the quotient of
$\mathbb{C}\times\mathbb{R}^{2}_{+\hbar}\times\mathbb{R}^{2}_{-\hbar}$ such
that translations on $\mathbb{C}$ by $\check{R}_{1}$ and
$\mathrm{i}\check{R}_{2}$ are accompanied by the action of the elements
$\exp(\mathop{\mathrm{Re}}\nolimits\hbar)$ and
$\exp(\mathop{\mathrm{Im}}\nolimits\hbar)$ of $\mathrm{U}(1)_{\hbar}$,
respectively.) Last, we apply T-duality on $\check{T}^{2}$ and decompactify
$T^{2}$ to $\mathbb{C}$ by taking $R_{1}$, $R_{2}\to\infty$. This last
T-duality yields a certain B-field due to the twist in the product between
$\check{T}^{2}$ and $\mathbb{R}^{2}_{+\hbar}\times\mathbb{R}^{2}_{-\hbar}$
introduced earlier.

From the point of view of the gauge theory, the first step amounts to lifting
the two-dimensional theory on $\mathbb{R}\times\check{S}^{1}$ to a four-
dimensional theory on $\mathbb{R}\times\check{S}^{1}\times\check{T}^{2}$.
Then, the second step turns on a holonomy for the background gauge field for
$\mathrm{U}(1)_{\hbar}$. The last step dimensionally reduces the four-
dimensional theory back to two dimensions. Since the components of a four-
dimensional gauge field along $\check{T}^{2}$ become the complex scalar field
for the corresponding two-dimensional gauge field, this procedure induces the
twisted masses for $\mathrm{U}(1)_{\hbar}$.

###  3.2 Four-dimensional Chern–Simons theory

We can convert the brane configuration (105) to a configuration realizing line
defects in four-dimensional Chern–Simons theory with gauge group
$\mathrm{GL}(m|n)$. To do so, we apply T-duality along $\check{S}^{1}$ and
then S-duality.

Under these dualities, $\check{S}^{1}$ is mapped to the dual circle $S^{1}$,
the NS5-branes are mapped to D5-branes, the D4-branes are mapped to D3-branes,
and the D2-branes are mapped to F1-branes (fundamental strings). Thus we
obtain the following type IIB setup:

$\begin{array}[]{r@{\colon\ }c@{\ \times\ }c@{\ \times\ }c@{\ \times\ }c@{\
\times\ }c@{\ \times\ }c@{\ \times\
}c}\mathrm{Spacetime}&\mathbb{R}&S^{1}&\mathbb{C}&\mathbb{R}_{X}&\mathbb{R}_{Y}&\mathbb{R}^{2}_{+\hbar}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{D5}_{i}\
([i]=\bar{0})&\mathbb{R}&S^{1}&\mathbb{C}&\\{X_{i}\\}&\\{Y_{i}\\}&\mathbb{R}^{2}_{+\hbar}&\\{0\\}\\\
\mathrm{D5}_{i}\
([i]=\bar{1})&\mathbb{R}&S^{1}&\mathbb{C}&\\{X_{i}\\}&\\{Y_{i}\\}&\\{0\\}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{D3}_{i}^{\ell}\
([i]=\bar{0})&\mathbb{R}&\\{y_{i}^{\ell}\\}&\\{\mu_{i}^{\ell}\\}&\\{X_{i}\\}&[Y_{i},\infty)&\mathbb{R}^{2}_{+\hbar}&\\{0\\}\\\
\mathrm{D3}_{i}^{\ell}\
([i]=\bar{1})&\mathbb{R}&\\{y_{i}^{\ell}\\}&\\{\mu_{i}^{\ell}\\}&\\{X_{i}\\}&[Y_{i},\infty)&\\{0\\}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{F1}_{r}^{a_{r}}&\mathbb{R}&\\{\tilde{y}_{r}^{a_{r}}\\}&\\{\sigma_{r}^{a_{r}}\\}&[X_{r},X_{r+1}]&\\{{\widetilde{Y}}_{r}\\}&\\{0\\}&\\{0\\}\end{array}$
(114)

The positions of the D3-branes and the F1-branes on $S^{1}$ are given by the
holonomies around $\check{S}^{1}$ of the gauge fields on their counterparts in
the type IIA setup. The B-field inducing the twisted masses for
$\mathrm{U}(1)_{\hbar}$ becomes a Ramond–Ramond (RR) two-form field in the new
setup.

As in the two-dimensional theory discussed before, the vacuum sector of the
theory on the D5-branes, with $\mathbb{R}$ taken to be the time direction, is
captured by the cohomology with respect to a certain supercharge. This
supercharge is dual to the supercharge $Q$ of the two-dimensional theory, and
we will use the same symbol to denote it.

We claim that the $Q$-invariant sector of the theory, which governs the
$Q$-cohomology, is equivalent to four-dimensional Chern–Simons theory with
gauge group $\mathrm{GL}(m|n)$.

Before demonstrating this equivalence, let us remark that related brane
constructions have appeared in the literature.444To relate these brane
constructions to ours, we endow
$\mathbb{R}^{2}_{+\hbar}\times\mathbb{R}^{2}_{-\hbar}$ with a Taub–NUT metric.
(The $Q$-invariant sector is independent of the choice of metric as long as it
preserves the rotational symmetries of $\mathbb{R}^{2}_{+\hbar}$ and
$\mathbb{R}^{2}_{-\hbar}$.) If we regard the Taub–NUT space as a circle
fibration over $\mathbb{R}^{3}$, then $\mathbb{R}^{2}_{+\hbar}\times\\{0\\}$
and $\\{0\\}\times\mathbb{R}^{2}_{-\hbar}$ are two semi-infinite cigar-shaped
subspaces extending in the opposite directions such that their tips touch at
the origin of $\mathbb{R}^{3}$; see [38], appendix B. T-duality in the
direction of the circle fibers produces an NS5-brane which sits at the origin
of $\mathbb{R}^{3}$ and extends in the directions transverse to the Taub–NUT
space. The D5-branes wrapping the two cigars are turned into D4-branes ending
on the NS5-brane from two sides. Considering the case with $n=0$, we reproduce
the construction of [39]. The field theory counterpart of this T-duality was
analyzed in [40]. From the D4–NS5 brane configuration we obtain the brane
configuration of [41], roughly speaking, by further replacing $\mathbb{C}$
with a cylinder, taking T-duality in the circumferential direction of the
cylinder, and decompactifying the dual cylinder. (Such T-duality was
considered in [42].) In [41], Mikhaylov and Witten gave a brane construction
of three-dimensional Chern–Simons theory with gauge group $\mathrm{GL}(m|n)$,
extending the construction for gauge group $\mathrm{GL}(m)$ given in [43].
Their construction uses $m$ D4-branes and $n$ D4-branes ending on an NS5-brane
from opposite sides. The supergroup Chern–Simons theory appears at the
intersection of the three kinds of branes. In [39], a construction of four-
dimensional Chern–Simons theory with gauge group $\mathrm{GL}(m)$ was
proposed. In this construction, $m$ D4-branes end on an NS5-brane.

####  3.2.1 Case with $m=0$ or $n=0$

In the case in which all D5-branes are of even type ($n=0$) or of odd type
($m=0$), the result just described was derived in [16]. Let us briefly review
the derivation in [16].

For $n=0$, the worldvolume theory on the D5-branes is a deformation of six-
dimensional ${\mathcal{N}}=(1,1)$ super Yang–Mills theory with gauge group
$\mathrm{U}(m)$, placed on $\mathbb{R}\times
S^{1}\times\mathbb{C}\times\mathbb{R}^{2}_{+\hbar}$. The deformation is what
is often called an _$\Omega$ -deformation_ and controlled by $\hbar$: we have

$Q^{2}=\hbar F_{\hbar}\,,$ (115)

where $F_{\hbar}$ is the generator of $\mathrm{U}(1)_{\hbar}$. (Here we are
considering the situation in which there are no D3-branes, hence no
$\mathrm{U}(L)$ flavor symmetries.) In the six-dimensional theory,
$\mathrm{U}(1)_{\hbar}$ is the antidiagonal subgroup of the rotation group
$\mathrm{U}(1)_{\mathbb{R}^{2}_{+\hbar}}$ on $\mathbb{R}^{2}_{+\hbar}$ and the
subgroup $\mathrm{U}(1)_{\mathbb{R}^{2}_{-\hbar}}$ of the R-symmetry group
$\mathrm{Spin}(4)$ coming from the rotation symmetry of
$\mathbb{R}^{2}_{-\hbar}$.

Six-dimensional (Euclidean) ${\mathcal{N}}=(1,1)$ super Yang–Mills theory on
$\mathbb{R}\times S^{1}\times\mathbb{C}\times\mathbb{R}^{2}_{+\hbar}$ reduces
to two-dimensional ${\mathcal{N}}=(8,8)$ super Yang–Mills theory on
$\mathbb{R}^{2}_{+\hbar}$ by dimensional reduction. In the undeformed case
(when $\hbar=0$), the supercharge $Q$ belongs to an ${\mathcal{N}}=(2,2)$
subalgebra of the ${\mathcal{N}}=(8,8)$ supersymmetry algebra. Accordingly,
${\mathcal{N}}=(1,1)$ super Yang–Mills theory on $\mathbb{R}\times
S^{1}\times\mathbb{C}\times\mathbb{R}^{2}_{+\hbar}$ may be thought of as an
${\mathcal{N}}=(2,2)$ supersymmetric gauge theory on
$\mathbb{R}^{2}_{+\hbar}$, with infinite-dimensional target space and gauge
group. The $\Omega$-deformation of the six-dimensional theory induces an
$\Omega$-deformation of the two-dimensional theory.

In general, the $Q$-invariant sector of an $\Omega$-deformed
${\mathcal{N}}=(2,2)$ supersymmetric gauge theory on $\mathbb{R}^{2}$ is
equivalent to a zero-dimensional theory [23, 24, 16]. Let ${\mathcal{G}}$ be
the gauge group of the theory and ${\mathcal{G}}_{\mathbb{C}}$ be its
complexification. By ${\mathcal{N}}=(2,2)$ supersymmetry, the chiral
multiplets take values in a Kähler manifold ${\mathcal{X}}$ with
${\mathcal{G}}_{\mathbb{C}}$-action. The superpotential is a
${\mathcal{G}}_{\mathbb{C}}$-invariant holomorphic function ${\mathcal{W}}$ on
${\mathcal{X}}$. The path integral with insertion of $Q$-invariant observables
localizes to a ${\mathcal{G}}_{\mathbb{C}}$-invariant submanifold $\gamma$ of
${\mathcal{X}}$. This submanifold is essentially a Lefschetz thimble: $\gamma$
is the union of all gradient flows generated by the real part of
${\mathcal{W}}/\hbar$, terminating on the ${\mathcal{G}}_{\mathbb{C}}$-orbit
of a chosen critical point of ${\mathcal{W}}$. (For simplicity, we assume that
the critical points of ${\mathcal{W}}$ are nondegenerate up to the
${\mathcal{G}}_{\mathbb{C}}$-action.) The localized path integral takes the
form

$\int_{\gamma/{\mathcal{G}}_{\mathbb{C}}}\exp\biggl{(}\frac{2\pi}{\hbar}{\mathcal{W}}\biggr{)}{\mathcal{O}}\,,$
(116)

where ${\mathcal{O}}$ descends from the $Q$-invariant observables inserted in
the path integral. This is the path integral for a zero-dimensional gauge
theory with gauge group ${\mathcal{G}}_{\mathbb{C}}$ and target space
$\gamma$. The action functional is $-2\pi{\mathcal{W}}/\hbar$.

The remarkable aspect of this localization phenomenon is that the gauge group
gets complexified. In the localization process, some fermionic fields have
zero modes. They may be regarded as ghost fields for partial gauge fixing that
breaks ${\mathcal{G}}_{\mathbb{C}}$ down to ${\mathcal{G}}$. Even though the
action functional is holomorphic and its real part is not bounded from below,
the integral (116) can converge since
$\mathop{\mathrm{Re}}\nolimits({\mathcal{W}}/\hbar)$ gets smaller and smaller
along the backward gradient flows in $\gamma$.

For the six-dimensional ${\mathcal{N}}=(1,1)$ super Yang–Mills theory on
$\mathbb{R}\times S^{1}\times\mathbb{C}\times\mathbb{R}^{2}_{+\hbar}$, the
gauge group ${\mathcal{G}}$ is the space of maps from $\mathbb{R}\times
S^{1}\times\mathbb{C}$ to $\mathrm{U}(m)$. In addition to the vector
multiplet, the theory has three chiral multiplets in the adjoint
representation of ${\mathcal{G}}$. Their scalar fields are $Q$-invariant and
can be combined into a one-form on $\mathbb{R}\times S^{1}\times\mathbb{C}$:

${\mathcal{A}}=(A_{x}+\mathrm{i}X)\mathrm{d}x+(A_{y}+\mathrm{i}Y)\mathrm{d}y+A_{\bar{z}}\,\mathrm{d}{\bar{z}}\,.$
(117)

Here, $A_{x}$, $A_{y}$ are the components of the gauge field along
$\mathbb{R}\times S^{1}$, $A_{\bar{z}}$ is the antiholomorphic component of
the gauge field along $\mathbb{C}$, and $X$, $Y$ are two of the four scalar
fields of the six-dimensional theory associated to motions along
$\mathbb{R}_{X}$ and $\mathbb{R}_{Y}$, respectively. The superpotential is
given by

${\mathcal{W}}=-\frac{\mathrm{i}}{e^{2}}\int_{\mathbb{R}\times
S^{1}\times\mathbb{C}}\mathrm{d}z\wedge\mathop{\mathrm{tr}}\nolimits\biggl{(}{\mathcal{A}}\wedge\mathrm{d}{\mathcal{A}}+\frac{2}{3}{\mathcal{A}}\wedge{\mathcal{A}}\wedge{\mathcal{A}}\biggr{)}\,,$
(118)

where $e$ is the gauge coupling and $\mathop{\mathrm{tr}}\nolimits$ is an
invariant symmetric bilinear form on the Lie algebra of $\mathrm{U}(m)$, which
we can take to be the trace in the defining representation.

According to the localization argument, the $\Omega$-deformation of the six-
dimensional theory is equivalent to a zero-dimensional gauge theory. This
zero-dimensional theory has infinite-dimensional target space and gauge group,
and can be more naturally interpreted as a four-dimensional gauge theory. Its
action is

$-\frac{2\pi\mathrm{i}}{\hbar
e^{2}}\int_{\Sigma\times\mathbb{C}}\mathrm{d}z\wedge\mathop{\mathrm{tr}}\nolimits_{\mathbb{C}^{m}}\biggl{(}{\mathcal{A}}^{00}\wedge\mathrm{d}{\mathcal{A}}^{00}+\frac{2}{3}{\mathcal{A}}^{00}\wedge{\mathcal{A}}^{00}\wedge{\mathcal{A}}^{00}\biggr{)}\,.$
(119)

Here, we have written the partial $\mathfrak{gl}(m)$ connection (117) as
${\mathcal{A}}^{00}$ to emphasize its place in the Lie superalgebra
$\mathfrak{gl}(m|n)$ that will arise later. This is the action for four-
dimensional Chern–Simons theory. Thus we conclude that the $Q$-invariant
sector of the $\Omega$-deformed six-dimensional ${\mathcal{N}}=(1,1)$ super
Yang–Mills theory on $\mathbb{R}\times
S^{1}\times\mathbb{C}\times\mathbb{R}^{2}_{+\hbar}$ with gauge group
$\mathrm{U}(m)$ is equivalent to four-dimensional Chern–Simons theory on
$\mathbb{R}\times S^{1}\times\mathbb{C}$ with gauge group $\mathrm{GL}(m)$ and
coupling given by $\hbar$.

Similarly, if we consider the case $m=0$, the worldvolume theory on the
D5-branes is an $\Omega$-deformed six-dimensional ${\mathcal{N}}=(1,1)$ super
Yang–Mills theory on $\mathbb{R}\times
S^{1}\times\mathbb{C}\times\mathbb{R}^{2}_{-\hbar}$ with gauge group
$\mathrm{U}(n)$. Its $Q$-invariant sector is equivalent to four-dimensional
Chern–Simons theory on $\mathbb{R}\times S^{1}\times\mathbb{C}$ with gauge
group $\mathrm{GL}(n)$ and action

$+\frac{2\pi\mathrm{i}}{\hbar e^{2}}\int_{\mathbb{R}\times
S^{1}\times\mathbb{C}}\mathrm{d}z\wedge\mathop{\mathrm{tr}}\nolimits_{\mathbb{C}^{n}}\biggl{(}{\mathcal{A}}^{11}\wedge\mathrm{d}{\mathcal{A}}^{11}+\frac{2}{3}{\mathcal{A}}^{11}\wedge{\mathcal{A}}^{11}\wedge{\mathcal{A}}^{11}\biggr{)}\,,$
(120)

with the partial $\mathfrak{gl}(n)$ connection ${\mathcal{A}}^{11}$ defined in
the same way as ${\mathcal{A}}^{00}$.

####  3.2.2 Case with nonzero $m$ and $n$

Let us turn to the case in which $m$ and $n$ are both nonzero. In this case,
the two sets of D5-branes lead to two copies of four-dimensional Chern–Simons
theory on $\mathbb{R}\times S^{1}\times\mathbb{C}$, one with gauge group
$\mathrm{GL}(m)$ and the other with gauge group $\mathrm{GL}(n)$, with
opposite couplings. Arranging ${\mathcal{A}}^{00}$ and ${\mathcal{A}}^{11}$
into a matrix

${\mathcal{A}}^{0}=\begin{pmatrix}{\mathcal{A}}^{00}&0\\\
0&{\mathcal{A}}^{11}\end{pmatrix}\,,$ (121)

we can write the sum of their action functionals as

$-\frac{2\pi\mathrm{i}}{\hbar e^{2}}\int_{\mathbb{R}\times
S^{1}\times\mathbb{C}}\mathrm{d}z\wedge\mathop{\mathrm{str}}\nolimits_{\mathbb{C}^{m|n}}\biggl{(}{\mathcal{A}}^{0}\wedge\mathrm{d}{\mathcal{A}}^{0}+\frac{2}{3}{\mathcal{A}}^{0}\wedge{\mathcal{A}}^{0}\wedge{\mathcal{A}}^{0}\biggr{)}\,.$
(122)

The two copies are coupled through strings stretched between the two sets of
D5-branes. These strings produce a four-dimensional ${\mathcal{N}}=2$
hypermultiplet on $\mathbb{R}\times S^{1}\times\mathbb{C}$ in the
bifundamental representation of $\mathrm{GL}(m)\times\mathrm{GL}(n)$. It
consists of bosonic complex scalars

$\displaystyle q$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m},\mathbb{C}^{n})\,,$
(123) $\displaystyle\tilde{q}^{\dagger}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m},\mathbb{C}^{n})\,,$
(124) $\displaystyle q^{\dagger}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{n},\mathbb{C}^{m})\,,$
(125) $\displaystyle\tilde{q}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{n},\mathbb{C}^{m})\,,$
(126)

and fermionic Weyl spinors

$\displaystyle\psi$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m},\mathbb{C}^{n})\,,$
(127) $\displaystyle{\tilde{\psi}}^{\dagger}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m},\mathbb{C}^{n})\,,$
(128) $\displaystyle\psi^{\dagger}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{n},\mathbb{C}^{m})\,,$
(129) $\displaystyle{\tilde{\psi}}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{n},\mathbb{C}^{m})\,.$
(130)

In the absence of coupling to the two copies of four-dimensional Chern–Simons
theory, the bifundamental hypermultiplet preserves eight supercharges. The
supercharge $Q$ is a linear combination of these supercharges such that the
generators of translations on $\mathbb{R}\times S^{1}$ and antiholomorphic
translations on $\mathbb{C}$ are $Q$-exact. By redefining fields if necessary,
we can take $Q$ to be the supercharge used in the holomorphic–topological
twist studied in [44], with the parameter $t=i$.

It turns out that most of the action for the hypermultiplet is $Q$-exact. The
remaining part of the action can be expressed in a suggestive form. Endow the
cylinder $\mathbb{R}\times S^{1}$ with a complex coordinate $w$, and define

$\displaystyle{\mathcal{A}}^{10}$
$\displaystyle=-{\tilde{\psi}}^{\dagger}_{-}\mathrm{d}w+\psi_{-}\mathrm{d}{\bar{w}}-\frac{1}{2}({\tilde{\psi}}_{+}^{\dagger}-\psi_{+})\mathrm{d}{\bar{z}}\,,$
(131) $\displaystyle{\mathcal{A}}^{01}$
$\displaystyle=\psi^{\dagger}_{-}\mathrm{d}w+{\tilde{\psi}}_{-}\mathrm{d}{\bar{w}}+\frac{1}{2}({\tilde{\psi}}_{+}^{\dagger}+\psi^{\dagger}_{+})\mathrm{d}{\bar{z}}$
(132)

and

$\displaystyle c^{10}$ $\displaystyle=4\mathrm{i}q^{\dagger}\,,$ (133)
$\displaystyle c^{01}$ $\displaystyle=4\tilde{q}^{\dagger}\,,$ (134)
$\displaystyle b^{10}$ $\displaystyle=\tilde{q}\,,$ (135) $\displaystyle
b^{01}$ $\displaystyle=-\mathrm{i}q\,,$ (136) $\displaystyle B^{10}$
$\displaystyle=\mathrm{i}({\tilde{\psi}}_{+}-\psi^{\dagger}_{+})\,,$ (137)
$\displaystyle B^{01}$
$\displaystyle=-\mathrm{i}(\psi_{+}+{\tilde{\psi}}^{\dagger}_{+})\,.$ (138)

We introduce a matrix

${\mathcal{A}}^{1}=\begin{pmatrix}0&{\mathcal{A}}^{01}\\\
{\mathcal{A}}^{10}&0\end{pmatrix}$ (139)

and matrices $c^{1}$, $b^{1}$, $B^{1}$ defined likewise. On these matrices $Q$
acts by

$\displaystyle Q\cdot{\mathcal{A}}^{1}$
$\displaystyle=-\mathrm{d}^{\prime}c^{1}\,,$ (140) $\displaystyle Q\cdot
c^{1}$ $\displaystyle=0\,,$ (141) $\displaystyle Q\cdot b^{1}$
$\displaystyle=B^{1}\,,$ (142) $\displaystyle Q\cdot B^{1}$
$\displaystyle=0\,,$ (143)

where

$\mathrm{d}^{\prime}=\mathrm{d}-\mathrm{d}z\,\partial_{z}=\mathrm{d}w\,\partial_{w}+\mathrm{d}{\bar{w}}\,\partial_{\bar{w}}+\mathrm{d}{\bar{z}}\,\partial_{\bar{z}}\,.$
(144)

The non-$Q$-exact part of the action is

$-\frac{2\pi\mathrm{i}}{\hbar e^{2}}\int_{\mathbb{R}\times
S^{1}\times\mathbb{C}}\mathrm{d}z\wedge\mathop{\mathrm{str}}\nolimits_{\mathbb{C}^{m|n}}({\mathcal{A}}^{1}\wedge\mathrm{d}{\mathcal{A}}^{1})\,.$
(145)

(Since this is quadratic in fermions, the prefactor is inessential.)

To describe the intersecting D5-branes, we couple this bifundamental
hypermultiplet to the two copies of four-dimensional Chern–Simons theory by
identifying the flavor groups $\mathrm{GL}(m)$ and $\mathrm{GL}(n)$ with the
gauge groups of the latter. Concretely, we replace the de Rham differential
that appears in the above formulas with the gauge-covariant differential

$\mathrm{d}_{{\mathcal{A}}^{0}}=\mathrm{d}+{\mathcal{A}}^{0}\,.$ (146)

Thus, the action of $Q$ on the fields is modified to

$\displaystyle Q\cdot{\mathcal{A}}^{1}$
$\displaystyle=-\mathrm{d}^{\prime}_{{\mathcal{A}}^{0}}c^{1}\,,$ (147)
$\displaystyle Q\cdot c^{1}$ $\displaystyle=0\,,$ (148) $\displaystyle Q\cdot
b^{1}$ $\displaystyle=B^{1}\,,$ (149) $\displaystyle Q\cdot B^{1}$
$\displaystyle=0\,,$ (150)

and the action functional for the bifundamental hypermultiplet becomes

$-\frac{2\pi\mathrm{i}}{\hbar e^{2}}\int_{\mathbb{R}\times
S^{1}\times\mathbb{C}}\mathrm{d}z\wedge\mathop{\mathrm{str}}\nolimits_{\mathbb{C}^{m|n}}({\mathcal{A}}^{1}\wedge\mathrm{d}_{{\mathcal{A}}^{0}}{\mathcal{A}}^{1})\,.$
(151)

Combining ${\mathcal{A}}^{0}$ and ${\mathcal{A}}^{1}$ into a single matrix

${\mathcal{A}}={\mathcal{A}}^{0}+{\mathcal{A}}^{1}=\begin{pmatrix}{\mathcal{A}}^{00}&{\mathcal{A}}^{01}\\\
{\mathcal{A}}^{10}&{\mathcal{A}}^{11}\end{pmatrix}\,,$ (152)

we can write the total action, which is the sum of the actions (122) and
(151), as

$-\frac{2\pi\mathrm{i}}{\hbar e^{2}}\int_{\mathbb{R}\times
S^{1}\times\mathbb{C}}\mathrm{d}z\wedge\mathop{\mathrm{str}}\nolimits_{\mathbb{C}^{m|n}}\biggl{(}{\mathcal{A}}\wedge\mathrm{d}{\mathcal{A}}+\frac{2}{3}{\mathcal{A}}\wedge{\mathcal{A}}\wedge{\mathcal{A}}\biggr{)}\,.$
(153)

This is the action for four-dimensional Chern–Simons theory with gauge group
$\mathrm{GL}(m|n)$.

Before concluding that we have obtained the desired theory, we need to solve
two problems. First, although the above action is invariant under
$\mathrm{GL}(m|n)$ gauge transformation, the gauge group of the theory is
still $\mathrm{GL}(m)\times\mathrm{GL}(n)$, not $\mathrm{GL}(m|n)$. Second,
the gauge-invariant action (151) for the bifundamental hypermultiplet is not
$Q$-invariant due to the coupling to ${\mathcal{A}}^{0}$. Its $Q$-variation
gives

$-\frac{4\pi\mathrm{i}}{\hbar e^{2}}\int_{\mathbb{R}\times
S^{1}\times\mathbb{C}}\mathrm{d}z\wedge\mathop{\mathrm{str}}\nolimits_{\mathbb{C}^{m|n}}({\mathcal{A}}^{1}\wedge[{\mathcal{F}}^{0},c^{1}])\,,$
(154)

where ${\mathcal{F}}^{0}$ is the curvature of ${\mathcal{A}}^{0}$.

The two problems are solved simultaneously if we correct the $Q$-action on
${\mathcal{A}}^{0}$ and $B^{1}$ to

$\displaystyle Q\cdot{\mathcal{A}}^{0}$
$\displaystyle=\\{{\mathcal{A}}^{1},c^{1}\\}\,,$ (155) $\displaystyle Q\cdot
B^{1}$ $\displaystyle=\frac{1}{2}[\\{c^{1},c^{1}\\},b^{1}]\,.$ (156)

With this modification, the $Q$-variation of the bosonic action (122) cancels
that of the fermionic action (151). At the same time, $c^{1}$, $b^{1}$ and
$B^{1}$ can now be interpreted as a ghost, an antighost and an auxiliary field
used in the Becchi–Rouet–Stora–Tyutin (BRST) procedure for partial gauge
fixing of $\mathrm{GL}(m|n)$ down to $\mathrm{GL}(m)\times\mathrm{GL}(n)$
[45].

To make the last point more explicit, let us introduce a ghost $c^{0}$, an
antighost $b^{0}$ and an auxiliary field $B^{0}$ for gauge fixing of
$\mathrm{GL}(m)\times\mathrm{GL}(n)$. The BRST charge $Q_{\mathrm{B}}$ acts on
the fields by

$\displaystyle Q_{\mathrm{B}}\cdot{\mathcal{A}}^{0}$
$\displaystyle=-\mathrm{d}_{{\mathcal{A}}_{0}}^{\prime}c^{0}\,,$ (157)
$\displaystyle Q_{\mathrm{B}}\cdot c^{0}$
$\displaystyle=\frac{1}{2}\\{c^{0},c^{0}\\}\,,$ (158) $\displaystyle
Q_{\mathrm{B}}\cdot b^{0}$ $\displaystyle=B^{0}\,,$ (159) $\displaystyle
Q_{\mathrm{B}}\cdot B^{0}$ $\displaystyle=0$ (160)

and

$\displaystyle Q_{\mathrm{B}}\cdot{\mathcal{A}}^{1}$
$\displaystyle=\\{c^{0},{\mathcal{A}}^{1}\\}\,,$ (161) $\displaystyle
Q_{\mathrm{B}}\cdot c^{1}$ $\displaystyle=[c^{0},c^{1}]\,,$ (162)
$\displaystyle Q_{\mathrm{B}}\cdot b^{1}$ $\displaystyle=[c^{0},b^{1}]\,,$
(163) $\displaystyle Q_{\mathrm{B}}\cdot B^{1}$
$\displaystyle=\\{c^{0},B^{1}\\}\,.$ (164)

Let us postulate that

$\displaystyle Q\cdot c^{0}$ $\displaystyle=-\frac{1}{2}\\{c^{1},c^{1}\\}\,,$
(165) $\displaystyle Q\cdot b^{0}$ $\displaystyle=0\,,$ (166) $\displaystyle
Q\cdot B^{0}$ $\displaystyle=0\,.$ (167)

Then, the modified BRST charge

${\widehat{Q}}=Q_{\mathrm{B}}+Q$ (168)

satisfies ${\widehat{Q}}^{2}=0$ and

$\displaystyle{\widehat{Q}}\cdot{\mathcal{A}}$
$\displaystyle=-\mathrm{d}_{\mathcal{A}}^{\prime}c\,,$ (169)
$\displaystyle{\widehat{Q}}\cdot c$
$\displaystyle=\frac{1}{2}\\{c^{0},c^{0}\\}-\frac{1}{2}\\{c^{1},c^{1}\\}+[c^{0},c^{1}]\,,$
(170) $\displaystyle{\widehat{Q}}\cdot b$ $\displaystyle=B\,,$ (171)
$\displaystyle{\widehat{Q}}\cdot B$ $\displaystyle=0\,,$ (172)

where

$\displaystyle c$ $\displaystyle=c^{0}+c^{1}\,,$ (173) $\displaystyle b$
$\displaystyle=b^{0}+b^{1}\,,$ (174) $\displaystyle B$
$\displaystyle=B^{0}+B^{1}+[c^{0},b^{1}]\,.$ (175)

The ${\widehat{Q}}$-cohomology computes the mixed Lie superalgebra cohomology
defined in [46].

Thus, we conclude that the $Q$-invariant sector of the theory on the
intersecting D5-branes is four-dimensional Chern–Simons theory with gauge
group $\mathrm{GL}(m|n)$.

###  3.3 Emergence of the spin chain

The D3-branes and the F1-branes in the type IIB setup (114) intersect the
D5-branes along lines in $\mathbb{R}\times S^{1}\times\mathbb{C}$. As such,
they create line defects in the four-dimensional Chern–Simons theory on
$\mathbb{R}\times S^{1}\times\mathbb{C}$, extending along $\mathbb{R}$ and
supported at points in $\mathbb{C}$. Such a configuration of line defects in
four-dimensional Chern–Simons theory is naturally identified with an
integrable spin chain [17, 18, 19]. We now show that this spin chain is
precisely the one that appears in the Bethe/gauge correspondence.

####  3.3.1 Line defects and spin chains

Let us first explain the relation between line defects in four-dimensional
Chern–Simons theory and integrable spin chains.

Consider four-dimensional Chern–Simons theory on
$\mathbb{R}\times\mathbb{R}\times\mathbb{C}$, with gauge group $G$ which we
take to be a complex simple Lie supergroup. Its field is a partial
$G$-connection of the form

${\mathcal{A}}={\mathcal{A}}_{x}\,\mathrm{d}x+{\mathcal{A}}_{y}\,\mathrm{d}y+{\mathcal{A}}_{\bar{z}}\,\mathrm{d}{\bar{z}}\,.$
(176)

We insert line defects

${\mathcal{L}}^{\ell}\,,\quad\ell=1,\dotsc,L\,,$ (177)

extending in the $x$-direction, which we regard as the time direction. Along
the $y$-axis, we arrange ${\mathcal{L}}^{1}$, $\dotsc$, ${\mathcal{L}}^{L}$ in
the ascending order. They are supported at points $\zeta^{1}$, $\dotsc$,
$\zeta^{L}$ in $\mathbb{C}$.

Solutions of the equation of motion for four-dimensional Chern–Simons theory,
away from the line defects, are flat connections. Away from the line defects,
flat connections can be gauged away. Then, all information about the state of
the theory is localized in the neighborhoods of the line defects, and the
Hilbert space $V$ factorises into the tensor product of the spaces attached to
the line defects:

$V=V^{1}\otimes\dotsb\otimes V^{L}\,.$ (178)

This is identified with the Hilbert space of an open spin chain with $L$
sites. The space $V^{\ell}$ supported on ${\mathcal{L}}^{\ell}$ is the state
space for the $\ell$th spin.

The four-dimensional Chern–Simons theory is topological on
$\mathbb{R}\times\mathbb{R}$ and holomorphic on $\mathbb{C}$. Due to the
topological invariance on $\mathbb{R}\times\mathbb{R}$, the Hamiltonian is
zero. To change the state, we can insert a Wilson line extending in the
$y$-direction, crossing the $L$ line defects introduced earlier. This Wilson
line is a non-gauge-invariant operator acting on $V$ and interpreted as a
monodromy matrix $T(\sigma)$ in the spin chain. The spectral parameter
$\sigma$ is the position of the Wilson line in $\mathbb{C}$, and the
holomorphy on $\mathbb{C}$ implies that $T(\sigma)$ is holomorphic in
$\sigma$.

If we introduce two Wilson lines and make them intersect in
$\mathbb{R}\times\mathbb{R}$, we get an R-matrix at the intersection. The two
sides of the RTT relation (33) correspond to two different configurations of
two open Wilson lines crossing each other and the line defects
${\mathcal{L}}^{1}$, $\dotsc$, ${\mathcal{L}}^{L}$. The topological invariance
on $\mathbb{R}\times\mathbb{R}$ and the existence of the extra dimensions
$\mathbb{C}$ imply the equivalence of the two configurations.

This R-matrix can be computed by perturbation theory [19], and it was found to
be the R-matrix for the rational spin chain with $G$ symmetry. Therefore, this
setup produces an open rational spin chain. For $G=\mathrm{GL}(m|n)$, the
R-matrix is the one given in (34), up to some equivalence relations.

To obtain a closed spin chain, we simply compactify the $y$-axis $\mathbb{R}$
to $S^{1}$. Now, flat connections have global gauge-invariant information,
namely the holonomy around $S^{1}$. The holonomy is fixed by the boundary
condition at infinity and becomes a parameter of the spin chain. Flat
connections can still be gauged away almost everywhere. Away from the line
defects, we can make them vanish except on a single line parallel to the line
defects and placed between ${\mathcal{L}}^{L}$ and ${\mathcal{L}}^{1}$, say.
The holonomy is then identified with the twist parameter $g$ of the periodic
boundary condition in the spin chain. A Wilson loop winding around $S^{1}$
gives a transfer matrix $t(g,\sigma)$ evaluated in the representation of the
Wilson loop.

####  3.3.2 Line defects created by D3-branes

Let us return to the setup for the Bethe/gauge correspondence.

By the topological invariance on $\mathbb{R}\times S^{1}$, the positions of
the D3-branes on $S^{1}$ do not matter. For each $\ell$, we gather the $m+n$
D3-branes $\mathrm{D3}_{i}^{\ell}$, $i=1$, $\dotsc$, $m+n$, to the same
position $y^{\ell}$ on $S^{1}$:

$y_{1}^{\ell}=\dotsb=y_{m+n}^{\ell}=y^{\ell}\,.$ (179)

Since they are also located at the same point $\zeta^{\ell}$ in $\mathbb{C}$
up to first order in $\hbar$, we can regard them as creating a single line
defect ${\mathcal{L}}^{\ell}$ supported on the line
$\mathbb{R}\times\\{y^{\ell}\\}\times\\{\zeta^{\ell}\\}$ in $\mathbb{R}\times
S^{1}\times\mathbb{C}$, treating the differences
$\mu_{i}^{\ell}-\zeta^{\ell}$, $i=1$, $\dotsc$, $m+n$, in the positions in
$\mathbb{C}$ as parameters of the line defect. The D3-branes thus create $L$
line defects ${\mathcal{L}}^{1}$, $\dotsc$, ${\mathcal{L}}^{L}$.

From the Bethe/gauge correspondence we know what the Hilbert space $V^{\ell}$
of ${\mathcal{L}}^{\ell}$ must be: it is the evaluation module of the Yangian
$Y(\mathfrak{gl}(m|n))$ with spectral parameter $\zeta^{\ell}$, obtained from
the Verma module $V_{\lambda^{\ell}}$ of $\mathfrak{gl}(m|n)$. The F1-branes
represent excitations in this Hilbert space.

Let us derive this Hilbert space from the point of view of brane dynamics. In
the four-dimensional Chern–Simons theory on $\mathbb{R}\times
S^{1}\times\mathbb{C}$, the positions of the D5-branes in $\mathbb{R}_{X}$ and
$\mathbb{R}_{Y}$ parametrize the vacuum expectation values of the gauge fields
along the topological directions. For the purpose of identifying the Hilbert
space of the line defect, we can consider the situation in which all D5-branes
are coincident, say $X_{i}=Y_{i}=0$ for all $i$.

The $Q$-invariant sector of the theory living on the D3-branes that create
${\mathcal{L}}^{\ell}$ is the BF theory with gauge group $G=\mathrm{GL}(m|n)$,
defined on $\mathbb{R}\times[0,\infty)$. This theory can be obtained from
four-dimensional Chern–Simons theory on
$\mathbb{R}\times[0,\infty)\times\mathbb{C}$ by dimensional reduction on
$\mathbb{C}$. It has the action

$\frac{1}{\hbar}\int_{\mathbb{R}\times[0,\infty)}\sigma(\mathrm{d}\eta+\eta\wedge\eta)\,,$
(180)

where $\eta$ is the gauge field and $\sigma$ is a scalar field valued in
$\mathfrak{g}^{*}$, the dual of the Lie algebra
$\mathfrak{g}=\mathfrak{gl}(m|n)$. The field $\sigma$ comes from the reduction
of the antiholomorphic component $A_{\bar{z}}$ of the four-dimensional gauge
field on $\mathbb{C}$.

One way to see that this is the right theory is to note that upon exchanging
the directions of $S^{1}$ and $\mathbb{R}_{Y}$, the D3-branes are T-dual to
D5-branes on $\mathbb{R}\times
S^{1}\times\mathbb{C}\times\mathbb{R}^{2}_{\pm\hbar}$ (after compactifying
$\mathbb{C}$ to a torus). Since the D5-branes are described by four-
dimensional Chern–Simons theory with gauge group $\mathrm{GL}(m|n)$, the
D3-branes are described by the BF theory. The exchange of two directions
amounts to swapping the role of the real and imaginary parts of $A_{y}$, and
does not alter the analysis in any essential way.

We view the BF theory as a Poisson sigma model [47, 48] with target space
$\mathfrak{g}^{*}$. The space of functions on $\mathfrak{g}^{*}$ is the
symmetric algebra $S(\mathfrak{g})$ of $\mathfrak{g}$. Linear functions are
elements of $\mathfrak{g}$, and the Poisson bracket between them is given by
the Lie bracket. Extending the Poisson bracket to $S(\mathfrak{g})$ by the
Leibniz rule, we endow $\mathfrak{g}^{*}$ with the Poisson structure. The
action (180) of the BF theory is related to the action of the Poisson sigma
model by integration by parts.

There are two boundaries in the spacetime
$\mathbb{R}\times[0,\infty)\subset\mathbb{R}\times\mathbb{R}_{Y}$, one at
$Y=0$ and the other at $Y=\infty$. It is more convenient to think of the
spacetime of the theory as the limit of

$\mathbb{R}\times[0,r]$ (181)

as $r\to\infty$. Physically, we can realize this setup by making the D3-brane
$\mathrm{D3}_{i}^{\ell}$ end on an NS5-brane $\mathrm{NS5}_{i}^{\ell}$ with
worldvolume

$\mathbb{R}\times\\{y^{\ell}\\}\times\\{\mu^{\ell}_{i}\\}\times\mathbb{R}_{X}\times\\{r\\}\times\mathbb{R}^{2}_{+\hbar}\times\mathbb{R}^{2}_{-\hbar}\subset\mathbb{R}\times
S^{1}\times\mathbb{C}\times\mathbb{R}_{X}\times\mathbb{R}_{Y}\times\mathbb{R}^{2}_{+\hbar}\times\mathbb{R}^{2}_{-\hbar}\,.$
(182)

Since the BF theory is topological, the value of $r$ does not matter. In
particular, we can take the limit $r\to 0$. In this limit the theory reduces
to a one-dimensional quantum mechanical system. This quantum mechanical system
describes the line defect after coupling to the four-dimensional Chern–Simons
theory.

The Hilbert space of the BF theory depends on the boundary conditions imposed
on the two boundaries $\mathbb{R}\times\\{0\\}$ and $\mathbb{R}\times\\{r\\}$.
On each boundary, we impose a boundary condition that defines what is known as
a coisotropic brane in the context of Poisson sigma models [49].

The boundary condition on $\mathbb{R}\times\\{0\\}$ is simple. The imaginary
part of the component $\eta_{x}$ of the gauge field along $\mathbb{R}$
parametrizes the positions of the D3-branes in $\mathbb{R}_{X}$. These are
necessarily fixed to the positions of the D5-branes on the boundary where the
D3-branes end on the D5-branes. By holomorphy, the real part must also obey
the Dirichlet boundary condition. Thus we have

$\eta|_{\mathbb{R}\times\\{0\\}}=0\,.$ (183)

The field $\sigma$ is unconstrained on the boundary. This boundary condition
completely breaks the gauge symmetry. The global symmetry $G$ on the boundary
is used for coupling to the four-dimensional Chern–Simons theory by gauging.

We propose that the boundary condition on $\mathbb{R}\times\\{r\\}$ is
determined by the positions of the NS5-branes in $\mathbb{C}$ as follows. The
diagonal part of $\sigma$ parametrizes these positions. Let

$\mathfrak{g}=\mathfrak{n_{-}}\oplus\mathfrak{h}\oplus\mathfrak{n_{+}}$ (184)

be the triangular decomposition of $\mathfrak{g}$ with respect to the chosen
basis; thus $\mathfrak{h}$ is spanned by diagonal matrices, and
$\mathfrak{n}_{+}$ and $\mathfrak{n}_{-}$ are spanned by strictly upper
triangular matrices and strictly lower triangular matrices, respectively.
Then, the boundary condition is

$\displaystyle\eta|_{\mathbb{R}\times\\{r\\}}$
$\displaystyle\in\mathfrak{b}\,,$ (185)
$\displaystyle\sigma|_{\mathbb{R}\times\\{r\\}}$
$\displaystyle\in\mathfrak{n}_{-}^{*}+\underline{\lambda}^{\ell}\,,$ (186)

where $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}_{+}$ is the Borel
subalgebra and $\underline{\lambda}^{\ell}$ is an element of
$\mathfrak{h}^{*}$. This condition breaks the gauge group on the boundary to
the Borel subgroup $B$ whose Lie algebra is $\mathfrak{b}$.555The choice of
Borel subgroup is determined by the positions of the D5-branes in
$\mathbb{R}_{X}$, which are in turn given by the vacuum expectation value of
the time component ${\mathcal{A}}_{x}=A_{x}+\mathrm{i}X$ of the gauge field;
the $\mathfrak{u}(m|n)$-valued field $X$ has the vacuum expectation value
$\langle X\rangle=\sum_{i=1}^{m+n}\mathrm{i}X_{i}{\mathcal{E}}_{ii}$. In this
background, a state evolving for duration $T$ is scaled by the factor
$\exp(T\langle{\mathcal{A}}_{x}\rangle)=\exp(-T\sum_{i=1}^{m+n}X_{i}{\mathcal{E}}_{ii})$.
(We have set $\langle A_{x}\rangle=0$ for simplicity.) Therefore, if we
compactify the time direction to a circle (say, of radius $1$), as one does
when computing the partition function of the lattice model equivalent to the
spin chain, then the periodic boundary condition is twisted by the action of
$\exp(-\sum_{i=1}^{m+n}X_{i}{\mathcal{E}}_{ii})$. In the magnon sector
$(M_{1},\dotsc,M_{m+n-1})$, this action is multiplication by
$\exp(-\sum_{\ell=1}^{L}\sum_{i=1}^{m+n}\lambda_{i}^{\ell}X_{i}+\sum_{r=1}^{m+n-1}M_{r}(X_{r}-X_{r+1}))$.
We must have the ordering (107) for the partition function to be a power
series in small variables.

By the state–operator correspondence, the Hilbert space is isomorphic to the
space of observables supported at the junction of the above two coisotropic
branes. This is a bimodule over the algebras of local observables on the two
boundaries.

Let $\\{T_{\alpha}\\}_{\alpha=1}^{\dim\mathfrak{n}_{-}}$ be a basis of
$\mathfrak{n}_{-}$ and extend it a basis
$\\{T_{a}\\}_{a=1}^{\dim\mathfrak{g}}$ of $\mathfrak{g}$. Let
$\sigma_{a}=\langle T_{a},\sigma\rangle$. Classically, the algebra of local
observables is the algebra of gauge-invariant polynomials in
$\\{\sigma_{a}\\}$.

On the boundary $\mathbb{R}\times\\{0\\}$, local observables are simply
polynomials in $\\{\sigma_{a}\\}$ since there is no gauge symmetry there. The
algebra of local observables on the boundary is therefore $S(\mathfrak{g})$ at
the classical level. Quantum corrections lead to a noncommutative deformation
of $S(\mathfrak{g})$. At the quantum level, the algebra is isomorphic to the
universal enveloping algebra of $\mathfrak{g}$ with bracket $\hbar[-,-]$; if
$[T_{a},T_{b}]=\sum_{c=1}^{\dim\mathfrak{g}}f_{ab}{}^{c}T_{c}$, then
$[\sigma_{a},\sigma_{b}]=\hbar\sum_{c=1}^{\dim\mathfrak{g}}f_{ab}{}^{c}\sigma_{c}$
quantum mechanically [50, 51]. In our setup $\hbar$ is a complex parameter
rather than a formal parameter, so the algebra is isomorphic to
$U(\mathfrak{g})$, with $\sigma_{a}$ mapped to $\hbar T_{a}$. (The
quantization map from $S(\mathfrak{g})$ to $U(\mathfrak{g})$ is complicated
for polynomials of higher degree.)

On the boundary $\mathbb{R}\times\\{r\\}$, the algebra of local observables is
trivial. The boundary condition says that local operators are constructed
entirely from $\\{\sigma_{\alpha}\\}$. Nontrivial polynomials cannot commute
with $\mathfrak{h}$ and, in particular, cannot be $B$-invariant. The only
local observables are multiples of the identity operator.

At the junction, the two boundary conditions combined imply that observables
are polynomials in $\\{\sigma_{\alpha}\\}$. As a vector space, the space of
observables is generated from the “highest-weight vector” $1$ by the action of
“creation operators” $\\{\sigma_{\alpha}\\}$. On this vector space the algebra
$U(\mathfrak{g})$ acts. Thus, the Hilbert space is a Verma module of
$U(\mathfrak{g})$, as expected.

It remains to show that the highest weight of the Verma module is determined
by the positions of the D3-branes. Classically, the boundary condition on
$\mathbb{R}\times\\{r\\}$ implies that the highest weight of the module is
$\underline{\lambda}^{\ell}$. There is, however, a quantum correction which
shifts the highest weight.666This shift can be understood as originating from
normal ordering of creation and annihilation operators, which correspond to
the positive and negative roots. We will see a similar shift in section 4.1.
The highest weight is actually

$\lambda^{\ell}=\underline{\lambda}^{\ell}-\rho\,,$ (187)

where $\rho$ is the Weyl vector defined in terms of the character of the
$\mathfrak{b}$-module $\mathfrak{g}/\mathfrak{b}$ as

$\rho(-)=-\frac{1}{2}\mathop{\mathrm{str}}\nolimits_{\mathfrak{g}/\mathfrak{b}}\mathop{\mathrm{ad}}\nolimits(-)\,.$
(188)

Since $\rho(\mathfrak{n}_{+})=\rho([\mathfrak{b},\mathfrak{b}])=0$, $\rho$ can
be regarded as an element of $\mathfrak{h}^{*}$. This is the graded half sum
of positive roots. When $\mathfrak{g}$ is an ordinary Lie algebra, $\rho$ is
the ordinary Weyl vector and the above shift was derived in [52] based on
results from [53, 54].

For $\mathfrak{g}=\mathfrak{gl}(m|n)$, we have

$\rho=\sum_{\begin{subarray}{c}k,l=1\\\
k<l\end{subarray}}^{m+n}\frac{1}{2}(-1)^{[k]+[l]}(\varepsilon_{k}-\varepsilon_{l})=\sum_{i=1}^{m+n}\frac{1}{2}(-1)^{[i]}(m-n+(-1)^{[i]}-2c_{i})\varepsilon_{i}\,.$
(189)

Comparing the relations (187) and (306), we find

$(-1)^{[i]}\biggl{(}\underline{\lambda}_{i}^{\ell}-\frac{1}{4}\biggr{)}=\frac{1}{\hbar}(\zeta^{\ell}-\mu_{i}^{\ell})+\frac{1}{2}(m-n-
c_{i})\,.$ (190)

####  3.3.3 Line defects for parabolic Verma modules of scalar type

There is a generalization of the above brane construction which produces line
defects in parabolic Verma modules of scalar type.

Let $(l_{1},\dotsc,l_{s})$ be an ordered partition of $m+n$:
$\sum_{\alpha=1}^{s}l_{\alpha}=m+n$. The partition specifies a parabolic
subalgebra $\mathfrak{p}$ of $\mathfrak{gl}(m|n)$, namely the subalgebra
consisting of upper-triangular block-diagonal matrices with diagonal blocks of
orders $l_{1}$, $\dotsc$, $l_{s}$. A character $\chi$ of $\mathfrak{p}$ is
determined by an $s$-tuple of complex numbers $(\chi_{1},\dotsc,\chi_{s})$ as

$\chi(-)=\mathop{\mathrm{str}}\nolimits(\chi^{\vee}-)\,,$ (191)

where the matrix $\chi^{\vee}$ is given by

$\chi^{\vee}=\mathop{\mathrm{diag}}\nolimits(\chi_{1},\dotsc,\chi_{1},\dotsc,\underbrace{\chi_{\alpha},\dotsc,\chi_{\alpha}}_{\text{$l_{\alpha}$
times}},\dotsc,\chi_{s},\dotsc,\chi_{s})\,.$ (192)

For each $\alpha$, we take $l_{\alpha}$ D3-branes and make them end on a
single NS5-brane on one side. On the other side, they end on separate
D5-branes as in the previous construction. In total, we have $m+n$ D3-branes
suspended between $m+n$ D5-branes and $s$ NS5-branes.

On the D3-branes we get the BF theory with gauge group $\mathrm{GL}(m|n)$. The
boundary condition on the D5-brane side is the same as before. On the
NS5-brane side, the boundary condition is

$\displaystyle\eta|_{\mathbb{R}\times\\{r\\}}$
$\displaystyle\in\mathfrak{p}\,,$ (193)
$\displaystyle\sigma|_{\mathbb{R}\times\\{r\\}}$
$\displaystyle\in\mathfrak{p}^{\perp}+\underline{\lambda}^{\ell}\,,$ (194)

where $\mathfrak{p}^{\perp}$ is the annihilator of $\mathfrak{p}$ in
$\mathfrak{g}^{*}$ and $\underline{\lambda}^{\ell}$ is a character of
$\mathfrak{p}$. Classically (that is to say, when $\underline{\lambda}^{\ell}$
is of order $\hbar^{-1}$ and quantities of order $\hbar^{0}$ are ignored), the
value of $\underline{\lambda}^{\ell}_{\alpha}$ is the position of the
$\alpha$th NS5-brane in $\mathbb{C}$.

We expect that the Hilbert space of the BF theory with these boundary
conditions is a parabolic Verma module of scalar type

$U(\mathfrak{g})\otimes_{U(\mathfrak{p})}\mathbb{C}_{\underline{\lambda}^{\ell}-\rho}\,.$
(195)

The character $\rho$ of $\mathfrak{p}$ is defined by

$\rho=-\frac{1}{2}\mathop{\mathrm{str}}\nolimits_{\mathfrak{g}/\mathfrak{p}}\mathop{\mathrm{ad}}\nolimits_{\mathfrak{p}}$
(196)

and $\mathbb{C}_{\underline{\lambda}^{\ell}-\rho}$ is the one-dimensional
$U(\mathfrak{p})$-module determined by the character
$\underline{\lambda}^{\ell}-\rho$.

###  3.4 Fermionic Dualities

As we have emphasized in our discussions, the Lie superalgebra
$\mathfrak{gl}(m|n)$ does not possess a unique Dynkin diagram. A Dynkin
diagram is specified by a choice of ordered basis of $\mathbb{C}^{m|n}$ (or a
choice of $\mathbb{Z}_{2}$-grading if we identify Dynkin diagrams related by
the action of the Weyl group), and different choices are related by a series
of certain adjacent transpositions, called odd reflections. Under odd
reflections, a highest-weight representation is mapped to a highest-weight
representation, but the highest weight is not preserved because the definition
of raising and lowering operators is altered.

Odd reflections change how we describe representations of
$\mathfrak{gl}(m|n)$, and the description of the Bethe vectors of the
superspin chain is changed accordingly. The map from the Bethe vectors for one
choice of ordered basis to another is known as a _fermionic duality_.

Fermionic dualities have been studied before purely from an algebraic
perspective [55, 56, 57, 58, 59, 60], with a notable exception of the work by
Orlando and Reffert [25] where they employed the point of view of string
theory to discuss the fermionic dualities for the supersymmetric $t$-$J$
model, which is the rational $\mathfrak{gl}(1|2)$ spin chain with spins valued
in the natural representation $\mathbb{C}^{1|2}$. Here we offer a string
theory explanation for important aspects of fermionic dualities for the
rational $\mathfrak{gl}(m|n)$ spin chain with Verma modules, namely their
action on highest weights and magnon numbers.

####  3.4.1 Odd reflections and fermionic dualities

Recall from section 2.1 that the definitions of positive and simple roots
depend on a choice of ordered basis $(e_{1},\dotsc,e_{m+n})$ of
$\mathbb{C}^{m|n}$, which is a permutation of
$(b_{1},\dotsc,b_{m},f_{1},\dotsc,f_{n})$ such that $(b_{1},\dotsc,b_{m})$ and
$(f_{1},\dotsc,f_{n})$ are the standard bases of $\mathbb{C}^{m}$ and
$\mathbb{C}^{n}$, respectively. There is a natural identification between
these basis vectors and their weights:

$\displaystyle(e_{1},\dotsc,e_{m+n})$
$\displaystyle\leftrightarrow(\varepsilon_{1},\dotsc,\varepsilon_{m+n})\,,$
(197) $\displaystyle(b_{1},\dotsc,b_{m})$
$\displaystyle\leftrightarrow(\epsilon_{1},\dotsc,\epsilon_{m})\,,$ (198)
$\displaystyle(f_{1},\dotsc,f_{n})$
$\displaystyle\leftrightarrow(\delta_{1},\dotsc,\delta_{n})\,.$ (199)

In the following discussion we will consider permutations of
$(\varepsilon_{1},\dotsc,\varepsilon_{m+n})$ induced by those of
$(e_{1},\dotsc,e_{m+n})$.

For a given choice of ordered basis
$(\varepsilon_{1},\dotsc,\varepsilon_{m+n})$ of the dual of the Cartan
subalgebra of $\mathfrak{gl}(m|n)$, the set of positive roots is

$\Phi^{+}=\\{\varepsilon_{i}-\varepsilon_{j}\mid i<j\\}$ (200)

and the set of simple roots is

$\Pi=\\{\varepsilon_{r}-\varepsilon_{r+1}\mid r=1,\dotsc,m+n-1\\}\,.$ (201)

A root $\varepsilon_{i}-\varepsilon_{j}$ is said to be _even_ if $[i]=[j]$ and
_odd_ if $[i]\neq[j]$.

Pick an odd simple root $\alpha_{s}=\varepsilon_{s}-\varepsilon_{s+1}$ and
apply to the ordered basis the adjacent transposition
$\sigma_{s}\colon\\{1,\dotsc,m+n\\}\to\\{1,\dotsc,m+n\\}$ interchanging
$\varepsilon_{s}$ and $\varepsilon_{s+1}$:

$(\varepsilon_{\sigma_{s}(1)},\dotsc,\varepsilon_{\sigma_{s}(s-1)},\varepsilon_{\sigma_{s}(s)},\varepsilon_{\sigma_{s}(s+1)},\varepsilon_{\sigma_{s}(s+2)},\dotsc,\varepsilon_{\sigma_{s}(m+n)})\\\
=(\varepsilon_{1},\dotsc,\varepsilon_{s-1},\varepsilon_{s+1},\varepsilon_{s},\varepsilon_{s+2},\dotsc,\varepsilon_{m+n})\,.$
(202)

The adjacent transposition alters the notion of positive and simple roots. For
the new ordered basis
$(\varepsilon_{\sigma_{s}(1)},\dotsc,\varepsilon_{\sigma_{s}(m+n)})$, the set
of positive roots is

$\Phi^{+}_{\alpha_{s}}=\\{\varepsilon_{\sigma_{s}(i)}-\varepsilon_{\sigma_{s}(j)}\mid
i<j\\}=\\{-\alpha_{s}\\}\cup\Phi^{+}\setminus\\{\alpha_{s}\\}$ (203)

and the set of simple roots is

$\begin{split}\Pi_{\alpha_{s}}&=\\{\varepsilon_{\sigma_{s}(r)}-\varepsilon_{\sigma_{s}(r+1)}\mid
r=1,\dotsc,m+n-1\\}\\\
&=\\{\varepsilon_{s-1}-\varepsilon_{s+1},\varepsilon_{s+1}-\varepsilon_{s},\varepsilon_{s}-\varepsilon_{s+2}\\}\cup\Pi\setminus\\{\alpha_{s-1},\alpha_{s},\alpha_{s+1}\\}\,.\end{split}$
(204)

This automorphism of the root system which transforms the positive and simple
roots is called the _odd reflection_ with respect to the odd simple root
$\alpha_{s}$.

As an example, take $(m|n)=(3|2)$ and
$(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4},\varepsilon_{5})=(\epsilon_{1},\epsilon_{2},\delta_{1},\delta_{2},\epsilon_{3})$.
This choice of ordered basis gives the Dynkin diagram (15). There are two odd
simple roots, $\alpha_{2}=\epsilon_{2}-\delta_{1}$ and
$\alpha_{4}=\delta_{2}-\epsilon_{3}$, represented by the crossed nodes.
Reflection with respect to $\alpha_{2}$ swaps $\epsilon_{2}$ and $\delta_{1}$,
leading to the new ordered basis
$(\epsilon_{1},\delta_{1},\epsilon_{2},\delta_{2},\epsilon_{3})$. The Dynkin
diagram corresponding to the reflected simple roots is

$\epsilon_{1}-\delta_{1}$$\delta_{1}-\epsilon_{2}$$\epsilon_{2}-\delta_{2}$$\delta_{2}-\epsilon_{3}$
(205)

We see that all simple roots are now odd. Reflection of the original ordered
basis with respect to $\alpha_{4}$ results in the ordered basis
$(\epsilon_{1},\epsilon_{2},\delta_{1},\epsilon_{3},\delta_{2})$ and the
Dynkin diagram

$\epsilon_{1}-\epsilon_{2}$$\epsilon_{2}-\delta_{1}$$\delta_{1}-\epsilon_{3}$$\epsilon_{3}-\delta_{2}$
(206)

Odd reflections change the characterization of highest weights. Let us see how
Verma modules are transformed. Fix an ordered basis of $\mathbb{C}^{m|n}$ and
consider the Verma module $M(\lambda)$, with the highest-weight vector
$|\Omega_{\lambda}\rangle$. Let $\alpha_{s}$ be an odd root. After the odd
reflection about $\alpha_{s}$, the roles of the raising operator
${\mathcal{E}}_{s,s+1}$ and the lowering operator ${\mathcal{E}}_{s+1,s}$ are
exchanged, while all other lowering operators remain unchanged. Consequently,
the state

$|\Omega_{\lambda^{\prime}}\rangle={\mathcal{E}}_{s+1,s}|\Omega_{\lambda}\rangle$
(207)

is annihilated by all elements of the new set of raising operators, that is,
it is a highest-weight state with respect to the new ordered basis. According
to the PBW theorem (23), the states of the form

$x_{1}^{n_{1}}\dotsm
x_{p-1}^{n_{p-1}}{\mathcal{E}}_{s+1,s}^{n_{p}}|\Omega_{\lambda}\rangle=x_{1}^{n_{1}}\dotsm
x_{p-1}^{n_{p-1}}{\mathcal{E}}_{s,s+1}^{1-n_{p}}|\Omega_{\lambda^{\prime}}\rangle\,,$
(208)

form a basis of the Fock space $V_{\lambda}$ for $M(\lambda)$, where
$(x_{1},\dotsc,x_{p-1},{\mathcal{E}}_{s+1,s})$ is an ordered set of lowering
operators in the original ordered basis. (Note that $n_{p}$ is either $0$ or
$1$.) By the PBW theorem, we see that $M(\lambda)$ is the Verma module
$M(\lambda^{\prime})$ with respect to the new ordered basis, with

$\lambda^{\prime}=\lambda-\alpha_{s}\,.$ (209)

In the spin chain, the highest weights of the Verma modules placed at the spin
sites are transformed by an odd reflection. The weight of each state of the
spin chain remains the same, so the magnon numbers must be transformed as

$\sum_{\ell=1}^{L}(\lambda^{\ell})^{\prime}-\sum_{r=1}^{m+n-1}M_{r}^{\prime}\alpha_{r}^{\prime}=\sum_{\ell=1}^{L}\lambda^{\ell}-\sum_{r=1}^{m+n-1}M_{r}\alpha_{r}\,,\quad\alpha_{r}^{\prime}=\varepsilon_{\sigma_{s}(r)}-\varepsilon_{\sigma_{s}(r+1)}\,,$
(210)

or more explicitly,

$M_{r}^{\prime}=\begin{cases}L+M_{s-1}+M_{s+1}-M_{s}&(r=s)\,;\\\ M_{r}&(r\neq
s)\,.\end{cases}$ (211)

The transformations of the highest weights and magnon numbers change the Bethe
equations. Of course, this is merely a change in the description of the spin
chain states, so the solutions of the new Bethe equations are in one-to-one
correspondence with the solutions of the original Bethe equations. This
correspondence is called the fermionic duality generated by the odd
reflection.

####  3.4.2 Fermionic duality from string theory

In the $\mathrm{D1}$–$\mathrm{D3}$–$\mathrm{NS5}$ duality frame that is S-dual
to (114), the choice of ordered basis of $\mathbb{C}^{m|n}$ is reflected in
the ordering of NS5-branes along $\mathbb{R}_{X}$; they are ordered as
$\mathrm{NS5}_{1}$, $\mathrm{NS5}_{2}$, $\dotsc$, $\mathrm{NS5}_{m+n}$ from
left to right. Therefore, the string theory interpretation of the reflection
about an odd root $\alpha_{s}$ is clear: it swaps the positions of
$\mathrm{NS5}_{s}$ and $\mathrm{NS5}_{s+1}$. We wish to understand the effect
of the exchange of positions on the highest weights and magnon numbers.

Recall that one of the boundary conditions for the BF theory that emerges on
the D3-branes creating the line defect ${\mathcal{L}}^{\ell}$ is specified by
the parameters $\underline{\lambda}^{\ell}_{i}$, $i=1$, $\dotsc$, $m+n$. The
position of $\mathrm{D3}_{i}^{\ell}$ in $\mathbb{C}$ is given by

$\zeta^{\ell}-(-1)^{[i]}\underline{\lambda}^{\ell}_{i}\,.$ (212)

Indeed, if we take $\underline{\lambda}_{i}^{\ell}$ to be of order
$\hbar^{-1}$, by the relation (190) this quantity coincides to order
$\hbar^{0}$ with the twisted mass $\mu_{i}^{\ell}$, which is identified with
the classical location of $\mathrm{D3}_{i}^{\ell}$. Swapping the positions of
$\mathrm{NS5}_{s}$ and $\mathrm{NS5}_{s+1}$ also exchanges $\mathrm{D3}_{s}$
and $\mathrm{NS5}_{s+1}$ while keeping their locations in $\mathbb{C}$ fixed.
Thus, $\\{\underline{\lambda}_{i}^{\ell}\\}$ are transformed to new values
$\\{(\underline{\lambda}^{\ell}_{i})^{\prime}\\}$ such that
$(\underline{\lambda}^{\ell}_{s})^{\prime}=\underline{\lambda}^{\ell}_{s+1}$
and
$(\underline{\lambda}^{\ell}_{s+1})^{\prime}=\underline{\lambda}^{\ell}_{s}$
in the new ordered basis. This simply means that we have

$(\underline{\lambda}^{\ell})^{\prime}=\sum_{i=1}^{m+n}(\underline{\lambda}^{\ell})^{\prime}_{i}\varepsilon_{\sigma_{s}(i)}=\underline{\lambda}^{\ell}\,.$
(213)

Although $\underline{\lambda}^{\ell}$ is invariant under the odd reflection,
the Weyl vector $\rho$ is transformed to a new Weyl vector $\rho^{\prime}$.
Since $\rho$ is the half sum of even positive roots minus the half sum of odd
positive roots, from the relation (203) between $\Phi^{+}$ and
$\Phi^{+}_{\alpha_{s}}$ we see

$\rho^{\prime}=\rho+\alpha_{s}\,.$ (214)

Then, the relation (187) between $\underline{\lambda}^{\ell}$ and the highest
weight $\lambda^{\ell}$ shows that $\lambda^{\ell}$ is transformed to
$(\lambda^{\ell})^{\prime}$ according to the formula (209).

Exchanging the pairs $(\mathrm{NS5}_{s},\mathrm{D3}_{s})$ and
$(\mathrm{NS5}_{s+1},\mathrm{D3}_{s+1})$ does not only transform the highest
weights, but also change the magnon numbers. In the brane picture, we can
understand this phenomenon as creation and annihilation of
$\mathrm{D1}$-branes due to the Hanany–Witten effect [61].777In general,
Hanany–Witten processes for type IIA brane configurations for two-dimensional
${\mathcal{N}}=(2,2)$ supersymmetric gauge theories classically suffer from
ambiguities, which are only resolved if one takes brane bending into account
or lifts the configurations to M-theory. [33]. In the present case, such
ambiguities do not arise because the relevant gauge group has the same number
of fundamental and antifundamental chiral multiplets. In order to exchange the
positions of the brane pairs, we first need to move each $\mathrm{D1}$-brane
between $\mathrm{NS5}_{s}$ and $\mathrm{NS5}_{s+1}$ so that one of its end is
attached to one of the NS5-branes, say $\mathrm{NS5}_{s+1}$. Then, we displace
$\mathrm{NS5}_{s+1}$ into the page and start moving it to the left. At one
point the D3-branes ending on $\mathrm{NS5}_{s+1}$ pass through
$\mathrm{NS5}_{s}$. As a result, the $\mathrm{D1}$-branes ending on these
D3-branes are annihilated and a new $\mathrm{D1}$-brane is created on each of
those D3-branes that did not have $\mathrm{D1}$-brane ending on it. In the
case in which $[s]=\bar{0}$, $[s+1]=\bar{1}$ and
$(M_{s-1},M_{s},M_{s+1})=(1,2,1)$, the process looks as follows:

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The dotted lines indicate the annihilation of $\mathrm{D1}$-branes. We see
that the numbers of $\mathrm{D1}$-branes between NS5-branes transform as in
the formula (211).

In the D2–D4–NS5 duality frame, the above manipulation is expected to lead to
an infrared duality of ${\mathcal{N}}=(2,2)$ supersymmetric gauge theories.
Indeed, there is a known duality transformation that sends a theory with
$\mathrm{U}(N_{c})$ gauge group, $N_{f}$ fundamental chiral multiplets and
$N_{a}$ antifundamental chiral multiplets to a theory with
$\mathrm{U}(N_{c}^{\prime})$ gauge group, $N_{a}$ fundamental chiral
multiplets and $N_{f}$ antifundamental chiral multiplets, plus mesons
transforming in the bifundamental representation of the flavor group
$\mathrm{U}(N_{a})\times\mathrm{U}(N_{f})$ [62]. The rank of the dual gauge
group is $N_{c}^{\prime}=\max(N_{f},N_{a})-N_{c}$. This is consistent with
what we have found since $N_{c}=M_{s}$ and $N_{f}=N_{a}=L+M_{s-1}+M_{s+1}$ in
our case. However, it appears that the mesons are absent from our final brane
configuration. Fortunately, the mesons, being neutral under the gauge
symmetry, do not affect the Bethe equations.

##  4 Bethe/gauge correspondence for compact superspin chains

The superspin chains that appear in the Bethe/gauge correspondence discussed
in the previous sections are noncompact, meaning that they carry spins valued
in infinite-dimensional representations of the Yangian
$Y(\mathfrak{gl}(m|n))$. Spin chains whose spins are valued in finite-
dimensional representations are said to be _compact_.

In this section we discuss the Bethe/gauge correspondence for compact rational
$\mathfrak{gl}(m|n)$ spin chains. We will follow a line of reasoning similar
to our treatment of the noncompact case, but in the reverse direction: we
start with the construction of line defects for finite-dimensional
representations in four-dimensional Chern–Simons theory, then identify their
brane realization and apply dualities to deduce the corresponding two-
dimensional quiver gauge theories.

###  4.1 Covariant and contravariant representations of $\mathfrak{gl}(m|n)$

Finite-dimensional representations of $\mathfrak{gl}(m|n)$ are most easily
discussed in the _distinguished grading_ , in which

$[i]=\begin{cases}\bar{0}&(i\leq m)\,;\\\ \bar{1}&(i>m)\,.\end{cases}$ (216)

For this reason, in this section we exclusively use the distinguished grading.
We will write a weight $\lambda=\sum_{i=1}^{m+n}\lambda_{i}\varepsilon_{i}$ as
$(\lambda_{1},\dotsc,\lambda_{m}|\lambda_{m+1},\dotsc,\lambda_{m+n})$.

The Verma module $M(\lambda)$ with highest weight $\lambda$ contains a unique
maximal submodule. In the distinguished grading, the corresponding simple
quotient module $L(\lambda)$ is finite-dimensional if and only if

$\lambda_{i}-\lambda_{i+1}\in\mathbb{Z}_{\geq 0}\,,\quad
i=1,\dotsc,\dotsc,m+n-1\,,\quad i\neq m\,,$ (217)

in other words, if and only if $(\lambda_{1},\dotsc,\lambda_{m})$ and
$(\lambda_{m+1},\dotsc,\lambda_{m+n})$ are highest weights of finite-
dimensional irreducible representations of $\mathfrak{gl}(m)$ and
$\mathfrak{gl}(n)$, respectively. Any finite-dimensional irreducible
representation of $\mathfrak{gl}(m|n)$ is isomorphic to $L(\lambda)$ for some
$\lambda$.

We will consider two classes of finite-dimensional irreducible representations
of $\mathfrak{gl}(m|n)$, called covariant representations and contravariant
representations. Covariant representations appear in tensor products of copies
of the natural representation

$\mathbb{C}^{m|n}=L\bigl{(}(1,0,\dotsc,0|0,\dotsc,0)\bigr{)}\,,$ (218)

whereas contravariant representations arise from tensor products of copies of
the dual representation888The dual $\pi^{*}$ of a representation $\pi$ is
given by $\pi^{*}=\tau\circ\pi$, where $\tau(X)=-X^{\mathrm{st}}$ is the
Chevalley automorphism. The supertranspose $X^{\mathrm{st}}$ of $X$ is defined
by $X^{\mathrm{st}}_{ij}=(-1)^{([i]+[j])[j]}X_{ji}$. Our definition of
supertranspose differs from a commonly used definition by a factor of
$(-1)^{[i]+[j]}$. With this definition, the quantum mechanical action (225) is
invariant under the natural action of
$\mathrm{GL}(m|n)\times\mathrm{GL}(K|{\overline{K}})$.

$(\mathbb{C}^{m|n})^{*}=L\bigl{(}(0,\dotsc,0|0,\dotsc,0,-1)\bigr{)}\,.$ (219)

Both covariant and contravariant representations are indexed by the so-called
$(m|n)$-hook partitions.

A partition $Y=(Y_{1},\dotsc,Y_{l(Y)})$ of size $|Y|$ and length $l(Y)$ is an
$l(Y)$-tuple of positive integers such that $Y_{1}\geq\dotsb\geq Y_{l(Y)}$ and
$Y_{1}+\dotsb+Y_{l(Y)}=|Y|$. It can be represented by a Young diagram with
$l(Y)$ rows, with the $\alpha$th row consisting of $Y_{\alpha}$ boxes. The
conjugate partition $Y^{\prime}$ has the Young diagram that is the transpose
of the Young diagram for $Y$.

A partition $Y$ is said to be _$(m|n)$ -hook_ if $Y_{m+1}\leq n$. If $Y$ is an
$(m|n)$-hook partition, then $Y^{\prime}$ is an $(n|m)$-hook partition,
$Y^{\prime}_{n+1}\leq m$. We let ${\mathcal{H}}_{m|n}$ denote the set of all
$(m|n)$-hook partitions.

For an $(m|n)$-hook partition $Y$, we define the integral weight

$Y^{\natural}=(Y_{1},\dotsc,Y_{m}|\langle
Y^{\prime}_{1}-m\rangle,\dotsc,\langle Y^{\prime}_{n}-m\rangle)\,,$ (220)

where $\langle a\rangle=\max(0,a)$. The even part of $Y^{\natural}$ is
represented by the Young diagram formed by the first $m$ rows of $Y$. The
Young diagram for the odd part of $Y^{\natural}$ is the transpose of the
remainder of $Y$, and its length is less than or equal to $n$ by the
$(m|n)$-hook condition.

Let $Y$ be an $(m|n)$-hook partition. The _covariant representation labeled by
$Y$_ is the highest-weight representation $L(Y^{\natural})$. The
_contravariant representation labeled by $Y$_ is the dual representation
$L(Y^{\natural})^{*}=L({\widetilde{Y}}^{\natural})$. Its highest weight
${\widetilde{Y}}^{\natural}$ equals the minus of the lowest weight of
$L(Y^{\natural})$ and is given by

${\widetilde{Y}}^{\natural}=(-\langle Y_{m}-n\rangle,\dotsc,-\langle
Y_{1}-n\rangle|-Y^{\prime}_{n},\dotsc,-Y^{\prime}_{1}\bigr{)}\,.$ (221)

###  4.2 Line defects in covariant and contravariant representations

Now we construct quantum mechanical systems whose Hilbert spaces are covariant
and contravariant representations of $\mathfrak{gl}(m|n)$. Coupled to four-
dimensional Chern–Simons theory with gauge group $\mathrm{GL}(m|n)$, they
describe line defects valued in these finite-dimensional irreducible
representations.

Let $K$, ${\overline{K}}$ be nonnegative integers and consider a pair of
fields

$\displaystyle\varphi$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{K|{\overline{K}}},\mathbb{C}^{m|n})\,,$
(222) $\displaystyle{\tilde{\varphi}}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m|n},\mathbb{C}^{K|{\overline{K}}})$
(223)

transforming in the bifundamental representations of
$\mathrm{GL}(m|n)\times\mathrm{GL}(K|{\overline{K}})$. Their components are
$\mathbb{Z}_{2}$-graded, with the grading given by

$[\varphi^{i}_{\alpha}]=[{\tilde{\varphi}}^{\alpha}_{i}]=[i]+[\alpha]\,.$
(224)

where $i$ and $\alpha$ are indices for $\mathbb{C}^{m|n}$ and
$\mathbb{C}^{K|{\overline{K}}}$, respectively. The even components are bosonic
and the odd ones are fermionic. The action of the theory is

$\frac{1}{\hbar}\int_{\mathbb{R}}\mathop{\mathrm{str}}\nolimits_{\mathbb{C}^{m|n}}(\varphi\,\mathrm{d}{\tilde{\varphi}})\,.$
(225)

We will find it convenient to define

$\displaystyle\eta^{i}_{\alpha}$
$\displaystyle=\frac{\mathrm{i}}{\hbar}\varphi^{i}_{\alpha}\,,$ (226)
$\displaystyle\chi_{i}^{\alpha}$
$\displaystyle=(-1)^{[\alpha]}{\tilde{\varphi}}_{i}^{\alpha}\,,$ (227)

and

$\displaystyle{\tilde{\chi}}^{i}_{\alpha}$
$\displaystyle=(-1)^{[i][\alpha]+[\alpha]}\eta^{i}_{\alpha}\,,$ (228)
$\displaystyle{\tilde{\eta}}_{i}^{\alpha}$
$\displaystyle=(-1)^{[i][\alpha]+[\alpha]}\frac{\mathrm{i}}{\hbar}\chi_{i}^{\alpha}\,.$
(229)

Then, the canonical commutation relations read

$[\eta_{\alpha}^{i},\chi_{j}^{\beta}]=[{\tilde{\chi}}_{\alpha}^{i},{\tilde{\eta}}_{j}^{\beta}]=\delta^{\beta}_{\alpha}\delta_{j}^{i}\,.$
(230)

The theory has a $\mathrm{GL}(m|n)\times\mathrm{GL}(K|{\overline{K}})$ global
symmetry. The associated conserved charges are

$\displaystyle q_{ij}$
$\displaystyle=\sum_{\alpha=1}^{K+{\overline{K}}}\chi_{i}^{\alpha}\eta_{\alpha}^{j}+(-1)^{[i]}c\delta_{ij}\,,$
(231) $\displaystyle Q_{\alpha\beta}$
$\displaystyle=-\sum_{i=1}^{m+n}(-1)^{([\alpha]+[\beta])[\alpha]}\chi_{i}^{\beta}\eta_{\alpha}^{i}+(-1)^{[\alpha]}C\delta_{\alpha\beta}$
(232)

and satisfy the $\mathfrak{gl}(m|n)\oplus\mathfrak{gl}(K|{\overline{K}})$
commutation relations:

$\displaystyle[q_{ij},q_{kl}]$
$\displaystyle=\delta_{jk}q_{il}-(-1)^{([i]+[j])([k]+[l])}\delta_{li}q_{kj}\,,$
(233) $\displaystyle[Q_{\alpha\beta},Q_{\gamma\delta}]$
$\displaystyle=\delta_{\beta\gamma}Q_{\alpha\delta}-(-1)^{([\alpha]+[\beta])([\gamma]+[\delta])}\delta_{\delta\alpha}Q_{\gamma\beta}\,,$
(234) $\displaystyle[q_{ij},Q_{\alpha\beta}]$ $\displaystyle=0\,.$ (235)

The constants $c$ and $C$ account for the ambiguity in operator ordering and
will be fixed in a moment. Under
$\mathrm{GL}(m|n)\times\mathrm{GL}(K|{\overline{K}})$, the sets of fields
$\\{\chi_{i}^{\alpha}\\}$ and $\\{{\tilde{\chi}}^{i}_{\alpha}\\}$ transform as
the standard basis vectors for
$\mathbb{C}^{m|n}\otimes(\mathbb{C}^{K|{\overline{K}}})^{*}$ and
$(\mathbb{C}^{m|n})^{*}\otimes\mathbb{C}^{K|{\overline{K}}}$, respectively:

$\displaystyle[q_{ij},\chi_{k}^{\alpha}]$
$\displaystyle=\delta_{jk}\chi_{i}^{\alpha}=\sum_{l=1}^{m+n}(E_{ij})_{lk}\chi_{l}^{\alpha}\,,$
(236) $\displaystyle[Q_{\alpha\beta},\chi_{i}^{\gamma}]$
$\displaystyle=-(-1)^{([\alpha]+[\beta])[\alpha]}\delta_{\alpha\gamma}\chi_{i}^{\beta}=\sum_{\delta=1}^{K+{\overline{K}}}(-E_{\alpha\beta}^{\mathrm{st}})_{\delta\gamma}\chi_{i}^{\delta}\,.$
(237) $\displaystyle[q_{ij},{\tilde{\chi}}^{k}_{\alpha}]$
$\displaystyle=-(-1)^{([i]+[j])[i]}\delta_{ki}{\tilde{\chi}}^{j}_{\alpha}=\sum_{l=1}^{m+n}(-E_{ij}^{\mathrm{st}})_{lk}{\tilde{\chi}}^{l}_{\alpha}\,,$
(238) $\displaystyle[Q_{\alpha\beta},{\tilde{\chi}}^{i}_{\gamma}]$
$\displaystyle=\delta_{\beta\gamma}{\tilde{\chi}}^{i}_{\alpha}=\sum_{\delta=1}^{K+{\overline{K}}}(E_{\alpha\beta})_{\delta\gamma}{\tilde{\chi}}^{i}_{\delta}\,.$
(239)

For the construction of line defects we actually break the
$\mathrm{GL}(K|{\overline{K}})$ symmetry. Let us gauge the Borel subgroup of
$\mathrm{GL}(K|{\overline{K}})$. We introduce an associated gauge field

${\mathcal{B}}=\sum_{\begin{subarray}{c}\alpha,\beta=1\\\
\alpha\leq\beta\end{subarray}}^{K+{\overline{K}}}{\mathcal{B}}^{\alpha\beta}E_{\alpha\beta}$
(240)

and couple it to the theory by replacing the de Rham differential $\mathrm{d}$
with $\mathrm{d}+{\mathcal{B}}$. For the moment we treat ${\mathcal{B}}$ as a
background field and give it a diagonal value

${\mathcal{B}}=\mathop{\mathrm{diag}}\nolimits(b_{1},\dotsc,b_{K+{\overline{K}}})\,.$
(241)

In this background, the action becomes

$\frac{1}{\hbar}\int_{\mathbb{R}}\sum_{i=1}^{m+n}\sum_{\alpha=1}^{K+{\overline{K}}}(-1)^{[i]}(\varphi_{\alpha}^{i}\mathrm{d}{\tilde{\varphi}}^{\alpha}_{i}+b_{\alpha}\varphi_{\alpha}^{i}{\tilde{\varphi}}^{\alpha}_{i})\,,$
(242)

and the $\mathrm{GL}(K|{\overline{K}})$ symmetry is broken to the stabilizer
of the gauge field, which is generically the maximal torus.

The coupling to the gauge field does not affect the canonical commutation
relations, but modifies the Hamiltonian. Before the introduction of the gauge
field, the theory was topological and the Hamiltonian was zero. It is now
given by

$H=\mathrm{i}\hbar\sum_{\alpha=1}^{K+{\overline{K}}}b_{\alpha}Q_{\alpha\alpha}\,.$
(243)

The Hilbert space of the theory is a $\mathbb{Z}_{2}$-graded Fock space
constructed from a vacuum state $|0\rangle$ by the action of the creation
operators. The action of $\varphi_{\alpha}^{i}$ changes $H$ by
$\mathrm{i}\hbar b_{\alpha}$ while ${\tilde{\varphi}}^{\alpha}_{i}$ changes
$H$ by $-\mathrm{i}\hbar b_{\alpha}$. Those component fields that increase
$\mathop{\mathrm{Re}}\nolimits(\mathrm{i}H/\hbar)$ are creation operators, and
those that decrease it are annihilation operators.999If the time axis is
compactified to a circle of radius $1$, then the partition function involves
trace twisted by $\exp(-\mathrm{i}H/\hbar)$. Creation operators should make
this factor smaller. We can think of
$\mathop{\mathrm{Re}}\nolimits(\mathrm{i}H/\hbar)$ as energy.

Suppose that we give the background value such that

$0<\mathop{\mathrm{Re}}\nolimits
b_{K+{\overline{K}}}<\mathop{\mathrm{Re}}\nolimits
b_{K+{\overline{K}}-1}<\dotsb<\mathop{\mathrm{Re}}\nolimits b_{1}\,.$ (244)

Then, $\chi^{\alpha}_{i}$ is a creation operator and $\eta_{\alpha}^{i}$ is an
annihilation operator. Requiring the vacuum to be invariant under (the maximal
torus of) $\mathrm{GL}(K|{\overline{K}})$, we find $c=C=0$. Let
${\mathcal{F}}$ be the corresponding Fock space.

The Fock space ${\mathcal{F}}$ decomposes into tensor products of covariant
representations of $\mathfrak{gl}(m|n)$ and contravariant representations of
$\mathfrak{gl}(K|{\overline{K}})$ [63, 64]:

${\mathcal{F}}=\bigoplus_{Y\in{\mathcal{H}}_{m|n}\cap{\mathcal{H}}_{K|{\overline{K}}}}L(Y^{\natural}_{m|n})\otimes
L(Y^{\natural}_{K|{\overline{K}}})^{*}\,.$ (245)

(We use subscripts to distinguish weights for $\mathfrak{gl}(m|n)$ and
$\mathfrak{gl}(K|{\overline{K}})$.) For example, the first excited states take
the form

$\sum_{i=1}^{m+n}\sum_{\alpha=1}^{K+{\overline{K}}}c^{i}_{\alpha}\chi^{\alpha}_{i}|0\rangle$
(246)

and span a subspace isomorphic to

$\mathbb{C}^{m|n}\otimes(\mathbb{C}^{K|{\overline{K}}})^{*}\,,$ (247)

as can be seen from the commutation relations (236) and (237).

This Hilbert space is too large, and we need to reduce it to a single
covariant representation $L(Y^{\natural}_{m|n})$ of $\mathfrak{gl}(m|n)$. To
do so, we impose constraints that singles out the summand
$L(Y^{\natural}_{m|n})\otimes L(Y^{\natural}_{K|{\overline{K}}})^{*}$ and
further projects it to the subspace of lowest-energy states. Since the raising
operator $Q_{\alpha\beta}$, $\alpha<\beta$, changes the energy
$\mathop{\mathrm{Re}}\nolimits(\mathrm{i}H/\hbar)$ by
$-\mathop{\mathrm{Re}}\nolimits b_{\alpha}+\mathop{\mathrm{Re}}\nolimits
b_{\beta}<0$, the lowest-energy states have the highest weight with respect to
$\mathfrak{gl}(K|{\overline{K}})$.

We implement this projection by making ${\mathcal{B}}$ dynamical. The vacuum
expectation value of ${\mathcal{B}}$ is given by the diagonal matrix (241).
Let us add to the action the Chern–Simons term

$-\mathrm{i}\int_{\mathbb{R}}\underline{{\widetilde{Y}}}^{\natural}_{K|{\overline{K}}}({\mathcal{B}})\,.$
(248)

Then, the equations of motion for ${\mathcal{B}}$ are

$Q_{\alpha\beta}=0\,,\quad\alpha<\beta$ (249)

and101010The Weyl quantization of the classical expression of
$Q_{\alpha\alpha}$ equals $Q_{\alpha\alpha}+(-1)^{[\alpha]}(m-n)/2$.

$Q_{\alpha\alpha}=(\underline{{\widetilde{Y}}}^{\natural}_{K|{\overline{K}}})_{\alpha}+\frac{m-n}{2}\,.$
(250)

The former equations restrict the Fock space to the subspace of states that
contains highest-weight vectors of covariant representations of
$\mathfrak{gl}(K|{\overline{K}})$:

$\bigoplus_{Y\in{\mathcal{H}}_{m|n}\cap{\mathcal{H}}_{K|{\overline{K}}}}L(Y^{\natural}_{m|n})\otimes|\Omega_{{\widetilde{Y}}^{\natural}_{K|{\overline{K}}}}\rangle\,.$
(251)

With the choice

${\widetilde{Y}}^{\natural}_{K|{\overline{K}}}=\underline{{\widetilde{Y}}}^{\natural}_{K|{\overline{K}}}-\frac{m-n}{2}\sum_{\alpha=1}^{K+{\overline{K}}}\varepsilon_{\alpha}\,,$
(252)

the second equation selects the highest weight
${\widetilde{Y}}^{\natural}_{K|{\overline{K}}}$, thereby reducing the Hilbert
space to the covariant representation
$L(Y^{\natural}_{m|n})\otimes|\Omega_{{\widetilde{Y}}^{\natural}_{K|{\overline{K}}}}\rangle$
of $\mathfrak{gl}(m|n)$.

In order to construct line defects in contravariant representations, we take

$\mathop{\mathrm{Re}}\nolimits
b_{K+{\overline{K}}}<\mathop{\mathrm{Re}}\nolimits
b_{K+{\overline{K}}-1}<\dotsb<\mathop{\mathrm{Re}}\nolimits b_{1}<0\,.$ (253)

In this case, ${\tilde{\chi}}_{i}^{\alpha}$ is a creation operator and
${\tilde{\eta}}^{i}_{\alpha}$ is an annihilation operator. The corresponding
Fock space ${\widetilde{{\mathcal{F}}}}$ decomposes as

${\widetilde{{\mathcal{F}}}}=\bigoplus_{Y\in{\mathcal{H}}_{m|n}\cap{\mathcal{H}}_{K|{\overline{K}}}}L(Y^{\natural}_{m|n})^{*}\otimes
L(Y^{\natural}_{K|{\overline{K}}})\,.$ (254)

Making ${\mathcal{B}}$ dynamical and adding the Chern–Simons term

$-\mathrm{i}\int_{\mathbb{R}}\underline{Y}^{\natural}_{K|{\overline{K}}}({\mathcal{B}})$
(255)

to the action, we can reduce the Hilbert space to the contravariant
representation $L(Y^{\natural})^{*}$, with

$Y^{\natural}_{K|{\overline{K}}}=\underline{Y}^{\natural}_{K|{\overline{K}}}-\frac{m-n}{2}\sum_{\alpha=1}^{K+{\overline{K}}}\varepsilon_{\alpha}\,.$
(256)

###  4.3 Brane construction of line defects

The quantum mechanical system discussed above can be constructed with
D3-branes and D5-branes. Let us remove from the brane system (114) the semi-
infinite D3-branes and the F1-branes stretched between the D5-branes, and
instead introduce infinite D3-branes $\mathrm{D3}_{\alpha}$, $\alpha=1$,
$\dotsc$, $K+{\overline{K}}$:

$\begin{array}[]{r@{\colon\ }c@{\ \times\ }c@{\ \times\ }c@{\ \times\ }c@{\
\times\ }c@{\ \times\ }c@{\ \times\
}c}\mathrm{Spacetime}&\mathbb{R}&S^{1}&\mathbb{C}&\mathbb{R}_{X}&\mathbb{R}_{Y}&\mathbb{R}^{2}_{+\hbar}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{D5}_{i}\ (i\leq
m)&\mathbb{R}&S^{1}&\mathbb{C}&\\{X_{i}\\}&\\{Y_{i}\\}&\mathbb{R}^{2}_{+\hbar}&\\{0\\}\\\
\mathrm{D5}_{i}\
(i>m)&\mathbb{R}&S^{1}&\mathbb{C}&\\{X_{i}\\}&\\{Y_{i}\\}&\\{0\\}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{D3}_{\alpha}\ (\alpha\leq
K)&\mathbb{R}&\\{y\\}&\\{\zeta\\}&\\{-\mathop{\mathrm{Re}}\nolimits
b_{\alpha}\\}&\mathbb{R}_{Y}&\mathbb{R}^{2}_{+\hbar}&\\{0\\}\\\
\mathrm{D3}_{\alpha}\
(\alpha>K)&\mathbb{R}&\\{y\\}&\\{\zeta\\}&\\{-\mathop{\mathrm{Re}}\nolimits
b_{\alpha}\\}&\mathbb{R}_{Y}&\\{0\\}&\mathbb{R}^{2}_{-\hbar}\end{array}$ (257)

We claim that strings stretched between the D3-branes and the D5-branes give
rise to the quantum mechanical system in question.

The $K$ D3-branes $\mathrm{D3}_{\alpha}$, $\alpha\leq K$, and the $m$
D5-branes $\mathrm{D5}_{i}$, $i\leq m$, share the three-dimensional spacetime
$\mathbb{R}\times\mathbb{R}^{2}_{+\hbar}$. Strings stretched between them
produce an ${\mathcal{N}}=4$ hypermultiplet in the bifundamental
representation of $\mathrm{U}(K)\times\mathrm{U}(m)$. Let

$\displaystyle\varphi^{00}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{K},\mathbb{C}^{m})\,,$
(258) $\displaystyle{\tilde{\varphi}}^{00}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m},\mathbb{C}^{K})$
(259)

be the scalar fields of this multiplet. We are looking at the sector of this
theory that is invariant under the supercharge $Q$ for the
holomorphic–topological twist. There is an $\Omega$-deformation induced by the
background RR two-form, and it has the effect of localizing the hypermultiplet
to the quantum mechanical model with action [23]

$\frac{1}{\hbar}\int_{\mathbb{R}}\mathop{\mathrm{tr}}\nolimits_{\mathbb{C}^{m}}(\varphi^{00}\,\mathrm{d}{\tilde{\varphi}}^{00})\,.$
(260)

Here we are using $\varphi^{00}$, ${\tilde{\varphi}}^{00}$ to denote the one-
dimensional fields that descend from the three-dimensional scalar fields.

Similarly, from strings stretched between the ${\overline{K}}$ D3-branes
$\mathrm{D3}_{\alpha}$, $\alpha>K$, and the $n$ D5-branes $\mathrm{D5}_{i}$,
$i>m$, we get an ${\mathcal{N}}=4$ hypermultiplet in the bifundamental
representation of $\mathrm{U}({\overline{K}})\times\mathrm{U}(n)$ on the
three-dimensional spacetime $\mathbb{R}\times\mathbb{R}^{2}_{-\hbar}$. By an
$\Omega$-deformation, the theory localizes to a quantum mechanical system with
action

$-\frac{1}{\hbar}\int_{\mathbb{R}}\mathop{\mathrm{tr}}\nolimits_{\mathbb{C}^{n}}(\varphi^{11}\,\mathrm{d}{\tilde{\varphi}}^{11})\,,$
(261)

where

$\displaystyle\varphi^{11}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{\overline{K}},\mathbb{C}^{n})\,,$
(262) $\displaystyle{\tilde{\varphi}}^{11}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{n},\mathbb{C}^{\overline{K}})\,.$
(263)

The branes $\mathrm{D3}_{\alpha}$, $\alpha\leq K$, and $\mathrm{D5}_{i}$,
$i>m$, intersect along the time axis $\mathbb{R}$, and from strings stretched
between them we get fermions

$\displaystyle\varphi^{10}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{K},\mathbb{C}^{n})\,,$
(264) $\displaystyle{\tilde{\varphi}}^{10}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{n},\mathbb{C}^{K})\,.$
(265)

They are described by the action

$\frac{1}{\hbar}\int_{\mathbb{R}}\mathop{\mathrm{tr}}\nolimits_{\mathbb{C}^{n}}(\varphi^{10}\,\mathrm{d}{\tilde{\varphi}}^{10})\,.$
(266)

This is the dimensional reduction of the two-dimensional chiral fermions that
arise from an intersection of D4-branes and D6-branes [65, 66].

In the same way, from strings stretched between $\mathrm{D3}_{\alpha}$,
$\alpha>K$, and $\mathrm{D5}_{i}$, $i\leq m$, we get fermionic fields

$\displaystyle\varphi^{01}\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{\overline{K}},\mathbb{C}^{m})\,,$
(267)
$\displaystyle{\tilde{\varphi}}^{01}\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m},\mathbb{C}^{\overline{K}})\,,$
(268)

described by the action

$-\frac{1}{\hbar}\int_{\mathbb{R}}\mathop{\mathrm{tr}}\nolimits_{\mathbb{C}^{m}}(\varphi^{01}\,\mathrm{d}{\tilde{\varphi}}^{01})\,.$
(269)

These four quantum mechanical systems can be combined into the single quantum
mechanical system described by the action (225), with the fields

$\displaystyle\varphi$
$\displaystyle=\begin{pmatrix}\varphi^{00}&\varphi^{01}\\\
\varphi^{10}&\varphi^{11}\end{pmatrix}\,,$ (270)
$\displaystyle{\tilde{\varphi}}$
$\displaystyle=\begin{pmatrix}{\tilde{\varphi}}^{00}&{\tilde{\varphi}}^{01}\\\
{\tilde{\varphi}}^{10}&{\tilde{\varphi}}^{11}\end{pmatrix}\,.$ (271)

The creation operator $\chi^{\alpha}_{i}$ adds a string stretched between
$\mathrm{D3}_{\alpha}$ and $\mathrm{D5}_{i}$. The annihilation operator
$\eta_{\alpha}^{i}$ removes a string between them.

This quantum mechanical system is coupled to the four-dimensional Chern–Simons
theory that arises from the D5-branes and to the BF theory that arises on the
D3-branes. As in the construction of line defects valued in Verma modules, the
boundary conditions on the BF theory (at infinity, or at finite distance if we
make the D3-branes end on NS5-branes) breaks the
$\mathrm{GL}(K|{\overline{K}})$ gauge symmetry to a Borel subgroup. Which
Borel subgroup is selected is determined by the ordering of the D3-branes on
$\mathbb{R}_{X}$. For the ordering (244) for $\\{\mathop{\mathrm{Re}}\nolimits
b_{\alpha}\\}$, it is the standard Borel subgroup.

The situation in which there are no strings stretched between the D3-branes
and the D5-branes corresponds to the vacuum of the Fock space ${\mathcal{F}}$.
Here is how the vacuum looks like for $(m|n)=(1|2)$ and
$(K|{\overline{K}})=(2|1)$:

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