus{r_{i}}}_{{\mathbb{C}}^{3}_{i}},$

where $\iota_{i}:{\mathbb{C}}^{3}_{i}\hookrightarrow\Delta$ are the inclusions
of the irreducible components. Our main moduli space of interest is
Grothendieck’s _Quot scheme_

${\mathcal{M}}_{\overline{r},n}:=\operatorname{Quot}_{\Delta}({\mathcal{E}}_{\overline{r}},n),$

which parametrizes isomorphism classes of quotients
$[{\mathcal{E}}_{\overline{r}}\twoheadrightarrow Q]$, where we identify two
such quotients if they have the same kernel. We refer to
${\mathcal{M}}_{\overline{r},n}$ as the _moduli space of tetrahedron
instantons_. The term _tetrahedron_ comes from the fact that the singular
variety $\Delta$ appears as the central fibre of a toric degeneration, whose
associated polyhedral complex is a tetrahedron111We thank Francesca Carocci
for explaining this to us from the point of view of Tropical Geometry (cf.
also [62, Fig. 1])..

The geometry of ${\mathcal{M}}_{\overline{r},n}$ is subtle and hard to
explicitly describe scheme-theoretically. In fact, already in the easiest
example $r=(1,0,0,0)$, the moduli space of tetrahedron instantons is
isomorphic to the _Hilbert scheme of points_
$\operatorname{Hilb}^{n}({\mathbb{C}}^{3})$, whose actual dimension and number
of irreducible components are not currently known.

Our first result is that the moduli space of tetrahedron instantons
${\mathcal{M}}_{\overline{r},n}$ can be realized as the zero-locus of an
isotropic section of a _special orthogonal bundle_ (cf. Sec. 2.1) inside a
smooth ambient space.

######  Theorem 1.1 (Theorem 2.3).

Let $\overline{r}=(r_{1},r_{2},r_{3},r_{4})$ and $n$ be non-negative integers.
Then

${{\mathcal{L}}}$${{\mathcal{M}}_{\overline{r},n}\cong
Z(s)}$${{\mathcal{M}}^{\rm
nc}_{\overline{r},n},}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\iota}$$\leftarrow$$\rightarrow$$\scriptstyle{s}$

where ${\mathcal{M}}^{\rm nc}_{\overline{r},n}$ is a smooth variety,
${\mathcal{L}}$ is a special orthogonal bundle, and $s\in
H^{0}({\mathcal{M}}_{\overline{r},n}^{\rm nc},{\mathcal{L}})$ is an isotropic
section.

We denote the smooth ambient space ${\mathcal{M}}^{\rm nc}_{\overline{r},n}$
by _non-commutative Quot scheme_ , in accordance with [65], who computed its
motivic classes in $K_{0}(\operatorname{Var}_{\mathbf{k}})$ in any dimension
and over any algebraically closed field $\mathbf{k}$. Similar structural
results — exploiting non-commutative Quot schemes — appeared in [5, 42]. The
non-commutative Quot scheme ${\mathcal{M}}^{\rm nc}_{\overline{r},n}$ is
realised as the moduli space of (framed) representations of the four-loop
quiver (cf. Fig. 1(b)) via GIT. Our realisation of the moduli space of
tetrahedron instantons inside the non-commutative Quot scheme in Theorem 1.1
should be seen as the representation-theoretic analogue of the _symplectic
reduction_ techniques employed in [62], starting from ADHM data (describing
non-commutative instantons). Granting this analogy, the special orthogonal
bundle ${\mathcal{L}}$ (and its section) are naturally determined by some
moment maps coming from gauge theory (cf. Sec. 3.8).

By the seminal work of Oh-Thomas [59] (cf. Sec. 2.1), the zero-locus model in
Theorem 1.1 naturally induces a _virtual fundamental class_ and a _virtual
structure sheaf_ of virtual dimension zero

$\displaystyle[{\mathcal{M}}_{\overline{r},n}]^{\operatorname{\mathrm{vir}}}$
$\displaystyle\in
A_{0}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right),$
$\displaystyle\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{{\mathcal{M}}_{\overline{r},n}}$
$\displaystyle\in
K_{0}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right).$

We will use these virtual classes to define invariants of
${\mathcal{M}}_{\overline{r},n}$ which match the ones studied by Pomoni-Yan-
Zhang [62] in string theory. However, due to the non-properness of the moduli
space ${\mathcal{M}}_{\overline{r},n}$, we need to define the invariants via
equivariant residues, with respect to a torus action on
${\mathcal{M}}_{\overline{r},n}$ with proper fixed locus.

###  1.2. Partition function

The moduli space of tetrahedron instantons ${\mathcal{M}}_{\overline{r},n}$ is
endowed with an action of an algebraic torus $\mathbf{T}$, whose _equivariant
parameters_ we denote by $t=(t_{1},\dots,t_{4}),w=(w_{11},\dots,w_{4r_{4}})$.
We remark that the first four variables obey the relation
$t_{1}t_{2}t_{3}t_{4}=1$, a technical assumption required at the very heart of
the constructions in [59]. We define the _tetrahedron instantons partition
function_ as the generating series of $K$-theoretic invariants

(1.1) $\displaystyle{\mathcal{Z}}_{\overline{r}}(q)=\sum_{n\geq
0}q^{n}\cdot\chi\left(\mathcal{M}_{\overline{r},n},\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{\mathcal{M}_{\overline{r},n}}\right)\in\frac{{\mathbb{Q}}(t_{1}^{\frac{1}{2}},t_{2}^{\frac{1}{2}},t_{3}^{\frac{1}{2}},t_{4}^{\frac{1}{2}},w^{\frac{1}{2}})}{(t_{1}t_{2}t_{3}t_{4}-1)}[\\![q]\\!].$

The moduli spaces ${\mathcal{M}}_{\overline{r},n}$ are not proper, therefore
the invariants need to be defined $\mathbf{T}$-equivariantly. The partition
function (1.1) reproduces the one of Pomoni-Yan-Zhang [62] (cf. Sec. 1.4), and
interpolates between Donaldson-Thomas theory in dimension three and four (cf.
Sec. 1.3). Morally, the invariants (1.1) should be seen as the algebro-
geometric analogue of the $\widehat{A}$-genus of the moduli space of
tetrahedron instantons.

Our second result is the explicit computation of the tetrahedron instantons
partition function222As this work was being finalised, a preprint of Pomoni-
Yan-Zhang [63] appeared, where a formula similar to the one in Theorem 1.2 is
independently conjectured.  For a formal variable $x$, set
$[x]=x^{\frac{1}{2}}-x^{-\frac{1}{2}}$. Letting $e_{i}$ be the $4$-tuple with
$1$ in the $i$-th entry and $0$ in all the others, we set
${\mathcal{Z}}^{(i)}(q)={\mathcal{Z}}_{e_{i}}(q)$.

######  Theorem 1.2 (Theorem 3.14, 3.15).

Let $\overline{r}=(r_{1},r_{2},r_{3},r_{4})$ and $r=\sum_{i=1}^{4}r_{i}$. We
have

${\mathcal{Z}}_{\overline{r}}((-1)^{r}q)=\operatorname{Exp}\left(-\frac{[t_{1}t_{2}][t_{1}t_{3}][t_{2}t_{3}]}{[t_{1}][t_{2}][t_{3}][t_{4}]}\frac{[\kappa_{\overline{r}}]}{[\kappa_{\overline{r}}^{\frac{1}{2}}q][\kappa_{\overline{r}}^{\frac{1}{2}}q^{-1}]}\right),$

where we set the weight $\kappa_{\overline{r}}=\prod_{i=1}^{4}t_{i}^{-r_{i}}$
and $\operatorname{Exp}$ is the plethystic exponential (3.14). Moreover, the
instanton partition function admits the factorisation

$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)=\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}{\mathcal{Z}}^{(i)}\left((-1)^{r+1}q\kappa_{i}^{\frac{-r_{i}-1}{2}+l}\prod_{j=1}^{4}\kappa_{j}^{\frac{r_{j}\cdot{\mathrm{sgn}}(i-j)}{2}}\right),$

where we set the weights $\kappa_{i}=t_{i}^{-1}$.

Theorem 1.2 generalizes the main result of [33], which deals with the Quot
scheme of the three-dimensional affine space. The factorisation of the
invariants into _rank 1 theories_ has already been observed in a similar
situation for $K$-theoretic invariants [33, 42, 19], motivic invariants [48,
21, 20], in String Theory [57, 27] and could find a conceptual answer in the
recent (but unrelated) work of Feyzbakhsh-Thomas [35, 34] on generalized
Donaldson-Thomas invariants, who exploit wall-crossing techniques to show the
rank reduction.

####  1.2.1. Strategy of the proof

We explain here the main ideas in the proof of Theorem 1.2. The torus fixed
locus ${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$ is reduced, zero-
dimensional and in bijection with tuples
$\overline{\pi}=(\overline{\pi}_{1},\dots,\overline{\pi}_{4})$, where each
$\overline{\pi}_{i}$ is an $r_{i}$-tuple of _plane partitions_. Therefore by
Oh-Thomas virtual localisation Theorem [59], the tetrahedron instantons
partition function can be expressed as

(1.2)
$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)=\sum_{\overline{\pi}}(-1)^{\sigma_{\overline{\pi}}}[-\mathsf{v}_{\overline{\pi}}]\cdot
q^{|\overline{\pi}|}.$

Let us pause for a moment to explain the notation in (1.2). The sum runs over
all tuples of plane partitions $\overline{\pi}$, and $|\overline{\pi}|$
denotes their _size_. To each $\overline{\pi}$, we attach two items: a Laurent
polynomial in the equivariant parameters $\mathsf{v}_{\overline{\pi}}$ (the
_square root_ of the virtual tangent space) and an (a priori non-explicit)
sign $(-1)^{\sigma_{\overline{\pi}}}$. By Oh-Thomas virtual localisation, each
tuple of plane partitions $\overline{\pi}$ contributes to the localised
invariants with rational functions in the equivariant parameters, obtained by
applying the operator $[\cdot]$ to the _vertex term_
$\mathsf{v}_{\overline{\pi}}$, but only up to a _sign_. Despite this situation
being very similar to the original treatment of the topological vertex in
[44], the presence of such a sign in our context is what makes the computation
of the partition function hard and cumbersome. We remark that the choice of
square root $\mathsf{v}_{\overline{\pi}}$ is not canonical; rather, one should
think of the pair
$(\mathsf{v}_{\overline{\pi}},(-1)^{\sigma_{\overline{\pi}}})$ to be canonical
in a suitable sense, cf. [46].

In order to prove Theorem 1.2, we need the following two crucial technical
results. The first one is of local nature and shows that for our particular
choice of square root, the corresponding sign is constantly
$(-1)^{\sigma_{\overline{\pi}}}=1$ (Theorem 3.7). The proof of these results
follows by a purely combinatorial study of the vertex terms applied to a
structural result of Kool-Rennemo [42] (cf. Appendix A).

The second result is of global nature, and asserts that the tetrahedron
instantons partition function does not depend on the _framing parameters_
$w_{ij}$, for all $i=1,\dots,4$ and $j=1,\dots,r_{i}$ (Theorem 3.11). This
independence is the incarnation of a _rigidity principle_ and ultimately
relies on the properness of the Quot-to-Chow morphism, and can already be
spotted by the explicit formula for the partition function in Theorem 1.2. We
remark that this framing independence is peculiar to the $K$-theoretic
invariants here considered, and fails for the _elliptic_ refinement proposed
in Section 3.7 (cf. Remark 3.20).

Granting this framing independence, we can set the framing parameters $w_{ij}$
to arbitrary values and take arbitrary limits. A suitable choice of such
limits yields a (non-trivial!) combinatorial expression of the partition
function where the factorisation property becomes evident. Finally, the
explicit expression in terms of the plethystic exponential is obtained by
combining the factorised form of the partition function with the explicit
formula for the _rank 1_ case, which had been initially conjectured by
Nekrasov in String Theory [53] and later proved by Okounkov [60].

####  1.2.2. Cohomological limit

A second source of interesting invariants comes from integrating over the
_virtual fundamental class_ in equivariant Chow cohomology, rather than in
$K$-theory, as classically considered in [44]. We define the _cohomological
tetrahedron instantons partition function_ as

(1.3)
$\displaystyle{\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)=\sum_{n\geq
0}q^{n}\cdot\int_{[{\mathcal{M}}_{\overline{r},n}]^{\operatorname{\mathrm{vir}}}}1\in\frac{{\mathbb{Q}}(s_{1},s_{2},s_{3},s_{4},v)}{(s_{1}+s_{2}+s_{3}+s_{4})}[\\![q]\\!],$

where $s_{1},\dots,s_{4},v_{11},\dots,v_{4r_{4}}$ are the generators of
$\mathbf{T}$-equivariant Chow cohomology $A^{\mathbf{T}}_{*}({\mathsf{pt}})$
and the integral makes sense only equivariantly. The partition function (1.3)
carries less refined information with respect to its $K$-theoretic counterpart
(1.1), and can in fact be realized as a _limit_ of the latter. This limiting
procedure at the level of the partition functions in Supersymmetric String
Theory corresponds to a suitable _dimensional reduction_ of the two-
dimensional low energy effective theory. Indeed, starting from the system of
D1-D7-branes in type IIB String Theory on $T^{2}\times{\mathbb{C}}^{4}$
engineering the moduli space of tetrahedron instantons [62], one can shrink a
circle in $T^{2}=S^{1}\times S^{1}$ and T-dualise along the small $S^{1}$. The
original brane system in the T-dualised picture then reduces to a system of
D0-D6 branes on $S^{1}$, corresponding to the quantum mechanics of the
tetrahedron instantons quiver in Fig. 2(a). An additional shrinking to a point
of the remaining $S^{1}$, followed by S-duality, leaves us with a
D(-1)-D5-brane system, whose low-energy partition function is exactly (1.3).

Applying such a limit, we prove a closed formula for the partition function
(1.3).

######  Corollary 1.3 (Corollary 3.18).

Let $\overline{r}=(r_{1},r_{2},r_{3},r_{4})$ and $r=\sum_{i=1}^{4}r_{i}$. We
have

${\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)=\mathrm{M}((-1)^{r}q)^{-\frac{(s_{1}+s_{2})(s_{1}+s_{3})(s_{2}+s_{3})(r_{1}s_{1}+r_{2}s_{2}+r_{3}s_{3}+r_{4}s_{4})}{s_{1}s_{2}s_{3}s_{4}}},$

where $\mathrm{M}(q)$ is the MacMahon series (3.1).

####  1.2.3. Witten genus

Pomoni-Yan-Zhang [62, Sec. 6] provide an alternative computation of the
tetrahedron instanton partition function — again, in string theory — from the
elliptic genus of the low-energy worldvolume theory on D1-branes.
Mathematically, we realize this as an _elliptic refinement_ of the invariants
in (1.1), which we call _virtual Witten genus_ of
${\mathcal{M}}_{\overline{r},n}$

${\mathcal{Z}}^{\mathrm{ell}}_{\overline{r}}=\sum_{n\geq
0}q^{n}\cdot\chi\left({\mathcal{M}}_{\overline{r},n},\widehat{\operatorname{\mathcal{O}}}_{{\mathcal{M}}_{\overline{r},n}}^{\operatorname{\mathrm{vir}}}\otimes\bigotimes_{k\geq
1}\operatorname{Sym}^{\bullet}_{p^{k}}\left(T^{\operatorname{\mathrm{vir}}}_{{\mathcal{M}}_{\overline{r},n}}\right)\right)\in\frac{{\mathbb{Q}}(t_{1}^{\frac{1}{2}},t_{2}^{\frac{1}{2}},t_{3}^{\frac{1}{2}},t_{4}^{\frac{1}{2}},w^{\frac{1}{2}})}{(t_{1}t_{2}t_{3}t_{4}-1)}[\\![p,q]\\!],$

where $T^{\operatorname{\mathrm{vir}}}_{{\mathcal{M}}_{\overline{r},n}}$
denotes the virtual tangent bundle of ${\mathcal{M}}_{\overline{r},n}$ and $p$
is the _elliptic parameter_. This elliptic genus is a generalization of the
_virtual chiral elliptic genus_ of Fasola-Monavari-Ricolfi [33, Sec. 8.1], and
appeared in a similar context in Donaldson-Thomas theory of Calabi-Yau
fourfolds [11, Def. 5.8].

As already mentioned, the virtual Witten genus in general depends on the
framing parameters, thus preventing us from computing its generating series
using the factorisation of the invariants. Nevertheless, under some suitable
specialisations of the equivariant parameters $t_{1},\dots,t_{4}$, we expect
that the generating series enjoys some rigidity behaviour of the elliptic
parameter (see Remark 3.21).

###  1.3. Relation to Donaldson-Thomas Theory

Our description of the moduli space of tetrahedron instantons and its
partition function nicely fits into the realm of Donaldson-Thomas (DT) theory.
In fact, as already pointed out by Pomoni-Yan-Zhang [62], tetrahedron
instantons interpolate between (higher rank) DT theory of ${\mathbb{C}}^{3}$
and ${\mathbb{C}}^{4}$, whose relevant moduli spaces are
$\operatorname{Quot}_{{\mathbb{C}}^{3}}({\operatorname{\mathcal{O}}^{r},n})$
and $\operatorname{Quot}_{{\mathbb{C}}^{4}}(\operatorname{\mathcal{O}}^{r},n)$
(cf. [33, 18, 42]). The interesting aspect of our construction is that it
shares common features with both DT theories, in dimension three and four. In
fact, despite ${\mathcal{M}}_{\overline{r},n}$ is a moduli space of sheaves on
a threefold and possesses a _zero-dimensional_ virtual fundamental class — as
in the classical DT theory of threefolds — it seems that its virtual structure
cannot be defined directly with the machinery of (two-terms) _perfect
obstruction theories_ of Behrend-Fantechi and Li-Tian [6, 43], but crucially
needs a symmetric (three-term) obstruction theory as in the work of Oh-Thomas
[59], which was specifically designed for DT theory of Calabi-Yau fourfolds.
Nevertheless, for suitable choices of the vector $\overline{r}$, the moduli
space ${\mathcal{M}}_{\overline{r},n}$ reproduces the Quot scheme
$\operatorname{Quot}_{{\mathbb{C}}^{3}}({\operatorname{\mathcal{O}}^{r},n})$
and its virtual structure, which _does_ come from a perfect obstruction theory
[5]. This shows that the natural class one should consider is always the
_twisted_ virtual structure sheaf, which is natural from the point of view of
DT theory of Calabi-Yau fourfolds, while in DT theory of threefolds, one
should _twist_ the virtual structure sheaf by hand to make the invariants more
relevant to study, following the circle of ideas initiated by Nekrasov-
Okounkov [56]. On the other hand, Theorem 1.2 shows that the tetrahedron
instantons partition function is a specialisation of the partition function of
higher rank Donaldson-Thomas invariants of ${\mathbb{C}}^{4}$, whose explicit
expression was proposed by Nekrasov-Piazzalunga [57] and which is conjectured
to be the mother of all instanton partition functions [54].

From the point of view of DT theory, one may ask if the zero-locus
construction of Theorem 1.1 — and its induced virtual structure — could have
an intrinsic deformation-theoretic interpretation, which does not require the
construction of a smooth ambient space via a quiver with potential as in our
work (cf. [64]). This is actually the case when our moduli space of
tetrahedron instantons reproduces the Hilbert scheme of points
$\operatorname{Hilb}^{n}({\mathbb{C}}^{3})$ (cf. [66], where it is shown that
the perfect obstruction theory induced by being a critical locus coincides
with the one coming by deformation theory). From this deformation-theoretic
point of view, it would be natural to replace $\Delta\subset{\mathbb{C}}^{4}$
with an arbitrary pair $(X,Y)$, where $X$ is a quasi-projective Calabi-Yau
fourfold and $Y$ is a boundary divisor, such that each irreducible component
is smooth. From this point of view, we would find it natural to expect a
virtual pullback formula to hold, generalizing the one of Park [61], which
would explain why the partition function of the magnificent four model [57]
specialises to the tetrahedron instanton partition function333Physically, this
would correspond to the fact that our configuration of intersecting D6 branes
should come from the annihilation of D8 and anti-D8 branes through tachyon
condensation.. Finally, suitably describing the virtual structure on
${\mathcal{M}}_{\overline{r},n}$ via deformation theory could shed some light
on its possible relation to _degeneration phenomena_. A piece of strong
evidence in this direction is that the invariants obtained on a moduli space
of sheaves over $\Delta$ are reduced to a product of invariants over moduli
spaces of sheaves over the irreducible components of $\Delta$, and that the
virtual classes are (morally) deformation invariant444Deformation invariance
would be expected if our moduli spaces were proper, for instance, if we
started with a pair $(X,Y)$ such that $X$ is a smooth _projective_ Calabi-Yau
fourfold, and $Y$ a divisor.. We plan to pursue these questions and
investigate their interpretation in string theory in the future.

###  1.4. String Theory

The moduli space of tetrahedron instantons was introduced in the context of
Supersymmetric String Theory in [62]. It describes the low energy theory of a
system of D1-branes probing some general configuration of intersecting
D7-branes. This in turn is interpreted as capturing instanton dynamics when
codimension-two supersymmetric defects are present. On the geometric side,
Supersymmetric String Theory in general, and D-branes systems specifically,
have proved useful to provide a physical realisation of interesting moduli
spaces and to compute several generating series of their (virtual) invariants.
The point of contact between the mathematical and physical realms can be found
in the study of the BPS sector of string theories. Indeed, this is a
particularly amenable task on the physical side, as it consists of quantities
which are known to be protected from quantum corrections and can therefore be
studied non-perturbatively. Also, BPS-bound state counting can usually be
interpreted in terms of counting/classification problems in geometry, and the
technique of localisation often available in supersymmetric settings makes it
feasible to compute relevant partition functions exactly. Here the connection
to enumerative geometry is made even clearer, as supersymmetric localisation
may be viewed as an infinite-dimensional analogue of equivariant localisation
in different settings (eg., cohomological, cf. [2], or K-theoretic
localisation, cf. [68]).

One particular class of moduli spaces which can be obtained in physics from
systems of D-branes are moduli spaces of (generalised) instantons. The ADHM
construction of moduli spaces of anti-self dual connections as symplectic
quotients shed light on their physical realisation either in the language of
Sigma Models [70], or as moduli spaces of BPS vacua of systems of D1-D5-branes
in type IIB Supersymmetric String Theory [28]. Similarly, a resolution of
singularities of the moduli space of ideal instantons on ${\mathbb{R}}^{4}$,
realised by moduli of framed torsion-free sheaves on ${\mathbb{P}}^{2}$ [52],
can be obtained from D-brane configurations in superstring theory, modelling
instantons on noncommutative ${\mathbb{R}}^{4}$. Partition functions of such
theories are then computed explicitly (and non-perturbatively) via equivariant
localisation [51, 58]. Several interesting generalisations of these
constructions have been studied over the past years, e.g. the construction of
instantons as systems of D-branes on orbifolds or in the presence of surface
defects [29, 40, 12, 13], as well as on other four-dimensional manifolds [22,
71, 16, 36, 15]. Moreover, ADHM-type constructions can be exploited to
construct instanton-like objects in higher dimensions [4, 50, 37], thus
providing a physical realisation of Donaldson-Thomas theory and its
generalisation on fourfolds. The relation between Donaldson-Thomas theory on
threefolds and string theoretic constructions in physics, such as quiver
matrix models, instantons in orbifold or non-commutative backgrounds and
elliptic generalisations, have been studied, for example, in [3, 24, 25, 26,
7, 23], and is rigorously understood in mathematics as in [44, 45, 33]. On the
other hand, the rather difficult task of interpreting higher-dimensional DT
theory can be similarly undertaken in the context of String Theory as a system
of D0-D8(-$\overline{\rm D8}$)-branes wrapping the affine four-space [55, 57].
Also, it has been shown to enjoy a realisation as a Topological Field Theory
localising on non-commutative $\operatorname{Spin}(7)$-instanton
configurations on ${\mathbb{R}}^{8}$, leading moreover to an ADHM-like
construction [14], and a natural generalisation to more general backgrounds,
such as orbifolds (cf. [14, 19, 67]). The construction of tetrahedron
instantons as moduli spaces of BPS solutions in the background of intersecting
D-branes also sits nicely in these lines of research. Indeed, the low-energy
D-brane dynamics is captured by a gauge theory whose moduli space of vacua can
be understood as the moduli space of stable representations of a suitable
quiver, which is then interpreted as a Quot scheme on a singular threefold,
which is suggestive of a DT-like interpretation. Also, the study of moduli
spaces of instantons on threefolds with the inclusion of codimension-two
defects naturally involves structures which are inherently fourfold in nature,
thus interpolating between two different regimes in DT theory.

We point out that, during the very final stage of finishing this paper, a
preprint of Pomoni-Yan-Zhang [63] appeared where they independently conjecture
the explicit formula for the tetrahedron instanton partition function, which
we rigorously prove in Theorem 1.2. Their main focus is its relation with the
index in $M$-theory, following the circle of ideas of Nekrasov-Okounkov [56].

### Acknowledgments

We are grateful to Francesca Carocci and Woonam Lim for useful discussion, to
Martijn Kool and Jørgen Rennemo for sharing an early version of their
manuscript [42], and to Andrea Ricolfi for previous collaborations and
enlightening discussions on Quot schemes over the years. N.F. was supported by
the Engineering and Physical Sciences Research Council through the grant no.
EP/S003657/2. S.M. was supported by the Chair of Arithmetic Geometry, EPFL.

##  2\. Moduli space of tetrahedron instantons

###  2.1. Virtual fundamental classes

Recently Oh-Thomas [59] devised a machinery to construct virtual fundamental
classes on moduli spaces of sheaves on Calabi-Yau fourfolds. We will use a toy
model of their work to define a virtual fundamental class on the moduli space
of tetrahedron instantons. For the construction of the virtual fundamental
class à la Oh-Thomas in a more general setting, we refer to Park [61].

Following the conventions of [59, Sec. 2], we say that a rank $2r$555For
simplicity, we assume here that the rank is even; cf. [59, Sec. 1] for the
definition in the general case. vector bundle ${\mathcal{E}}$ on a quasi-
projective scheme ${\mathcal{A}}$ is a _special orthogonal bundle_ (or an _$\
SO(2r,{\mathbb{C}})$ -bundle_) if ${\mathcal{E}}$ is endowed with a non-
degenerate quadratic pairing

$q:{\mathcal{E}}\otimes{\mathcal{E}}\to\operatorname{\mathcal{O}}_{{\mathcal{A}}},$

together with a choice of trivialization (an _orientation_)

$o:\det{\mathcal{E}}\xrightarrow{\sim}\operatorname{\mathcal{O}}_{\mathcal{A}},$

whose square666See [59, Def. 2.1] for the precise sign conventions of
orientations. is $o^{\otimes 2}=\pm\det q$. We say that a subbundle
$\Lambda\subset{\mathcal{E}}$ is _maximal isotropic_ if
$\operatorname{rk}\Lambda=r$ and $q|_{\Lambda}=0$. Any choice of maximal
isotropic subbundle naturally induces an orientation of ${\mathcal{E}}$. In
fact, taking determinants of the short exact sequence

$0\to\Lambda\to{\mathcal{E}}\to\Lambda^{*}\to 0,$

yields
$\det{\mathcal{E}}\cong\det\Lambda\otimes\det\Lambda^{*}\cong\operatorname{\mathcal{O}}_{\mathcal{A}}$,
which gives the required isomorphism (cf. [59, Sec. 2.3] for the precise sign
conventions).

We say that a section $s\in H^{0}({\mathcal{A}},{\mathcal{E}})$ is _isotropic_
if $q(s,s)=0$. Let a scheme $Z$

${{\mathcal{E}}}$${Z:=Z(s)}$${{\mathcal{A}}}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\iota}$$\leftarrow$$\rightarrow$$\scriptstyle{s}$

be the zero locus of an isotropic section
$s\in\Gamma({\mathcal{A}},{\mathcal{E}})$, where ${\mathcal{E}}$ is a
$SO(2r,{\mathbb{C}})$-bundle over a smooth quasi-projective variety
${\mathcal{A}}$. Then there exists an induced obstruction theory on $Z$

(2.1)
${\big{[}\Omega^{*}_{{\mathcal{A}}|Z}}$${{\mathcal{E}}^{*}|_{Z}}$${\Omega_{{\mathcal{A}}|Z}\big{]}}$${{\mathbb{E}}}$${\big{[}{\mathcal{I}}/{\mathcal{I}}^{2}}$${\Omega_{{\mathcal{A}}|Z}\big{]}}$${{\mathbb{L}}_{Z},}$$\leftarrow$$\rightarrow$$\scriptstyle{ds}$$\leftarrow$$\rightarrow$$\scriptstyle{s}$$\leftarrow$$\rightarrow$$\scriptstyle{(ds)^{*}}$${\parallel}$${\cong}$$\leftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{d}$${\cong}$

in $\mathbf{D}^{[-2,0]}(Z)$, where
${\mathcal{I}}\subset\operatorname{\mathcal{O}}_{A}$ is the ideal sheaf of the
inclusion $Z\hookrightarrow{\mathcal{A}}$ and ${\mathbb{L}}_{Z}$ denotes the
truncated cotangent complex. By the work of [59], there exist a _virtual
fundamental class_ and a _virtual structure sheaf_ on $Z$ satisfying

$\displaystyle\iota_{*}[Z]^{\operatorname{\mathrm{vir}}}$
$\displaystyle=\sqrt{e}({\mathcal{E}})\cap[{\mathcal{A}}]\in
A_{\dim{\mathcal{A}}-r}\left({\mathcal{A}},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right),$
$\displaystyle\iota_{*}\widehat{\operatorname{\mathcal{O}}}_{Z}^{\operatorname{\mathrm{vir}}}$
$\displaystyle=\sqrt{\mathfrak{e}}({\mathcal{E}})\otimes
K^{1/2}_{\mathcal{A}}\in
K_{0}\left({\mathcal{A}},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right),$

where $K^{1/2}_{\mathcal{A}}$ is a square root of the canonical bundle of
${\mathcal{A}}$ and $\sqrt{e},\sqrt{\mathfrak{e}}$ are the Edidin-Graham
square root Euler classes in Chow cohomology and $K$-theory (cf. [59, Sec.
3.1, 5.1]). We call the integer $\dim{\mathcal{A}}-r$ the _virtual dimension_
of $Z$. Exchanging the orientation with its opposite orientation results in
changing the sign of the induced virtual classes.

###  2.2. Quot scheme

Let $\overline{r}=(r_{1},\dots,r_{4})$ and set $r=\sum_{i=1}^{4}r_{i}$. We
defined the moduli space of _tetrahedron instantons_ as Grothendieck’s Quot
scheme

${\mathcal{M}}_{\overline{r},n}:=\operatorname{Quot}_{\Delta}({\mathcal{E}}_{\overline{r}},n),$

where $\Delta=\bigcup_{i=1}^{4}{\mathbb{C}}^{3}_{i}\subset{\mathbb{C}}^{4}$ is
the union of the four coordinate hyperplanes of ${\mathbb{C}}^{4}$ and

${\mathcal{E}}_{\overline{r}}=\bigoplus\iota_{i,*}\operatorname{\mathcal{O}}^{\oplus{r_{i}}}_{{\mathbb{C}}^{3}_{i}}$

is a torsion sheaf on $\Delta$, where
$\iota_{i}:{\mathbb{C}}^{3}_{i}\hookrightarrow\Delta$ are the inclusions of
the irreducible components. We have natural closed immersions among Quot
schemes

(2.2)
$\displaystyle{\mathcal{M}}_{\overline{r},n}\hookrightarrow\operatorname{Quot}_{\Delta}(\operatorname{\mathcal{O}}_{\Delta}^{r},n)\hookrightarrow\operatorname{Quot}_{{\mathbb{C}}^{4}}(\operatorname{\mathcal{O}}_{{\mathbb{C}}^{4}}^{r},n),$

obtained by precomposing the quotients
$[{\mathcal{E}}_{\overline{r}}\twoheadrightarrow Q]$ with

$\operatorname{\mathcal{O}}_{{\mathbb{C}}^{4}}^{r}\twoheadrightarrow\operatorname{\mathcal{O}}_{\Delta}^{r}\twoheadrightarrow{\mathcal{E}}_{\overline{r}}.$

Denote by $\operatorname{Sym}^{n}\Delta$ the _$n$ -th symmetric power_ of
$\Delta$. Attached to ${\mathcal{M}}_{\overline{r},n}$ there is the _Quot-to-
Chow morphism_

$\rho:{\mathcal{M}}_{\overline{r},n}\to\operatorname{Sym}^{n}\Delta,$

which, at the level of closed points, sends a quotient to its 0-dimensional
support, counted with multiplicity.

######  Example 2.1.

We work out explicitly the example of $\overline{r}=(1,1,0,0)$ and $n=1$. The
moduli space of tetrahedron instantons ${\mathcal{M}}_{(1,1,0,0),1}$ admits a
locally closed stratification, obtained by taking the preimages of a locally
closed stratification of $\operatorname{Sym}^{1}\Delta\cong\Delta$

$\displaystyle{\mathcal{M}}_{(1,1,0,0),1}$
$\displaystyle=\rho^{-1}({\mathbb{C}}_{1}^{3}\setminus{\mathbb{C}}_{12}^{2})\sqcup\rho^{-1}({\mathbb{C}}_{2}^{3}\setminus{\mathbb{C}}_{12}^{2})\sqcup\rho^{-1}({\mathbb{C}}_{12}^{2})$
$\displaystyle={\mathbb{C}}_{1}^{3}\setminus{\mathbb{C}}_{12}^{2}\sqcup{\mathbb{C}}_{2}^{3}\setminus{\mathbb{C}}_{12}^{2}\sqcup{\mathbb{C}}_{12}^{2}\times{\mathbb{P}}^{1},$

where ${\mathbb{C}}^{2}_{12}:={\mathbb{C}}_{1}^{3}\cap{\mathbb{C}}_{2}^{3}$ is
the intersection of the two coordinate hyperplanes and we used that

$\rho^{-1}({\mathbb{C}}_{12}^{2})\cong\operatorname{Quot}_{{\mathbb{C}}^{2}}(\operatorname{\mathcal{O}}_{{\mathbb{C}}^{2}}^{2},1)\cong{\mathbb{C}}_{12}^{2}\times{\mathbb{P}}^{1},$

where the last isomorphism follows by [49, Rem. 2.2]. Furthermore, there is a
better understanding of the geometry of ${\mathcal{M}}_{(1,1,0,0),1}$ in terms
of blow-ups. Let
$Y={\mathbb{C}}^{3}_{1}\cup{\mathbb{C}}^{3}_{2}\subset{\mathbb{C}}^{4}$ the
union of the first two coordinate hyperplanes, and consider the blow-up of
${\mathbb{C}}^{4}$ centred in ${\mathbb{C}}^{2}_{12}$. Then the moduli space
of tetrahedron instantons identifies with the proper transform of $Y$

${{\mathcal{M}}_{(1,1,0,0),1}\cong
p^{-1}(Y)}$${Bl_{{\mathbb{C}}_{12}^{2}}{\mathbb{C}}^{4}}$${{\mathbb{C}}^{4}\times{\mathbb{P}}^{1}\cong\operatorname{Quot}_{{\mathbb{C}}^{4}}(\operatorname{\mathcal{O}}_{{\mathbb{C}}^{4}}^{2},1)}$${Y}$${{\mathbb{C}}^{4}}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\scriptstyle{p}$$\hookleftarrow$$\rightarrow$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$

Here, the composition of the first two horizontal maps is the closed embedding
(2.2), and shows that the closure of the two strata
${\mathbb{C}}_{i}^{3}\setminus{\mathbb{C}}_{12}^{2}$ in
${\mathcal{M}}_{(1,1,0,0),1}$ are the strict transform of $Y$, while the
stratum ${\mathbb{C}}_{12}^{2}\times{\mathbb{P}}^{1}$ coincides with the
exceptional divisor of the blow-up.

######  Remark 2.2.

The situation in Example 2.1 readily generalizes to arbitrary $\overline{r}$
and $n$, where a stratification of ${\mathcal{M}}_{\overline{r},n}$ is induced
by a natural stratification of $\operatorname{Sym}^{n}\Delta$, thus
generalizing the stratification described for $n=1$ in [62, Eqn. (125)] via
gauge theory. Similarly, we expect in the general case a description of the
moduli space of tetrahedron instantons in terms of blow-ups, which
intriguingly hints at an interpretation as a degeneration phenomenon.

###  2.3. Non-commutative Quot scheme

Consider then the quiver $\mathsf{L_{4}}$ in Fig. 1(a) with one vertex and
four loops, labelled $x_{1},x_{2},x_{3},x_{4}$. In the following we will be
interested in studying representations of the framed quiver
$\widetilde{\mathsf{L}}_{4}$ in Fig. 1(b).

2.5pt

$0$white$x_{a}$ (a) Four-loop quiver $\mathsf{L_{4}}$

$\infty_{1}$$\infty_{4}$$0$$\vdots$white$\varphi_{1}$white$\varphi_{4}$white$x_{a}$
(b) Framed quiver $\widetilde{\mathsf{L}}_{4}$

Figure 1.

In particular, letting $\overline{r}=(r_{1},\dots,r_{4})$ be the dimension
vector associated to the framing nodes $(\infty_{i})_{i=1,\dots,4}$, we will
study representations
$X\in\mathsf{Rep}_{{\mathbb{C}}}(\widetilde{\mathsf{L}}_{4})$ with $\dim
X=(\overline{r},n)$. This amounts to assigning a tuple of vector spaces
$(W_{1},\dots,W_{4},V)$, one to each node of $\widetilde{\mathsf{L}}_{4}$, in
such a way that $\dim W_{i}=r_{i}$, $\dim V=n$, and homomorphisms
$I_{i}\in\operatorname{Hom}(W_{i},V)$, $B_{a}\in\operatorname{End}(V)$
according the the arrows of $\widetilde{\mathsf{L}}_{4}$. Schematically, a
representation of $\widetilde{\mathsf{L}}_{4}$ is shown in Fig. 2(a). Studying
framed representations of this sort is then equivalent to studying
representations of a different framing of $\mathsf{L_{4}}$, shown pictorially
in Fig. 2(b).

2.5pt

$r_{1}$$r_{4}$$n$$\vdots$white$I_{1}$white$I_{4}$white$B_{a}$ (a)
Representation of $\widetilde{\mathsf{L}}_{4}$

$1$$n$$\vdots$white$I_{1}^{(1)}$white$I_{r_{4}}^{(4)}$white$B_{a}$ (b)
Equivalent quiver representation

Figure 2.

In the following, we will often use the notation $r=r_{1}+\cdots+r_{4}$. Let
then $V$ be a given $n$-dimensional vector space, and consider the affine
$(4n^{2}+nr)$-dimensional representation space

$\mathsf{R}_{\overline{r},n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}={\mathbb{C}}^{4}\otimes\operatorname{End}(V)\oplus\operatorname{Hom}({\mathbb{C}},V)^{\oplus
r}.$

An element $X\in\mathsf{R}_{\overline{r},n}$, together with the choice of $V$,
forms a representation as in Fig. 2(b). In the following, we will refer to
$X\in\mathsf{R}_{\overline{r},n}$ itself as a representation. Also, a
representation $X\in\mathsf{R}_{\overline{r},n}$ induces a representation as
in Fig. 2(a) by defining

$I_{i}\left(\overbrace{0,\dots,0}^{j-1},1,\overbrace{0,\dots,0}^{r_{i}-j}\right)=I_{i}^{(j)}(1),$

and $W_{i}={\mathbb{C}}^{r_{i}}$. Conversely, any representation
$(W_{1},\dots,W_{4},V,B_{1},\dots,B_{4},I_{1},\dots,I_{4})$ as in Fig. 2(a)
induces a representation as in Fig. 2(b) by choosing a basis
$\\{e^{(i)}_{j}\\}\subset W_{i}$ and setting

$I_{i}^{(j)}(1)=I_{i}\left(e_{j}^{(i)}\right).$

The group $\operatorname{GL}(V)$ acts naturally on the representation space by
simultaneously conjugating the endomorphisms and scaling the vectors. Consider
then then the open subscheme
$\mathsf{U}_{\overline{r},n}\subset\mathsf{R}_{\overline{r},n}$ parametrising
those quiver representations
$(B_{1},\dots,B_{4},I_{1}^{(1)},\dots,I_{4}^{(r_{4})})$ such that the
morphisms $(I_{i}^{(j)})_{i,j}$ jointly generate $V\cong{\mathbb{C}}^{n}$ as a
${\mathbb{C}}\langle B_{1},B_{2},B_{3},B_{4}\rangle$-module, that is

(2.3) $\bigoplus_{i=1}^{4}\bigoplus_{j=1}^{r_{i}}{\mathbb{C}}\langle
B_{1},B_{2},B_{3},B_{4}\rangle I_{i}^{(j)}(1)\cong V.$

The $\operatorname{GL}(V)$-action is free on $\mathsf{U}_{\overline{r},n}$, so
the quotient $\mathcal{M}^{\rm
nc}_{\overline{r},n}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathsf{U}_{\overline{r},n}/\operatorname{GL}(V)$
exists as a smooth $(3n^{2}+nr)$-dimensional quasiprojective variety, cf. [65,
Lemma 2.1]. The moduli space of representations thus constructed is known as
the non-commutative Quot scheme, and parametrises quotients
$[\mathbb{C}\langle x_{1},\dots,x_{4}\rangle^{\oplus r}\twoheadrightarrow Q]$
of the free rank $r$ algebra $\mathbb{C}\langle
x_{1},\dots,x_{4}\rangle^{\oplus r}$ on four generators, cf. [5, Thm. 2.5] and
[65, Prop. 2.3], and can be furthermore identified with the GIT quotient of
$\mathsf{R}_{\overline{r},n}$ with respect to King stability [41].

###  2.4. Zero-locus construction

We prove in this section Theorem 1.1 from the Introduction. Over the affine
space $\mathsf{R}_{\overline{r},n}$, consider the trivial vector bundle of
rank $6n^{2}+2rn$

$\mathcal{L}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left(\Lambda^{2}{\mathbb{C}}^{4}\otimes\operatorname{End}(V)\right)\oplus\bigoplus_{i=1}^{4}\left(\operatorname{Hom}(W_{i},V)\oplus\operatorname{Hom}(W_{i},V)^{\ast}\right).$

We endow ${\mathcal{L}}$ with a non-degenerate quadratic pairing
$q:\mathcal{L}^{\otimes
2}\to\operatorname{\mathcal{O}}_{\mathsf{R}_{\overline{r},n}}$, defined
fiberwise by

$q\left((\phi_{1}\otimes f_{1},\bm{w}_{1},\bm{\omega}_{1}),(\phi_{2}\otimes
f_{2},\bm{w}_{2},\bm{\omega}_{2})\right)=\left(\phi_{1}\wedge\phi_{2}\right)\otimes\operatorname{tr}(f_{1}\circ
f_{2})+\frac{1}{2}\left(\bm{\omega}_{1}(\bm{w}_{2})+\bm{\omega}_{2}(\bm{w}_{1})\right),$

where $\phi_{i}\in\Lambda^{2}{\mathbb{C}}^{4}$, $f_{i}\in\operatorname{End}V$,
$\bm{w}_{i}\in\bigoplus_{j=1}^{4}\operatorname{Hom}(W_{j},V)$,
$\bm{\omega}_{i}\in\bigoplus_{j=1}^{4}\operatorname{Hom}(W_{j},V)^{\ast}$. The
quadratic bundle $({\mathcal{L}},q)$ restricts to the open locus
$\mathsf{U}_{\overline{r},n}\subset\mathsf{R}_{\overline{r},n}$ and is
naturally $\operatorname{GL}(V)$-equivariant, with respect to the
$\operatorname{GL}(V)$-action on $U_{\overline{r},n}$. Therefore,
$({\mathcal{L}},q)$ descends to a vector bundle with a quadratic pairing on
the quotient $\mathcal{M}^{\rm nc}_{\overline{r},n}$. With a slight abuse of
notation, we keep denoting the induced vector bundle by $(\mathcal{L},q)$.

######  Theorem 2.3.

Let $\overline{r}=(r_{1},\dots,r_{4})$ and $n$ be non-negative integers. There
exists an isotropic section $s\in H^{0}({\mathcal{M}}_{\overline{r},n}^{\rm
nc},{\mathcal{L}})$ such that ${\mathcal{M}}_{\overline{r},n}$ is realised as
the zero locus $Z(s)$

${{\mathcal{L}}}$${{\mathcal{M}}_{\overline{r},n}\cong
Z(s)}$${{\mathcal{M}}^{\rm
nc}_{\overline{r},n}.}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\iota}$$\leftarrow$$\rightarrow$$\scriptstyle{s}$

###### Proof.

We start by constructing an isotropic section of $({\mathcal{L}},q)$ on the
affine space $R_{\overline{r},n}$. Let
$X=(B_{1},\dots,B_{4},I_{1},\dots,I_{4})\in R_{\overline{r},n}$ be a quiver
representation and let $(e_{i})_{i=1,\dots,4}$ be a basis of
${\mathbb{C}}^{4}$. Define the section

$\displaystyle s(X)=\sum_{a,b=1}^{4}(e_{a}\wedge e_{b})\otimes
B_{a}B_{b}+\sum_{i=1}^{4}B_{i}I_{i}.$

In particular, the restriction of the section to the last summand of
${\mathcal{L}}$ vanishes

$\displaystyle s|_{\bigoplus_{i=1}^{4}\operatorname{Hom}(W_{i},V)^{\ast}}=0.$

The section $s$ is clearly isotropic with respect to the quadratic pairing
$q$, i.e. $q(s,s)=0$. Since the restriction of the section to the open locus
$\mathsf{U}_{\overline{r},n}$ is $\operatorname{GL}(V)$-equivariant, it
descends to an isotropic section on the quotient ${\mathcal{M}}^{\rm
nc}_{\overline{r},n}$, which we keep denoting by $s\in
H^{0}({\mathcal{M}}^{\rm nc}_{\overline{r},n},{\mathcal{L}})$.

There is a natural closed embedding

$\displaystyle\iota:{\mathcal{M}}_{\overline{r},n}\hookrightarrow{\mathcal{M}}^{\rm
nc}_{\overline{r},n},$

which is obtained by the closed embedding (2.2), followed by the closed
embedding of the Quot scheme
$\operatorname{Quot}_{{\mathbb{C}}^{4}}(\operatorname{\mathcal{O}}_{{\mathbb{C}}^{4}}^{r},n)$
into its non-commutative counterpart, defined via their moduli functors. We
claim that

${\mathcal{M}}_{\overline{r},n}\cong Z(s).$

Let $[\alpha:{\mathcal{E}}_{\overline{r}}\twoheadrightarrow
Q]\in{\mathcal{M}}_{\overline{r},n}$ be a closed point. Recall that the
embedding into the non-commutative Quot scheme identifies this quotient with
the quiver representation
$(B_{1},\dots,B_{4},I_{1},\dots,I_{4})\in{\mathcal{M}}^{\rm
nc}_{\overline{r},n}$, where

$\displaystyle B_{i}:Q\to Q,\quad i=1,\dots,4,$ $\displaystyle
I^{(j)}_{i}=\alpha(0,\dots,0,1,0,\dots,0)\in Q,\quad
i=1,\dots,4,\,j=1,\dots,r_{i}.$

Via this correspondence, the matrices $B_{i}$ are the module multiplication by
$x_{i}$, and the cyclic vectors $I^{(j)}_{i}$ are the images of the constant
1, each coming from a summand of ${\mathcal{E}}_{\overline{r}}$.

By construction, if
$(B_{1},\dots,B_{4},I_{1},\dots,I_{4})\in{\mathcal{M}}^{\rm
nc}_{\overline{r},n}$ is a closed point in the image of $\iota$, then it
satisfies the relations

(2.4) $\displaystyle\begin{split}&B_{a}B_{b}-B_{b}B_{a}=0,\quad
a,b=1,\dots,4,\\\ &B_{i}I_{i}=0,\quad i=1,\dots,4.\end{split}$

On the other hand, consider a quotient $[\mathbb{C}\langle
x_{1},\dots,x_{4}\rangle^{\oplus r}\twoheadrightarrow Q]\in{\mathcal{M}}^{\rm
nc}_{\overline{r},n}$ satisfying the relations (2.4). Then the quotient
factors through

${\mathbb{C}\langle x_{1},\dots,x_{4}\rangle^{\oplus
r}}$${Q}$${{\mathcal{E}}_{\overline{r}}}$$\leftarrow$$\twoheadrightarrow$$\leftarrow$$\twoheadrightarrow$$\leftarrow$$\twoheadrightarrow$

thus defining a closed point in ${\mathcal{M}}_{\overline{r},n}$, as desired.
We conclude by noticing that the same argument carries over flat families,
which yields the desired isomorphism. ∎

###  2.5. The maximal isotropic subbundle

We endow the vector bundle ${\mathcal{L}}$ with a structure of special
orthogonal bundle, by showing it admits a maximal isotropic subbundle.
Consider the subbundles

$\displaystyle\tilde{\Lambda}$ $\displaystyle:=\left(\langle
e_{4}\rangle\wedge{\mathbb{C}}^{3}\right)\otimes\operatorname{End}(V)\subset\left(\Lambda^{2}{\mathbb{C}}^{4}\otimes\operatorname{End}(V)\right)$
$\displaystyle\Lambda$
$\displaystyle=\tilde{\Lambda}\oplus\bigoplus_{i=1}^{4}\operatorname{Hom}(W_{i},V)^{*}\subset{\mathcal{L}},$

on the affine space $\mathsf{R}_{\overline{r},n}$. Then $\Lambda$ is a maximal
isotropic subbundle. In fact, $\Lambda$ has rank $3n^{2}+rn$ and the
restriction of the quadratic pairing vanishes. The maximal isotropic
subbundles $\tilde{\Lambda},\Lambda$ descend to maximal isotropic subbundles
on the non-commutative Quot scheme ${\mathcal{M}}^{\rm nc}_{\overline{r},n}$,
which we keep denoting by $\tilde{\Lambda},\Lambda$.

######  Remark 2.4.

The non-commutative Quot scheme ${\mathcal{M}}^{\rm nc}_{\overline{r},n}$ and
a summand of its special orthogonal bundle ${\mathcal{L}}$ (with its maximal
isotropic subbundle $\tilde{\Lambda}$) appear also in the work of Kool-Rennemo
[42], who exploit them to study Donaldson-Thomas theory of ${\mathbb{C}}^{4}$.

Since ${\mathcal{L}}$ is a special orthogonal bundle, Theorem 2.3 immediately
implies the following corollary.

######  Corollary 2.5.

The moduli space of tetrahedron instantons ${\mathcal{M}}_{\overline{r},n}$ is
endowed with a virtual fundamental class and a virtual structure sheaf of
virtual dimension zero

$\displaystyle[{\mathcal{M}}_{\overline{r},n}]^{\operatorname{\mathrm{vir}}}$
$\displaystyle\in
A_{0}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right),$
$\displaystyle\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{{\mathcal{M}}_{\overline{r},n}}$
$\displaystyle\in
K_{0}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right).$

We will use these virtual classes to define invariants of
${\mathcal{M}}_{\overline{r},n}$ which match the ones studied by Pomoni-Yan-
Zhang [62] in string theory.

##  3\. The partition function

###  3.1. Torus representations and their weights

Let $\mathbf{T}=({\mathbb{C}}^{*})^{g}$ be an algebraic torus, with character
lattice
$\widehat{\mathbf{T}}=\operatorname{Hom}(\mathbf{T},{\mathbb{C}}^{\ast})\cong{\mathbb{Z}}^{g}$.
Let $K^{0}_{\mathbf{T}}({\mathsf{pt}})$ be the Grothendieck group of the
category of $\mathbf{T}$-representations. Any finite-dimensional
$\mathbf{T}$-representation $V$ splits as a sum of $1$-dimensional
representations called the _weights_ of $V$. Each weight corresponds to a
character $\mu\in\widehat{\mathbf{T}}$, and in turn each character corresponds
to a monomial
$\operatorname{\mathfrak{t}}^{\mu}=\operatorname{\mathfrak{t}}_{1}^{\mu_{1}}\cdots\operatorname{\mathfrak{t}}_{g}^{\mu_{g}}$
in the coordinates of $\mathbf{T}$. In other words, there is an isomorphism

$K^{0}_{\mathbf{T}}({\mathsf{pt}})\cong{\mathbb{Z}}\left[\operatorname{\mathfrak{t}}^{\mu}\mid\mu\in\widehat{\mathbf{T}}\right],$

identifying the class of a $\mathbf{T}$-representation with its decomposition
into weight spaces. We will therefore sometimes identify a (virtual)
$\mathbf{T}$-representation with its character. With a slight abuse of
notation, we still denote by $\mathbf{T}$ any finite cover of the algebraic
torus $\mathbf{T}$ where the square roots $t^{\frac{\mu}{2}}$ of some
irreducible characters $t^{\mu}$ are well-defined. This will be needed as most
of our constructions need naturally to extract square roots.

###  3.2. Torus action

Let $\overline{r}=(r_{1},\dots,r_{4})$ and $r=\sum_{i=0}^{4}r_{i}$ and define
the algebraic tori

$\mathbf{T}=\mathbf{T}_{0}\times\mathbf{T}_{1},\quad\mathbf{T}_{0}=\\{t_{1}t_{2}t_{3}t_{4}=1\\}\subset({\mathbb{C}}^{*})^{4},\quad\mathbf{T}_{1}=({\mathbb{C}}^{*})^{r}.$

The torus $\mathbf{T}_{0}$ acts on ${\mathbb{C}}^{4}$ by

$(t_{1},\dots,t_{4})\cdot(x_{1},\dots,x_{4})=(t_{1}x_{1},\dots,t_{4}x_{4})$

preserving the Calabi-Yau volume form. In particular, it restricts to the
union of the coordinate hyperplanes $\Delta\subset{\mathbb{C}}^{4}$ and
naturally lifts to the moduli space of tetrahedron instantons
${\mathcal{M}}_{\overline{r},n}$, by moving the support of the quotients. The
torus $\mathbf{T}_{1}$ acts on ${\mathcal{M}}_{\overline{r},n}$ by scaling the
fibers of ${\mathcal{E}}_{\overline{r}}$. Therefore, we have an induced
$\mathbf{T}$-action on ${\mathcal{M}}_{\overline{r},n}$. Equivalently, the
$\mathbf{T}$-action on ${\mathcal{M}}_{\overline{r},n}$ is the restriction of
the $\mathbf{T}$-action on the moduli space of non-commutative Quot scheme
${\mathcal{M}}^{\mathrm{nc}}_{\overline{r},n}$ given on points by

$(t,w)\cdot(B_{1},\dots,B_{4},I^{(1)}_{1},\dots,I_{4}^{(r_{4})})=(t_{1}^{-1}B_{1},\dots,t_{4}^{-1}B_{4},w_{11}^{-1}I_{1}^{(1)},\dots,w_{4r_{4}}^{-1}I^{(r_{4})}_{4}),$

where $w_{ij}$ are the coordinates of $\mathbf{T}_{1}$, for $i=1,\dots,4$ and
$j=1\dots,r_{i}$.

######  Remark 3.1.

The fixed locus ${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}_{0}}$ is proper.
Indeed, a $\mathbf{T}_{0}$-invariant surjection
${\mathcal{E}}_{\overline{r}}\twoheadrightarrow Q$ necessarily has the
quotient $Q$ entirely supported at the origin $0\in\Delta$. Hence

${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}_{0}}\hookrightarrow\rho^{-1}(n\cdot[0])$

sits as a closed subscheme inside the fiber over $n\cdot[0]$ of the Quot-to-
Chow morphism
$\rho:{\mathcal{M}}_{\overline{r},n}\to\operatorname{Sym}^{n}\Delta$. But
$\rho^{-1}(n\cdot[0])$ is proper, since $\rho$ is a proper morphism.

The torus fixed locus of the moduli space of tetrahedron instantons is
classified in terms of _plane partitions_.

######  Definition 3.2.

A _plane partition_ $\pi$ is a finite subset of ${\mathbb{Z}}^{3}_{\geq 0}$
such that if $(i,j,k)\in\pi$ with $(i,j,k)\neq 0$, then
$(i-1,j,k),(i,j-1,k),(i,j,k-1)\in\pi$, and we denote by $|\pi|$ its _size_.
Alternatively, we can view the plane partition as boxes stacked in
${\mathbb{R}}^{3}$, where the lattice points are determined by the _interior_
corners of the boxes, the corners closest to the origin. For a tuple of plane
partitions $\overline{\pi}=(\pi_{1},\dots,\pi_{r})$ we denote its _size_ by
$|\overline{\pi}|=\sum_{i=1}^{r}|\pi_{i}|$.

It is well-known that plane partitions are in bijection with monomial ideals
of ${\mathbb{C}}[x,y,z]$

$\pi\subset{\mathbb{Z}}^{3}_{\geq 0}\longleftrightarrow
I_{\pi}=(x^{i}y^{j}z^{k})_{(i,j,k)\notin\pi}\subset{\mathbb{C}}[x,y,z],$

where the lattice points of ${\mathbb{Z}}^{3}_{\geq 0}$ correspond to
monomials in three variables (see e.g. [33, Sec. 3.4]).

######  Proposition 3.3.

Let $\overline{r}=(r_{1},\dots,r_{4})$. The $\mathbf{T}$-fixed locus
$\mathcal{M}_{\overline{r},n}^{\mathbf{T}}$ is reduced, zero-dimensional and
in bijection with

$\left\\{\overline{\pi}=(\overline{\pi}_{1},\dots,\overline{\pi}_{4}):\overline{\pi}_{i}=(\pi_{il})_{1\leq
l\leq r_{4}},\mbox{$\pi_{il}$ plane partitions such that
}|\overline{\pi}|=n\right\\}.$

###### Proof.

By the same arguments as in [10] its $\mathbf{T}_{1}$-fixed locus is scheme-
theoretically isomorphic to

$\mathcal{M}_{\overline{r},n}^{\mathbf{T}_{1}}\cong\bigsqcup_{n=\sum_{i=1}^{4}\sum_{j=1}^{r_{i}}n_{ij}}\prod_{i=1}^{4}\prod_{j=1}^{r_{i}}\operatorname{Quot}_{\Delta}(\iota_{i,*}\operatorname{\mathcal{O}}_{{\mathbb{C}}^{3}_{i}},n_{ij}).$

Moreover, we have an identification of each factor with a Hilbert scheme of
points

$\operatorname{Quot}_{\Delta}(\iota_{i,*}\operatorname{\mathcal{O}}_{{\mathbb{C}}^{3}_{i}},n_{ij})\cong\operatorname{Hilb}^{n_{ij}}({\mathbb{C}}_{i}^{3}),$

such that the natural $({\mathbb{C}}^{*})^{3}$-action on
${\mathbb{C}}^{3}_{i}$ lifts on
$\operatorname{Hilb}^{n_{ij}}({\mathbb{C}}_{i}^{3})$ and coincides with the
induced $\mathbf{T}_{0}$-action on the left-hand-side. The
$({\mathbb{C}}^{*})^{3}$-fixed locus
$\operatorname{Hilb}^{n_{ij}}({\mathbb{C}}_{i}^{3})^{({\mathbb{C}}^{*})^{3}}$
is reduced and zero-dimensional, which implies that the same holds for
$\mathcal{M}_{\overline{r},n}^{\mathbf{T}}$ and that every $\mathbf{T}$-fixed
point in $\mathcal{M}_{\overline{r},n}^{\mathbf{T}}$ corresponds to a quotient

$\bigoplus_{i=1}^{4}\bigoplus_{j=1}^{r_{i}}I_{ij}\hookrightarrow\bigoplus_{i=1}^{4}\iota_{i,*}\operatorname{\mathcal{O}}_{{\mathbb{C}}^{3}_{i}}^{r_{i}}\twoheadrightarrow\bigoplus_{i=1}^{4}\bigoplus_{j=1}^{r_{i}}\operatorname{\mathcal{O}}_{Z_{ij}},$

where each $Z_{ij}\subset{\mathbb{C}}^{3}_{i}$ is a closed zero-dimensional
subscheme with monomial ideal sheaf $I_{ij}$, each of which corresponds to a
plane partition. Reversing the correspondence concludes the argument. ∎

Define the _MacMahon series_

(3.1) $\displaystyle\mathrm{M}(q)=\prod_{n\geq 1}\frac{1}{(1-q^{n})^{n}}.$

######  Corollary 3.4.

Let $\overline{r}=(r_{1},\dots,r_{4})$ and $r=\sum_{i=1}^{4}r_{i}$. The
generating series of topological Euler characteristics is

$\sum_{n\geq 0}e(\mathcal{M}_{\overline{r},n})\cdot q^{n}=\mathrm{M}(q)^{r}.$

###### Proof.

The Euler characteristic of a scheme coincides with the one of its
$\mathbf{T}$-fixed locus, therefore the result follows by Proposition 3.3
combined with the classical fact that

$\sum_{\pi}q^{|\pi|}=\mathrm{M}(q),$

where the sum is over all plane partitions. ∎

###  3.3. Invariants

For any $\overline{r}=(r_{1},\dots,r_{4})$ and $n\in{\mathbb{Z}}_{\geq 0}$,
denote by $t_{1},\dots,t_{4}$ the irreducible characters of the
$\mathbf{T}_{0}$-action on $\mathcal{M}_{\overline{r},n}$ and by $w_{ij}$ the
irreducible characters of the $\mathbf{T}_{1}$-action, where $i=1,\dots,4$ and
$j=1,\dots r_{i}$. Since all ingredients in Theorem 2.3 are naturally
$\mathbf{T}$-equivariant and in particular the $\mathbf{T}$-action preserves
the quadratic pairing777The requirement to restrict to the Calabi-Yau
condition $t_{1}t_{2}t_{3}t_{4}=1$ is needed precisely to preserve the
quadratic pairing equivariantly. on ${\mathcal{L}}$, the virtual structure
sheaf
$\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{\mathcal{M}_{\overline{r},n}}$
naturally lifts to a $\mathbf{T}$-equivariant class in equivariant $K$-theory.
We define the $\mathbf{T}$-equivariant $K$-theoretic invariants

(3.2)
$\displaystyle{\mathcal{Z}}_{\overline{r},n}=\chi\left(\mathcal{M}_{\overline{r},n},\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{\mathcal{M}_{\overline{r},n}}\right)\in\frac{{\mathbb{Q}}(t_{1}^{\frac{1}{2}},t_{2}^{\frac{1}{2}},t_{3}^{\frac{1}{2}},t_{4}^{\frac{1}{2}},w^{\frac{1}{2}})}{(t_{1}t_{2}t_{3}t_{4}-1)}.$

Since ${\mathcal{M}}_{\overline{r},n}$ is not proper, one cannot directly
define $K$-theoretic invariants via proper push-forward in $K$-theory.
Instead, since the fixed locus ${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$
is proper, we define invariants as the composition

(3.3)
$\displaystyle\chi\left(\mathcal{M}_{\overline{r},n},\cdot\right):K^{\mathbf{T}}_{0}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right)\to
K^{\mathbf{T}}_{0}({\mathcal{M}}_{\overline{r},n})_{\mathrm{loc}}\xrightarrow{\sim}K^{\mathbf{T}}_{0}({\mathcal{M}}^{\mathbf{T}}_{\overline{r},n})_{\mathrm{loc}}\to
K_{0}^{\mathbf{T}}({\mathsf{pt}})_{\mathrm{loc}}.$

Here, the first map is a suitable localisation of
$K^{\mathbf{T}}_{0}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right)$,
the second map is Thomason’s abstract localisation [68] and the third map is
the proper pushforward on ${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$
(extended to the localisation).

We define the _tetrahedron instanton partition function_ as the generating
series

$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)=\sum_{n\geq
0}{\mathcal{Z}}_{\overline{r},n}\cdot
q^{n}\in\frac{{\mathbb{Q}}(t_{1}^{\frac{1}{2}},t_{2}^{\frac{1}{2}},t_{3}^{\frac{1}{2}},t_{4}^{\frac{1}{2}},w^{\frac{1}{2}})}{(t_{1}t_{2}t_{3}t_{4}-1)}[\\![q]\\!].$

Let $T_{{\mathcal{M}}_{\overline{r},n}}^{\operatorname{\mathrm{vir}}}$ be the
_virtual tangent bundle_ of $\mathcal{M}_{\overline{r},n}$, that is the dual
of the obstruction theory ${\mathbb{E}}$ in (2.1)

$\displaystyle
T_{{\mathcal{M}}_{\overline{r},n}}^{\operatorname{\mathrm{vir}}}=[{T_{{\mathcal{A}}}}|_{\mathcal{M}_{\overline{r},n}}\to{\mathcal{L}}|_{\mathcal{M}_{\overline{r},n}}\to{T^{*}_{{\mathcal{A}}}}|_{\mathcal{M}_{\overline{r},n}}],$

and denote by $N^{\operatorname{\mathrm{vir}}}$ the _virtual normal bundle_ ,
which is the $\mathbf{T}$-movable part of
$T_{{\mathcal{M}}_{\overline{r},n}}^{\operatorname{\mathrm{vir}}}$. We will
denote by
$T_{{\mathcal{M}}_{\overline{r},n}}^{\operatorname{\mathrm{vir}}},N^{\operatorname{\mathrm{vir}}}$
also their classes in $K$-theory, whenever it is clear from the context.

By Oh-Thomas localisation theorem in K-theory [59] we can compute the
invariants via $\mathbf{T}$-equivariant residues on the fixed locus

$\displaystyle\chi\left(\mathcal{M}_{\overline{r},n},\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{\mathcal{M}_{\overline{r},n}}\right)=\chi\left(\mathcal{M}_{\overline{r},n}^{\mathbf{T}},\frac{\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{\mathcal{M}_{\overline{r},n}^{\mathbf{T}}}}{\sqrt{\mathfrak{e}}(N^{\operatorname{\mathrm{vir}}})}\right),$

where $\mathfrak{e}$ is the ($\mathbf{T}$-equivariant) $K$-theoretic Euler
class, which is defined as follows. Let ${\mathcal{E}}$ be a
$\mathbf{T}$-equivariant locally free sheaf on
${\mathcal{M}}_{\overline{r},n}$. We define

$\mathfrak{e}({\mathcal{E}}):=\Lambda^{\bullet}{\mathcal{E}}^{*}=\sum_{i\geq
0}(-1)^{i}\Lambda^{i}{\mathcal{E}}^{*}\in
K^{0}_{\mathbf{T}}({\mathcal{M}}_{\overline{r},n}),$

and extend it by linearity to any class in
$K^{0}_{\mathbf{T}}({\mathcal{M}}_{\overline{r},n})$. Finally,
$\sqrt{\mathfrak{e}}(\cdot)$ is the ($\mathbf{T}$-equivariant) $K$-theoretic
square-root Euler class; the complete description and construction of this
class is in [59, Sec. 5.1]. We remark that in the proof of the localisation
theorem, Oh-Thomas [59] prove that the fixed locus
$\mathcal{M}_{\overline{r},n}^{\mathbf{T}}$ is naturally endowed with an
_orientation_ (in the sense of [17]) – and therefore a virtual structure sheaf
– and that the virtual normal bundle $N^{\operatorname{\mathrm{vir}}}$ is a
self-dual complex (with a natural induced orientation), which implies that its
square-root Euler class is well-defined (cf. [59, Eqn. (112)]). By Proposition
3.3 the fixed locus $\mathcal{M}_{\overline{r},n}^{\mathbf{T}}$ consists of
finitely many reduced points labelled by tuples of plane partitions. Thus, if
we let $Z\in\mathcal{M}_{\overline{r},n}^{\mathbf{T}}$ be a fixed point, the
induced virtual fundamental class satisfies
$[Z]^{\operatorname{\mathrm{vir}}}=\pm[Z]$, for a well-determined sign.
Therefore the tetrahedron partition functions is given by

$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)=\sum_{\overline{\pi}}(-1)^{o_{\overline{\pi}}}\sqrt{\mathfrak{e}}(-T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}})\cdot
q^{|\overline{\pi}|},$

where the sum runs over tuples
$\overline{\pi}=(\overline{\pi}_{1},\dots,\overline{\pi}_{4})$, where each
$\overline{\pi}_{i}$ is an $r_{i}$-tuple of plane partitions,
$T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}$ is the virtual tangent
bundle at the fixed point corresponding to $\overline{\pi}$ via the
correspondence in Proposition 3.3 and $(-1)^{o_{\overline{\pi}}}$ is the sign
of the induced virtual fundamental class (with respect to the standard
fundamental class). We used the fact that the virtual tangent bundle is
$\mathbf{T}$-movable by Proposition 3.8.

The definition of the square-root Euler class is a priori non-explicit and
depends on the chosen orientation of $\mathcal{M}_{\overline{r},n}$. To ease
our life, we define the _square root_ of a $\mathbf{T}$-representation.

######  Definition 3.5 ([46, Def. 1.2]).

Let $V\in K^{0}_{\mathbf{T}}({\mathsf{pt}})$ be a virtual
$\mathbf{T}$-representation. We say that $T\in
K^{0}_{\mathbf{T}}({\mathsf{pt}})$ is a _square root_ of $V$ if

$V=T+\overline{T}\in K_{\mathbf{T}}^{0}({\mathsf{pt}}),$

where $\overline{(\cdot)}$ denotes the dual $\mathbf{T}$-representation.

Let $V$ be a $\mathbf{T}$-representation with a square root $T$ in
$K^{0}_{\mathbf{T}}({\mathsf{pt}})$. Its $K$-theoretic square root Euler class
satisfies

(3.4) $\displaystyle\sqrt{\mathfrak{e}}(V)$
$\displaystyle=\pm\mathfrak{e}(T)\otimes({\det}T)^{\frac{1}{2}}\in
K^{0}_{\mathbf{T}}\left({\mathsf{pt}},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right),$

for a well-defined (but non-explicit) sign. For an irreducible
$\mathbf{T}$-representation $t^{\mu}$, define

$[t^{\mu}]=t^{\frac{\mu}{2}}-t^{-\frac{\mu}{2}}\in
K^{0}_{\mathbf{T}}({\mathsf{pt}})$

and extend it by linearity to any $T\in K_{\mathbf{T}}^{0}({\mathsf{pt}})$ by
setting $[t^{\mu}\pm t^{\nu}]=[t^{\mu}][t^{\nu}]^{\pm 1}$, as long as $\nu$ is
not the trivial weight. It is shown in [33, Sec. 6.1] that

$\displaystyle\mathfrak{e}(T)\otimes({\det}T)^{\tfrac{1}{2}}=[T],$

for any virtual $\mathbf{T}$-representation $T\in
K_{\mathbf{T}}^{0}({\mathsf{pt}})$. Therefore we can express (3.4) explicitly
as

(3.5) $\displaystyle\sqrt{\mathfrak{e}}(V)=\pm[T].$

We derive in the next section a vertex formalism that computes the partition
function of tetrahedron instantons, by choosing suitable square roots of the
virtual tangent bundle and the associated signs.

###  3.4. Vertex formalism

Let $r=(r_{1},\dots r_{4})$ and
$\overline{\pi}=(\overline{\pi}_{1},\dots,\overline{\pi}_{4})$ correspond to a
$\mathbf{T}$-fixed quotient $[{\mathcal{E}}_{\overline{r}}\twoheadrightarrow
Q_{\overline{\pi}}]\in{\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$, where each
$\overline{\pi}_{i}=(\pi_{i1},\dots,\pi_{ir_{i}})$ is a tuple of plane
partitions. We denote by $T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}$
the virtual tangent space at the fixed point $\overline{\pi}$. Recall that
$Q_{\overline{\pi}}$ decomposes as

$Q_{\overline{\pi}}=\bigoplus_{i=1}^{4}\bigoplus_{j=1}^{r_{i}}\operatorname{\mathcal{O}}_{Z_{{\pi_{ij}}}},$

where each $Z_{\pi_{ij}}\subset{\mathbb{C}}^{3}_{i}$ is a zero-dimensional
closed subscheme defined by a monomial ideal corresponding to a plane
partition $\pi_{ij}$. We keep denoting their global sections by $Z_{\pi_{ij}}$
(resp. $Q_{\overline{\pi}}$) which are given, as a $\mathbf{T}_{0}$ (resp.
$\mathbf{T}$)-representation, by

(3.6)
$\displaystyle\begin{split}Z_{\pi_{ij}}&=\sum_{(a,b,c)\in\pi_{ij}}t_{i_{1}}^{a}t_{i_{2}}^{b}t_{i_{3}}^{c},\\\
Q_{\overline{\pi}_{i}}&=\sum_{j=1}^{r_{i}}w_{ij}Z_{\pi_{ij}}\\\
Q_{\overline{\pi}}&=\sum_{i=1}^{4}Q_{\overline{\pi}_{i}},\end{split}$

where $\\{i_{1},i_{2},i_{3}\\}$ are the three indices in $\\{1,2,3,4\\}$
different from $i$. We denote by $\overline{(\cdot)}$ the involution on
$K^{0}_{\mathbf{T}}({\mathsf{pt}})$ which acts as
$\overline{t^{\mu}}=t^{-\mu}$ on irreducible representations and by
$P_{I}=\prod_{l\in I}(1-t_{l})$, for any set of indices $I$. For $i=1,\dots,4$
set the $\mathbf{T}_{1}$-representations

$\displaystyle K_{i}$ $\displaystyle=\sum_{j=1}^{r_{i}}w_{ij},$ $\displaystyle
K$ $\displaystyle=\sum_{i=1}^{4}K_{i}.$

######  Proposition 3.6.

. We have an identification of virtual $\mathbf{T}$-representations

$\displaystyle
T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}=\overline{K}Q_{\overline{\pi}}+K\overline{Q_{\overline{\pi}}}-P_{1234}Q_{\overline{\pi}}\overline{Q_{\overline{\pi}}}-\sum_{i=1}^{4}K_{i}t_{i}\overline{Q_{\overline{\pi}}}-\sum_{i=1}^{4}\overline{K_{i}}t^{-1}_{i}Q_{\overline{\pi}}.$

###### Proof.

The virtual tangent space at a $\mathbf{T}$-fixed point corresponding to a
tuple of plane partitions $\overline{\pi}$ is computed as a class in
$K$-theory by

(3.7) $\displaystyle
T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}=T_{{\mathcal{M}}^{\mathrm{nc}}_{\overline{r},n}}|_{\overline{\pi}}-{\mathcal{L}}|_{\overline{\pi}}+T^{*}_{{\mathcal{M}}^{\mathrm{nc}}_{\overline{r},n}}|_{\overline{\pi}}\in
K_{\mathbf{T}}^{0}({\mathsf{pt}}).$

The tangent space
$T_{{\mathcal{M}}^{\mathrm{nc}}_{\overline{r},n}}|_{\overline{\pi}}$ at the
fixed quotient corresponding to
$\overline{\pi}\in{\mathcal{M}}^{\mathrm{nc}}_{\overline{r},n}$ sits in an
exact sequence of $\mathbf{T}$-representations

$\displaystyle
0\to\operatorname{Hom}(Q_{\overline{\pi}},Q_{\overline{\pi}})\to{\mathbb{C}}^{4}\otimes\operatorname{Hom}(Q_{\overline{\pi}},Q_{\overline{\pi}})\oplus\bigoplus_{i=1}^{4}\bigoplus_{j=1}^{r_{i}}\operatorname{Hom}({\mathbb{C}}\cdot
w_{ij},Q_{\overline{\pi}})\to
T_{{\mathcal{M}}^{\mathrm{nc}}_{\overline{r},n}}|_{\overline{\pi}}\to 0,$

where the middle representation is the tangent space of the affine space
$\mathsf{R}_{\overline{r},n}$ and the first representation records the free
action of $\operatorname{GL}({\mathbb{C}}^{n})$. The weight decomposition of
${\mathbb{C}}^{4}$ is

${\mathbb{C}}^{4}=t_{1}^{-1}+t_{2}^{-1}+t_{3}^{-1}+t_{4}^{-1},$

by which one readily computes in $K$-theory

$T_{{\mathcal{M}}^{\mathrm{nc}}_{\overline{r},n}}|_{\overline{\pi}}=(t_{1}^{-1}+t_{2}^{-1}+t_{3}^{-1}+t_{4}^{-1}-1)Q_{\overline{\pi}}\overline{Q_{\overline{\pi}}}+\overline{K}Q_{\overline{\pi}}.$

The vector bundle ${\mathcal{L}}$ at the point $\overline{\pi}$ satisfies

${\mathcal{L}}|_{\overline{\pi}}=\Lambda^{2}{\mathbb{C}}^{4}\otimes\operatorname{Hom}(Q_{\overline{\pi}},Q_{\overline{\pi}})\oplus\bigoplus_{i=1}^{4}\operatorname{Hom}(K_{i}\cdot
t_{i},Q_{\overline{\pi}})\oplus\bigoplus_{i=1}^{4}\operatorname{Hom}(K_{i}\cdot
t_{i},Q_{\overline{\pi}})^{*},$

where the weight decomposition of $\Lambda^{2}{\mathbb{C}}^{4}$ is

$\Lambda^{2}{\mathbb{C}}^{4}=t_{1}^{-1}t_{2}^{-1}+t_{1}^{-1}t_{3}^{-1}+t_{1}^{-1}t_{4}^{-1}+t_{2}^{-1}t_{3}^{-1}+t_{2}^{-1}t_{4}^{-1}+t_{3}^{-1}t_{4}^{-1},$

which implies that

${\mathcal{L}}|_{\overline{\pi}}=(t_{1}^{-1}t_{2}^{-1}+t_{1}^{-1}t_{3}^{-1}+t_{1}^{-1}t_{4}^{-1}+t_{2}^{-1}t_{3}^{-1}+t_{2}^{-1}t_{4}^{-1}+t_{3}^{-1}t_{4}^{-1})Q_{\overline{\pi}}\overline{Q_{\overline{\pi}}}+\sum_{i=1}^{4}K_{i}t_{i}\overline{Q_{\overline{\pi}}}+\sum_{i=1}^{4}\overline{K_{i}}t^{-1}_{i}Q_{\overline{\pi}}.$

Plugging the computations in (3.7) and exploiting the relation
$t_{1}t_{2}t_{3}t_{4}=1$ concludes the proof. ∎

We exhibit a square root of the virtual tangent bundle at each fixed point.
Let $\overline{\pi}$ be a tuple of plane partitions. Set the _vertex term_

(3.8)
$\displaystyle\mathsf{v}_{\overline{\pi}}=\overline{K}Q_{\overline{\pi}}-\sum_{j=1}^{4}K_{j}t_{j}\overline{Q_{\overline{\pi}}}-\sum_{j=1}^{4}\overline{P_{j_{1}j_{2}j_{3}}}Q_{\overline{\pi}_{j}}\overline{Q_{\overline{\pi}_{j}}}-\sum_{1\leq
i<j\leq
4}\overline{P_{j_{1}j_{2}j_{3}}}\left(Q_{\overline{\pi}_{j}}\overline{Q_{\overline{\pi}_{i}}}+Q_{\overline{\pi}_{i}}\overline{Q_{\overline{\pi}_{j}}}\right),$

where $\\{j_{1},j_{2},j_{3}\\}$ are the three indices in $\\{1,2,3,4\\}$
different from $j$. It is easily seen to satisfy

$\displaystyle
T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}=\mathsf{v}_{\overline{\pi}}+\overline{\mathsf{v}_{\overline{\pi}}},$

thanks to the identity

$P_{j_{1}j_{2}j_{3}}+\overline{P_{j_{1}j_{2}j_{3}}}=P_{1234},$

which exploits the Calabi-Yau condition $t_{1}t_{2}t_{3}t_{4}=1$. Thus, by
(3.5) we can compute the partition function of tetrahedron instantons as

$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)=\sum_{\overline{\pi}}(-1)^{\sigma_{\overline{\pi}}}[-\mathsf{v}_{\overline{\pi}}]\cdot
q^{|\overline{\pi}|},$

where the sign $(-1)^{\sigma_{\overline{\pi}}}$ receives a contribution both
from $(-1)^{{o_{\overline{\pi}}}}$ (coming from the orientation of the fixed
locus) and from the choice of the square root of the virtual tangent bundle.

We postpone to Appendix A the proof of the correct sign rule associated with
our choice of square root (cf. Corollary A.4).

######  Theorem 3.7.

We have that

$(-1)^{\sigma_{\overline{\pi}}}=1.$

Thanks to Theorem 3.7, we can explicitly express the partition function as

(3.9)
$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)=\sum_{\overline{\pi}}[-\mathsf{v}_{\overline{\pi}}]\cdot
q^{|\overline{\pi}|}.$

where the sum runs over tuples
$\overline{\pi}=(\overline{\pi}_{1},\dots,\overline{\pi}_{4})$, where each
$\overline{\pi}_{i}$ is an $r_{i}$-tuple of plane partitions.

###  3.5. factorisation

Let $\overline{r}=(r_{1},\dots,r_{4})$ and
$\overline{\pi}=(\overline{\pi}_{1},\dots,\overline{\pi}_{4})$, where each
$\overline{\pi}_{i}=(\pi_{il})_{il}$ is an $r_{i}$-tuple of plane partitions,
for $i=1,\dots,4$, $l=1,\dots,r_{i}$. The vertex term can be written as a sum

$\displaystyle\mathsf{v}_{\overline{\pi}}$ $\displaystyle=\sum_{1\leq i\leq
j\leq 4}\mathsf{v}^{(ij)}_{\overline{\pi}},$
$\displaystyle\mathsf{v}^{(ij)}_{\overline{\pi}}$
$\displaystyle=\begin{cases}\overline{K}_{j}Q_{\overline{\pi}_{j}}-K_{j}t_{j}\overline{Q_{\overline{\pi}_{j}}}-\overline{P_{j_{1}j_{2}j_{3}}}Q_{\overline{\pi}_{j}}\overline{Q_{\overline{\pi}_{j}}}&i=j,\\\
\overline{K}_{i}Q_{\overline{\pi}_{j}}-K_{j}t_{j}\overline{Q_{\overline{\pi}_{i}}}+\overline{K}_{j}Q_{\overline{\pi}_{i}}-K_{i}t_{i}\overline{Q_{\overline{\pi}_{j}}}-\overline{P_{j_{1}j_{2}j_{3}}}(Q_{\overline{\pi}_{j}}\overline{Q_{\overline{\pi}_{i}}}+Q_{\overline{\pi}_{i}}\overline{Q_{\overline{\pi}_{j}}})&i<j.\end{cases}$

If $i=j$, the vertex term $\mathsf{v}^{(ii)}_{\overline{\pi}}$ recovers the
vertex term of the higher rank Donaldson-Thomas theory of
${\mathbb{C}}^{3}_{i}$, for the tuple of plane partitions $\overline{\pi}_{i}$
(cf. [33, Sec. 5.1]). For this identification, we crucially use that
$t_{i}=(t_{i_{1}}t_{i_{2}}t_{i_{3}})^{-1}$ where $\\{i_{1},i_{2},i_{3}\\}$ are
the three indices in $\\{1,2,3,4\\}$ different from $i$. To ease the notation,
we define the _Calabi-Yau weight_ of the coordinate hyperplane
${\mathbb{C}}^{3}_{i}$ to be

$\kappa_{i}:=t_{i_{1}}t_{i_{2}}t_{i_{3}}=t_{i}^{-1}.$

By (3.6) we can further identify the terms
$\mathsf{v}^{(ij)}_{\overline{\pi}}$ as sums

$\displaystyle\mathsf{v}^{(ij)}_{\overline{\pi}}$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq l\leq r_{i}\\\ 1\leq k\leq
r_{j}\end{subarray}}\mathsf{v}^{(ij,lk)}_{\overline{\pi}},$
$\displaystyle\mathsf{v}^{(ij,lk)}_{\overline{\pi}}$
$\displaystyle=\begin{cases}w_{jl}^{-1}w_{jk}\left(Z_{\pi_{jk}}-\kappa_{j}^{-1}\overline{Z_{\pi_{jl}}}-\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{jk}}\overline{Z_{\pi_{jl}}}\right)&i=j\\\
w_{il}^{-1}w_{jk}\left(Z_{\pi_{jk}}-\kappa^{-1}_{j}\overline{Z_{\pi_{il}}}-\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}}\right)+w_{il}w_{jk}^{-1}\left(Z_{\pi_{il}}-\kappa^{-1}_{i}\overline{Z_{\pi_{jk}}}-\overline{P_{j_{1}j_{2}j_{3}}}(Z_{\pi_{il}}\overline{Z_{\pi_{jk}}})\right)&i<j,\end{cases}$

where each $Z_{il}$ are the global sections of a zero-dimensional closed
subscheme of ${\mathbb{C}}^{3}_{i}$, whose monomial ideal corresponds to the
plane partition $\pi_{il}$. If $i=j$ and $l=k$, the vertex term
$\mathsf{v}^{(ii,ll)}_{\overline{\pi}}$ recovers the vertex term of the rank
one Donaldson-Thomas theory of ${\mathbb{C}}^{3}_{i}$, for the plane partition
$\pi_{il}$ (cf. [44, Eqn. (12)]).

As already anticipated, the vertex term is $\mathbf{T}$-movable, which in
particular implies that the rational function $[-\mathsf{v}_{\overline{\pi}}]$
is well-defined and non-zero.

######  Proposition 3.8.

The vertex term $\mathsf{v}_{\overline{\pi}}$ is $\mathbf{T}$-movable.

###### Proof.

By the decomposition of the vertex term as

(3.10) $\displaystyle\mathsf{v}_{\overline{\pi}}=\sum_{1\leq i\leq j\leq
4}\sum_{\begin{subarray}{c}1\leq l\leq r_{i}\\\ 1\leq k\leq
r_{j}\end{subarray}}\mathsf{v}^{(ij,lk)}_{\overline{\pi}},$

it is clear that the possible contributions to the $\mathbf{T}$-fixed part
have to come from the _diagonal_ terms
$\mathsf{v}^{(ii,ll)}_{\overline{\pi}}$, which are $\mathbf{T}$-movable by
[44, Lemma 8] (see also [33, Prop. 3.14]). ∎

####  3.5.1. Rigidity

Analogously to the arguments of [1, Sec. 3.1], we establish a general
criterion which determines the possible poles that can appear in the
denominator after $K$-theoretic localisation. Let ${\mathcal{M}}$ be a quasi-
projective scheme, acted by an algebraic torus ${\mathbb{T}}$, such that the
fixed locus ${\mathcal{M}}^{\mathbb{T}}$ is proper.

######  Definition 3.9 ([1, Def. 3.1]).

Given a weight $w\in\widehat{\mathbb{T}}$, denote by
${\mathbb{T}}_{w}\subset{\mathbb{T}}$ the maximal torus inside the subgroup
$\ker(w)\subset{\mathbb{T}}$. A weight $w\in\widehat{{\mathbb{T}}}$ is called
a _compact_ weight of ${\mathcal{M}}$ if ${\mathcal{M}}^{{\mathbb{T}}_{w}}$ is
proper, and _non-compact_ otherwise.

Let ${\mathcal{F}}\in K^{\mathbb{T}}_{0}({\mathcal{M}})$ be a
${\mathbb{T}}$-equivariant class in $K$-theory. As in (3.3), by Thomason
$K$-theoretic localisation [68], we have that

$\displaystyle\chi({\mathcal{M}},{\mathcal{F}})\in{\mathbb{Q}}[t_{1}^{\pm
1},\dots,t_{m}^{\pm 1}]\left[\frac{1}{1-w}:w\in\widehat{\mathbb{T}}\right],$

where $m=\operatorname{rk}{\mathbb{T}}$ is the rank of the torus. In other
words, the pushforward $K_{0}^{\mathbb{T}}({\mathcal{M}}^{\mathbb{T}})\to
K_{0}^{\mathbb{T}}({\mathcal{M}})$ becomes an isomorphism after inverting
virtual characters of the form $1-t^{\mu}$. We generalize [1, Prop. 3.2] –
spelled out in loc. cit. for virtual structures à la Behrend-Fantechi [6].

######  Proposition 3.10.

We have that

$\chi({\mathcal{M}},{\mathcal{F}})=\frac{p(t_{1},\dots,t_{m})}{\prod_{w}(1-w)},$

where $p(t)$ is a Laurent polynomial and the product is over some non-compact
weights $w$ of ${\mathcal{M}}$ (with possible multiplicities).

###### Proof.

By definition of $K$-theoretic invariants (3.3), we have that

$\chi^{\mathbb{T}}({\mathcal{M}},{\mathcal{F}})=\frac{p(t_{1},\dots,t_{m})}{q(t_{1},\dots,t_{m})},$

where $p(t),q(t)$ are Laurent polynomial and the superscript ${\mathbb{T}}$
records that we defined the invariants ${\mathbb{T}}$-equivariantly. Let $w$
be a compact weight. Up to replacing ${\mathbb{T}}$ with a finite cover, we
can assume that $w=(w_{0})^{e}$, where $w_{0}=t_{1}^{\mu_{1}}\cdots
t_{m}^{\mu_{m}}$ is weight such that at least one $|\mu_{i}|=1$. Then the
subtorus ${\mathbb{T}}_{w}$ coincides with $\ker(w_{0})$. Since the
${\mathbb{T}}_{w}$-fixed locus ${\mathcal{M}}^{{\mathbb{T}}_{w}}$ is proper,
we can define the invariants ${\mathbb{T}}_{w}$-equivariantly, and get

$\chi^{{\mathbb{T}}_{w}}({\mathcal{M}},{\mathcal{F}})=\frac{\tilde{p}(t_{1},\dots,t_{m})}{\tilde{q}(t_{1},\dots,t_{m})},$

where $\tilde{p}(t),\tilde{q}(t)$ are the Laurent polynomials coming by the
restricting $p(t),q(t)$ as ${\mathbb{T}}$-representations to
${\mathbb{T}}_{w}$-representations. But the invariants

$\chi^{{\mathbb{T}}_{w}}({\mathcal{M}},{\mathcal{F}})\in\frac{{\mathbb{Q}}[t_{1}^{\pm
1},\dots,t_{m}^{\pm
1}]}{(1-w_{0})}\left[\frac{1}{1-v}:v\in\widehat{\mathbb{T}},v\neq
w_{0}^{n}\right]$

are well-defined, which implies that $(1-w_{0}^{n})$ cannot appear as factors
of $q(t)$, for all $n\in{\mathbb{Z}}$. ∎

####  3.5.2. Framing limits

We prove in this section Theorem 1.2 from the Introduction. We first show that
the partition function of tetrahedron instantons does not depend on the
framing parameters $w_{ij}$, exploiting a suitable rigidity argument. Granting
this independence, we specialise the framing parameters $w_{ij}$ to arbitrary
values and send them to infinity, which recovers the formula in Theorem 1.2.

######  Theorem 3.11.

The partition function of tetrahedron instantons
${\mathcal{Z}}_{\overline{r}}(q)$ does not depend on the weights $w_{il}$, for
$i=1,\dots,4$ and $l=1,\dots,r_{i}$.

###### Proof.

We prove this result by exploiting similar arguments to [33, Thm. 6.5]. The
$n$-th coefficient of ${\mathcal{Z}}_{\overline{r}}(q)$ is a sum of
contributions

$[-\mathsf{v}_{\overline{\pi}}],\qquad\lvert\overline{\pi}\rvert=n.$

A simple manipulation shows that

(3.11)
$[-\mathsf{v}_{\overline{\pi}}]=A(t)\prod_{(i,l)\neq(j,k)}\frac{\prod_{\mu_{ij,lk}}(1-w_{il}^{-1}w_{jk}t^{\mu_{ij,lk}})}{\prod_{\nu_{ij,lk}}(1-w_{il}^{-1}w_{jk}t^{\nu_{ij,lk}})},$

where the product is over all $(i,l)\neq(j,k)$ such that $i,j=1,\dots,4$ and
$l=1,\dots,r_{i}$ and $j=1,\dots,r_{j}$ while
$A(t)\in{\mathbb{Q}}(\\!(t_{1}^{\frac{1}{2}},t_{2}^{\frac{1}{2}},t_{3}^{\frac{1}{2}},t_{4}^{\frac{1}{2}})\\!)/(t_{1}t_{2}t_{3}t_{4}-1)$
and the number of weights $\mu_{ij,lk}$ and $\nu_{ij,lk}$ is the same. Thus,
${\mathcal{Z}}_{\overline{r}}(q)$ is a homogeneous rational expression of
total degree 0 with respect to the variables $w_{il}$. We aim to show that
${\mathcal{Z}}_{\overline{r}}(q)$ has no poles of the form
$1-w_{il}^{-1}w_{jk}t^{\nu}$, implying that it is a degree $0$ polynomial in
the $w_{il}$, hence constant in the $w_{il}$.

Set $\mathsf{w}=w_{il}^{-1}w_{jk}t^{\nu}$ for fixed $(i,l)\neq(j,k)$ and
$\nu\in\widehat{\mathbf{T}}_{0}$. To see that $1-\mathsf{w}$ is not a pole, we
show that $\mathsf{w}$ is a non-compact weight and use Proposition 3.10. In
other words, we need to prove that the fixed locus of the torus
$\mathbf{T}_{\mathsf{w}}=\ker(\mathsf{w})$ is proper.

Consider, consider the automorphism
$\tau_{\nu}\colon\mathbf{T}\,\widetilde{\to}\,\mathbf{T}$ defined by

$(t_{1},t_{2},t_{3},t_{4},w_{11},\dots,w_{4r_{4}})\mapsto(t_{1},t_{2},t_{3},t_{4},w_{11},\ldots,w_{il}t^{-\nu},\ldots,w_{jk},\ldots,w_{4r_{4}}).$

It maps $\mathbf{T}_{\mathsf{w}}\subset\mathbf{T}$ isomorphically onto the
subtorus $\mathbf{T}_{0}\times\set{w_{il}=w_{jk}}\subset\mathbf{T}$. This
yields an inclusion of tori

(3.12)
$\mathbf{T}_{0}\,\widetilde{\to}\,\mathbf{T}_{0}\times\Set{(1,\ldots,1)}\hookrightarrow\tau(\mathbf{T}_{\mathsf{w}}).$

We consider the action
$\sigma_{\nu}\colon\mathbf{T}\times{\mathcal{M}}_{\overline{r},n}\to{\mathcal{M}}_{\overline{r},n}$
where $\mathbf{T}_{0}$ translates the support of the quotient sheaf in the
usual way, the $l$-th summand of $\iota_{i,*}\mathscr{O}^{\oplus r_{i}}$ gets
scaled by $w_{il}t^{\nu}$ and all other summands by $w_{jk}$ for
$(j,k)\neq(i,l)$. We have a commutative diagram

${\mathbf{T}_{\mathsf{w}}\times{\mathcal{M}}_{\overline{r},n}}$${{\mathcal{M}}_{\overline{r},n}}$${\tau_{\nu}(\mathbf{T}_{\mathsf{w}})\times{\mathcal{M}}_{\overline{r},n}}$${{\mathcal{M}}_{\overline{r},n}}$$\leftarrow$$\rightarrow$$\scriptstyle{\sigma}$$\leftarrow$$\rightarrow$$\scriptstyle{\tau_{\nu}\times\operatorname{id}}$$\Leftarrow$$\Leftarrow$$\leftarrow$$\rightarrow$$\scriptstyle{\sigma_{\nu}}$

where $\sigma$ is the restriction of the usual $\mathbf{T}$-action on
${\mathcal{M}}_{\overline{r},n}$. This diagram induces a natural isomorphism
${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}_{\mathsf{w}}}\,\widetilde{\to}\,{\mathcal{M}}_{\overline{r},n}^{\tau_{\nu}(\mathbf{T}_{\mathsf{w}})}$,
which combined with (3.12) yields an inclusion

${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}_{\mathsf{w}}}\,\widetilde{\to}\,{\mathcal{M}}_{\overline{r},n}^{\tau_{\nu}(\mathbf{T}_{\mathsf{w}})}\hookrightarrow{\mathcal{M}}_{\overline{r},n}^{\mathbf{T}_{0}},$

where ${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}_{0}}$ is the fixed locus
with respect to the action $\sigma_{\nu}$. But by the same reasoning as in
Remark 3.1, this fixed locus is proper (because, again, a
$\mathbf{T}_{0}$-fixed surjection
${\mathcal{E}}_{\overline{r},n}\twoheadrightarrow T$ necessarily has the
quotient $T$ entirely supported at the origin $0\in\Delta$). Thus $\mathsf{w}$
is a compact weight, and the result follows. ∎

By Theorem 3.11 the partition function of tetrahedron instantons does not
depend on the framing parameters $w_{il}$, therefore we can compute it by
specialising them to arbitrary values. We set $w_{il}=L^{N_{il}}$, where
$N_{il}\gg 0$ are large integers satisfying $N_{il}>N_{ik}$ for $k>l$ and
$N_{jk}\gg N_{il}$ for $j>i$, and take limits $L\to\infty$. We totally order
the indices $(i,j)$ lexicographically, setting $(j,k)>(i,l)$ if $j>i$ or $i=j$
and $k>l$.

######  Proposition 3.12.

Let $(j,k)>(i,l)$. If $i=j$, we have

$\displaystyle\lim_{L\to\infty}[-\mathsf{v}^{(ij,lk)}_{\overline{\pi}}][-\mathsf{v}^{(ji,kl)}_{\overline{\pi}}]|_{w_{il}=L^{N_{il}}}=(-\kappa_{i}^{\frac{1}{2}})^{|\pi_{jk}|}(-\kappa_{j}^{-\frac{1}{2}})^{|\pi_{il}|}.$

If $i<j$, we have

$\lim_{L\to\infty}[-\mathsf{v}^{(ij,lk)}_{\overline{\pi}}]|_{w_{il}=L^{N_{il}}}=(-\kappa_{i}^{\frac{1}{2}})^{|\pi_{jk}|}(-\kappa_{j}^{-\frac{1}{2}})^{|\pi_{il}|}.$

###### Proof.

For all monomials $t^{\mu}$, we have the limits

$\displaystyle\lim_{L\to\infty}[w_{il}^{-1}w_{jk}t^{\mu}]|_{w_{il}=L^{N_{il}}}$
$\displaystyle=t^{\frac{\mu}{2}}\cdot\lim_{L\to\infty}L^{\frac{N_{jk}-N_{il}}{2}},$
$\displaystyle\lim_{L\to\infty}[w_{jk}^{-1}w_{il}t^{\mu}]|_{w_{il}=L^{N_{il}}}$
$\displaystyle=(-t^{-\frac{\mu}{2}})\cdot\lim_{L\to\infty}L^{\frac{N_{jk}-N_{il}}{2}}.$

Set $Z_{\pi_{jk}}=\sum_{\nu}t^{\nu}$ and $Z_{\pi_{il}}=\sum_{\mu}t^{\mu}$.
Taking limits, we have

$\displaystyle\lim_{L\to\infty}[-\mathsf{v}^{(ij,lk)}_{\overline{\pi}}]|_{w_{il}=L^{N_{il}}}$
$\displaystyle=\lim_{L\to\infty}[w_{il}^{-1}w_{jk}(-Z_{\pi_{jk}}+\kappa_{j}^{-1}\overline{Z_{\pi_{il}}}+\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}})]|_{w_{il}=L^{N_{il}}}$
$\displaystyle=\lim_{L\to\infty}L^{\frac{N_{jk}-N{il}}{2}(|\pi_{jk}|-|\pi_{il}|)}\cdot\frac{\prod_{\mu}t^{-\frac{\mu}{2}}}{\prod_{\nu}t^{\frac{\nu}{2}}}\cdot\kappa_{j}^{-\frac{|\pi_{il}|}{2}}.$

A similar computation yields

$\displaystyle\lim_{L\to\infty}[-\mathsf{v}^{(ji,kl)}_{\overline{\pi}}]|_{w_{il}=L^{N_{il}}}=(-1)^{|\pi_{jk}|-|\pi_{il}|}\lim_{L\to\infty}L^{\frac{N_{jk}-N_{il}}{2}}\cdot\frac{\prod_{\nu}t^{\frac{\nu}{2}}}{\prod_{\mu}t^{-\frac{\mu}{2}}}\cdot\kappa_{i}^{\frac{|\pi_{jk}|}{2}},$

by which we conclude that

$\displaystyle\lim_{L\to\infty}[-\mathsf{v}^{(ij,lk)}_{\overline{\pi}}][-\mathsf{v}^{(ji,kl)}_{\overline{\pi}}]|_{w_{il}=L^{N_{il}}}=(-\kappa_{i}^{\frac{1}{2}})^{|\pi_{jk}|}(-\kappa_{j}^{-\frac{1}{2}})^{|\pi_{il}|}.$

The second claim follows by an analogous computation. ∎

For an integer $m\in{\mathbb{Z}}$, we set the _sign function_

${\mathrm{sgn}}(m)=\begin{cases}1&m>0\\\ -1&m<0\\\ 0&m=0.\end{cases}$

######  Lemma 3.13.

Let $x_{i}$ be variables and $c_{i}\in{\mathbb{Z}}$ be integers, for
$i=1,\dots,r$. Then we have

$\prod_{1\leq i<j\leq
r}x_{i}^{c_{j}}x_{j}^{-c_{i}}=\prod_{i=1}^{r}\prod_{j=1}^{r}x_{j}^{{\mathrm{sgn}}(i-j)c_{i}}.$

###### Proof.

For $r=1$ the statement holds trivially. By induction, suppose that it holds
for $r-1$. Then we have

$\displaystyle\prod_{1\leq i<j\leq r}x_{i}^{c_{j}}x_{j}^{-c_{i}}$
$\displaystyle=\prod_{1\leq i<j\leq
r-1}x_{i}^{c_{j}}x_{j}^{-c_{i}}\cdot\prod_{i=1}^{r-1}x_{i}^{c_{r}}x_{r}^{-c_{i}}$
$\displaystyle=\prod_{i,j=1}^{r-1}x_{j}^{{\mathrm{sgn}}(i-j)c_{i}}\cdot\prod_{i=1}^{r-1}x_{i}^{c_{r}}x_{r}^{-c_{i}}$
$\displaystyle=\prod_{i,j=1}^{r}x_{j}^{{\mathrm{sgn}}(i-j)c_{i}}.$

∎

If $\overline{r}=(0,0,0,1)$ the partition function of tetrahedron instantons
recovers888A direct way to see this is that if $\overline{r}=(0,0,0,1)$ then
the vertex term (3.8) coincides with the one used by Okounkov [60, Eqn.
(3.4.30)]. the partition function of _K_ -theoretic Donaldson-Thomas theory of
${\mathbb{C}}^{3}$, whose explicit formula had been proved by Okounkov [60,
Thm. 3.3.6]

(3.13)
$\displaystyle{\mathcal{Z}}_{(0,0,0,1)}(-q)=\operatorname{Exp}\left(\frac{[t_{1}t_{2}][t_{1}t_{3}][t_{2}t_{3}]}{[t_{1}][t_{2}][t_{3}]}\frac{1}{[\kappa_{4}^{\frac{1}{2}}q][\kappa_{4}^{\frac{1}{2}}q^{-1}]},\right),$

where the plethystic exponential is defined as follows. For any formal power
series $f(p_{1},\ldots,p_{r};q_{1},\ldots,q_{s})$ in
${\mathbb{Q}}(p_{1},\ldots,p_{r})[\\![q_{1},\ldots,q_{s}]\\!]$, such that
$f(p_{1},\ldots,p_{r};0,\ldots,0)=0$, its _plethystic exponential_ is defined
as

(3.14)
$\displaystyle\operatorname{Exp}(f(p_{1},\ldots,p_{r};q_{1},\ldots,q_{s}))$
$\displaystyle:=\exp\Big{(}\sum_{n=1}^{\infty}\frac{1}{n}f(p_{1}^{n},\ldots,p_{r}^{n};q_{1}^{n},\ldots,q_{s}^{n})\Big{)},$

viewed as an element of
${\mathbb{Q}}(p_{1},\ldots,p_{r})[\\![q_{1},\ldots,q_{s}]\\!]$.

Notice that ${\mathcal{Z}}_{(0,0,0,1)}(q)$ is symmetric in the variables
$t_{1},t_{2},t_{3}$, and invariant under $q\to q^{-1}$. We define the
partition function

$\displaystyle{\mathcal{Z}}^{(i)}(q)={\mathcal{Z}}_{(0,0,0,1)}(q)|_{t_{1}=t_{i_{1}},t_{2}=t_{i_{2}},t_{3}=t_{i_{3}}},\quad
i=1,\dots,4,$

which is just the partition function (3.13) in the three variables
$\\{i_{1},i_{2},i_{3}\\}$ different than $i$. In other words,
${\mathcal{Z}}^{(i)}(q)$ is the tetrahedron partition function for the tuple
$e_{i}$, which is $4$-tuple with $1$ in the $i$-th entry and $0$ in all the
others zero.

######  Theorem 3.14.

Let $\overline{r}=(r_{1},\dots,r_{4})$ and $r=\sum_{i=1}^{4}r_{i}$. We have
the factorisation

$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)=\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}{\mathcal{Z}}^{(i)}\left((-1)^{r+1}q\kappa_{i}^{\frac{-r_{i}-1}{2}+l}\prod_{j=1}^{4}\kappa_{j}^{\frac{r_{j}\cdot{\mathrm{sgn}}(i-j)}{2}}\right).$

###### Proof.

Set $w_{il}=L^{N_{il}}$. By Theorem 3.11 the partition function
${\mathcal{Z}}_{\overline{r}}(q)$ can be computed in the limit $L\to\infty$.
By Proposition 3.12, Lemma 3.13 and (3.9) we have

$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)$
$\displaystyle=\lim_{L\to\infty}\sum_{\overline{\pi}}q^{|\overline{\pi}|}\prod_{1\leq
i\leq j\leq 4}\prod_{\begin{subarray}{c}1\leq l\leq r_{i}\\\ 1\leq k\leq
r_{j}\end{subarray}}[-\mathsf{v}^{(ij,lk)}_{\overline{\pi}}]$
$\displaystyle=\lim_{L\to\infty}\sum_{\overline{\pi}}\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}q^{|\pi_{il}|}[-\mathsf{v}_{\overline{\pi}}^{(ii,ll)}]\prod_{i=1}^{4}\prod_{1\leq
k<l\leq
r_{i}}[-\mathsf{v}^{(ii,lk)}_{\overline{\pi}}][-\mathsf{v}^{(ii,kl)}_{\overline{\pi}}]\prod_{1\leq
i<j\leq 4}\prod_{\begin{subarray}{c}1\leq l\leq r_{i}\\\ 1\leq k\leq
r_{j}\end{subarray}}[-\mathsf{v}^{(ij,lk)}_{\overline{\pi}}]$
$\displaystyle=\sum_{\overline{\pi}}\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}q^{|\pi_{il}|}[-\mathsf{v}_{\overline{\pi}}^{(ii,ll)}]\prod_{(i,l)<(j,k)}(-\kappa_{i}^{\frac{1}{2}})^{|\pi_{jk}|}(-\kappa_{j}^{-\frac{1}{2}})^{|\pi_{il}|}$
$\displaystyle=\sum_{\overline{\pi}}\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}q^{|\pi_{il}|}[-\mathsf{v}_{\overline{\pi}}^{(ii,ll)}]\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}\left(\prod_{j=1}^{4}\prod_{k=1}^{r_{j}}(-\kappa_{j}^{\frac{1}{2}})^{{\mathrm{sgn}}((i,l)-(j,k))}\right)^{|\pi_{il}|},$

where the sign function is defined as

${\mathrm{sgn}}((i,l)-(j,k))=\begin{cases}1&(i,l)<(j,k)\\\ -1&(i,l)>(j,k)\\\
0&(i,l)=(j,k).\end{cases}$

For a fixed $(i,l)$, counting the number of $(j,k)$ such that $(i,l)<(j,k)$
with respect to the lexicographic order yields the identity

$\prod_{j=1}^{4}\prod_{k=1}^{r_{j}}(-\kappa_{j}^{\frac{1}{2}})^{{\mathrm{sgn}}((i,l)-(j,k))}=(-1)^{r+1}\kappa_{i}^{\frac{-r_{i}-1}{2}+l}\prod_{j=1}^{4}\kappa_{j}^{\frac{r_{j}\cdot{\mathrm{sgn}}(i-j)}{2}}.$

Therefore we have

$\displaystyle{\mathcal{Z}}_{\overline{r}}(q)$
$\displaystyle=\sum_{\overline{\pi}}\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}[-\mathsf{v}_{\overline{\pi}}^{(ii,ll)}]\left((-1)^{r+1}q\kappa_{i}^{\frac{-r_{i}-1}{2}+l}\prod_{j=1}^{4}\kappa_{j}^{\frac{r_{j}\cdot{\mathrm{sgn}}(i-j)}{2}}\right)^{|\pi_{il}|}$
$\displaystyle=\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}\sum_{\pi_{il}}[-\mathsf{v}_{\overline{\pi}}^{(ii,ll)}]\left((-1)^{r+1}q\kappa_{i}^{\frac{-r_{i}-1}{2}+l}\prod_{j=1}^{4}\kappa_{j}^{\frac{r_{j}\cdot{\mathrm{sgn}}(i-j)}{2}}\right)^{|\pi_{il}|}$
$\displaystyle=\prod_{i=1}^{4}\prod_{l=1}^{r_{i}}{\mathcal{Z}}^{(i)}\left((-1)^{r+1}q\kappa_{i}^{\frac{-r_{i}-1}{2}+l}\prod_{j=1}^{4}\kappa_{j}^{\frac{r_{j}\cdot{\mathrm{sgn}}(i-j)}{2}}\right).$

∎

Combining Theorem 3.14 with the explicit expression (3.13), we prove a closed
formula for the partition function of tetrahedron instantons. We set

$\kappa_{\overline{r}}:=\prod_{i=1}^{4}\kappa_{i}^{r_{i}}.$

######  Theorem 3.15.

Let $\overline{r}=(r_{1},\dots,r_{4})$ and $r=\sum_{i=1}^{4}r_{i}$. We have

${\mathcal{Z}}_{\overline{r}}((-1)^{r}q)=\operatorname{Exp}\left(-\frac{[t_{1}t_{2}][t_{1}t_{3}][t_{2}t_{3}]}{[t_{1}][t_{2}][t_{3}][t_{4}]}\frac{[\kappa_{\overline{r}}]}{[\kappa_{\overline{r}}^{\frac{1}{2}}q][\kappa_{\overline{r}}^{\frac{1}{2}}q^{-1}]}\right).$

###### Proof.

By Theorem 3.14 and taking plethystic logarithm, i.e. the inverse of the
plethystic exponential $\operatorname{Exp}$, we just need show that

(3.15)
$\displaystyle\sum_{i=1}^{4}\sum_{l=1}^{r_{i}}\frac{[t_{i_{1}}t_{i_{2}}][t_{i_{1}}t_{i_{3}}][t_{i_{2}}t_{i_{3}}]}{[t_{i_{1}}][t_{i_{2}}][t_{i_{3}}]}\frac{1}{[\kappa_{i}^{\frac{1}{2}}q_{il}][\kappa_{i}^{\frac{1}{2}}q^{-1}_{il}]}=-\frac{[t_{1}t_{2}][t_{1}t_{3}][t_{2}t_{3}]}{[t_{1}][t_{2}][t_{3}][t_{4}]}\frac{[\kappa_{\overline{r}}]}{[\kappa_{\overline{r}}^{\frac{1}{2}}q][\kappa_{\overline{r}}^{\frac{1}{2}}q^{-1}]},$

where $\\{i_{1},i_{2},i_{3}\\}$ are the three indices in $\\{1,2,3,4\\}$
different from $i$ and we define

$q_{il}=q\kappa_{i}^{\frac{-r_{i}-1}{2}+l}\prod_{j=1}^{4}\kappa_{j}^{\frac{r_{j}\cdot{\mathrm{sgn}}(i-j)}{2}}.$

By the relations $t_{1}t_{2}t_{3}t_{4}=1$ and $\kappa_{i}=t_{i}^{-1}$, the
claim (3.15) is equivalent to

(3.16)
$\displaystyle\sum_{i=1}^{4}\sum_{l=1}^{r_{i}}\frac{[\kappa_{i}]}{[\kappa_{i}^{\frac{1}{2}}q_{il}][\kappa_{i}^{\frac{1}{2}}q^{-1}_{il}]}=\frac{[\kappa_{\overline{r}}]}{[\kappa_{\overline{r}}^{\frac{1}{2}}q][\kappa_{\overline{r}}^{\frac{1}{2}}q^{-1}]}.$

We prove a slightly more general statement: let $x_{1},\dots,x_{r}$ be
variables, and let
$q_{i}=q\prod_{j=1}^{r}x_{j}^{\frac{{\mathrm{sgn}}(i-j)}{2}}$. Then we claim
that

(3.17)
$\displaystyle\sum_{i=1}^{r}\frac{[x_{i}]}{[x_{i}^{\frac{1}{2}}q_{i}][x_{i}^{\frac{1}{2}}q^{-1}_{i}]}=\frac{[\prod_{i=1}^{r}x_{i}]}{[\prod_{i=1}^{r}x_{i}^{\frac{1}{2}}q][\prod_{i=1}^{r}x_{i}^{\frac{1}{2}}q^{-1}]}.$

Clearly, (3.16) follows by (3.17) by setting
$x_{1},\dots,x_{r_{1}}=\kappa_{1},\dots,x_{r_{1}+r_{2}+r_{3}+1}=\dots=x_{r}=\kappa_{4}$.

We prove (3.17) by induction on $r$. If $r=1$, the claim is trivial, and if
$r=2$, it is easy to check it holds. So let $r\geq 3$ and assume it holds for
$r-2$. For $a=1,\dots,r-2$ set

$\displaystyle y_{a}$ $\displaystyle=x_{a+1},$ $\displaystyle p$
$\displaystyle=qx_{1}^{\frac{1}{2}}x_{r}^{-\frac{1}{2}},$ $\displaystyle
p_{a}$
$\displaystyle=p\prod_{b=1}^{r-2}y_{b}^{\frac{{\mathrm{sgn}}(a-b)}{2}}.$

Then we have

$\displaystyle\sum_{i=1}^{r}\frac{[x_{i}]}{[x_{i}^{\frac{1}{2}}q_{i}][x_{i}^{\frac{1}{2}}q^{-1}_{i}]}$
$\displaystyle=\sum_{a=1}^{r-2}\frac{[y_{a}]}{[y_{a}^{\frac{1}{2}}p_{a}][y_{a}^{\frac{1}{2}}p^{-1}_{a}]}+\frac{[x_{1}]}{[x_{1}^{\frac{1}{2}}q_{1}][x_{1}^{\frac{1}{2}}q^{-1}_{1}]}+\frac{[x_{r}]}{[x_{r}^{\frac{1}{2}}q_{r}][x_{r}^{\frac{1}{2}}q^{-1}_{r}]}$
$\displaystyle=\frac{[\prod_{a=1}^{r-2}y_{a}]}{[\prod_{a=1}^{r-2}y_{a}^{\frac{1}{2}}p][\prod_{a=1}^{r-2}y_{a}^{\frac{1}{2}}p^{-1}]}+\frac{[x_{1}]}{[x_{1}^{\frac{1}{2}}q_{1}][x_{1}^{\frac{1}{2}}q^{-1}_{1}]}+\frac{[x_{r}]}{[x_{r}^{\frac{1}{2}}q_{r}][x_{r}^{\frac{1}{2}}q^{-1}_{r}]},$

where in the last step we used the inductive step. Notice, in particular, that
the last expression just depends on
$x_{1},x_{r},\tilde{x}=\prod_{i=2}^{r-1}x_{i},q$, and therefore (3.17) reduces
to

$\displaystyle\frac{[\prod_{a=1}^{r-2}y_{a}]}{[\prod_{a=1}^{r-2}y_{a}^{\frac{1}{2}}p][\prod_{a=1}^{r-2}y_{a}^{\frac{1}{2}}p^{-1}]}+\frac{[x_{1}]}{[x_{1}^{\frac{1}{2}}q_{1}][x_{1}^{\frac{1}{2}}q^{-1}_{1}]}+\frac{[x_{r}]}{[x_{r}^{\frac{1}{2}}q_{r}][x_{r}^{\frac{1}{2}}q^{-1}_{r}]}=\frac{[\prod_{i=1}^{r}x_{i}]}{[\prod_{i=1}^{r}x_{i}^{\frac{1}{2}}q][\prod_{i=1}^{r}x_{i}^{\frac{1}{2}}q^{-1}]},$

which is a formal identity depending on the four variables
$x_{i},x_{r},\tilde{x},q$, and can therefore be directly checked by comparing
left-hand-side and right-hand-side999To be precise, we checked that this
identity holds with the help of Mathematica.. ∎

######  Corollary 3.16.

Let $\overline{r}=(r,r,r,r)$. Then for $n>0$ there is a vanishing

${\mathcal{Z}}_{\overline{r},n}=0.$

###### Proof.

From $\overline{r}=(r,r,r,r)$ and $t_{1}t_{2}t_{3}t_{4}=1$ it follows that
$\kappa_{\overline{r}}=1$. Therefore

$\frac{[\kappa_{\overline{r}}]}{[\kappa_{\overline{r}}^{\frac{1}{2}}q][\kappa_{\overline{r}}^{\frac{1}{2}}q^{-1}]}=0,$

and by Theorem 3.15 it follows that the partition function of tetrahedron
instantons is ${\mathcal{Z}}_{\overline{r}}(q)=1$. ∎

######  Remark 3.17.

The vanishing of the instanton partition function for tuples of the form
$(r,r,r,r)$ yields non-trivial relations among the coefficients of
${\mathcal{Z}}^{(i)}(q)$. For instance, if we choose $\overline{r}=(1,1,1,1)$
and denote by ${\mathcal{Z}}^{(i)}_{n}$ the $n$-th coefficient of
${\mathcal{Z}}^{(i)}(q)$, we obtain the relation

$t_{1}^{-\frac{1}{2}}{\mathcal{Z}}^{(1)}_{1}+t_{1}^{-1}t_{2}^{-\frac{1}{2}}{\mathcal{Z}}^{(2)}_{1}+t_{1}^{-1}t_{2}^{-1}t_{3}^{-\frac{1}{2}}{\mathcal{Z}}^{(3)}_{1}+t_{1}^{-1}t_{2}^{-1}t_{3}^{-1}t_{4}^{-\frac{1}{2}}{\mathcal{Z}}^{(4)}_{1}=0.$

This provides a new method to generate identities among classical rank 1
Donaldson-Thomas invariants of ${\mathbb{C}}^{3}$.

###  3.6. Cohomological limit

Let $\overline{r}=(r_{1},\dots,r_{4})$ and $n\geq 0$. By Corollary 2.5, the
moduli space of tetrahedron instantons is endowed with a
$\mathbf{T}$-equivariant _virtual fundamental class_
$[{\mathcal{M}}_{\overline{r},n}]^{\operatorname{\mathrm{vir}}}\in
A^{\mathbf{T}}_{*}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right)$.
We define the $\mathbf{T}$-equivariant invariants

$\displaystyle{\mathcal{Z}}^{\operatorname{coh}}_{\overline{r},n}=\int_{[{\mathcal{M}}_{\overline{r},n}]^{\operatorname{\mathrm{vir}}}}1\in\frac{{\mathbb{Q}}(s_{1},s_{2},s_{3},s_{4},v)}{(s_{1}+s_{2}+s_{3}+s_{4})},$

where $s_{1},\dots,s_{4},v_{11},\dots,v_{4r_{4}}$ are the generators of
$\mathbf{T}$-equivariant Chow cohomology $A^{\mathbf{T}}_{*}({\mathsf{pt}})$.
In other words, they can be realized as
$s_{i}=c_{1}(t_{i}),v_{il}=c_{i}(w_{il})$, i.e. the first Chern classes of the
irreducible representations generating $K^{0}_{\mathbf{T}}({\mathsf{pt}})$.

Since ${\mathcal{M}}_{\overline{r},n}$ is not proper, one cannot directly
define invariants via proper push-forward in equivariant cohomology. Instead,
since the fixed locus ${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$ is proper,
we define invariants as the composition

$\displaystyle\int_{[{\mathcal{M}}_{\overline{r},n}]^{\operatorname{\mathrm{vir}}}}:A^{\mathbf{T}}_{*}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right)\to
A^{\mathbf{T}}_{*}({\mathcal{M}}_{\overline{r},n})_{\mathrm{loc}}\xrightarrow{\sim}A^{\mathbf{T}}_{*}({\mathcal{M}}^{\mathbf{T}}_{\overline{r},n})_{\mathrm{loc}}\to
A_{*}^{\mathbf{T}}({\mathsf{pt}})_{\mathrm{loc}}.$

Here, the first map is a suitable localisation of
$A^{\mathbf{T}}_{*}\left({\mathcal{M}}_{\overline{r},n},{\mathbb{Z}}\left[\tfrac{1}{2}\right]\right)$,
the second map is Edidin-Graham’s abstract localisation [31] and the third map
is the proper pushforward on ${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$
(extended to the localisation).

We define the _cohomological tetrahedron instanton partition function_ as the
generating series

$\displaystyle{\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)=\sum_{n\geq
0}{\mathcal{Z}}^{\operatorname{coh}}_{\overline{r},n}\cdot
q^{n}\in\frac{{\mathbb{Q}}(s_{1},s_{2},s_{3},s_{4},v)}{(s_{1}+s_{2}+s_{3}+s_{4})}[\\![q]\\!].$

By Oh-Thomas localisation theorem in equivariant cohomology [59] we can
compute the invariants via $\mathbf{T}$-equivariant residues on the fixed
locus

$\displaystyle\int_{[{\mathcal{M}}_{\overline{r},n}]^{\operatorname{\mathrm{vir}}}}1=\int_{[{\mathcal{M}}^{\mathbf{T}}_{\overline{r},n}]^{\operatorname{\mathrm{vir}}}}\frac{1}{\sqrt{e}(N^{\operatorname{\mathrm{vir}}})},$

where $e(\cdot)$ is the ($\mathbf{T}$-equivariant) Euler class and
$\sqrt{e}(\cdot)$ is the ($\mathbf{T}$-equivariant) square-root Euler class of
Edidin-Graham [30] (cf. also [59, Sec. 3.1]). Therefore the tetrahedron
partition function is given by

$\displaystyle{\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)=\sum_{\overline{\pi}}(-1)^{o_{\overline{\pi}}}\sqrt{e}(-T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}})\cdot
q^{|\overline{\pi}|},$

where the sum runs over tuples
$\overline{\pi}=(\overline{\pi}_{1},\dots,\overline{\pi}_{4})$, where each
$\overline{\pi}_{i}$ is an $r_{i}$-tuple of plane partitions,
$T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}$ is the virtual tangent
bundle at the fixed point corresponding to $\overline{\pi}$ via the
correspondence in Proposition 3.3 and $(-1)^{o_{\overline{\pi}}}$ is the sign
of the induced virtual fundamental class.

Similarly to the $K$-theoretic setting, we can compute the square root Euler
class of $T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}$ using the square
root $\mathsf{v}_{\overline{\pi}}$ and obtain

$\displaystyle{\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)=\sum_{\overline{\pi}}(-1)^{\sigma_{\overline{\pi}}}e(-\mathsf{v}_{\overline{\pi}})\cdot
q^{|\overline{\pi}|}.$

We explicitly describe how the Euler class $e(\cdot)$ acts on a virtual
$\mathbf{T}$-representation. For an irreducible representation
$t_{1}^{\mu_{1}}\cdots t_{4}^{\mu_{4}}w_{11}^{\mu_{11}}\cdots
w_{4r_{4}}^{\mu_{4r_{4}}}$, we have

$e(t_{1}^{\mu_{1}}\cdots t_{4}^{\mu_{4}}w_{11}^{\mu_{11}}\cdots
w_{4r_{4}}^{\mu_{4r_{4}}})=\mu_{1}s_{1}+\dots+\mu_{4}s_{4}+\mu_{11}v_{11}+\dots+\mu_{4r_{4}}v_{4r_{4}},$

and is linear $e(t^{\mu}\pm t^{\nu})=e(t^{\mu})e(t^{\nu})^{\pm 1}$, as long as
$\nu$ is not the trivial weight. To ease the notation we set
$\mu=(\mu_{1},\dots,\mu_{4},\mu_{11},\dots,\mu_{4r_{4}})$ and
$s=(s_{1},\dots,s_{4},v_{11},\dots,s_{4r_{4}})$, and write
$e(t^{\mu})=\mu\cdot s$ where the product is the usual scalar product.

As explained in [33, Sec. 7.1], one should think of $e(\cdot)$ as the
_linearization_ of $[\cdot]$, since

$\displaystyle[t^{\mu}]|_{t_{i}=e^{bs_{i}},w_{il}=e^{bv_{il}}}=e^{\frac{b\mu\cdot
s}{2}}(1-e^{-b\mu\cdot s})=be(t^{\mu})+O(b^{2}).$

In particular, if $V$ is a $\mathbf{T}$-representation of rank 0, the limit
$b\to 0$ is well-defined and yields

(3.18) $\displaystyle\lim_{b\to
0}[V]|_{t_{i}=e^{bs_{i}},w_{il}=e^{bv_{il}}}=e(V).$

Since the vertex term $\mathsf{v}_{\overline{\pi}}$ has rank 0, this identity
tells us that the cohomological tetrahedron instanton partition function
${\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)$ is a
limit101010Alternatively, we could show that the cohomological invariants are
the limit of the $K$-theoretic invariants using virtual Riemann-Roch [59, Thm.
6.1] (cf. [19, Thm. 6.4]).  of the tetrahedron instanton partition function
${\mathcal{Z}}_{\overline{r}}(q)$.

######  Corollary 3.18.

Let $\overline{r}=(r_{1},\dots,r_{4})$ and $r=\sum_{i=1}^{4}r_{i}$. We have

${\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)=\mathrm{M}((-1)^{r}q)^{-\frac{(s_{1}+s_{2})(s_{1}+s_{3})(s_{2}+s_{3})(r_{1}s_{1}+r_{2}s_{2}+r_{3}s_{3}+r_{4}s_{4})}{s_{1}s_{2}s_{3}s_{4}}},$

where $\mathrm{M}(q)$ is the MacMahon series (3.1).

###### Proof.

By the limit (3.18), we can compute the cohomological instanton partition
function as

${\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)=\lim_{b\to
0}{\mathcal{Z}}_{\overline{r}}(q)|_{t_{i}=e^{bs_{i}}}.$

Notice in particular that the independence of
${\mathcal{Z}}_{\overline{r}}(q)$ from the framing parameters $w_{il}$
directly implies the independence of
${\mathcal{Z}}^{\operatorname{coh}}_{\overline{r}}(q)$ from the variables
$v_{il}$. Set $\overline{r}\cdot
s=r_{1}s_{1}+r_{2}s_{2}+r_{3}s_{3}+r_{4}s_{4}$. We have

$\displaystyle\lim_{b\to
0}{\mathcal{Z}}_{\overline{r}}(q)|_{t_{i}=e^{bs_{i}}}$
$\displaystyle=\lim_{b\to 0}\exp\left(\sum_{n\geq
1}\frac{1}{n}\frac{[e^{bns_{1}}e^{bns_{2}}][e^{bns_{1}}e^{bns_{3}}][e^{bns_{2}}e^{bns_{3}}]}{[e^{bns_{1}}][e^{bns_{2}}][e^{bns_{3}}][e^{bns_{3}}]}\frac{[e^{bn\overline{r}\cdot
s}]}{[e^{\frac{bn\overline{r}\cdot s}{2}}q][e^{\frac{bn\overline{r}\cdot
s}{2}}q^{-1}]}\right)$
$\displaystyle=\exp\left(\frac{(s_{1}+s_{2})(s_{1}+s_{3})(s_{2}+s_{3})(r_{1}s_{1}+r_{2}s_{2}+r_{3}s_{3}+r_{4}s_{4})}{s_{1}s_{2}s_{3}s_{4}}\sum_{n\geq
1}\frac{1}{n}\frac{-q^{n}}{(1-q^{n})^{2}}\right).$

Recall the plethystic expression for the MacMahon series

$\mathrm{M}(q)=\operatorname{Exp}\left(\frac{q}{(1-q)^{2}}\right),$

by which we conclude that

$\lim_{b\to
0}{\mathcal{Z}}_{\overline{r}}(q)|_{t_{i}=e^{bs_{i}}}=\mathrm{M}((-1)^{r}q)^{-\frac{(s_{1}+s_{2})(s_{1}+s_{3})(s_{2}+s_{3})(r_{1}s_{1}+r_{2}s_{2}+r_{3}s_{3}+r_{4}s_{4})}{s_{1}s_{2}s_{3}s_{4}}}.$

∎

###  3.7. Witten genus

For a scheme endowed with a perfect obstruction theory in the sense of
Behrend-Fantechi [6], the authors [33] introduced the _virtual chiral elliptic
genus_ , inspired by the work of Benini-Bonelli-Poggi-Tanzini in String Theory
[7]. In the context of [33], the _chirality_ refers to the novel feature of
extracting square roots of the virtual canonical bundle, as opposed to the
more classical _virtual elliptic genus_ of Fantechi-Göttsche [32]. From the
complex-analytic point of view, the virtual chiral elliptic genus is the
virtual analogue of the _Witten genus_ , which refines the $\widehat{A}$-genus
(cf. [69]), while the (virtual) elliptic genus refines the (virtual)
$\chi_{y}$-genus.

More recently, for a moduli space of sheaves on a Calabi-Yau fourfold, Bojko
[11, Def. 5.8] defined the _DT4 Witten genus_ , which can be seen as the
generalization of the virtual chiral elliptic genus to virtual structures à la
Oh-Thomas. We adapt his notion to our setting to define the _virtual Witten
genus_ of the moduli space of tetrahedron instantons.

For a vector bundle $F$ on a scheme $X$, define the _total symmetric power_
and _total exterior power_

$\displaystyle\operatorname{Sym}^{\bullet}_{p}F$
$\displaystyle=\bigoplus_{n\geq 0}p^{n}\operatorname{Sym}^{n}F\in
K^{0}(X)[\\![p]\\!],$ $\displaystyle\Lambda^{\bullet}_{p}F$
$\displaystyle=\bigoplus_{n\geq 0}p^{n}\Lambda^{n}F\in K^{0}(X)[p],$

and extend them by linearity to $K^{0}(X)$. They satisfy

$\displaystyle\operatorname{Sym}^{\bullet}_{p}F=\frac{1}{\Lambda^{\bullet}_{-p}F}.$

######  Definition 3.19.

Let $\overline{r}=(r_{1},\dots,r_{4})$. The _virtual Witten genus_ of
${\mathcal{M}}_{\overline{r},n}$ is defined as

${\mathcal{Z}}^{\mathrm{ell}}_{\overline{r},n}=\chi\left({\mathcal{M}}_{\overline{r},n},\widehat{\operatorname{\mathcal{O}}}_{{\mathcal{M}}_{\overline{r},n}}^{\operatorname{\mathrm{vir}}}\otimes\bigotimes_{n\geq
1}\operatorname{Sym}^{\bullet}_{p^{n}}\left(T^{\operatorname{\mathrm{vir}}}_{{\mathcal{M}}_{\overline{r},n}}\right)\right)\in\frac{{\mathbb{Q}}(t_{1}^{\frac{1}{2}},t_{2}^{\frac{1}{2}},t_{3}^{\frac{1}{2}},t_{4}^{\frac{1}{2}},w^{\frac{1}{2}})}{(t_{1}t_{2}t_{3}t_{4}-1)}[\\![p]\\!].$

We refer to $p$ as the _elliptic parameter_ and define the _elliptic
tetrahedron instanton partition function_ as the generating series

$\displaystyle{\mathcal{Z}}^{\mathrm{ell}}_{\overline{r}}(q)=\sum_{n\geq
0}{\mathcal{Z}}^{\mathrm{ell}}_{\overline{r},n}\cdot
q^{n}\in\frac{{\mathbb{Q}}(t_{1}^{\frac{1}{2}},t_{2}^{\frac{1}{2}},t_{3}^{\frac{1}{2}},t_{4}^{\frac{1}{2}},w^{\frac{1}{2}})}{(t_{1}t_{2}t_{3}t_{4}-1)}[\\![p,q]\\!].$

As for the case of $K$-theoretic invariants, the moduli space
${\mathcal{M}}_{\overline{r},n}$ is not proper, but its torus fixed locus
${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$ is, so we define the invariants
via equivariant residues (3.3).

The _Jacobi theta function_ $\theta(p;y)$ and the _Dedekind eta function_
$\eta(p)$ are defined as

$\displaystyle\theta(p;y)$
$\displaystyle=-ip^{1/8}(y^{1/2}-y^{-1/2})\prod_{n=1}^{\infty}(1-p^{n})(1-yp^{n})(1-y^{-1}p^{n}),$
$\displaystyle\eta(p)$
$\displaystyle=p^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-p^{n}),$

and denote $\theta(\tau|z):=\theta(e^{2\pi i\tau};e^{2\pi iz})$. If we set
$p=e^{2\pi i\tau}$, with
$\tau\in\mathbb{H}=\set{\tau\in{\mathbb{C}}}{\mathrm{Im}(\tau)>0}$, then
$\theta$ enjoys the modular behaviour

$\theta(\tau|z+a+b\tau)=(-1)^{a+b}e^{-2\pi ibz}e^{-i\pi
b^{2}\tau}\theta(\tau|z),\quad a,b\in{\mathbb{Z}}.$

See [33, Sec. 8.1] and [32, Sec. 6] for a related discussion on the modularity
of these functions. For an irreducible $\mathbf{T}$-representation $t^{\mu}$,
define (cf. [33, Sec. 8.1])

$\theta[t^{\mu}]=(i\cdot\eta(p))^{-1}\theta(p;t^{\mu})\in
K^{0}_{\mathbf{T}}({\mathsf{pt}})\llbracket p\rrbracket[p^{\pm\frac{1}{12}}]$

and extend it by linearity to any $V\in K_{\mathbf{T}}^{0}({\mathsf{pt}})$.
Notice that $\theta[t^{\mu}]$ satisfies $\theta[t^{-\mu}]=-\theta[t^{\mu}]$
and if $V$ is a virtual representation such that $\operatorname{rk}V=0$, the
elliptic measure refines both $[\cdot]$ and $e(\cdot)$

$\theta[V]\xrightarrow{p\to 0}[V]\xrightarrow{b\to 0}e(V).$

Let $\mathsf{v}_{\overline{\pi}}$ our preferred square root of the virtual
tangent space $T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}$ at the
$\mathbf{T}$-fixed point corresponding to $\overline{\pi}$ and recall that the
sign appearing in the localisation satisfies
$(-1)^{\sigma_{\overline{\pi}}}=1$. By Oh-Thomas localisation [59], we have

$\displaystyle{\mathcal{Z}}^{\mathrm{ell}}_{\overline{r}}(q)$
$\displaystyle=\sum_{\overline{\pi}}\frac{\prod_{n\geq
1}\operatorname{Sym}^{\bullet}_{p^{n}}\left(T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}\right)}{\sqrt{\mathfrak{e}}(T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}})}\cdot
q^{|\overline{\pi}|}$ $\displaystyle=\sum_{\overline{\pi}}\frac{\prod_{n\geq
1}\operatorname{Sym}^{\bullet}_{p^{n}}\left(\mathsf{v}_{\overline{\pi}}+\overline{\mathsf{v}_{\overline{\pi}}}\right)}{\mathfrak{e}(\mathsf{v}_{\overline{\pi}})\cdot({\det}\mathsf{v}_{\overline{\pi}})^{\frac{1}{2}}}\cdot
q^{|\overline{\pi}|}$
$\displaystyle=\sum_{\overline{\pi}}\theta[-\mathsf{v}_{\overline{\pi}}]\cdot
q^{|\overline{\pi}|},$

where the last line follows from the explicit computation in [33, pag. 34].

######  Remark 3.20.

The elliptic tetrahedron instanton partition function
${\mathcal{Z}}^{\mathrm{ell}}_{\overline{r}}(q)$ does in general depend on the
framing parameters $w_{il}$, differently from its $K$-theoretical and
cohomological analogues, where a rigidity principle applies. See [33, Ex. 8.7]
for the case of $r=(0,0,0,3)$ and $n=1$.

######  Remark 3.21.

We discussed in [33, Sec. 8.3] — in the context of higher rank Donaldson-
Thomas theory — how specialising the equivariant parameters to roots of unity
yields, once more, a rigidity phenomenon for which the dependence on the
elliptic parameter disappears and the partition function is computed
explicitly [33, Thm. 8.11]. We expect a similar phenomenon to hold for the
elliptic tetrahedron instanton partition function, where one suitably
specialises the equivariant parameters $t_{1},\dots,t_{4}$ to $r_{i}$-roots of
unity. See [33, Rem. 8.12] for its physical role from the point of view of
supersymmetry.

###  3.8. Pomoni-Yan-Zhang

As we pointed out in § 1.1, our work aims to give a geometrisation of the
moduli spaces originally considered in [62], to get a better understanding of
their structure and the relation they bear to enumerative theories already
extensively studied. In loc. cit., the authors gave a description of a moduli
space of BPS solutions to the D1-brane low-energy effective theory of a system
of D1-D7-branes on the type IIB supersymmetric background
$X={\mathbb{R}}^{1,1}\times{\mathbb{R}}^{8}$. In particular, a stack of $n$
coincident D1-branes probes a configuration $r=r_{1}+r_{2}+r_{3}+r_{4}$ of
intersecting D7-branes, where each stack of $r_{i}$ coincident D7-branes wraps
an irreducible component of the boundary toric divisor
$\Delta\subset{\mathbb{C}}^{4}\cong{\mathbb{R}}^{8}$. The resulting low-energy
effective theory on the D1-branes is a so-called $2d$ $\mathcal{N}=(0,2)$
quiver Gauged Linear Sigma Model, whose partition function can be explicitly
computed by means of supersymmetric localisation as in [8, 9]. The advantage
of this procedure is that it does not require an in-depth understanding of the
moduli space at hand, as the partition function of the theory is only
dependent on the field content (which is entirely encoded in the datum of the
quiver and possibly the relations), and it is expressed in terms of so-called
Jeffrey-Kirwan residues [39], via a well-defined procedure. In this section,
we will briefly describe the constructions in loc. cit., and explain how the
connection to geometry is drawn.

In [62, Sec. 2], the moduli space of tetrahedron instantons is introduced via
the study of vacua of the low-energy theory of the stack of D1-branes in the
background of the D7-branes. In particular, the effective theory has a family
of classical vacua, which are characterised by the vanishing of the scalar
potential and are parametrised by the symplectic quotient by the unitary group
$\operatorname{U}_{n}$

$\mathfrak{M}_{\overline{r},n}(\zeta)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mu_{{\mathbb{R}}}^{-1}(\zeta)\cap\mu_{{\mathbb{C}}}^{-1}(0)\cap\sigma^{-1}(0)/\operatorname{U}_{n},\quad\zeta\in\mathfrak{u}_{n}^{\ast},$

where the moment map
$\mu_{\mathbb{R}}:\mathsf{R}_{\overline{r},n}\to\mathfrak{u}_{n}$ is defined
by

$\mu_{{\mathbb{R}}}(B_{1},\dots,B_{4},I_{1},\dots,I_{4})=\frac{\sqrt{-1}}{2}\left(\sum_{a=1}^{4}[B_{a},B_{a}^{\dagger}]+\sum_{i=1}^{4}I_{i}I_{i}^{\dagger}\right),$

while the maps $\mu_{{\mathbb{C}}}=(\mu_{{\mathbb{C}}})_{a,b}$, for
$a,b=1,\dots,4$ and $\sigma=(\sigma)_{i}$, for $i=1,\dots,4$, act on
representations in $\mathsf{R}_{\overline{r},n}$ as

$\displaystyle\mu_{{\mathbb{C}}}(B_{1},\dots,B_{4},I_{1},\dots,I_{4})$
$\displaystyle=\left([B_{a},B_{b}]\right)_{a,b=1,\dots,4},$
$\displaystyle\sigma(B_{1},\dots,B_{4},I_{1},\dots,I_{4})$
$\displaystyle=\left(B_{i}I_{i}\right)_{i=1,\dots,4}.$

By Theorem 2.3 and [52, Cor. 3.22], there is a bijection between the
symplectic quotient $\mathfrak{M}_{\overline{r},n}(\sqrt{-1}d\chi)$ and (the
closed points of) the Quot scheme
$\operatorname{Quot}_{\Delta}(\mathcal{E},n)$, the latter being realised as a
GIT quotient with respect to the $\operatorname{GL}_{n}$-character $\chi$.
Moreover, in [62, § 4.3.2, § 4.3.3], a stratification of the one-instanton
moduli space ${\mathcal{M}}_{\overline{r},1}$ is obtained from the gauge-
theoretic description derived from physics. This stratification agrees with
our Example 2.1, where it is also interpreted geometrically in terms of blow-
ups.

The $n$-instantons contribution to the partition function of the two-
dimensional low-energy theory on the worldvolume of the D1-branes is then
captured by a refinement of the Witten index. This is known in physics as
elliptic genus, and is defined as in [62, Eq. (163)]. Notably, this refined
index can be computed exactly for two-dimensional Gauged Linear Sigma Models
via supersymmetric localisation (cf. [8, Eq. (3.7)] for rank-1 gauge groups
and [9, Eq. (2.27)] for general rank), and its evaluation involves the
computation of certain equivariant residues, the Jeffrey-Kirwan residues [39],
which are only dependent on the (bosonic) field content of the theory through
their 1-loop contribution to the path integral formulation of the elliptic
genus. When the moduli space engineered by the BPS vacua of the $2d$
$\mathcal{N}=(0,2)$ Gauged Linear Sigma Model is equipped with a perfect
obstruction theory in the sense of Li-Tian [43] and Behrend-Fantechi [6], the
physical elliptic genus has a geometric realisation as the (virtual) chiral
elliptic genus, first computed in the context of DT theory in [7], and later
defined in [33]. In the case of tetrahedron instantons, however, a perfect
obstruction theory is not available, but virtual structures à la Oh-Thomas are
induced by the zero-locus construction of Thm. 2.3. Then the elliptic genus
computed in [62, Eqs. (211)-(214)] is recovered exactly from our computation
of the virtual Witten genus in § 3.7. In this sense, one might view
supersymmetric localisation as nothing but an infinite-dimensional analogue of
equivariant localisation. However, despite the undoubted advantage that an in-
depth knowledge of the structure of the moduli space of BPS vacua is not
required for the calculation of the elliptic invariants in Def. 3.19, carried
out as in [62, § 5], it nevertheless overshadows some interesting features
which might be worth pointing out. Indeed, the fact that the virtual structure
on ${\mathcal{M}}_{\overline{r},n}$ is induced, via the seminal work of Oh-
Thomas [59], by the construction in Thm. 2.3, means that the invariants
computed in K-theory by localisation will depend on a choice of orientation
via a sign, cf. § 3.3. This sign, however, is not manifestly present in [62,
§§ 5-6]. Moreover, the choice of a square root of the virtual tangent bundle
as in § 3.3 is not unique, but the couple square root/sign is somewhat
canonical (cf. Thm. A.1 and [46]). Therefore, while the character
$\chi_{\overline{\pi}}$ in [62, Eq. (147)] is a good choice for a square root
of the virtual tangent space at the points in the fixed locus
${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$, it is not the only possible one
(a different choice being given by our vertex term
$\mathsf{v}_{\overline{\pi}}$ in Eq. 3.8), and in the computation of the
invariants [62, Eq. (153)], it should in principle be accompanied by the
additional choice of an appropriate sign. Similarly, the observation in loc.
cit. that the partition function thereby computed is invariant only up to a
sign under permutations of the $\kappa_{i}$, is a shadow of the same
phenomenon.

##  Appendix A The sign rule

We provide in this appendix a proof of Theorem 3.7, which computes the correct
sign for each localised contribution of the tetrahedron instantons partition
function. Our proof relies on two results of Kool-Rennemo, whose proofs will
appear in their forthcoming paper [42].

Let a scheme $Z$

${{\mathcal{E}}}$${Z:=Z(s)}$${{\mathcal{A}}}$$\leftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\scriptstyle{\iota}$$\leftarrow$$\rightarrow$$\scriptstyle{s}$

be the zero locus of an isotropic section
$s\in\Gamma({\mathcal{A}},{\mathcal{E}})$, where ${\mathcal{E}}$ is a
$SO(2r,{\mathbb{C}})$-bundle over a smooth quasi-projective variety
${\mathcal{A}}$, so that $Z$ is endowed with the obstruction theory (2.1).
Assume that the orientation of ${\mathcal{E}}$ is induced by the choice of a
maximal isotropic subbundle $\Lambda\subset{\mathcal{E}}$, which induces a
projection

$p_{\Lambda}:{\mathcal{E}}^{*}\to\Lambda.$

Let ${\mathcal{A}}$ be acted by an algebraic torus ${\mathbb{T}}$, such that
$({\mathcal{E}},q)$ is ${\mathbb{T}}$-equivariant and the
${\mathbb{T}}$-action restricts to $Z$, which implies that the obstruction
theory ${\mathbb{E}}\to{\mathbb{L}}_{Z}$ is ${\mathbb{T}}$-equivariant and $Z$
is endowed with a ${\mathbb{T}}$-equivariant virtual structure sheaf
$\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{Z}$.
Finally, we assume that the fixed locus $Z^{\mathbb{T}}$ is reduced, zero-
dimensional, and such that the fibre of the obstruction theory
${\mathbb{E}}|_{p}$ is ${\mathbb{T}}$-movable, for every fixed point $p\in
Z^{\mathbb{T}}$.

######  Theorem A.1 (Kool-Rennemo [42]).

We have that

$\displaystyle\chi\left(Z,\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{Z}\right)=\sum_{p\in
Z^{\mathbb{T}}}(-1)^{\psi_{p}}[-T_{{\mathcal{A}}}|_{p}+\Lambda|_{p}],$

where the sign is defined as

$\psi_{p}=\dim\left(\operatorname{coker}(T_{{\mathcal{A}}}|_{p}\xrightarrow{(p_{\Lambda}\circ
ds)|_{p}}\Lambda|_{p})^{\mathsf{fix}}\right)\mod 2.$

Notice that the sign $(-1)^{\psi_{p}}$ in Theorem A.1 depends on the chosen
maximal isotropic subbundle $\Lambda$, which affects the orientation on $Z$
and therefore the one on the fixed locus $Z^{\mathbb{T}}$.

The assumptions of Theorem A.1 are clearly satisfied by the
$\mathbf{T}$-action on the moduli space of tetrahedron instantons
${\mathcal{M}}_{\overline{r},n}$, whose orientation is induced by the maximal
isotropic subbundle $\Lambda\subset{\mathcal{L}}$ as constructed in Section
2.4. Let $\overline{\pi}$ be a tuple of plane partitions corresponding to a
fixed point of ${\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}$. Then

${\mathcal{L}}|_{\overline{\pi}}=\Lambda^{2}{\mathbb{C}}^{4}\otimes\operatorname{Hom}(Q_{\overline{\pi}},Q_{\overline{\pi}})\oplus\bigoplus_{i=1}^{4}\operatorname{Hom}(K_{i}\cdot
t_{i},Q_{\overline{\pi}})\oplus\bigoplus_{i=1}^{4}\operatorname{Hom}(K_{i}\cdot
t_{i},Q_{\overline{\pi}})^{*}$

is a vector space, endowed with a non-degenerate quadratic pairing, with
maximal isotropic subbundle

$\displaystyle\Lambda|_{\overline{\pi}}$
$\displaystyle=\tilde{\Lambda}|_{\overline{\pi}}\oplus\bigoplus_{i=1}^{4}\operatorname{Hom}(K_{i}\cdot
t_{i},Q_{\overline{\pi}})^{*}$
$\displaystyle\tilde{\Lambda}|_{\overline{\pi}}$ $\displaystyle=\left(\langle
e_{4}\rangle\wedge{\mathbb{C}}^{3}\right)\otimes\operatorname{Hom}(Q_{\overline{\pi}},Q_{\overline{\pi}}),$

and isotropic section

$s|_{\overline{\pi}}=(\tilde{s}|_{\overline{\pi}},0):\Lambda|_{\overline{\pi}}\to{\mathcal{L}}|_{\overline{\pi}}.$

By Theorem A.1, we have that

$\displaystyle\chi\left({\mathcal{M}}_{\overline{r},n},\widehat{\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}}_{{\mathcal{M}}_{\overline{r},n}}\right)=\sum_{\overline{\pi}}(-1)^{\psi_{\overline{\pi}}}[-T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{\overline{\pi}}+\Lambda|_{\overline{\pi}}],$

where

$\displaystyle\psi_{\overline{\pi}}=\dim\left(\operatorname{coker}(T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{p}\xrightarrow{(p_{\Lambda}\circ
ds)|_{\overline{\pi}}}\Lambda|_{\overline{\pi}})^{\mathsf{fix}}\right)\mod 2.$

By Proposition 3.8 the vertex term $\mathsf{v}_{\overline{\pi}}$ is
$\mathbf{T}$-movable, which implies that the virtual tangent space
$T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}$ is $\mathbf{T}$-movable as
well. In particular, any other square root of
$T^{\operatorname{\mathrm{vir}}}_{\overline{\pi}}$ has to be
$\mathbf{T}$-movable. Therefore, $T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{\overline{\pi}}-\Lambda|_{\overline{\pi}}$ must be a
$\mathbf{T}$-movable virtual representation.

######  Lemma A.2.

We have

$\displaystyle\psi_{\overline{\pi}}=\dim\left(\operatorname{coker}(T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{p}\xrightarrow{(p_{\tilde{\Lambda}}\circ
d\tilde{s})|_{\overline{\pi}}}\tilde{\Lambda}|_{\overline{\pi}})^{\mathsf{fix}}\right)\mod
2.$

###### Proof.

The differential of the isotropic section satisfies $ds=(d\tilde{s},0)$,
therefore

$\displaystyle\operatorname{coker}(T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{p}\xrightarrow{(p_{\Lambda}\circ
ds)|_{\overline{\pi}}}\Lambda|_{\overline{\pi}})=\operatorname{coker}(T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{p}\xrightarrow{(p_{\tilde{\Lambda}}\circ
d\tilde{s})|_{\overline{\pi}}}\tilde{\Lambda}|_{\overline{\pi}})\oplus\bigoplus_{i=1}^{4}\operatorname{Hom}(K_{i}\cdot
t_{i},Q_{\overline{\pi}})^{*}.$

The proof follows by the vanishing of the fixed part

$\displaystyle\left(\operatorname{Hom}(K_{i}\cdot
t_{i},Q_{\overline{\pi}})\right)^{\mathsf{fix}}$
$\displaystyle=\left(\overline{K_{i}}\kappa_{i}Q_{\overline{\pi}}\right)^{\mathsf{fix}}$
$\displaystyle=\left(\sum_{j=1}^{4}\sum_{\begin{subarray}{c}1\leq l\leq
r_{i}\\\ 1\leq k\leq
r_{j}\end{subarray}}w_{il}^{-1}w_{jk}\kappa_{i}Z_{\pi_{jk}}\right)^{\mathsf{fix}}$
$\displaystyle=\left(\sum_{l=1}^{r_{i}}\kappa_{i}Z_{\pi_{il}}\right)^{\mathsf{fix}}$
$\displaystyle=0,$

since $Z_{\pi_{il}}$ is a polynomial in the variables
$t_{i_{1}},t_{i_{2}},t_{i_{3}}$, where $\\{i_{1},i_{2},i_{3}\\}$ are the three
indices in $\\{1,2,3,4\\}$ different from $i$. ∎

The vanishing in the proof of Lemma A.2 readily implies that
$T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{\overline{\pi}}-\tilde{\Lambda}|_{\overline{\pi}}$ is a
$\mathbf{T}$-movable virtual representation.

For every tuple of plane partitions $\overline{\pi}$, define the Laurent
polynomial

$\displaystyle\tilde{\mathsf{v}}_{\overline{\pi}}=\overline{K}Q_{\overline{\pi}}-\overline{P_{123}}Q_{\overline{\pi}}\overline{Q_{\overline{\pi}}}.$

The vertex term $\tilde{\mathsf{v}}_{\overline{\pi}}$ is the higher rank
version of the vertex term of Cao-Kool-Monavari [18, Eqn. (18)], used in loc.
cit. to compute the Donaldson-Thomas invariants of ${\mathbb{C}}^{4}$, and is
easily seen to be $\mathbf{T}$-movable by [46, Lemma 2.4]. Kool-Rennemo [42]
exploited this vertex term to prove the closed formula for Donaldson-Thomas
invariants of ${\mathbb{C}}^{4}$ conjectured by Nekrasov-Piazzalunga [57], by
explicitly computing the corresponding sign for every localised contribution.
Notice that the immersion

$\displaystyle{\mathcal{M}}_{\overline{r},n}\hookrightarrow\operatorname{Quot}_{{\mathbb{C}}^{4}}(\operatorname{\mathcal{O}}_{{\mathbb{C}}^{4}}^{r},n)$

induces a closed immersion of their $\mathbf{T}$-fixed locus

$\displaystyle{\mathcal{M}}_{\overline{r},n}^{\mathbf{T}}\hookrightarrow\operatorname{Quot}_{{\mathbb{C}}^{4}}(\operatorname{\mathcal{O}}_{{\mathbb{C}}^{4}}^{r},n)^{\mathbf{T}}.$

The right-hand-side is reduced, zero-dimensional and in bijection with tuples
of _solid partitions_ of size $n$, which can be seen as _box arrangements_ in
${\mathbb{Z}}^{4}_{\geq 0}$ (cf. [46, Sec. 2.1] for the rank 1 case),
generalizing plane partitions to one dimension higher. Under this
identification, a tuple of plane partitions
$\overline{\pi}=(\overline{\pi}_{i})_{i=1,\dots,4}$ can be seen as a tuple of
solid partitions, such that each plane partition $\pi_{il}$ – for
$i=1,\dots,4$ and $l=1,\dots,r_{i}$ – is seen as a solid partition with the
$i$-coordinate equal to zero.

A key step of the computation of Kool-Rennemo [42] yields the following
combinatorial result.

######  Proposition A.3 (Kool-Rennemo [42]).

Let $\overline{\pi}=(\overline{\pi}_{1},\dots\overline{\pi}_{4})$, where
$\overline{\pi}_{i}=(\pi_{i1},\dots,\pi_{ir_{i}})$ are tuples of plane
partitions. Then

$(-1)^{\dim\left(\operatorname{coker}(T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{p}\xrightarrow{(p_{\tilde{\Lambda}}\circ
d\tilde{s})|_{\overline{\pi}}}\tilde{\Lambda}|_{\overline{\pi}})^{\mathsf{fix}}\right)}[-T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{\overline{\pi}}+\tilde{\Lambda}|_{\overline{\pi}}]=(-1)^{\rho_{\overline{\pi}}}[-\tilde{\mathsf{v}}_{\overline{\pi}}],$

where the sign is defined as

$\displaystyle\rho_{\overline{\pi}}=\sum_{i=1}^{4}\sum_{l=1}^{r_{i}}\left|\\{(a,a,a,d)\in\pi_{il}:a<d\\}\right|\mod
2.$

The sign $\rho_{\overline{\pi}}$ was firstly conjecture to be of this form by
Nekrasov-Piazzalunga [57] (cf. [18, 46] for the rank 1 theory). We remark once
more that we are seeing here the plane partition $\pi_{il}$ as a solid
partition, via the embedding ${\mathbb{Z}}^{3}_{\geq
0}\subset{\mathbb{Z}}^{4}_{\geq 0}$ given by imposing the $i$-th coordinate to
be zero.

######  Corollary A.4.

We have

$\displaystyle(-1)^{\psi_{\overline{\pi}}}[-T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{\overline{\pi}}+\Lambda|_{\overline{\pi}}]=[-\mathsf{v}_{\overline{\pi}}].$

In particular, the sign $(-1)^{\sigma_{\overline{\pi}}}=1$ is constant.

###### Proof.

Proposition A.3 yields

$\displaystyle(-1)^{\psi_{\overline{\pi}}}[-T_{{\mathcal{M}}_{\overline{r},n}^{\rm
nc}}|_{\overline{\pi}}+\Lambda|_{\overline{\pi}}]=(-1)^{\rho_{\overline{\pi}}}[-\tilde{\mathsf{v}}_{\overline{\pi}}+\sum_{i=1}^{4}K_{i}t_{i}\overline{Q_{\overline{\pi}}}].$

To conclude the argument, we need to show that

(A.1)
$\displaystyle(-1)^{\rho_{\overline{\pi}}}[-\tilde{\mathsf{v}}_{\overline{\pi}}+\sum_{i=1}^{4}K_{i}t_{i}\overline{Q_{\overline{\pi}}}]=[-\mathsf{v}_{\overline{\pi}}].$

By the decomposition of the vertex term (3.10) and Proposition A.3, Equation
(A.1) holds if, equivalently, the families of equations (A.2), (A.3) hold. The
first one asks that for every $(i,l)$

(A.2)
$\displaystyle(-1)^{\rho_{{\pi_{il}}}}[Z_{il}-\overline{P_{123}}Z_{il}\overline{Z_{il}}]$
$\displaystyle=[Z_{il}-\overline{P_{i_{1}i_{2}i_{3}}}Z_{il}\overline{Z_{il}}],$

where the sign is defined as

$\displaystyle\rho_{\pi_{il}}=\left|\\{(a,a,a,d)\in\pi_{il}:a<d\\}\right|\mod
2.$

The second asks that for every $(i,l)<(j,k)$

(A.3)
$[(w_{il}^{-1}w_{jk}\left(Z_{\pi_{jk}}-\kappa^{-1}_{j}\overline{Z_{\pi_{il}}}-\overline{P_{123}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}}\right)+w_{il}w_{jk}^{-1}\left(Z_{\pi_{il}}-\kappa^{-1}_{i}\overline{Z_{\pi_{jk}}}-\overline{P_{123}}Z_{\pi_{il}}\overline{Z_{\pi_{jk}}}\right)]\\\
=[w_{il}^{-1}w_{jk}\left(Z_{\pi_{jk}}-\kappa^{-1}_{j}\overline{Z_{\pi_{il}}}-\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}}\right)+w_{il}w_{jk}^{-1}\left(Z_{\pi_{il}}-\kappa^{-1}_{i}\overline{Z_{\pi_{jk}}}-\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{il}}\overline{Z_{\pi_{jk}}}\right)].$

Equation (A.3) is equivalent to

(A.4)
$[w_{il}^{-1}w_{jk}\overline{P_{123}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}}+w_{il}w_{jk}^{-1}\overline{P_{123}}Z_{\pi_{il}}\overline{Z_{\pi_{jk}}}]\\\
=[w_{il}^{-1}w_{jk}\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}}+w_{il}w_{jk}^{-1}\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{il}}\overline{Z_{\pi_{jk}}}].$

To prove (A.4), we just need to show that

$w_{il}^{-1}w_{jk}\overline{P_{123}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}}+w_{il}w_{jk}^{-1}\overline{P_{123}}Z_{\pi_{il}}\overline{Z_{\pi_{jk}}}\\\
-\left(w_{il}^{-1}w_{jk}\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}}+w_{il}w_{jk}^{-1}\overline{P_{j_{1}j_{2}j_{3}}}Z_{\pi_{il}}\overline{Z_{\pi_{jk}}}\right)=U-\overline{U},$

for a certain Laurent polynomial $U$ such that
$(-1)^{\operatorname{rk}U^{\mathsf{mov}}}=1$. In fact, in this case we would
have that

$[w_{il}^{-1}w_{jk}\overline{P_{123}}Z_{\pi_{jk}}\overline{Z_{\pi_{il}}}+w_{il}w_{jk}^{-1}\overline{P_{123}}Z_{\pi_{il}}\overline{Z_{\pi_{jk}}}\\\