$\displaystyle\frac{\theta\left(\varepsilon_{12}\right)\theta\left(\varepsilon_{14}\right)\theta\left(\varepsilon_{24}\right)}{\theta\left(\varepsilon_{1}\right)\theta\left(\varepsilon_{2}\right)\theta\left(\varepsilon_{4}\right)}\frac{\theta\left(\delta-\varepsilon_{4}\right)}{\theta\left(\delta+\varepsilon_{4}\right)}\times$
(224)
$\displaystyle\times\frac{\theta\left(2\varepsilon_{4}+\varepsilon_{1}\right)\theta\left(2\varepsilon_{4}+\varepsilon_{2}\right)\theta\left(\varepsilon_{12}-\varepsilon_{4}\right)}{\theta\left(2\varepsilon_{2}\right)\theta\left(\varepsilon_{2}-\varepsilon_{1}\right)\theta\left(\varepsilon_{2}-\varepsilon_{4}\right)},$

  3. 3.

If $\left|\pi^{(123)}\right|=\left|\pi^{(124)}\right|=1$, the genuine pole can
only be at $\left(\mathtt{a}_{123},\mathtt{a}_{124}\right)$, and the
corresponding residue is

$\displaystyle z_{\left(\mathtt{a}_{123},\mathtt{a}_{124}\right)}$
$\displaystyle=$
$\displaystyle\frac{\theta^{2}\left(\varepsilon_{12}\right)\theta^{2}\left(\varepsilon_{13}\right)\theta^{2}\left(\varepsilon_{23}\right)}{\theta^{2}\left(\varepsilon_{1}\right)\theta^{2}\left(\varepsilon_{2}\right)\theta\left(\varepsilon_{3}\right)\theta\left(\varepsilon_{4}\right)}\times$
(225)
$\displaystyle\times\frac{\theta\left(\pm\delta+\varepsilon_{12}\right)\theta\left(\pm\delta+\varepsilon_{13}\right)\theta\left(\pm\delta+\varepsilon_{23}\right)}{\theta\left(\pm\delta+\varepsilon_{1}\right)\theta\left(\pm\delta+\varepsilon_{2}\right)\theta\left(-\delta+\varepsilon_{3}\right)\theta\left(\delta+\varepsilon_{4}\right)}.$

In the above calculations, we have taken a particular ordering of $\phi_{1}$
and $\phi_{2}$ to cancel the factor of $2$ in the denominator of (215).
Summing up these contributions, we get

$Z_{2}=\sum_{a\in\left(123\right)}z_{\left(\mathtt{a}_{123},\mathtt{a}_{123}-\varepsilon_{a}\right)}+\sum_{a\in\left(124\right)}z_{\left(\mathtt{a}_{124},\mathtt{a}_{124}-\varepsilon_{a}\right)}+z_{\left(\mathtt{a}_{123},\mathtt{a}_{124}\right)}.$
(226)

One can check that (226) matches the general expression given in the previous
subsection.

###  6.4 Expectation value of real codimension-two defects

Up to now, we treat all D$7_{A}$-branes on equal footing, but the string
theory construction of tetrahedron instantons and the geometric interpretation
of the moduli space suggest a different point of view of the instanton
partition function. We choose the physical spacetime to be
$\mathbb{R}^{1,1}\times\mathbb{C}_{123}^{3}$, so that the bound states of
D$1$\- and D$7_{123}$-branes give rise to instantons on
$\mathbb{C}_{123}^{3}$. The remaining D$7_{A}$-branes for
$A\in\underline{4}^{\vee}\setminus\left\\{(123)\right\\}$ will produce real
codimension-two defects from the viewpoint of the physical spacetime. This
provides the physical realization of the projection of the moduli space
$\mathfrak{M}_{\vec{n},k}$ of tetrahedron instantons to the moduli spaces
$\mathfrak{M}_{\left(n_{123},0,0,0\right),k^{\prime}}$ of instantons on
$\mathbb{C}_{123}^{3}$ discussed in section 4. Thus we identify the instanton
partition function as the expectation value of real codimension-two defects
$\mathcal{O}_{A}$ in the instanton partition function of the Donaldson-Thomas
theory,

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\mathtt{q}^{k}}{k!}\int\prod_{i=1}^{k}d\phi_{i}\left[\left(Z_{k}^{1-1}Z_{k}^{1-7_{123}}\right)\left(\prod_{A\in\underline{4}^{\vee}\setminus\left\\{\left(123\right)\right\\}}Z_{k}^{1-7_{A}}\right)\right]$
(227) $\displaystyle=$
$\displaystyle\left\langle\prod_{A\in\underline{4}^{\vee}\setminus\left\\{\left(123\right)\right\\}}\mathcal{O}_{A}\right\rangle_{\mathrm{DT}},$

where the bracket denotes the unnormalized vacuum expectation value in the
Donaldson-Thomas theory on $\mathbb{C}_{123}^{3}$ whose instanton partition
function is given by

$Z_{\mathrm{DT}}=\sum_{k=0}^{\infty}\frac{\mathtt{q}^{k}}{k!}\int\prod_{i=1}^{k}d\phi_{i}Z_{k}^{1-1}Z_{k}^{1-7_{123}}.$
(228)

###  6.5 Dimensional reductions

We now briefly discuss dimensional reductions of the system.

Performing a T-duality along $x^{9}$ of the configuration in Table 1, we get
D$0$-branes probing a configuration of intersecting D$6$-branes in type IIA
superstring theory. The generating function of the generalized Witten indices
of the supersymmetric gauged quantum mechanical models on D$0$-branes is the
K-theoretical version of the instanton partition function of tetrahedron
instantons. Since there are no anomalies of large gauge transformations, we no
longer impose the constraint (172). Taking the limit $q\to 0$ of $Z$, we get
the dimensionally reduced instanton partition function $Z^{\downarrow}$,

$\displaystyle Z^{\downarrow}$ $\displaystyle=$
$\displaystyle\sum_{\vec{\pi}}\mathtt{q}^{\left|\vec{\pi}\right|}Z_{\vec{\pi}}^{\downarrow}$
(229) $\displaystyle=$
$\displaystyle\sum_{\vec{\pi}}\mathtt{q}^{\left|\vec{\pi}\right|}\left(\prod_{\mathcal{A}\in\underline{n}}Z_{\vec{\pi}}^{\downarrow(\mathcal{A})}\right)\left(\prod_{\mathcal{A}\neq\mathcal{B}\in\underline{n}}Z_{\vec{\pi}}^{\downarrow(\mathcal{A},\mathcal{B})}\right),$

where $Z_{\vec{\pi}}^{\downarrow(\mathcal{A})}$ and
$Z_{\vec{\pi}}^{\downarrow(\mathcal{A},\mathcal{B})}$ are obtained from
$Z_{\vec{\pi}}^{(\mathcal{A})}$ and
$Z_{\vec{\pi}}^{(\mathcal{A},\mathcal{B})}$ by substituting

$\theta\left(z\right)\to 2\sinh\left(\frac{\beta z}{2}\right),$ (230)

and $\beta$ is the circumference of the circle of the supersymmetric quantum
mechanics. The instanton partition function $Z^{\downarrow}$ matches
$\mathcal{Z}$ in (155) with the K-theoretical version (157) of the operator
$\mathbb{E}$.

We can further perform a T-duality along $x^{0}$ direction to get D-instantons
probing a configuration of intersecting D$5$-branes in type IIB superstring
theory. The instanton partition function $Z^{\Downarrow}$ is obtained by

$Z^{\Downarrow}=\sum_{\vec{\pi}}\mathtt{q}^{\left|\vec{\pi}\right|}Z_{\vec{\pi}}^{\Downarrow}=\sum_{\vec{\pi}}\mathtt{q}^{\left|\vec{\pi}\right|}\left(\prod_{\mathcal{A}\in\underline{n}}Z_{\vec{\pi}}^{\Downarrow(\mathcal{A})}\right)\left(\prod_{\mathcal{A}\neq\mathcal{B}\in\underline{n}}Z_{\vec{\pi}}^{\Downarrow(\mathcal{A},\mathcal{B})}\right).$
(231)

Here $Z_{\vec{\pi}}^{\Downarrow(\mathcal{A})}$ and
$Z_{\vec{\pi}}^{\Downarrow(\mathcal{A},\mathcal{B})}$ are obtained from
$Z_{\vec{\pi}}^{(\mathcal{A})}$ and
$Z_{\vec{\pi}}^{(\mathcal{A},\mathcal{B})}$ by substituting
$\theta\left(z\right)\to z$,

$\displaystyle
Z_{\vec{\pi}}^{\Downarrow\left(\mathcal{A}=\left(A,\alpha\right)\right)}$
$\displaystyle=$
$\displaystyle\varepsilon_{\check{A}}\left(\frac{\prod_{a<b\in
A}\varepsilon_{ab}}{\prod_{a\in\underline{4}}\varepsilon_{a}}\right)^{\left|\pi^{(\mathcal{A})}\right|}\left(\prod_{s\neq
t\in\pi^{(\mathcal{A})}}\mathcal{R}\left\\{\mathcal{D}_{\mathcal{A},t}^{\mathcal{A},s}\right\\}\right)\times$
(232)
$\displaystyle\times\left(\prod_{s\in\pi^{(\mathcal{A})}\setminus(1,1,1)}\frac{\mathcal{C}_{\mathcal{A},s}-a_{\mathcal{A}}-\varepsilon_{\mathcal{A}}}{\mathcal{C}_{\mathcal{A},s}-a_{\mathcal{A}}}\right),$
$\displaystyle Z_{\vec{\pi}}^{\Downarrow(\mathcal{A},\mathcal{B})}$
$\displaystyle=$
$\displaystyle\left(\prod_{s\in\pi^{(\mathcal{A})}}\prod_{t\in\pi^{(\mathcal{B})}}\mathcal{R}\left\\{\mathcal{D}_{\mathcal{B},t}^{\mathcal{A},s}\right\\}\right)\left(\prod_{s\in\pi^{(\mathcal{A})}}\frac{\mathcal{C}_{\mathcal{A},s}-a_{\mathcal{B}}-\varepsilon_{\mathcal{B}}}{\mathcal{C}_{\mathcal{A},s}-a_{\mathcal{B}}}\right),$
(233)

where

$\mathcal{R}\left\\{x\right\\}=\frac{x\left(x+\varepsilon_{12}\right)\left(x+\varepsilon_{13}\right)\left(x+\varepsilon_{23}\right)}{\left(x+\varepsilon_{1}\right)\left(x+\varepsilon_{2}\right)\left(x+\varepsilon_{3}\right)\left(x+\varepsilon_{4}\right)}.$
(234)

The partition function $Z^{\Downarrow}$ matches $\mathcal{Z}$ in (155)
exactly.

##  7 Free field representation

Following Kazakov:1998ji ; Nekrasov:2002qd ; Losev:2003py ; Nekrasov:2003rj ;
Benini:2018hjy , we give a free field representation of the instanton
partition function. This is in the general spirit of the BPS/CFT
correspondence Nekrasov:2015wsu .

Recall that the torus propagator for a free massless $r$-component scalar
field $\varphi=\left(\varphi_{1},\cdots,\varphi_{r}\right)$ is given by
DiFrancesco:1997nk

$\displaystyle G_{i,j}\left(z,\bar{z}\right)$ $\displaystyle=$
$\displaystyle\left\langle\varphi_{i}\left(z,\bar{z}\right)\varphi_{j}\left(0,0\right)\right\rangle_{\mathbb{T}^{2}}$
(235) $\displaystyle=$
$\displaystyle-\log\left|\frac{\theta_{1}\left(\left.z\right|\tau\right)}{2\pi\eta(\tau)^{3}}\exp\left(-\frac{\pi\left(\mathrm{Im}z\right)^{2}}{\mathrm{Im}\tau}\right)\right|^{2}\delta_{i,j},\quad
i,j=1,\cdots,r,$

where the torus $\mathbb{T}^{2}$ is described by a complex $z$-plane with the
identification $z\cong z+1\cong z+\tau$, and $G_{i,j}\left(z,\bar{z}\right)$
is the normalized doubly periodic solution of the Laplacian on
$\mathbb{T}^{2}$,

$-\Delta
G_{i,j}\left(z,\bar{z}\right)=\left(2\pi\delta^{2}(z)-\frac{4\pi}{\mathrm{Im}\tau}\right)\delta_{i,j}.$
(236)

The basic vertex operators of the theory are the exponential fields
parameterized by a $r$-component vector parameter
$\alpha=\left(\alpha_{1},\cdots,\alpha_{r}\right)$,

$\mathbb{V}_{\alpha}\left(z,\bar{z}\right)=:e^{\mathrm{i}\sum_{i=1}^{r}\alpha_{i}\varphi_{i}\left(z,\bar{z}\right)}:.$
(237)

We require that the complex structure $\tau$ of $\mathbb{T}^{2}$ is the same
as that in the definition of the elliptic genus (161). Then we will use the
abbreviation (168) in the following.

We shall take $r=7$ and introduce a slightly deformed vertex operator

$\mathscr{V}_{\alpha,\rho}\left(z,\bar{z}\right)=:e^{\mathrm{i}\sum_{i=1}^{7}\alpha_{i}\varphi_{i}\left(z+\rho_{i},\bar{z}+\rho_{i}\right)}::e^{-\mathrm{i}\sum_{i=1}^{7}\alpha_{i}\varphi_{i}\left(z-\rho_{i},\bar{z}-\rho_{i}\right)}:,$
(238)

where

$\alpha=\left(\mathrm{i},\mathrm{i},\mathrm{i},\mathrm{i},1,1,1\right),\quad\rho=\frac{1}{2}\left(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4},\varepsilon_{12},\varepsilon_{13},\varepsilon_{23}\right).$
(239)

It is an important fact that when $\sum_{a\in\underline{4}}\varepsilon_{a}=0$
we have

$\sum_{i=1}^{7}\alpha_{i}^{2}\left(\mathrm{Im}\left(\rho_{i}\right)\right)^{2}=0.$
(240)

Performing the Wick contraction, we can get

$\mathscr{V}_{\alpha,\rho}\left(z,\bar{z}\right)=\left|\frac{2\pi\eta(\tau)^{3}\prod_{1\leq
a<b\leq
3}\theta\left(\varepsilon_{ab}\right)}{\prod_{a\in\underline{4}}\theta\left(\varepsilon_{a}\right)}\right|^{2}:\mathscr{V}_{\alpha,\rho}\left(z,\bar{z}\right):,$
(241)

and

$\displaystyle\left\langle:\mathscr{V}_{\alpha,\rho}\left(z,\bar{z}\right)::\mathscr{V}_{\alpha,\rho}\left(w,\bar{w}\right):\right\rangle_{\mathbb{T}^{2}}$
(242) $\displaystyle=$
$\displaystyle\left|\frac{\theta^{2}\left(z-w\right)\prod_{1\leq a<b\leq
3}\theta\left(z-w\pm\varepsilon_{ab}\right)}{\prod_{a\in\underline{4}}\theta\left(z-w\pm\varepsilon_{a}\right)}\right|^{2},$

where
$\theta\left(z\pm\tilde{\varepsilon}\right)=\theta\left(z+\tilde{\varepsilon}\right)\theta\left(z-\tilde{\varepsilon}\right)$.
Since (242) takes the form of an absolute square, we can define the
holomorphic part as

$\displaystyle\left\langle:\mathscr{V}_{\alpha,\rho}\left(z,\bar{z}\right)::\mathscr{V}_{\alpha,\rho}\left(w,\bar{w}\right):\right\rangle_{\mathbb{T}^{2}}^{\mathrm{hol}}$
(243) $\displaystyle=$
$\displaystyle\frac{\theta^{2}\left(z-w\right)\prod_{1\leq a<b\leq
3}\theta\left(z-w\pm\varepsilon_{ab}\right)}{\prod_{a\in\underline{4}}\theta\left(z-w\pm\varepsilon_{a}\right)}.$

We further introduce a linear source operator,

$\varUpsilon=\frac{1}{2\pi\mathrm{i}}\oint_{\Gamma}dz\sum_{A\in\underline{4}^{\vee}}\varpi_{A}(z)\partial_{z}\varphi_{\check{A}}(z),$
(244)

where the contour $\Gamma$ is chosen to be a loop around $z=0$ encircling all
$\pm\rho_{i}$ for $i=1,\cdots,7$, and $\varpi_{A}(z)$ is a locally analytic
function inside $\Gamma$,

$\varpi_{A}(z)=\sum_{\alpha=1}^{n_{A}}\log\theta\left(z-\mathtt{a}_{A,\alpha}-\frac{1}{2}\varepsilon_{A}\right).$
(245)

Then

$\left\langle
e^{\varUpsilon}:\mathscr{V}_{\alpha,\rho}(z):\right\rangle_{\mathbb{T}^{2}}=\prod_{\mathcal{A}=\left(A,\alpha\right)\in\underline{n}}\frac{\theta\left(z-\mathtt{a}_{\mathcal{A}}-\varepsilon_{A}\right)}{\theta\left(z-\mathtt{a}_{\mathcal{A}}\right)},$
(246)

which is already holomorphic.

Therefore, we have the expansion

$\displaystyle\left\langle
e^{\varUpsilon}e^{\mathtt{q}\oint_{\mathcal{C}}\mathscr{V}_{\alpha,\rho}(z)dz}\right\rangle_{\mathbb{T}^{2}}^{\mathrm{hol}}$
(247) $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{\mathtt{q}^{k}}{k!}\left[\frac{2\pi\eta(\tau)^{3}\prod_{1\leq
a<b\leq
3}\theta\left(\varepsilon_{ab}\right)}{\prod_{a\in\underline{4}}\theta\left(\varepsilon_{a}\right)}\right]^{k}\times$
$\displaystyle\times\oint_{\mathcal{C}}dz_{1}\cdots\oint_{\mathcal{C}}dz_{k}\prod_{i=1}^{k}\prod_{\mathcal{A}=\left(A,\alpha\right)\in\underline{n}}\frac{\theta\left(z-\mathtt{a}_{\mathcal{A}}-\varepsilon_{A}\right)}{\theta\left(z-\mathtt{a}_{\mathcal{A}}\right)}\times$
$\displaystyle\times\prod_{\begin{subarray}{c}i,j=1\\\ i\neq
j\end{subarray}}^{k}\frac{\theta\left(z_{ij}\right)\prod_{1\leq a<b\leq
3}\theta\left(z_{ij}+\varepsilon_{ab}\right)}{\prod_{a\in\underline{4}}\theta\left(z_{ij}+\varepsilon_{a}\right)},$

which coincides with the instanton partition function (164) if the contour
$\mathcal{C}$ is chosen to give the Jeffrey-Kirwan residues. We see that the
contributions from the D$1$-D$1$ and D$1$-D$7$ strings are reproduced by the
Wick contractions within the exponentiated integrated vertex, and the Wick
contractions between the exponentiated integrated vertex and the linear
source, respectively.

##  8 Conclusions and future directions

In this paper, we introduced tetrahedron instantons and explained how to
construct them from string theory and from noncommutative field theory. We
analyzed the moduli space of tetrahedron instantons and discussed its
geometric interpretations. We computed the instanton partition function in two
different approaches: the infrared approach which computes the partition
function via equivariant localization on the moduli space of tetrahedron
instantons, and the ultraviolet approach which computes the partition function
as the elliptic genus of the worldvolume theory on the D$1$-branes probing a
configuration of intersecting D$7$-branes. Both approaches lead to the same
result. Our instanton partition function can also be viewed as the expectation
value of the most general real codimension-two defects in the instanton
partition function of the Donaldson-Thomas theory. Finally, we find a free
field representation of the instanton partition function, indicating the
existence of a novel kind of symmetry acting on the cohomology of the moduli
spaces of tetrahedron instantons.

There are still many interesting aspects of tetrahedron instantons that remain
to be better understood. Some of the future directions in which this work
could be continued are listed in the following.

  1. 1.

According to Witten:2000mf , a supersymmetric bound state can be formed by the
D$1$-D$9$ system if we turn on a constant $B$-field satisfying

$\sum_{a\in\underline{4}}v_{a}\geq 1.$ (248)

Therefore, we can generalize the tetrahedron instantons by adding a stack of
D$9$-branes without further breaking the supersymmetry if the $B$-field
satisfies the condition

$v_{1}=v_{2}=v_{3}=v_{4}\geq\frac{1}{4}.$ (249)

This generalization can also be viewed as instantons in the magnificent four
model with all possible real codimension-two defects. Furthermore, it is
fascinating to also incorporate in the system the spiked instantons, which can
be realized by D$1$-branes probing six stacks of (anti-)D$5$-branes in type
IIB superstring theory. As analyzed in Nekrasov:2016gud , supersymmetry is
completely broken when we have six stacks of D$5$-branes, and two supercharges
can be preserved when we have four stacks of D$5$-branes and two stacks of
anti-D$5$-branes in the presence of a $B$-field obeying

$v_{1}=-v_{2}=v_{3}=-v_{4}.$ (250)

In both cases, no supersymmetry will be preserved when we put together the
magnificent four model, the tetrahedron instantons and the spiked instantons.
On the other hand, the configuration of D$1$-branes with six stacks of
anti-D$5$-branes preserve two supercharges when the $B$-field obeys 333We
thank the reviewer for pointing to us this configuration.

$v_{1}=v_{2}=v_{3}=v_{4}.$ (251)

In this case, we can study the supersymmetric combination of the magnificent
four model, the tetrahedron instantons and the spiked instantons. This
combined system can be understood as instantons in the magnificent four model
with all possible real codimension-two and real codimension-four defects.

  2. 2.

It was proposed that the partition function of the magnificent four model is
the mother of all instanton partition functions Nekrasov:2017cih ;
Nekrasov:2018xsb . In particular, it was shown in Nekrasov:2018xsb  that the
partition function of the magnificent four model at a degenerate limit reduces
to the instanton partition function of the Donaldson-Thomas theory on
$\mathbb{C}^{3}$. The magnificent four model can be realized in string theory
using D$0$-branes probing a collection of D$8$\- and anti-D$8$-branes wrapping
a Calabi-Yau fourfold, with an appropriate $B$-field. Here the D$0$-D$8$
system gives an ADHM-type construction for instantons in the eight-dimensional
gauge theory, while the presence of the anti-D$8$-branes introduces certain
fundamental matter fields. The degenerate limit corresponds to a fine-tuned
position of the anti-D$8$-branes, and it was conjectured that anti-D$8$-branes
will annihilate the D$8$-branes, leaving a configuration of D$6$-branes after
the tachyon condensation. It is natural to imagine that by taking more general
degenerate limits, the instanton partition function of our model can always be
obtained from that of the magnificent four model. The matching of the
instanton partition function will then be a highly nontrivial test of the
tachyon condensation in nontrivial string backgrounds.

  3. 3.

It is well known that the partition function of the Donaldson-Thomas theory on
a toric Calabi-Yau threefold and the partition function of the magnificent
four model play important roles in the study of the compactification of
M-theory on Calabi-Yau fivefolds Haupt:2008nu ; Nekrasov:2014nea ;
Zotto:2021xah . An equivalence between the Donaldson-Thomas invariants and
Gromov-Witten invariants was conjectured maulik2006gromov1 ; maulik2006gromov2
; maulik2011gromov . Together with the Gopakumar-Vafa invariants
Gopakumar:1998ii ; Gopakumar:1998ki ; Gopakumar:1998jq , they arise from
different expansions of the same topological string amplitude. A fascinating
direction is to explore our model from this viewpoint. We consider the bound
state of $k$ D$0$-branes and $n_{A}$ D$6_{A}$-branes on
$\mathbb{S}^{1}\times\mathbb{C}_{A}^{3}$ for all $A\in\underline{4}^{\vee}$,
which can be lifted to M-theory as a bound state of $k$ graviton Kaluza-Klein
modes on $\mathbb{S}^{1}\times\mathscr{X}$, where $\mathscr{X}$ is a
noncompact Calabi-Yau fivefolds. When only one of the $n_{A}$ is nonzero,
$\mathscr{X}$ becomes $\mathbb{C}_{A}^{3}\times\mathrm{TN}_{n_{A}}$, where
$\mathrm{TN}_{n_{A}}$ is the $n_{A}$-centered Taub-NUT space. After
introducing the $\Omega$-deformation, the eleven-dimensional spacetime
$\mathbb{S}^{1}\times\mathscr{X}$ is replaced by a fiber bundle over
$\mathbb{S}^{1}$ with fiber $\mathscr{X}$, such that the fiber is rotated by
an element $g\in\mathrm{SU}(5)$ as we go around $\mathbb{S}^{1}$. The eleven-
dimensional supergravity partition function on this background is defined as
the twisted Witten index,

$\displaystyle
Z^{\mathrm{sugra}}\left[\mathbb{S}^{1}\rtimes_{g}\mathscr{X}\right]\left(\mathfrak{q}_{1},\cdots,\mathfrak{q}_{5}\right)$
$\displaystyle=$
$\displaystyle\mathrm{Tr}_{\mathcal{H}(\mathscr{X})}(-1)^{F}e^{-\beta\left\\{\mathfrak{Q},\mathfrak{Q}^{*}\right\\}}g$
(252) $\displaystyle=$
$\displaystyle\exp\left[\sum_{\ell=1}^{\infty}\frac{1}{\ell}\mathcal{F}^{\mathrm{sugra}}\left(\mathfrak{q}_{1}^{\ell},\cdots,\mathfrak{q}_{5}^{\ell}\right)\right],$

where $\mathcal{H}(\mathscr{X})$ is the Hilbert space of the supergravity
theory on $\mathscr{X}$, $F$ is the fermion number operator, $\beta$ is the
circumference of $\mathbb{S}^{1}$, $\mathfrak{Q}$ is a preserved supercharge
that commutes with $g$, and $\mathfrak{q}_{1},\cdots,\mathfrak{q}_{5}$
satisfying $\prod_{i=1}^{5}\mathfrak{q}_{i}=1$ are the fugacities associated
with the $\mathrm{SU}(5)$ action. We can decompose the single-particle index
$\mathcal{F}$ into two parts,

$\mathcal{F}^{\mathrm{sugra}}=\mathcal{F}^{\mathrm{sugra,pert}}+\mathcal{F}^{\mathrm{sugra,inst}},$
(253)

where $\mathcal{F}^{\mathrm{sugra,pert}}$ is the perturbative contribution
from D$6$-branes in the absence of D$0$-branes, and
$\mathcal{F}^{\mathrm{sugra,inst}}$ should coincide with the single-particle
index (147) of the instanton partition function. An extraordinary feature of
this correspondence is that the instanton counting parameter $\mathtt{q}$ in
the instanton partition function will be expressed in terms of the fugacities
$\mathfrak{q}_{1},\cdots,\mathfrak{q}_{5}$ in
$\mathcal{F}^{\mathrm{sugra,inst}}$.

  4. 4.

In this paper we only considered the simplest spacetime geometry
$\mathbb{R}^{1,1}\times\mathbb{C}^{4}$. It is definitely interesting to
generalize our analysis to $\mathbb{R}^{1,1}\times\mathscr{Y}$, where
$\mathscr{Y}$ is an arbitrary toric Calabi-Yau fourfold. For example, one can
consider the orbifold $\mathscr{Y}=\mathbb{C}^{4}/\Gamma$, where $\Gamma$ is a
finite subgroup of $\mathrm{SU}(4)$. The moduli space will be a generalization
of Nakajima quiver varieties Nakajima:1994nid ; Nakajima1998 ;
Nekrasov:2016qym ; Bonelli:2020gku  and the chain-saw quiver
feigin2011yangians ; finkelberg2014quantization ; Kanno:2011fw . The instanton
partition function on the orbifold can be obtained by projecting onto the
$\Gamma$-invariant part. Another nature choice is to blowup the origin of
$\mathbb{C}^{4}$ in the spirit of Nakajima:2003pg ; Nakajima:2003uh ;
Nakajima:2005fg , and it may be useful for the study of BPS/CFT correspondence
Nekrasov:2020qcq ; Jeong:2020uxz . These instanton partition functions should
lie between the Donaldson-Thomas invariants of toric Calabi-Yau threefolds
maulik2006gromov1 ; maulik2006gromov2  and fourfolds Cao:2017swr ; Cao:2018rbp
; Cao:2020otr . We can even generalize our computations by including extra
D-branes wrapping compact divisors.

  5. 5.

The instanton partition function of the Donaldson-Thomas theory was identified
with the classical statistical mechanics of melting crystal Okounkov:2003sp ,
and can be interpreted as a quantum gravitational path integral involving
fluctuations of geometry and topology Iqbal:2003ds . It would be wonderful if
one can provide a similar interpretation for the instanton partition function
of tetrahedron instantons, in particular from the expression (227).

  6. 6.

It would be interesting if we can have a better understanding of the free
field representation of the instanton partition function, generalizing the
discussion in Dijkgraaf:2007sw .

  7. 7.

We may consider the tetrahedron instantons with supergroups by adding negative
branes Vafa:2001qf ; Okuda:2006fb ; Dijkgraaf:2016lym  in our construction. We
can then calculate the instanton partition function as in Kimura:2019msw .

###### Acknowledgements.

We are grateful to Giulio Bonelli, Nikita Nekrasov, Mauricio Romo, Peng Shan,
and Dingxin Zhang for comments and discussions. We thank Jie Zhou in
particular for valuable comments and suggestions on the manuscript. WY would
also like to thank the education from the course _Geometric Representation
Theory_ at Tsinghua University. EP and XZ are partially supported by the GIF
Research Grant I-1515-303./2019. WY is supported by the Young overseas high-
level talents introduction plan, national key research and development program
of China (NO. 2020YFA0713000), and NNSF of China with Grant NO: 11847301 and
12047502.

##  Appendix A Open strings in the presence of a constant $B$-field

The closed string background on which the open strings propagate is the flat
spacetime $\mathbb{R}^{1,1}\times\mathbb{C}^{4}$ with metric
$G_{\mu\nu}=\eta_{\mu\nu}$ and a constant $B$-field whose nonzero components
are given by (10). The worldsheet action of the open string on such background
in the conformal gauge is

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\ell_{s}^{2}}\int
d\tau\int_{0}^{\pi}d\sigma\,G_{\mu\nu}\left(\partial_{\tau}X^{\mu}\partial_{\tau}X^{\nu}-\partial_{\sigma}X^{\mu}\partial_{\sigma}X^{\nu}+2\mathrm{i}\psi_{-}^{\mu}\partial_{+}\psi_{-}^{\nu}+2\mathrm{i}\psi_{+}^{\mu}\partial_{-}\psi_{+}^{\nu}\right)$
(254) $\displaystyle-\frac{1}{2\pi\ell_{s}^{2}}\int
d\tau\,\left[B_{\mu\nu}\left(\left(\partial_{\tau}X^{\mu}\right)X^{\nu}+\mathrm{i}\psi_{-}^{\mu}\psi_{-}^{\nu}+\mathrm{i}\psi_{+}^{\mu}\psi_{+}^{\nu}\right)\right]_{\sigma=0}^{\sigma=\pi},$

where $\ell_{s}$ is the string length, and $\sigma^{\pm}=\tau\pm\sigma$ are
the light-cone coordinates with
$\partial_{\pm}=\frac{1}{2}\left(\partial_{\tau}\pm\partial_{\sigma}\right)$.
From the variations of the action (254), we can obtain the equations of motion
for $X^{\mu}$ and $\psi_{\pm}$,

$\partial_{+}\partial_{-}X^{\mu}=0,\quad\partial_{+}\psi_{-}^{\mu}=\partial_{-}\psi_{+}^{\mu}=0,$
(255)

as well as the boundary conditions

$\displaystyle\left[\left(G_{\mu\nu}\partial_{\sigma}X^{\mu}+B_{\mu\nu}\partial_{\tau}X^{\mu}\right)\delta
X^{\nu}\right]_{\sigma=0}^{\sigma=\pi}$ $\displaystyle=$ $\displaystyle 0,$
(256)
$\displaystyle\left[\delta\psi_{-}^{\mu}\left(G_{\mu\nu}-B_{\mu\nu}\right)\psi_{-}^{\nu}-\delta\psi_{+}^{\mu}\left(G_{\mu\nu}+B_{\mu\nu}\right)\psi_{+}^{\nu}\right]_{\sigma=0}^{\sigma=\pi}$
$\displaystyle=$ $\displaystyle 0.$ (257)

Hence, there are two possible boundary conditions for the worldsheet bosons
$X^{\mu}$ at $\sigma=0$ or $\sigma=\pi$: the Dirichlet (D) boundary condition

$\left.\delta
X^{\mu}\right|_{\sigma=0,\pi}=0\Leftrightarrow\left.\partial_{\tau}X^{\mu}\right|_{\sigma=0,\pi}=0,$
(258)

and the twisted (T) boundary condition

$\left.\left(G_{\mu\nu}\partial_{\sigma}X^{\mu}+B_{\mu\nu}\partial_{\tau}X^{\mu}\right)\right|_{\sigma=0,\pi}=0.$
(259)

The boundary condition (259) becomes the Neumann (N) boundary condition for
$B=0$. The worldsheet supersymmetry transformations in the bulk are

$\delta
X^{\mu}=\mathrm{i}\epsilon_{+}\psi_{-}^{\mu}-\mathrm{i}\epsilon_{-}\psi_{+}^{\mu},\quad\delta\psi_{\pm}^{\mu}=\pm
2\epsilon_{\mp}\partial_{\pm}X^{\mu}.$ (260)

Since we introduce D$1$-branes along $\mathbb{R}^{1,1}$ and D$7_{A}$-branes
along $\mathbb{R}^{1,1}\times\mathbb{C}_{A}^{3}$ with
$A\in\underline{4}^{\vee}$, open strings always satisfy NN boundary conditions
along $\mathbb{R}^{1,1}$,

$\left.\partial_{\sigma}X^{0,9}\right|_{\sigma=0}=\left.\partial_{\sigma}X^{0,9}\right|_{\sigma=\pi}=0.$
(261)

For the remaining $8$ directions, let us introduce the complex bosons

$Z^{a}=X^{2a-1}+\mathrm{i}X^{2a},\quad\bar{Z}^{a}=X^{2a-1}-\mathrm{i}X^{2a},\quad
a\in\underline{4}.$ (262)

The general solution of the equation of motion of $Z^{a}$ is given by
$Z^{a}=Z_{L}^{a}\left(\sigma^{+}\right)+Z_{R}^{a}\left(\sigma^{-}\right)$,
where

$\displaystyle Z_{L}^{a}\left(\sigma^{+}\right)$ $\displaystyle=$
$\displaystyle\frac{z_{L}^{a}}{2}+\frac{\ell_{s}^{2}}{2}p_{L}^{a}\sigma^{+}+\frac{\mathrm{i}\ell_{s}}{\sqrt{2}}\sum_{n\neq
0}\frac{\alpha_{n}^{a}}{n}e^{-\mathrm{i}n\sigma^{+}},$ $\displaystyle
Z_{R}^{a}\left(\sigma^{-}\right)$ $\displaystyle=$
$\displaystyle\frac{z_{R}^{a}}{2}+\frac{\ell_{s}^{2}}{2}p_{R}^{a}\sigma^{-}+\frac{\mathrm{i}\ell_{s}}{\sqrt{2}}\sum_{n\neq
0}\frac{\tilde{\alpha}_{n}^{a}}{n}e^{-\mathrm{i}n\sigma^{-}},$ (263)

and the boundary condition can be written uniformly as

$\left.\left(\partial_{+}-e^{-2\pi\mathrm{i}\nu_{a}}\partial_{-}\right)Z^{a}\right|_{\sigma=0}=\left.\left(\partial_{+}-e^{-2\pi\mathrm{i}\nu_{a}^{\prime}}\partial_{-}\right)Z^{a}\right|_{\sigma=\pi}=0.$
(264)

Here $\nu_{a}=v_{a}$ ($\nu_{a}^{\prime}=v_{a}$) if the $\sigma=0$
($\sigma=\pi$) end of the open string is on D$7_{A}$-brane with $a\in A$, and
$\nu_{a}=\frac{1}{2}$ ($\nu_{a}^{\prime}=\frac{1}{2}$) otherwise. The mode
expansions of $Z^{a}$ when $\nu_{a}=\nu_{a}^{\prime}$ is

$Z^{a}=z_{a}+\ell_{s}^{2}p^{a}\left(\sigma^{+}+e^{2\pi\mathrm{i}\nu_{a}}\sigma^{-}\right)+\frac{\mathrm{i}\ell_{s}}{\sqrt{2}}\sum_{n\in\mathbb{Z}\setminus\left\\{0\right\\}}\frac{\alpha_{n}^{a}}{n}\left(e^{-\mathrm{i}n\sigma^{+}}+e^{2\pi\mathrm{i}\nu_{a}}e^{-\mathrm{i}n\sigma^{-}}\right),$
(265)

and when $\nu_{a}^{\prime}-\nu_{a}=\delta\neq 0$ is

$Z^{a}=z_{a}+\frac{\mathrm{i}\ell_{s}}{\sqrt{2}}\sum_{r\in\mathbb{Z}+\delta}\frac{\alpha_{r}^{a}}{r}\left(e^{-\mathrm{i}r\sigma^{+}}+e^{2\pi\mathrm{i}\nu_{a}}e^{-\mathrm{i}r\sigma^{-}}\right).$
(266)

Meanwhile, we introduce the complex combinations of fermions

$\Psi_{\pm}^{a}=\psi_{\pm}^{2a-1}+\mathrm{i}\psi_{\pm}^{2a},\quad\bar{\Psi}_{\pm}^{a}=\psi_{\pm}^{2a-1}-\mathrm{i}\psi_{\pm}^{2a}.$
(267)

The boundary conditions compatible with (264) can be chosen as

$\left.\left(\Psi_{+}^{a}-(-1)^{\xi}e^{-2\pi\mathrm{i}\nu_{a}}\Psi_{-}^{a}\right)\right|_{\sigma=0}=\left.\left(\Psi_{+}^{a}-e^{-2\pi\mathrm{i}\nu_{a}^{\prime}}\Psi_{-}^{a}\right)\right|_{\sigma=\pi}=0,$
(268)

with $\xi=0$ for the Ramond sector and $\xi=1$ for the Neveu-Schwarz sector.
The Ramond sector preserves half of the worldsheet supersymmetry (260) with
$\epsilon=\epsilon_{-}=-\epsilon_{+}$, while the Neveu-Schwarz sector breaks
all the worldsheet supersymmetry. We combine $\Psi_{+}^{a}$ and $\Psi_{-}^{a}$
into a single field $\Psi^{a}$ with the extended range $0\leq\sigma\leq 2\pi$,

$\Psi^{a}\left(\tau,\sigma\right)=\begin{cases}\Psi_{+}^{a}\left(\tau,\sigma\right)&0\leq\sigma\leq\pi\\\
e^{-2\pi\mathrm{i}\nu_{a}^{\prime}}\Psi_{-}^{a}\left(\tau,2\pi-\sigma\right)&\pi\leq\sigma\leq
2\pi\end{cases},$ (269)

whose field equation is $\partial_{-}\Psi^{a}=0$. The boundary condition (268)
at $\sigma=\pi$ ensures that $\Psi^{a}\left(\tau,\sigma\right)$ is continuous,
while the boundary condition (268) at $\sigma=0$ leads to

$\Psi^{a}\left(\tau,2\pi\right)=\exp\left(-2\pi\mathrm{i}\left(\delta-\frac{1}{2}\xi\right)\right)\Psi^{a}\left(\tau,0\right).$
(270)

Therefore, the mode expansion of $\Psi^{a}$ in the Ramond sector is

$\Psi^{a}\left(\tau,\sigma\right)=\ell_{s}\sum_{r\in\mathbb{Z}+\delta}d_{r}^{a}e^{-\mathrm{i}r\sigma^{+}},$
(271)

and that in the Neveu-Schwarz sector is

$\Psi^{a}\left(\tau,\sigma\right)=\ell_{s}\sum_{r\in\mathbb{Z}+\delta-\frac{1}{2}}b_{r}^{a}e^{-\mathrm{i}r\sigma^{+}}.$
(272)

The zero-point energy of $Z^{a}$ is

$\mathscr{V}_{Z}\left(\delta\right)=\sum_{n=0}^{\infty}\left(n+\left|\delta\right|\right)\overset{\mathrm{reg}}{=}\zeta_{H}\left(-1,\left|\delta\right|\right)=\frac{1}{24}-\frac{1}{2}\left(\left|\delta\right|-\frac{1}{2}\right)^{2},$
(273)

and that of $\Psi^{a}$ is

$\mathscr{V}_{\Psi}\left(\delta\right)=\begin{cases}-\sum_{n=0}^{\infty}\left(n+\left|\delta\right|\right)\overset{\mathrm{reg}}{=}-\zeta_{H}\left(-1,\left|\delta\right|\right)=-\frac{1}{24}+\frac{1}{2}\left(\left|\delta\right|-\frac{1}{2}\right)^{2}&\mathrm{R}\\\
-\sum_{n=0}^{\infty}\left(n+\left|\left|\delta\right|-\frac{1}{2}\right|\right)\overset{\mathrm{reg}}{=}-\zeta_{H}\left(-1,\left|\left|\delta\right|-\frac{1}{2}\right|\right)=-\frac{1}{24}+\frac{1}{2}\left(\left|\left|\delta\right|-\frac{1}{2}\right|-\frac{1}{2}\right)^{2}&\mathrm{NS}\end{cases},$
(274)

where $\zeta_{H}\left(s,a\right)=\sum_{n=0}^{\infty}(n+a)^{-s}$ is the Hurwitz
zeta function. The sum of the zero-point energy
$\mathscr{V}=\mathscr{V}_{Z}+\mathscr{V}_{\Psi}$ is

$\mathscr{V}\left(\delta\right)=\begin{cases}0,&\mathrm{R}\\\
\frac{1}{8}-\frac{1}{2}\left|\left|\delta\right|-\frac{1}{2}\right|,&\mathrm{NS}\end{cases}.$
(275)

The vanishing of the zero-point energy in the Ramond sector is guaranteed by
the unbroken worldsheet supersymmetry.

In the absence of a constant $B$-field, we have $\delta=0$ for DD or NN
directions, and $\left|\delta\right|=\frac{1}{2}$ for DN and ND directions.
The total zero-point energy of the D$p$-D$p^{\prime}$ strings in the Neveu-
Schwarz sector is given by

$E^{(0)}=\frac{\kappa}{2}\mathscr{V}\left(\frac{1}{2}\right)+\frac{8-\kappa}{2}\mathscr{V}\left(0\right)=-\frac{1}{2}+\frac{\kappa}{8},$
(276)

where $\kappa$ is the number of DN and ND directions.

For the $B$-field given by (10), the physical ground states of D$1$-D$1$ and
D$7_{A}$-D$7_{A}$ strings still have zero energy. The total zero-point energy
of the D$1$-D$7_{A}$ strings in the Neveu-Schwarz sector becomes

$E^{(0)}=\sum_{a\in
A}\mathscr{V}\left(\frac{1}{2}-v_{a}\right)+\mathscr{V}\left(0\right)=\frac{1}{4}-\frac{1}{2}\sum_{a\in
A}\left|v_{a}\right|,$ (277)

and that of the D$7_{(acd)}$-D$7_{(bcd)}$ string becomes

$E^{(0)}=\mathscr{V}\left(\frac{1}{2}-v_{a}\right)+\mathscr{V}\left(v_{b}-\frac{1}{2}\right)+2\mathscr{V}\left(0\right)=-\frac{1}{2}\left(\left|v_{a}\right|+\left|v_{b}\right|\right).$
(278)

##  Appendix B Two-dimensional supersymmetric gauge theory

In this appendix, we review two-dimensional $\mathcal{N}=\left(2,2\right)$ and
$\mathcal{N}=\left(0,2\right)$ supersymmetric gauge theories Witten:1993yc ;
Hori:2003ic .

###  B.1 $\mathcal{N}=\left(2,2\right)$ supersymmetry

The $\mathcal{N}=\left(2,2\right)$ supersymmetry algebra in two-dimensional
Minkowski spacetime $\mathbb{R}^{1,1}$ with coordinates $x^{\mu},\mu=0,1$ is
generated by four supercharges $Q_{\pm}$ and
$\bar{Q}_{\pm}=Q_{\pm}^{\dagger}$, spacetime translations $H$, $P$, the
Lorentz boost $M=M_{01}$, and $\mathrm{U}(1)_{V}$ and $\mathrm{U}(1)_{A}$
R-symmetries $F_{V}$ and $F_{A}$. They satisfy the (anti-)commutation
relations,

$\displaystyle Q_{\pm}^{2}=\bar{Q}_{\pm}^{2}=0,$
$\displaystyle\left\\{Q_{\pm},\bar{Q}_{\pm}\right\\}=2\left(H\mp P\right),$
$\displaystyle\left\\{\bar{Q}_{+},\bar{Q}_{-}\right\\}=2Z,$
$\displaystyle\left\\{Q_{+},Q_{-}\right\\}=2Z^{*},$
$\displaystyle\left\\{\bar{Q}_{+},Q_{-}\right\\}=2\tilde{Z},$
$\displaystyle\left\\{Q_{+},\bar{Q}_{-}\right\\}=2\tilde{Z}^{*},$
$\displaystyle\left[M,Q_{\pm}\right]=\mp Q_{\pm},$
$\displaystyle\left[M,\bar{Q}_{\pm}\right]=\mp\bar{Q}_{\pm},$
$\displaystyle\left[F_{V},Q_{\pm}\right]=-Q_{\pm},$
$\displaystyle\left[F_{V},\bar{Q}_{\pm}\right]=+\bar{Q}_{\pm},$
$\displaystyle\left[F_{A},Q_{\pm}\right]=\mp Q_{\pm},$
$\displaystyle\left[F_{A},\bar{Q}_{\pm}\right]=\pm\bar{Q}_{\pm},$ (279)

where $Z$ and $\tilde{Z}$ commute with all operators in the theory and are
called central charges. A central charge can be nonzero if there is a soliton
that interpolates different vacua or if the theory has a continuous abelian
symmetry. In superconformal field theory, both central charges must vanish.

In terms of the $\mathcal{N}=\left(2,2\right)$ superspace with coordinates
$\left(x^{\mu},\theta^{\pm},\bar{\theta}^{\pm}\right)$, the supercharges are
given by

$\displaystyle Q_{\pm}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\theta^{\pm}}+2\mathrm{i}\bar{\theta}^{\pm}\partial_{\pm},$
$\displaystyle\bar{Q}_{\pm}$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\bar{\theta}^{\pm}}-2\mathrm{i}\theta^{\pm}\partial_{\pm},$
(280)

where $\partial_{\pm}=\frac{1}{2}\left(\partial_{0}\pm\partial_{1}\right)$.
They anti-commute with the super-derivatives

$\displaystyle\mathrm{D}_{\pm}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\theta^{\pm}}-2\mathrm{i}\bar{\theta}^{\pm}\partial_{\pm},$
$\displaystyle\bar{\mathrm{D}}_{\pm}$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\bar{\theta}^{\pm}}+2\mathrm{i}\theta^{\pm}\partial_{\pm},$
(281)

which also obey anti-commutation relations

$\mathrm{D}_{\pm}^{2}=\bar{\mathrm{D}}_{\pm}^{2}=0,\quad\left\\{\mathrm{D}_{\pm},\bar{\mathrm{D}}_{\pm}\right\\}=4\mathrm{i}\partial_{\pm}.$
(282)

R-symmetries act on a superfield
$\mathcal{F}\left(x^{\mu},\theta^{\pm},\bar{\theta}^{\pm}\right)$ with vector
R-charge $q_{V}$ and axial R-charge $q_{A}$ as

$\displaystyle e^{\mathrm{i}\alpha
F_{V}}\mathcal{F}\left(x^{\mu},\theta^{\pm},\bar{\theta}^{\pm}\right)$
$\displaystyle=$ $\displaystyle e^{\mathrm{i}\alpha
q_{V}}\mathcal{F}\left(x^{\mu},e^{-\mathrm{i}\alpha}\theta^{\pm},e^{\mathrm{i}\alpha}\bar{\theta}^{\pm}\right),$
(283) $\displaystyle e^{\mathrm{i}\alpha
F_{A}}\mathcal{F}\left(x^{\mu},\theta^{\pm},\bar{\theta}^{\pm}\right)$
$\displaystyle=$ $\displaystyle e^{\mathrm{i}\alpha
q_{A}}\mathcal{F}\left(x^{\mu},e^{\mp\mathrm{i}\alpha}\theta^{\pm},e^{\pm\mathrm{i}\alpha}\bar{\theta}^{\pm}\right).$
(284)

There are three basic types of $\mathcal{N}=\left(2,2\right)$ superfields. A
chiral superfield $\Phi$ satisfies

$\bar{\mathrm{D}}_{\pm}\Phi=0,$ (285)

which can be expanded as

$\Phi\left(x^{\mu},\theta^{\pm},\bar{\theta}^{\pm}\right)=\phi\left(y^{\pm}\right)+\sqrt{2}\theta^{\alpha}\psi_{\alpha}\left(y^{\pm}\right)+2\theta^{+}\theta^{-}F\left(y^{\pm}\right),$
(286)

where $y^{\pm}=x^{\pm}-2\mathrm{i}\theta^{\pm}\bar{\theta}^{\pm}$, and $F$ is
a complex auxiliary field. The complex conjugate of $\Phi$ is an anti-chiral
superfield, $\mathrm{D}_{\pm}\bar{\Phi}=0$.

A twisted chiral superfield $\Lambda$ satisfies

$\bar{\mathrm{D}}_{+}\Lambda=\mathrm{D}_{-}\Lambda=0,$ (287)

which can be expanded as

$\Lambda=\varphi\left(\tilde{y}^{\pm}\right)+\sqrt{2}\theta^{+}\bar{\chi}_{+}\left(\tilde{y}^{\pm}\right)+\sqrt{2}\bar{\theta}^{-}\chi_{-}\left(\tilde{y}^{\pm}\right)+2\theta^{+}\bar{\theta}^{-}\widetilde{F}\left(\tilde{y}^{\pm}\right),$
(288)

where $\tilde{y}^{\pm}=x^{\pm}\mp 2\mathrm{i}\theta^{\pm}\bar{\theta}^{\pm}$,
and $\widetilde{F}$ is a complex auxiliary field. The complex conjugate of
$\Lambda$ is a twisted anti-chiral superfield,
$\bar{\mathrm{D}}_{-}\Lambda=\mathrm{D}_{+}\Lambda=0$.

We can also introduce a vector multiplet, which consists of a vector field
$A_{\pm}$, Dirac fermions $\lambda_{\pm}$, $\bar{\lambda}_{\pm}$ which are
conjugate to each other, and a complex scalar $\sigma$ in the adjoint
representation of the gauge group. The vector superfield $V$ is a real
superfield and can be expanded in the Wess-Zumino gauge as

$\displaystyle V$ $\displaystyle=$
$\displaystyle\theta^{-}\bar{\theta}^{-}\left(A_{0}-A_{1}\right)+\theta^{+}\bar{\theta}^{+}\left(A_{0}+A_{1}\right)-\theta^{-}\bar{\theta}^{+}\sigma-\theta^{+}\bar{\theta}^{-}\bar{\sigma}+$
(289)
$\displaystyle+\sqrt{2}\mathrm{i}\left(\theta^{-}\theta^{+}\bar{\theta}^{-}\bar{\lambda}_{-}+\theta^{-}\theta^{+}\bar{\theta}^{+}\bar{\lambda}_{+}+\bar{\theta}^{+}\bar{\theta}^{-}\theta^{-}\lambda_{-}+\bar{\theta}^{+}\bar{\theta}^{-}\theta^{+}\lambda_{+}\right)+$
$\displaystyle+2\theta^{-}\theta^{+}\bar{\theta}^{+}\bar{\theta}^{-}D,$

where $D$ is a real auxiliary field. To couple a matter superfield to the
gauge field, we simply replace the super-derivatives
$\mathrm{D}_{\pm}^{2},\bar{\mathrm{D}}_{\pm}^{2}$ by the gauge-covariant
super-derivatives

$\mathbb{D}_{\pm}=e^{-V}D_{\pm}e^{V},\quad\bar{\mathbb{D}}_{\pm}=e^{V}\bar{D}_{\pm}e^{-V}.$
(290)

The field strength of $V$ is given by

$\Sigma=\frac{1}{2}\left\\{\bar{\mathbb{D}}_{+},\mathbb{D}_{-}\right\\},$
(291)

which is a twisted chiral superfield
$\bar{\mathbb{D}}_{+}\Sigma=\mathbb{D}_{-}\Sigma=0$.

The supersymmetric Lagrangian can be written as

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\int
d^{4}\theta\mathcal{K}\left(\mathcal{F},\bar{\mathcal{F}}\right)+\frac{1}{2}\left(\int
d\theta^{-}d\theta^{+}\left.\mathcal{W}(\Phi)\right|_{\bar{\theta}^{\pm}=0}+c.c.\right)+$
(292) $\displaystyle+\frac{1}{2}\left(\int
d\bar{\theta}^{-}d\theta^{+}\left.\widetilde{\mathcal{W}}(\Lambda)\right|_{\theta^{-}=\bar{\theta}^{+}=0}+c.c.\right),$

where the first term involving an arbitrary real function
$\mathcal{K}\left(\mathcal{F},\bar{\mathcal{F}}\right)$ is the D-term
contribution, the second term involving a superpotential $\mathcal{W}$ is the
F-term contribution, and the third term involving a twisted superpotential
$\widetilde{\mathcal{W}}$ is the twisted F-term contribution. Here
$\mathcal{W}(\Phi)$ and $\widetilde{\mathcal{W}}(\Lambda)$ are required to be
holomorphic functions of chiral superfields and twisted chiral superfields,
respectively.

We are mainly interested in the gauged linear sigma model which describes a
vector superfield $V$ with gauge group $\mathrm{U}(N)$ and field strength
$\Sigma$, coupled with charged chiral multiplets $\Phi_{i}$. The Lagrangian is
given by (292), with

$\mathcal{K}=-\frac{1}{2e^{2}}\mathrm{Tr}\bar{\Sigma}\Sigma+\mathrm{Tr}\left(\sum_{i}\bar{\Phi}_{i}\Phi_{i}\right),\quad\mathcal{W}=0,\quad\widetilde{\mathcal{W}}=-t\Sigma,$
(293)

where $e$ is the gauge coupling constant, and $t=r-\mathrm{i}\vartheta$ is the
complex combination of the Fayet-Iliopoulos parameter $r$ and the theta angle
$\vartheta$.

###  B.2 $\mathcal{N}=\left(0,2\right)$ supersymmetry

We can get $\mathcal{N}=\left(0,2\right)$ supersymmetry from
$\mathcal{N}=\left(2,2\right)$ supersymmetry by dropping $Q_{-}$ and
$\bar{Q}_{-}$. There is only one $\mathrm{U}(1)_{\mathcal{R}}$ R-symmetry
$\mathcal{R}$ satisfying

$\left[\mathcal{R},Q_{+}\right]=-Q_{+},\quad\left[\mathcal{R},\bar{Q}_{+}\right]=+\bar{Q}_{+}.$
(294)

The $\mathcal{N}=\left(0,2\right)$ superspace with coordinates
$\left(x^{\mu},\theta^{+},\bar{\theta}^{+}\right)$ is the subspace of
$\mathcal{N}=\left(2,2\right)$ superspace with
$\theta^{-}=\bar{\theta}^{-}=0$.

There are three basic types of $\mathcal{N}=\left(0,2\right)$ superfields. An
$\mathcal{N}=\left(0,2\right)$ chiral superfield $\varPhi$ is a complex-valued
Lorentz scalar obeying

$\bar{\mathrm{D}}_{+}\varPhi=0,$ (295)

which can be expanded as

$\varPhi=\phi+\sqrt{2}\theta^{+}\psi_{+}-2\mathrm{i}\theta^{+}\bar{\theta}^{+}\partial_{+}\phi,$
(296)

where $\phi$ is a complex scalar and $\psi_{+}$ is a right-moving fermion.

An $\mathcal{N}=\left(0,2\right)$ Fermi superfield $\varPsi_{-}$ is a left-
moving spinor satisfying

$\bar{\mathrm{D}}_{+}\varPsi_{-}=\sqrt{2}E\left(\varPhi_{i}\right),$ (297)

which can be expanded as

$\varPsi_{-}=\psi_{-}-\sqrt{2}\theta^{+}G-2\mathrm{i}\theta^{+}\bar{\theta}^{+}\partial_{+}\psi_{-}-\sqrt{2}\bar{\theta}^{+}E\left(\phi_{i}\right)+2\theta^{+}\bar{\theta}^{+}\frac{\partial
E}{\partial\phi_{i}}\psi_{+,i},$ (298)

where $\psi_{-}$ is a left-moving fermion and $G$ is an auxiliary field.

The $\mathcal{N}=\left(0,2\right)$ vector superfield $U$ is a real superfield
with the expansion

$U=A_{0}-A_{1}-2\mathrm{i}\theta^{+}\bar{\lambda}_{-}-2\mathrm{i}\bar{\theta}^{+}\lambda_{-}+2\theta^{+}\bar{\theta}^{+}D,$
(299)

where $A_{\mu}$ is the gauge field, $\lambda_{-},\bar{\lambda}_{-}$ are left-
moving fermions, and $D$ is a real auxiliary field. All the fields are in the
adjoint representation of the gauge group. The gauge-covariant super-
derivatives $\mathbb{D}_{+}$ and $\bar{\mathbb{D}}_{+}$ are given by

$\displaystyle\mathbb{D}_{+}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\theta^{+}}-\mathrm{i}\bar{\theta}^{+}\left(\mathcal{D}_{0}+\mathcal{D}_{1}\right),$
$\displaystyle\bar{\mathbb{D}}_{+}$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\bar{\theta}^{+}}+\mathrm{i}\theta^{+}\left(\mathcal{D}_{0}+\mathcal{D}_{1}\right),$
(300)

where

$\displaystyle\mathcal{D}_{0}$ $\displaystyle=$
$\displaystyle\partial_{0}+\mathrm{i}A_{0}+\theta^{+}\bar{\lambda}_{-}+\bar{\theta}^{+}\lambda_{-}+\mathrm{i}\theta^{+}\bar{\theta}^{+}D,$
$\displaystyle\mathcal{D}_{1}$ $\displaystyle=$
$\displaystyle\partial_{1}+\mathrm{i}A_{1}-\theta^{+}\bar{\lambda}_{-}-\bar{\theta}^{+}\lambda_{-}-\mathrm{i}\theta^{+}\bar{\theta}^{+}D,$
(301)

are the gauge-covariant derivatives. We can organize $U$ in terms of the
gauge-invariant field strength
$\varUpsilon=\frac{1}{2}\left[\bar{\mathbb{D}}_{+},\mathcal{D}_{0}-\mathcal{D}_{1}\right]$,
which is a Fermi superfield.

We can write down the supersymmetric Lagrangian of an
$\mathcal{N}=\left(0,2\right)$ gauged linear sigma model with a vector
multiplet $V$ whose field strength is $\varUpsilon$ coupled to chiral
multiplets $\varPhi_{i}$ and the Fermi multiplets $\Psi_{a}$,

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\int
d\theta^{+}d\bar{\theta}^{+}\left(\frac{1}{2e^{2}}\mathrm{Tr}\bar{\varUpsilon}\varUpsilon-\frac{\mathrm{i}}{2}\mathrm{Tr}\sum_{i}\bar{\varPhi}_{i}\mathcal{D}_{-}\varPhi_{i}-\frac{1}{2}\mathrm{Tr}\sum_{a}\bar{\varPsi}_{-,a}\varPsi_{-,a}\right)+$
(302) $\displaystyle+\left(\frac{\mathrm{i}t}{2}\int
d\theta^{+}\left.\varUpsilon\right|_{\bar{\theta}^{+}=0}+\mathrm{c.c.}\right)-\frac{1}{\sqrt{2}}\left(\int
d\theta^{+}\left.\mathrm{Tr}\sum_{a}\varPsi_{-,a}J^{a}\right|_{\bar{\theta}^{+}=0}+c.c.\right),$

where $J^{a}\left(\varPhi_{i}\right)$ are holomorphic functions obeying

$\sum_{a}E_{a}\left(\varPhi_{i}\right)J^{a}\left(\varPhi_{i}\right)=0.$ (303)

It is sometimes useful to write a theory with $\mathcal{N}=\left(2,2\right)$
supersymmetry in the language of the $\mathcal{N}=\left(0,2\right)$
superspace. An $\mathcal{N}=\left(2,2\right)$ vector multiplet $V$ decomposes
into an $\mathcal{N}=\left(0,2\right)$ vector multiplet $U$ and an
$\mathcal{N}=\left(0,2\right)$ chiral multiplet
$\Sigma^{\prime}=\left.\Sigma\right|_{\theta^{-}=\bar{\theta}^{-}=0}$. An
$\mathcal{N}=\left(2,2\right)$ chiral multiplet $\Phi$ decomposes into an
$\mathcal{N}=\left(0,2\right)$ chiral multiplet
$\varPhi=\left.\Phi\right|_{\theta^{-}=\bar{\theta}^{-}=0}$ and an
$\mathcal{N}=\left(0,2\right)$ Fermi superfield
$\varPsi_{-}=\frac{1}{\sqrt{2}}\left.\mathbb{D}_{-}\Phi\right|_{\theta^{-}=\bar{\theta}^{-}=0}$,
with

$E=\frac{1}{2}\left.\bar{\mathbb{D}}_{+}\mathbb{D}_{-}\Phi\right|_{\theta^{-}=\bar{\theta}^{-}=0}=\frac{1}{2}\left.\left\\{\bar{\mathbb{D}}_{+},\mathbb{D}_{-}\right\\}\Phi\right|_{\theta^{-}=\bar{\theta}^{-}=0}=\Sigma^{\prime}\varPhi.$
(304)

The kinetic terms decompose naturally, while the F-term contribution specified
by the superpotential $\mathcal{W}(\Phi)$ is reduced to a collection of
functions $J^{a}$, one for each
$\Phi_{a}=\left(\varPhi_{a},\varPsi_{-,a}\right)$, with

$J^{a}=\frac{\partial\mathcal{W}}{\partial\varPhi_{a}}.$ (305)

The condition (303) is satisfied automatically.

##  Appendix C Elliptic genus of $\mathcal{N}=\left(0,2\right)$ theories

We consider the Euclidean path-integral of a two-dimensional
$\mathcal{N}=\left(0,2\right)$ supersymmetric theory on a torus
$\mathbb{T}^{2}$, in the presence of a background flat connection for the
flavor symmetry. Let $T_{a}$ be the Cartan generators of the flavor symmetry
group $G_{f}$. In the Hamiltonian formalism, the elliptic genus can be defined
by Gadde:2013wq ; Gadde:2013ftv ; Benini:2013nda ; Benini:2013xpa

$Z\left(x;q\right)=\mathrm{Tr}_{\mathrm{R}}(-1)^{F}q^{H_{L}}\bar{q}^{H_{R}}\prod_{a}e^{2\pi\mathrm{i}x_{a}T_{a}},$
(306)

where the trace is over the Hilbert space of the theory on the spatial circle,
with periodic boundary conditions for fermions. $F$ is the fermion number.
$q=e^{2\pi\mathrm{i}\tau}$ specifies the complex structure $\tau$ of $T^{2}$.
$H_{L}$ and $H_{R}$ are the left- and right-moving Hamiltonians, respectively.
Based on the standard argument in Witten:1982df , the elliptic genus is
independent of $\bar{q}$ if the theory has a discrete spectrum. 444Notice that
the elliptic genus can suffer from a holomorphic anomaly for noncompact
models. See Murthy:2013mya ; Ashok:2013pya ; Nian:2014fma  for examples with
$\mathcal{N}=\left(2,2\right)$ supersymmetry.

We consider a two-dimensional $\mathcal{N}=\left(0,2\right)$ supersymmetric
gauged linear sigma model which is described by a vector multiplet $V$ with
gauge group $G$ of rank $r$, chiral multiplets $\Phi_{i}$ transforming in the
representation $\mathfrak{R}\left(\Phi_{i}\right)$ of $G\times G_{f}$, and
Fermi multiplets $\Psi_{a}$ transforming in the representation
$\mathfrak{R}\left(\Psi_{a}\right)$ of $G\times G_{f}$. The elliptic genus has
been rigorously derived using the technique of path integral localization
Benini:2013nda ; Benini:2013xpa ,

$Z\left(x;q\right)=\frac{1}{\left|W_{G}\right|}\oint_{\mathrm{JK}}Z_{V}\prod_{i}Z_{\Phi_{i}}\prod_{a}Z_{\Psi_{a}},$
(307)

where $\left|W_{G}\right|$ is the order of the Weyl group of $G$, $Z_{V}$,
$Z_{\Phi_{i}}$, and $Z_{\Psi_{a}}$ are the contributions from $V$ without
zero-modes of the Cartan generators, $\Phi_{i}$, and $\Psi_{a}$, respectively.
The contour integral is evaluated using the Jeffrey-Kirwan residue
prescription jeffrey1995localization . In terms of the Dedekind eta function
$\eta(\tau)$ and the Jacobi theta function
$\theta_{1}\left(\left.z\right|\tau\right)$,

$\displaystyle\eta(\tau)$ $\displaystyle=$ $\displaystyle
q^{\frac{1}{24}}\prod_{n=1}^{\infty}\left(1-q^{n}\right),$ (308)
$\displaystyle\theta_{1}\left(\left.z\right|\tau\right)$ $\displaystyle=$
$\displaystyle\mathrm{i}\sum_{n\in\mathbb{Z}}\left(-1\right)^{n}e^{(2n+1)\pi\mathrm{i}z}q^{\frac{1}{2}\left(n+\frac{1}{2}\right)^{2}},$
(309)

the explicit expressions of $Z_{V}$, $Z_{\Phi_{i}}$, and $Z_{\Psi_{a}}$ are
given by

$\displaystyle Z_{V}$ $\displaystyle=$
$\displaystyle\left(\frac{2\pi\eta(\tau)^{2}}{\mathrm{i}}\right)^{r}\prod_{I=1}^{r}d\varphi_{I}\prod_{\alpha\in
G}\frac{\mathrm{i}\theta_{1}\left(\left.\alpha\cdot\varphi\right|\tau\right)}{\eta(\tau)},$
(310) $\displaystyle Z_{\Phi_{i}}$ $\displaystyle=$
$\displaystyle\prod_{\rho\in\mathfrak{R}\left(\Phi_{i}\right)}\frac{\mathrm{i}\eta(\tau)}{\theta_{1}\left(\left.\rho\cdot\zeta\right|\tau\right)},$
(311) $\displaystyle Z_{\Psi_{a}}$ $\displaystyle=$
$\displaystyle\prod_{\rho\in\mathfrak{R}\left(\Psi_{a}\right)}\frac{\mathrm{i}\theta_{1}\left(\left.\rho\cdot\zeta\right|\tau\right)}{\eta(\tau)},$
(312)

where $\varphi$ parametrizes a Cartan subalgebra of $G$, and $\zeta$ includes
both $\varphi$ and $x$. The function
$\theta_{1}\left(\left.z\right|\tau\right)$ has no poles, but there are simple
zeros at $z\in\mathbb{Z}+\tau\mathbb{Z}$, with residues of its inverse

$\frac{1}{2\pi\mathrm{i}}\oint_{z=a+b\tau}\frac{dz}{\theta_{1}\left(\left.z\right|\tau\right)}=\frac{(-1)^{a+b}e^{\mathrm{i}\pi\tau
b^{2}}}{2\pi\eta(\tau)^{3}},$ (313)

where we have used the identity

$2\pi\eta(\tau)^{3}=\partial_{z}\theta_{1}\left(\left.0\right|\tau\right).$
(314)

In this paper, we often use the abbreviation

$\theta\left(z\right)\equiv\theta_{1}\left(\left.z\right|\tau\right).$ (315)

By taking the degenerate limit $q\to 1$ and neglecting an overall
$x$-independent factor, we can reduce the elliptic genus of a two-dimensional
supersymmetric gauge theory to the Witten index of the one-dimensional
supersymmetric quantum mechanics obtained by the standard dimensional
reduction. The contributions of $Z_{V}$, $Z_{\Phi_{i}}$, and $Z_{\Psi_{a}}$
become

$\displaystyle Z_{V}$ $\displaystyle=$
$\displaystyle\prod_{I=1}^{r}d\varphi_{I}\prod_{\alpha\in
G}2\sinh\left(\frac{\beta\alpha\cdot\varphi}{2}\right)$ (316) $\displaystyle
Z_{\Phi_{i}}$ $\displaystyle=$
$\displaystyle\prod_{\rho\in\mathfrak{R}\left(\Phi_{i}\right)}\frac{1}{2\sinh\left(\frac{\beta\rho\cdot\zeta}{2}\right)},$
(317) $\displaystyle Z_{\Psi_{a}}$ $\displaystyle=$
$\displaystyle\prod_{\rho\in\mathfrak{R}\left(\Psi_{a}\right)}2\sinh\left(\frac{\beta\rho\cdot\zeta}{2}\right),$
(318)

where $\beta$ is the circumference of $\mathbb{S}^{1}$. If we further reduce
to zero dimension, the partition function of the corresponding supersymmetric
matrix model can be obtained by replacing $2\sinh\left(\frac{\beta
z}{2}\right)\to z$.

##  Appendix D Jeffrey-Kirwan residue formula

The Jeffrey-Kirwan residue formula introduced in jeffrey1995localization
gives a prescription for expressing multiple contour integrals as a sum of
iterated residues.

Let $\omega$ be a meromorphic $\left(k,0\right)$-form on a $k$-dimensional
complex manifold,

$\omega=\frac{A(u)}{B(u)}du_{1}\wedge\cdots\wedge du_{k},$ (319)

where $A(u)$ and $B(u)$ are two holomorphic functions of $k$ complex variables
$u=\left(u_{1},\cdots,u_{k}\right)$. We assume that $B(u)$ is a product of
linear factors,

$B(u)=\prod_{i}\left(\mathbf{Q}_{i}\cdot u+b_{i}\right),$ (320)

where $\mathbf{Q}_{i}$ is the charge vector associated with the singular
hyperplane $H_{i}$,

$H_{i}=\left\\{\left.u\in\mathbb{C}^{n}\right|\mathbf{Q}_{i}\cdot
u+b_{i}=0\right\\}.$ (321)

Using the standard basis $\left\\{\mathbf{e}_{j}\right\\}_{j=1,\cdots,k}$ of
$\mathbb{R}^{k}$,

$\mathbf{e}_{j}=\left(0,\cdots,0,\overset{j}{1},0,\cdots,\overset{k}{0}\right),$
(322)

we can write $\mathbf{Q}_{i}$ as

$\mathbf{Q}_{i}=\sum_{j=1}^{k}\mathbf{Q}_{i,j}\mathbf{e}_{j},$ (323)

and

$\mathbf{Q}_{i}\cdot u=\sum_{j=1}^{k}\mathbf{Q}_{i,j}u_{j}.$ (324)

Clearly, $\omega$ is holomorphic on the complement of
$\mathscr{M}^{\mathrm{sing}}=\bigcup_{i}H_{i}$. Let
$\mathscr{M}_{\ast}^{\mathrm{sing}}\subset\mathscr{M}^{\mathrm{sing}}$ be the
set of isolated points where $n\geq k$ linearly independent singular
hyperplanes meet. For $u_{\ast}\in\mathscr{M}_{\ast}^{\mathrm{sing}}$, we
denote by $\mathbf{Q}\left(u_{\ast}\right)$ the set of charge vectors of the
singular hyperplanes meeting at $u_{\ast}$,

$\mathbf{Q}\left(u_{\ast}\right)=\left\\{\left.\mathbf{Q}_{i}\right|u_{\ast}\in
H_{i},i=1,\cdots,n\right\\}.$ (325)

We assume that for each $u_{\ast}\in\mathscr{M}_{\ast}^{\mathrm{sing}}$, the
hyperplane arrangement is projective, which requires that the set
$\mathbf{Q}\left(u_{\ast}\right)$ is contained in a half-space of
$\mathbb{R}^{k}$. This assumption is automatically obeyed when the hyperplane
arrangement is nondegenerate, which means that the number of hyperplanes
meeting at every $u_{\ast}\in\mathscr{M}_{\ast}^{\mathrm{sing}}$ is exactly
$k$. Then the residue of $\omega$ at $u_{\ast}$ is given by its integral over
$\prod_{i=1}^{k}\mathcal{C}_{i}$, where $\mathcal{C}_{i}$ is a small circle
around $H_{i}$.

We denote the cone spanned by $\mathbf{Q}_{1},\cdots,\mathbf{Q}_{k}$ by

$\mathrm{Cone}\left(\mathbf{Q}_{1},\cdots,\mathbf{Q}_{k}\right)=\left\\{\left.\sum_{i=1}^{k}\lambda_{i}\mathbf{Q}_{i}=\bm{\eta}\right|\lambda_{i}>0\right\\}.$
(326)

Let $\mathrm{Cone}_{\mathrm{sing}}\left(\mathbf{Q}\right)$ be the union of the
cones generated by all subsets of $\mathbf{Q}$ with $k-1$ elements. The space
$\mathbb{R}^{k}\setminus\mathrm{Cone}_{\mathrm{sing}}\left(\mathbf{Q}\right)$
is a union of connected components, and we call each connected component a
chamber. We can specify a chamber by a generic nonzero vector
$\bm{\eta}\in\mathbb{R}^{k}\setminus\mathrm{Cone}_{\mathrm{sing}}\left(\mathbf{Q}\right)$.
Then the Jeffrey-Kirwan residue formula states that

$\int\omega\rightsquigarrow\sum_{u_{\ast}\in\mathscr{M}_{\ast}^{\mathrm{sing}}}\underset{u=u_{*}}{\mathrm{JKRes}}\left(\mathbf{Q}\left(u_{*}\right),\bm{\eta}\right)\omega,$
(327)

where the JK-residue operator is defined by the condition

$\underset{u=u_{*}}{\mathrm{JKRes}}\left(\mathbf{Q}\left(u_{*}\right),\bm{\eta}\right)\frac{du_{1}\wedge\cdots\wedge
du_{k}}{\prod_{i=1}^{k}\left(\mathbf{Q}_{i}\cdot\left(u-u_{\ast}\right)\right)}=\begin{cases}\frac{1}{\left|\det\left(\mathbf{Q}_{1},\cdots,\mathbf{Q}_{k}\right)\right|},&\bm{\eta}\in\mathrm{Cone}\left(\mathbf{Q}_{1},\cdots,\mathbf{Q}_{k}\right)\\\
0,&\mathrm{Otherwise}\end{cases},$ (328)

As $\bm{\eta}$ is varied, the JK-residue is locally constant but can jump when
$\bm{\eta}$ crosses the boundary of a chamber. In the simplest case of $k=1$,
we have

$\underset{u=u_{\ast}}{\mathrm{JKRes}}\left(\left\\{q\right\\},\eta\right)\frac{du}{u-u_{\ast}}=\begin{cases}\mathrm{sign}(q),&\eta
q>0\\\ 0,&\eta q<0\end{cases}.$ (329)

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