Superspin chains from superstring theory

Nafiz Ishtiaque1$\star$, Seyed F. Moosavian2, Surya Raghavendran3 and Junya
Yagi4

1 Institute for Advanced Study, Princeton, NJ 08540 USA

2 Department of Physics, McGill University, Montréal, QC H3A 2T8 Canada

3 Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5 Canada

4 Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084 China

⋆<EMAIL_ADDRESS>
## Abstract

We present a correspondence between two-dimensional ${\mathcal{N}}=(2,2)$
supersymmetric gauge theories and rational integrable $\mathfrak{gl}(m|n)$
spin chains with spin variables taking values in Verma modules. To explain
this correspondence, we realize the gauge theories as configurations of branes
in string theory and map them by dualities to brane configurations that
realize line defects in four-dimensional Chern–Simons theory with gauge group
$\mathrm{GL}(m|n)$. The latter configurations embed the superspin chains into
superstring theory. We also provide a string theory derivation of a similar
correspondence, proposed by Nekrasov, for rational $\mathfrak{gl}(m|n)$ spin
chains with spins valued in finite-dimensional representations.

###### Contents

  1. 1 Introduction
  2. 2 Bethe/gauge correspondence for noncompact superspin chains
    1. 2.1 $\mathfrak{gl}(m|n)$ and its Verma modules
      1. 2.1.1 Lie superalgebra $\mathfrak{gl}(m|n)$
      2. 2.1.2 Verma modules of $\mathfrak{gl}(m|n)$
      3. 2.1.3 Tensor products of Verma modules
    2. 2.2 Bethe side
      1. 2.2.1 Yangian
      2. 2.2.2 Representations of $Y(\mathfrak{gl}(m|n))$
      3. 2.2.3 The spin chain
      4. 2.2.4 Bethe equations
    3. 2.3 Gauge side
      1. 2.3.1 The gauge theories
      2. 2.3.2 Vacuum equations
      3. 2.3.3 Twisted chiral ring
    4. 2.4 The correspondence
  3. 3 String theory realization of the Bethe/gauge correspondence
    1. 3.1 Brane construction of the gauge theory vacua
      1. 3.1.1 Semiclassical type IIA configuration
      2. 3.1.2 Lift to M-theory
      3. 3.1.3 Turning on $\hbar$
    2. 3.2 Four-dimensional Chern–Simons theory
      1. 3.2.1 Case with $m=0$ or $n=0$
      2. 3.2.2 Case with nonzero $m$ and $n$
    3. 3.3 Emergence of the spin chain
      1. 3.3.1 Line defects and spin chains
      2. 3.3.2 Line defects created by D3-branes
      3. 3.3.3 Line defects for parabolic Verma modules of scalar type
    4. 3.4 Fermionic Dualities
      1. 3.4.1 Odd reflections and fermionic dualities
      2. 3.4.2 Fermionic duality from string theory
  4. 4 Bethe/gauge correspondence for compact superspin chains
    1. 4.1 Covariant and contravariant representations of $\mathfrak{gl}(m|n)$
    2. 4.2 Line defects in covariant and contravariant representations
    3. 4.3 Brane construction of line defects
    4. 4.4 Two-dimensional ${\mathcal{N}}=(2,2)$ supersymmetric gauge theories
    5. 4.5 Bethe/gauge correspondence for finite-dimensional representations
  5. A Four-dimensional Chern–Simons theory with gauge supergroup from twisted string theory
    1. A.1 Topological strings
    2. A.2 Topological open string field theory
    3. A.3 Topological closed string field theory
    4. A.4 Closed–open map
    5. A.5 Four-dimensional Chern–Simons theory with gauge supergroup from the $\mathrm{SU}(3)$-invariant twist of type IIB string theory

##  1 Introduction

The Bethe/gauge correspondence, discovered by Nekrasov and Shatashvili [1, 2]
in 2009, connects two seemingly unrelated areas of physics. The Bethe side of
the correspondence refers to one-dimensional integrable quantum spin chains.
The gauge side is supersymmetric gauge theories.

Arguably the most prominent example of the Bethe/gauge correspondence involves
Heisenberg’s XXX spin chain and its generalizations. In this example, the
eigenvectors of commuting conserved charges of a rational $\mathfrak{gl}(m)$
spin chain are identified with the vacua of a family of gauge theories whose
gauge group is the product of $m-1$ unitary gauge groups. The gauge theories
have ${\mathcal{N}}=(4,4)$ supersymmetry broken to ${\mathcal{N}}=(2,2)$
subgroup by mass deformations, and their gauge and matter contents are encoded
in quiver diagrams whose underlying graphs contain the Dynkin diagram of type
$A_{m-1}$ as a subgraph.

In 2018, Nekrasov [3] presented a generalization of the above correspondence
where the relevant spin chains carry superspins, namely rational
$\mathfrak{gl}(m|n)$ spin chains. The corresponding gauge theories are
essentially ${\mathcal{N}}=(2,2)$ supersymmetric, as opposed to having softly
broken ${\mathcal{N}}=(4,4)$ supersymmetry. One of the main results of this
paper is an explanation of the origin of this correspondence using superstring
theory.

In fact, the goal of the present work is much more ambitious: we wish to place
the Bethe/gauge correspondence for superspin chains into a large web of
dualities that relate diverse phenomena in which the same superspin chains
arise from different supersymmetric gauge theories in various spacetime
dimensions.

Many of the phenomena that are expected to constitute this web of dualities
are yet to be uncovered, but their specializations to the case of
$\mathfrak{gl}(m|0)=\mathfrak{gl}(m)$ are known and have been studied in
recent years. Besides the Bethe/gauge correspondence already described, the
structures of rational $\mathfrak{gl}(m)$ spin chains (and their trigonometric
and elliptic generalizations) have appeared in quantization of the
Seiberg–Witten geometries of four-dimensional ${\mathcal{N}}=2$ supersymmetric
gauge theories [4, 5, 6], the action of surface and line defects on
supersymmetric indices of four-dimensional supersymmetric gauge theories [7,
8, 9, 10, 11, 12], quantization of the Coulomb branches of three-dimensional
${\mathcal{N}}=4$ supersymmetric gauge theories [13, 14], and correlation
functions of local operators on interfaces in four-dimensional
${\mathcal{N}}=4$ super Yang–Mills theory [15], to name a few.

All of these gauge theory setups have realization in string theory, and one
suspects that they are related to each other in one way or another via string
dualities. This idea has turned out to be true. It was argued in [16] that
brane constructions of these setups (except for the last one which we expect
is also related) are all dual to brane configurations that realize line
defects in a four-dimensional analog of Chern–Simons theory [17, 18, 19] with
gauge group $\mathrm{GL}(m)$. This theory only has a bosonic gauge field, but
it is secretly supersymmetric. Indeed, it is equivalent to a
holomorphic–topological twist of six-dimensional ${\mathcal{N}}=(1,1)$ super
Yang–Mills theory with gauge group $\mathrm{U}(m)$ in the presence of
$\Omega$-deformation [20, 21, 22, 23, 24]. The six-dimensional theory
describes the low-energy dynamics of a stack of $m$ D5-branes, which comprise
part of the brane configurations.

Four-dimensional Chern–Simons theory placed on
$\mathbb{R}^{2}\times\mathbb{C}$ is topological on the plane $\mathbb{R}^{2}$
and holomorphic on the complex plane $\mathbb{C}$. Due to this
holomorphic–topological property, line defects extending along
$\mathbb{R}^{2}$ automatically satisfy the Yang–Baxter relation. Moreover,
each line defect carries one complex parameter, its position in $\mathbb{C}$.
These facts imply that line defects making up a square lattice in
$\mathbb{R}^{2}$ defines a two-dimensional classical integrable lattice model;
their correlation function equals the partition function of the lattice model.

Equivalently, with one of the lattice directions regarded as a time direction,
a lattice of line defects in four-dimensional Chern–Simons theory defines a
one-dimensional quantum integrable spin chain. For Wilson lines, this spin
chain is a rational $\mathfrak{gl}(m)$ spin chain [17, 18, 19]. (If one
replaces $\mathbb{C}$ in spacetime with $\mathbb{C}\setminus\\{0\\}$ or an
elliptic curve, then one obtains trigonometric or elliptic $\mathfrak{gl}(m)$
spin chain, respectively.) Thus, the fact that the gauge theory setups
mentioned above are all dual to line defects in four-dimensional Chern–Simons
theory explains the appearance of $\mathfrak{gl}(m)$ spin chains in these
setups.

Now, in view of the Bethe/gauge correspondence for rational
$\mathfrak{gl}(m|n)$ spin chains, one wonders how one can incorporate it into
the picture just described. If one could generalize the string theory
realization of the Bethe/gauge correspondence for bosonic spin chains to the
superspin chain case, one would generalize, implicitly by string dualities,
_all_ of the gauge theory phenomena mentioned above to their
$\mathfrak{gl}(m|n)$ versions. This is what we aim to achieve.

In this paper, we provide brane constructions of the gauge theories pertinent
to the Bethe/gauge correspondence for rational $\mathfrak{gl}(m|n)$ spin
chains, and show that they are related by dualities to line defects in four-
dimensional Chern–Simons theory with gauge group $\mathrm{GL}(m|n)$. The
latter theory is obtained from two copies of $\Omega$-deformed six-dimensional
${\mathcal{N}}=(1,1)$ super Yang–Mills theory, with gauge groups
$\mathrm{U}(m)$ and $\mathrm{U}(n)$, coupled by a four-dimensional
hypermultiplet in the bifundamental representation of
$\mathrm{U}(m)\times\mathrm{U}(n)$. In turn, this gauge theory setup arises
from a stack of $m$ D5-branes intersecting a stack of $n$ D5-branes. This
brane construction is another main result of the paper.

Actually, we present two versions of the Bethe/gauge correspondence, one for
compact spin chains and one for noncompact spin chains. The difference is
whether spin variables are valued in finite-dimensional or infinite-
dimensional representations.

We introduce the Bethe/gauge correspondence for noncompact rational
$\mathfrak{gl}(m|n)$ spin chains in section 2. The brane constructions of the
corresponding gauge theories, as well as the duality relating these theories
to line defects in four-dimensional Chern–Simons theory with gauge group
$\mathrm{GL}(m|n)$, are discussed in section 3. Discussions in this section
provide a string theory explanation for the Bethe/gauge correspondence. The
case of compact spin chains is treated in section 4, where we reproduce the
correspondence proposed in [3].

It should be remarked that in an inspiring paper [25] in 2010, Orlando and
Reffert found the Bethe/gauge correspondence for the rational
$\mathfrak{gl}(1|2)$ spin chain with spins taking values in the natural
$(1|2)$-dimensional representation $\mathbb{C}^{1|2}$. Furthermore, they gave
a string theory argument to explain dualities between different families of
gauge theories corresponding to different choices of Dynkin diagrams of
$\mathfrak{gl}(1|2)$. On the spin chain side, these dualities are known as
fermionic dualities. In section 3.4 we discuss the fermionic dualities for
rational $\mathfrak{gl}(m|n)$ spin chains from a similar point of view.

As we mentioned above, four-dimensional Chern–Simons theory with gauge
supergroup can be constructed from two copies of six-dimensional super
Yang–Mills theory coupled by four-dimensional matter fields. There is a
related construction in topological string theory, which may prove useful in
future attempts to put some of the physical arguments given in this work on a
rigorous mathematical footing. We describe the topological string construction
in appendix A.

The present work unveils only a small part of a collection of phenomena in
which superspin chains emerge from supersymmetric gauge theories. It will be
extremely interesting to study other, but ultimately related, phenomena whose
existence is predicted by string dualities and other tools. We conclude this
introduction by stating mathematical conjectures as examples of such
phenomena.

In section 2, we define a family of ${\mathcal{N}}=(2,2)$ supersymmetric gauge
theories labeled by the set of $(m+n-1)$-tuples of nonnegative integers
$\mathbb{Z}_{\geq 0}^{m+n-1}$, corresponding to a closed rational
$\mathfrak{gl}(m|n)$ spin chain of length $L$ with spins valued in Verma
modules. If we turn off all mass parameters and turn on appropriate
Fayet–Iliopoulos (FI) parameters, these theories are described in the infrared
by effective sigma models. The target space of the sigma model with label
$\mathbf{M}=(M_{1},\dotsc,M_{m+n-1})$ is a Calabi–Yau manifold
${\mathcal{M}}(\mathbf{M})$ with an action of
$\mathrm{GL}(L)^{m+n}\times\mathrm{GL}(1)$. The topological A-twist of this
sigma model, with mass parameters associated with the maximal torus $T$ of
$\mathrm{GL}(L)^{m+n}\times\mathrm{GL}(1)$ turned on, is equivalent to the
sector of the spin chain in which there are $M_{r}$ magnons of type $r$. The
highest weights of the Verma modules are determined by the mass parameters.

######  Conjecture 1.

The direct sum of equivariant cohomology groups

$\bigoplus_{\mathbf{M}\in\mathbb{Z}_{\geq
0}^{m+n-1}}H_{T}\bigl{(}{\mathcal{M}}(\mathbf{M})\bigr{)}$ (1)

is a module over $Y(\mathfrak{gl}(m|n))$, isomorphic to the tensor product of
$L$ evaluation modules obtained from the Verma modules.

######  Conjecture 2.

There is a homomorphism from the Bethe algebra of the Yangian
$Y(\mathfrak{gl}(m|n))$ to the direct sum of equivariant quantum cohomology
rings

$\bigoplus_{\mathbf{M}\in\mathbb{Z}_{\geq
0}^{m+n-1}}\mathit{QH}_{T}\bigl{(}{\mathcal{M}}(\mathbf{M})\bigr{)}\,.$ (2)

The first conjecture says that the Hilbert space of states of the A-model is
the same as that of the spin chain. For $n=0$ and $L=1$, the conjecture is
proved in [26]. The second conjecture means that the algebra of local
operators of the A-model includes the algebra generated by the commuting
conserved charges of the spin chain. For $n=0$, this conjecture follows from a
result of [27].

Similar conjectures can be made for the target spaces of effective sigma
models corresponding to compact rational $\mathfrak{gl}(m|n)$ spin chains. The
brane configurations in the compact case have been recently considered by
Rimanyi and Rozansky [28] from the perspective of geometric construction of
R-matrices [27], so we expect that the above conjectures also hold if the
target spaces are varieties defined in [28].

##  2 Bethe/gauge correspondence for noncompact superspin chains

The Bethe/gauge correspondence for superspin chains relates a closed spin
chain with $\mathrm{GL}(m|n)$ symmetry and two-dimensional gauge theories with
${\mathcal{N}}=(2,2)$ supersymmetry. In this section we present a version of
the correspondence in which the spin chain consists of spins taking values in
infinite-dimensional highest-weight representations of $\mathfrak{gl}(m|n)$.
After reviewing some basic facts about $\mathfrak{gl}(m|n)$ and its Verma
modules, we introduce the spin chain and its Bethe equations. Then, we
introduce the gauge theories and their vacuum equations, and explain in what
sense the two sides are equivalent.

###  2.1 $\mathfrak{gl}(m|n)$ and its Verma modules

To begin with, let us review the structures of $\mathfrak{gl}(m|n)$ and its
Verma modules, with emphasis on aspects that are important for the Bethe/gauge
correspondence.

####  2.1.1 Lie superalgebra $\mathfrak{gl}(m|n)$

Let $\mathbb{C}^{m|n}$ be the vector space graded by
$\mathbb{Z}_{2}=\\{\bar{0},\bar{1}\\}$ whose even subspace
$\mathbb{C}^{m|n}_{\bar{0}}=\mathbb{C}^{m}$ and odd subspace
$\mathbb{C}^{m|n}_{\bar{1}}=\mathbb{C}^{n}$. Let $(b_{1},\dotsc,b_{m})$ and
$(f_{1},\dotsc,f_{n})$ be the standard basis of $\mathbb{C}^{m}$ and that of
$\mathbb{C}^{n}$, respectively. Throughout this section and the next section
except section 3.4, we fix an ordered basis $(e_{1},\dotsc,e_{m+n})$ of
$\mathbb{C}^{m|n}$ that is a permutation of
$(b_{1},\dotsc,b_{m},f_{1},\dotsc,f_{n})$. The corresponding
$\mathbb{Z}_{2}$-grading $[-]\colon\\{1,\dotsc,m+n\\}\to\mathbb{Z}_{2}$ is
defined by

$[i]=\begin{cases}\bar{0}&(e_{i}\in\mathbb{C}^{m})\,;\\\
\bar{1}&(e_{i}\in\mathbb{C}^{n})\,.\end{cases}$ (3)

The space of endomorphisms $\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})$
of $\mathbb{C}^{m|n}$ is also a $\mathbb{Z}_{2}$-graded vector space, with the
even subspace

$\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})_{\bar{0}}=\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m},\mathbb{C}^{m})\oplus\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{n},\mathbb{C}^{n})$
(4)

and the odd subspace

$\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})_{\bar{1}}=\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{m},\mathbb{C}^{n})\oplus\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{n},\mathbb{C}^{m})\,.$
(5)

The elementary matrix $E_{ij}$, which has $1$ in the $(i,j)$th entry and $0$
elsewhere, has grading $[E_{ij}]=[i]+[j]$.

The Lie superalgebra $\mathfrak{gl}(m|n)$ is the $\mathbb{Z}_{2}$-graded
vector space $\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})$, endowed with
the graded commutator
$[-,-]\colon\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})\otimes\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})\to\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})$:
for elements $a$, $b$ with homogeneous $\mathbb{Z}_{2}$-grading,

$[a,b]=ab-(-1)^{[a][b]}ba\,.$ (6)

We will distinguish the elements of $\mathfrak{gl}(m|n)$ from those of
$\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})$ by writing them as
${\mathcal{E}}_{ij}$ rather than $E_{ij}$. They satisfy the commutation
relations

$[{\mathcal{E}}_{ij},{\mathcal{E}}_{kl}]=\delta_{jk}{\mathcal{E}}_{il}-(-1)^{([i]+[j])([k]+[l])}\delta_{li}{\mathcal{E}}_{kj}\,.$
(7)

The Cartan subalgebra of $\mathfrak{gl}(m|n)$ is generated by

$H_{r}=(-1)^{[r]}{\mathcal{E}}_{rr}-(-1)^{[r+1]}{\mathcal{E}}_{r+1,r+1}\,,\quad
r=1\,,\dotsc,m+n-1\,,$ (8)

and one more diagonal matrix, say ${\mathcal{E}}_{11}$. The elementary matrix
${\mathcal{E}}_{ij}$ has the root $\varepsilon_{i}-\varepsilon_{j}$, with

$\varepsilon_{i}={\mathcal{E}}_{ii}^{\vee}$ (9)

being the weight of $e_{i}$ in the natural $(m+n)$-dimensional representation.
The positive roots are $\varepsilon_{i}-\varepsilon_{j}$ with $i<j$. The
simple roots are

$\alpha_{r}=\varepsilon_{r}-\varepsilon_{r+1}\,,\quad r=1\,,\dotsc,m+n-1\,.$
(10)

The elements having the roots $\alpha_{r}$ and $-\alpha_{r}$ are
$E_{r}={\mathcal{E}}_{r,r+1}$ and $F_{r}={\mathcal{E}}_{r+1,r}$, respectively.
They satisfy

$\displaystyle[H_{r},E_{s}]$ $\displaystyle=a_{rs}E_{s}\,,$ (11)
$\displaystyle[H_{r},F_{s}]$ $\displaystyle=-a_{rs}F_{s}\,,$ (12)
$\displaystyle[E_{r},F_{s}]$ $\displaystyle=\delta_{rs}(-1)^{[r]}H_{r}\,,$
(13)

where

$a_{rs}=\alpha_{s}(H_{r})=\delta_{rs}\bigl{(}(-1)^{[r]}+(-1)^{[r+1]}\bigr{)}-\delta_{r+1,s}(-1)^{[r+1]}-\delta_{r,s+1}(-1)^{[r]}$
(14)

is the $(r,s)$th entry of the Cartan matrix.

The structure of $\mathfrak{gl}(m|n)$ can be encoded in a Dynkin diagram, in
which a simple root $\alpha_{r}$ is represented by a blank node if $a_{rr}=\pm
2$ and a crossed node if $a_{rr}=0$, and two nodes $\alpha_{r}$, $\alpha_{s}$
are connected by an edge if $a_{rs}\neq 0$. As an example, consider the case
with $(m|n)=(3|2)$ and
$(e_{1},e_{2},e_{3},e_{4},e_{5})=(b_{1},b_{2},f_{1},f_{2},b_{3})$. The
associated Dynkin diagram is

$\epsilon_{1}-\epsilon_{2}$$\epsilon_{2}-\delta_{1}$$\delta_{1}-\delta_{2}$$\delta_{2}-\epsilon_{3}$
(15)

where $\epsilon_{i}$ and $\delta_{i}$ are the weights of $b_{i}$ and $f_{i}$,
respectively.

Let us give a different presentation of the content of the Dynkin diagram in
terms of a quiver diagram, which makes the connection to gauge theory
transparent.

First, we represent $\varepsilon_{i}$ by a vertical line, of one of two colors
depending on its grading:

$\varepsilon_{i}=\begin{cases}\ \leavevmode\hbox to1.2pt{\vbox
to15.2pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.05859375,0.125,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.05859375,0.125,0.5}\pgfsys@color@rgb@stroke{0.05859375}{0.125}{0.5}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.05859375}{0.125}{0.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.05859375,0.125,0.5}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{14.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&([i]=\bar{0})\,;\\\ \
\leavevmode\hbox to1.2pt{\vbox to15.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.9609375,0.47265625,0.2265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.9609375,0.47265625,0.2265625}\pgfsys@color@rgb@stroke{0.9609375}{0.47265625}{0.2265625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.9609375}{0.47265625}{0.2265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.9609375,0.47265625,0.2265625}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{14.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&([i]=\bar{1})\,.\end{cases}$
(16)

The ordered set $(\varepsilon_{1},\dotsc,\varepsilon_{m+n})$ is then
represented graphically as $m$ vertical lines of one color and $n$ vertical
lines of the other color, placed in the order specified by the choice of the
$\mathbb{Z}_{2}$-grading:

(17)

Next, we put a circle node between each pair of adjacent vertical lines:

(18)

The $r$th node represents the simple root $\alpha_{r}$.

Finally, for each pair $(r,s)$ with $a_{rs}\neq 0$, we draw an arrow from the
$r$th node to the $s$th node and write the number $a_{rs}$ on the side. We can
erase the vertical lines at this stage:

$-1$$-1$$+1$$+1$$+1$$+1$$+2$$-2$ (19)

This quiver has the same content as the Dynkin diagram (15) modulo the action
of the Weyl group $\mathfrak{S}_{m}\times\mathfrak{S}_{n}$ which permutes the
basis vectors $(e_{1},\dotsc,e_{m+n})$ without changing the
$\mathbb{Z}_{2}$-grading.

####  2.1.2 Verma modules of $\mathfrak{gl}(m|n)$

A representation of $\mathfrak{gl}(m|n)$ in a $\mathbb{Z}_{2}$-graded vector
space $V$ is a Lie superalgebra homomorphism
$\pi\colon\mathfrak{gl}(m|n)\to\mathop{\mathrm{End}}\nolimits(V)$, where
$\mathop{\mathrm{End}}\nolimits(V)$ is given the structure of a Lie
superalgebra by the graded commutator.

The Verma module
$M(\lambda)\colon\mathfrak{gl}(m|n)\to\mathop{\mathrm{End}}\nolimits(V_{\lambda})$,
with highest weight

$\lambda=\sum_{i=1}^{m+n}\lambda_{i}\varepsilon_{i}\,,\quad\lambda_{i}\in\mathbb{C}\,,$
(20)

is a representation of $\mathfrak{gl}(m|n)$ constructed from a highest-weight
vector $|\Omega_{\lambda}\rangle$ that is an eigenstate of the diagonal
matrices:

$\displaystyle{\mathcal{E}}_{ii}|\Omega_{\lambda}\rangle$
$\displaystyle=\lambda_{i}|\Omega_{\lambda}\rangle\,,$ (21)
$\displaystyle{\mathcal{E}}_{ij}|\Omega_{\lambda}\rangle$
$\displaystyle=0\,,\quad i<j\,.$ (22)

The other vectors in $V_{\lambda}$ are created by the action of lowering
operators $\\{{\mathcal{E}}_{ij}\mid i>j\\}$ on $|\Omega_{\lambda}\rangle$,
and two vectors are identified if they are related by the commutation
relations (7).

More explicitly, the Fock space $V_{\lambda}$ can be described as follows. Let
us introduce an ordering among all lowering operators and name them $x_{1}$,
$\dotsc$, $x_{p}$. Then, by the Poincaré–Birkhoff–Witt (PBW) theorem, an
element of $V_{\lambda}$ is a linear combination of states of the form

$x_{1}^{n_{1}}\dotsm x_{p}^{n_{p}}|\Omega_{\lambda}\rangle\,,\quad
n_{i}\in\begin{cases}\mathbb{Z}_{\geq 0}&([a_{i}]=\bar{0})\,;\\\
\\{0,1\\}&([a_{i}]=\bar{1})\,.\end{cases}$ (23)

Verma modules are infinite-dimensional unless $(m|n)=(1|1)$, in which case
there is only one lowering operator and it is odd.

Since the lowering operator ${\mathcal{E}}_{ij}$ changes the weight by
$\varepsilon_{i}-\varepsilon_{j}=-\alpha_{j}-\alpha_{j+1}-\dotsb-\alpha_{i-1}$,
a state of $M(\lambda)$ has a weight of the form

$\lambda-\sum_{r=1}^{m+n-1}M_{r}\alpha_{r}\,,\quad M_{r}\in\mathbb{Z}_{\geq
0}\,.$ (24)

We can also represent this weight graphically. To represent the highest weight
$\lambda$, for each vertical line we draw a diagonal line ending on it and
write $\lambda_{i}$ next to the $i$th diagonal line; and to represent the
weight (24), we draw $M_{r}$ horizontal line segments between the $r$th and
$(r+1)$st vertical lines. Here is an example for
$(M_{1},M_{2},M_{3},M_{4})=(2,3,2,1)$:

$\lambda_{1}$$\lambda_{2}$$\lambda_{3}$$\lambda_{4}$$\lambda_{5}$ (25)

We convert this diagram into a quiver by replacing the diagonal lines with
square nodes and writing $M_{r}$ inside the $r$th circle node and $1$ inside
the square nodes:

$-1$$-1$$+1$$+1$$+1$$+1$$+2$$-2$$M_{1}$$M_{2}$$M_{3}$$M_{4}$$1$$1$$1$$1$$1$$\lambda_{1}$$\lambda_{2}$$\lambda_{3}$$\lambda_{4}$$\lambda_{5}$
(26)

To fix the horizontal positions of the square nodes, we have added arrows
connecting circle and square nodes.

####  2.1.3 Tensor products of Verma modules

If $V_{1}$ and $V_{2}$ are $\mathbb{Z}_{2}$-graded vector spaces, the tensor
product $V_{1}\otimes V_{2}$ is naturally $\mathbb{Z}_{2}$-graded. Given two
representations
$\pi_{1}\colon\mathfrak{gl}(m|n)\to\mathop{\mathrm{End}}\nolimits(V_{1})$ and
$\pi_{2}\colon\mathfrak{gl}(m|n)\to\mathop{\mathrm{End}}\nolimits(V_{2})$, the
tensor product representation
$\pi_{1}\otimes\pi_{2}\colon\mathfrak{gl}(m|n)\to\mathop{\mathrm{End}}\nolimits(V_{1}\otimes
V_{2})$ is defined by

$(\pi_{1}\otimes\pi_{2})(x)(v_{1}\otimes v_{2})=\pi_{1}(x)v_{1}\otimes
v_{2}+(-1)^{[x][v_{1}]}v_{1}\otimes\pi_{2}(x)v_{2}\,,$ (27)

where $v_{1}$, $v_{2}$ and $x$ are homogeneous in $\mathbb{Z}_{2}$-grading.
The tensor products of more than two representations can be defined
recursively.

The tensor product of $L$ Verma modules $M(\lambda^{1})$, $\dotsc$,
$M(\lambda^{L})$ has a highest-weight vector
$|\Omega_{\lambda^{1}}\rangle\otimes\dotsb\otimes|\Omega_{\lambda^{L}}\rangle$
with highest weight $\lambda^{1}+\dotsb+\lambda^{L}$. A weight of
$M(\lambda^{1})\otimes\dotsb\otimes M(\lambda^{L})$ takes the form

$\sum_{\ell=1}^{L}\lambda^{\ell}-\sum_{r=1}^{m+n-1}M_{r}\alpha_{r}\,,\quad
M_{r}\in\mathbb{Z}_{\geq 0}\,.$ (28)

Graphically, we represented it by a diagram similar to the diagram (25) for a
weight of a single Verma module, but with $L$ diagonal lines ending on each
vertical line. For example, the diagram

$\lambda^{3}_{1}$$\lambda^{3}_{2}$$\lambda^{3}_{3}$$\lambda^{3}_{4}$$\lambda^{3}_{5}$$\lambda^{2}_{1}$$\lambda^{2}_{2}$$\lambda^{2}_{3}$$\lambda^{2}_{4}$$\lambda^{2}_{5}$$\lambda^{1}_{1}$$\lambda^{1}_{2}$$\lambda^{1}_{3}$$\lambda^{1}_{4}$$\lambda^{1}_{5}$
(29)

represents a weight with $(M_{1},M_{2},M_{3},M_{4})=(2,1,0,2)$ in the
representation $M(\lambda^{1})\otimes M(\lambda^{2})\otimes M(\lambda^{3})$ of
$\mathfrak{gl}(3|2)$.

The corresponding quiver diagram is the same as before, except that the $i$th
square node is now labeled $L$ and accompanied by the $L$-tuple
$\vec{\lambda}_{i}=(\lambda^{1}_{i},\dotsc,\lambda^{L}_{i})$:

$-1$$-1$$+1$$+1$$+1$$+1$$+2$$-2$$M_{1}$$M_{2}$$M_{3}$$M_{4}$$L$$L$$L$$L$$L$$\vec{\lambda}_{1}$$\vec{\lambda}_{2}$$\vec{\lambda}_{3}$$\vec{\lambda}_{4}$$\vec{\lambda}_{5}$
(30)

The above quiver diagram will be identified with a quiver describing a two-
dimensional ${\mathcal{N}}=(2,2)$ supersymmetric gauge theory that appears on
the gauge theory side of the Bethe/gauge correspondence. The graphical
representation using lines will be interpreted as a diagram depicting a brane
configuration in string theory.

###  2.2 Bethe side

Now we explain the Bethe side of the Bethe/gauge correspondence. The spin
chains we consider in this paper are rational $\mathfrak{gl}(m|n)$ spin
chains, for which spins take values in representations of
$\mathfrak{gl}(m|n)$. More generally, spins in rational $\mathfrak{gl}(m|n)$
spin chains are valued in representations of the Yangian
$Y(\mathfrak{gl}(m|n))$ of $\mathfrak{gl}(m|n)$.

####  2.2.1 Yangian

The _Yangian_ $Y(\mathfrak{gl}(m|n))$ is a $\mathbb{Z}_{2}$-graded Hopf
algebra, which in particular is a unital associative $\mathbb{Z}_{2}$-graded
algebra. It is generated by elements

$T_{ij}^{(l)}\,,\quad i,j=1,\dotsc,m+n\,,\quad l\in\mathbb{Z}_{>0}\,,$ (31)

with grading $[T_{ij}^{(l)}]=[i]+[j]$. The level-$1$ generators $T_{ij}^{(1)}$
span a subalgebra isomorphic to $\mathfrak{gl}(m|n)$, with the identification
of generators being ${\mathcal{E}}_{ij}=(-1)^{[j]}T_{ji}^{(1)}$.

To describe the algebra relations for all generators in a compact form, let us
introduce a formal variable $\sigma$ and combine the generators into a single
$\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})\otimes
Y(\mathfrak{gl}(m|n))$-valued power series in $\sigma^{-1}$:

$T(\sigma)=\sum_{i,j=1}^{m+n}E_{ij}\otimes
T_{ij}(\sigma)=\sum_{i,j=1}^{m+n}\sum_{l=0}^{\infty}\frac{\hbar^{l}}{\sigma^{l}}E_{ij}\otimes
T_{ij}^{(l)}\,.$ (32)

Here $T_{ij}^{(0)}=\delta_{ij}$ and $\hbar$ is a complex parameter. We can
think of $T(\sigma)$ as an $(m+n)\times(m+n)$ matrix whose entries are
elements of $Y(\mathfrak{gl}(m|n))[[\sigma^{-1}]]$; it is called the
_monodromy matrix_ and is a function of _spectral parameter_ $\sigma$. In
terms of the monodromy matrix, the algebra relations for
$Y(\mathfrak{gl}(m|n))$ are encoded in the _RTT relation_

$R_{12}(\sigma_{1}-\sigma_{2})T_{1}(\sigma_{1})T_{2}(\sigma_{2})=T_{2}(\sigma_{2})T_{1}(\sigma_{1})R_{12}(\sigma_{1}-\sigma_{2})\,.$
(33)

This is a relation between elements in
$\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})\otimes\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})\otimes
Y(\mathfrak{gl}(m|n))[[\sigma^{-1}]]$, and the subscript(s) on an operator
indicate which factor(s) of $\mathbb{C}^{m|n}$ the operator acts on.

The operator
$R_{12}(\sigma)\in\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})\otimes\mathop{\mathrm{End}}\nolimits(\mathbb{C}^{m|n})$
that appears in the RTT relation is the _rational $\mathfrak{gl}(m|n)$
R-matrix_. It is given by

$R_{12}(\sigma)=\sigma I\otimes I+\hbar P_{12}\,,$ (34)

where $I$ is the identity matrix and

$P_{12}=\sum_{i,j=1}^{m+n}(-1)^{[j]}E_{ij}\otimes E_{ji}\,.$ (35)

The permutation operator $P_{12}$ swaps tensor factors as

$P_{12}(e_{i}\otimes e_{j})=(-1)^{[i][j]}e_{j}\otimes e_{i}\,.$ (36)

The R-matrix commutes with the automorphism group $\mathrm{GL}(m|n)$ of
$\mathbb{C}^{m|n}$:

$[g\otimes g,R_{12}(\sigma)]=0\,,\quad g\in\mathrm{GL}(m|n)\,.$ (37)

The dynamics of a closed rational $\mathfrak{gl}(m|n)$ spin chain is generated
by the _transfer matrix_

$t(g,\sigma)=\mathop{\mathrm{str}}\nolimits_{\mathbb{C}^{m|n}}\bigl{(}gT(\sigma)\bigr{)}=\sum_{i=1}^{m+n}(-1)^{[i]}g_{ij}T_{ji}(\sigma)\,,\quad
g\in\mathrm{GL}(m|n)\,.$ (38)

The supertrace taken over $\mathbb{C}^{m|n}$ corresponds to the topology of
the spin chain which is closed, and $g$ twists the periodic boundary
condition.

Multiplying both sides of the RTT relation (33) by $g\otimes g\otimes 1$ from
the left and $R_{12}(\sigma_{1}-\sigma_{2})^{-1}$ from the right, then using
the symmetry (37) of the R-matrix and taking the supertrace over
$\mathbb{C}^{m|n}\otimes\mathbb{C}^{m|n}$, we see that transfer matrices for a
fixed $g$ commute with each other:

$[t(g,\sigma_{1}),t(g,\sigma_{2})]=0\,.$ (39)

Therefore, if we expand $t(g,\sigma)$ in powers of $\sigma^{-1}$, the
coefficients are mutually commuting elements of $Y(\mathfrak{gl}(m|n))$. They
generate a commutative subalgebra called the _Bethe algebra_ (or the Baxter
algebra) of $Y(\mathfrak{gl}(m|n))$.

####  2.2.2 Representations of $Y(\mathfrak{gl}(m|n))$

While the Yangian $Y(\mathfrak{gl}(m|n))$ and its Bethe algebra are the
algebraic structures underlying rational $\mathfrak{gl}(m|n)$ spin chains, to
get a concrete physical realization of a spin chain we need to specify a
representation of $Y(\mathfrak{gl}(m|n))$.

A representation $\rho\colon
Y(\mathfrak{gl}(m|n))\to\mathop{\mathrm{End}}\nolimits(V)$ of
$Y(\mathfrak{gl}(m|n))$ maps the monodromy matrix to an
$\mathop{\mathrm{End}}\nolimits(V)$-valued matrix

$\rho\bigl{(}T(\sigma)\bigr{)}=\sum_{i,j=1}^{m+n}\sum_{l=0}^{\infty}\frac{\hbar^{l}}{\sigma^{l}}E_{ij}\otimes\rho(T_{ij}^{(l)})\,.$
(40)

Conversely, an $\mathop{\mathrm{End}}\nolimits(V)$-valued matrix satisfying
the RTT relation determines a representation of $Y(\mathfrak{gl}(m|n))$.

Given a representation
$\pi\colon\mathfrak{gl}(m|n)\to\mathop{\mathrm{End}}\nolimits(V)$ of
$\mathfrak{gl}(m|n)$, we obtain a one-parameter family of representations
$\pi_{\zeta}\colon Y(\mathfrak{gl}(m|n))\to\mathop{\mathrm{End}}\nolimits(V)$,
$\zeta\in\mathbb{C}$, of $Y(\mathfrak{gl}(m|n))$ by

$\pi_{\zeta}\bigl{(}T(\sigma)\bigr{)}=I\otimes\mathop{\mathrm{id}}\nolimits_{V}+\frac{\hbar}{\sigma-\zeta}\sum_{i,j=1}^{m+n}(-1)^{[j]}E_{ij}\otimes\pi({\mathcal{E}}_{ji})\,.$
(41)

This is known as the _evaluation module_ for $\pi$, and $\zeta$ is called the
_inhomogeneity parameter_. Note that we have

$\pi_{\zeta}\bigl{(}(-1)^{[j]}T_{ji}^{(1)}\bigr{)}=\pi({\mathcal{E}}_{ij})\,.$
(42)

For a representation
$\pi\colon\mathfrak{gl}(m|n)\to\mathop{\mathrm{End}}\nolimits(V)$ of
$\mathfrak{gl}(m|n)$, let us define a one-parameter family of representations
$\pi^{c}\colon\mathfrak{gl}(m|n)\to\mathop{\mathrm{End}}\nolimits(V)$,
$c\in\mathbb{C}$, by

$\pi^{c}({\mathcal{E}}_{ij})=\pi({\mathcal{E}}_{ij})+(-1)^{[i]}c\delta_{ij}\mathop{\mathrm{id}}\nolimits_{V}\,.$
(43)

In the associated Yangian representations, the parameter $c$ is related to a
shift in the inhomogeneity parameter. Suppose that $\rho\colon
Y(\mathfrak{gl}(m|n))\to\mathop{\mathrm{End}}\nolimits(V)$ is a representation
of $Y(\mathfrak{gl}(m|n))$. Then, $\rho(T(\sigma))$ satisfies the RTT
equation, and for any function $f$ of $\sigma$, the RTT equation is still
satisfied when $\rho(T(\sigma))$ is replaced by $f(\sigma)\rho(T(\sigma))$.
Therefore, if $f(\sigma)$ can be expanded in a power series in $\sigma^{-1}$
starting from $1$, then $f(\sigma)\rho(T(\sigma))$ defines a new
representation of $Y(\mathfrak{gl}(m|n))$. Since

$\pi^{c}_{\zeta}\bigl{(}T(\sigma)\bigr{)}=\pi_{\zeta}(T(\sigma))+\frac{c\hbar}{\sigma-\zeta}I\otimes\mathop{\mathrm{id}}\nolimits_{V}=\frac{\sigma-\zeta+c\hbar}{\sigma-\zeta}\pi_{\zeta-c\hbar}\bigl{(}T(\sigma)\bigr{)}\,,$
(44)

we see that $\pi^{c}_{\zeta}$ and $\pi_{\zeta-c\hbar}$ are related in this
manner.

To construct tensor product representations of $Y(\mathfrak{gl}(m|n))$, we use
the coproduct $\Delta\colon Y(\mathfrak{gl}(m|n))\linebreak[1]\to
Y(\mathfrak{gl}(m|n))\otimes Y(\mathfrak{gl}(m|n))$. The map $\Delta$ is
defined by the formula

$\Delta\bigl{(}T(\sigma)\bigr{)}=\sum_{i,j,k=1}^{m+n}(-1)^{([i]+[k])([k]+[j])}E_{ij}\otimes
T_{ik}(\sigma)\otimes T_{kj}(\sigma)\,.$ (45)

Given two representations $\rho_{1}\colon
Y(\mathfrak{gl}(m|n))\to\mathop{\mathrm{End}}\nolimits(V_{1})$ and
$\rho_{2}\colon
Y(\mathfrak{gl}(m|n))\to\mathop{\mathrm{End}}\nolimits(V_{2})$, the tensor
product representation $\rho_{1}\dot{\otimes}\rho_{2}\colon
Y(\mathfrak{gl}(m|n))\to\mathop{\mathrm{End}}\nolimits(V_{1}\otimes V_{2})$ is
defined by

$\rho_{1}\dot{\otimes}\rho_{2}=(\rho_{1}\otimes\rho_{2})\circ\Delta\,.$ (46)

A calculation shows that $(\rho_{1}\dot{\otimes}\rho_{2})(T(\sigma))$
satisfies the RTT relation.

####  2.2.3 The spin chain

Now, fix a positive integer $L$, and choose $L$ highest weights

$\vec{\lambda}=(\lambda^{1},\dotsc,\lambda^{L})$ (47)

of $\mathfrak{gl}(m|n)$ and $L$ inhomogeneity parameters

$\vec{\zeta}=(\zeta^{1},\dotsc,\zeta^{L})\,.$ (48)

Also, choose a diagonal element111The spin chain can be defined for any choice
of $g$, not necessarily diagonal ones. However, nondiagonalizable choices of
$g$ do not appear to have a clear interpretation on the gauge theory side of
the Bethe/gauge correspondence. of $\mathrm{GL}(m|n)$:

$g=\mathop{\mathrm{diag}}\nolimits(e^{\phi_{1}},\dotsc,e^{\phi_{m+n}})\,,\quad\phi_{i}\in\mathbb{C}\,.$
(49)

We consider the closed rational $\mathfrak{gl}(m|n)$ spin chain of length $L$,
with the spin at the $\ell$th site valued in the evaluation module
$M(\lambda^{\ell})_{\zeta^{\ell}}$ for the Verma module $M(\lambda^{\ell})$
and the periodic boundary condition twisted by $g$.

The Hilbert space of states of this spin chain is the tensor product

$V_{\vec{\lambda}}=\bigotimes_{\ell=1}^{L}V_{\lambda^{\ell}}\,,$ (50)

and the $L$ spins can be thought of as a single spin in the tensor product
representation

$M(\vec{\lambda})_{\vec{\zeta}}=M(\lambda^{1})_{\zeta^{1}}\dot{\otimes}\dotsb\dot{\otimes}M(\lambda^{L})_{\zeta^{L}}\,.$
(51)

The transfer matrix of the spin chain

$M(\vec{\lambda})_{\vec{\zeta}}\bigl{(}t(g,\sigma)\bigr{)}$ (52)

generates commuting conserved charges acting on $V_{\vec{\lambda}}$, making
the spin chain integrable. The Hamiltonian is a linear combination of these
charges.

####  2.2.4 Bethe equations

The Hilbert space of the spin chain is spanned by vectors that simultaneously
diagonalize the commuting conserved charges, or equivalently, diagonalize the
transfer matrix (52) for all values of the spectral parameter $\sigma$. These
eigenvectors, referred to as _Bethe vectors_ , are the main characters from
the Bethe side of the Bethe/gauge correspondence.

The Bethe vectors for the rational $\mathfrak{gl}(m|n)$ spin chain have been
constructed by Bethe ansatz methods [29, 30]. The construction starts with the
highest-weight vector $|\Omega_{\vec{\lambda}}\rangle$ of the tensor product
representation

$M(\vec{\lambda})=\bigotimes_{\ell=1}^{L}M(\lambda^{\ell})\,.$ (53)

This state is called the _pseudovacuum_ and satisfies

$\displaystyle
M(\vec{\lambda})_{\vec{\zeta}}\bigl{(}T_{ii}(\sigma)\bigr{)}|\Omega_{\vec{\lambda}}\rangle$
$\displaystyle=\Biggl{(}\prod_{\ell=1}^{L}\frac{\sigma-\zeta^{\ell}+(-1)^{[i]}\lambda_{i}^{\ell}\hbar}{\sigma-\zeta^{\ell}}\Biggr{)}|\Omega_{\vec{\lambda}}\rangle\,,$
(54) $\displaystyle
M(\vec{\lambda})_{\vec{\zeta}}\bigl{(}T_{ij}(\sigma)\bigr{)}|\Omega_{\vec{\lambda}}\rangle$
$\displaystyle=0\,,\quad i>j\,.$ (55)

According to our graphical notation, the pseudovacuum is represented by a
diagram with no horizontal segments:

$|\Omega_{\vec{\lambda}}\rangle=\leavevmode\hbox to267.06pt{\vbox
to62.93pt{\pgfpicture\makeatletter\hbox{\hskip
42.46048pt\lower-20.33034pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.05859375,0.125,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.05859375,0.125,0.5}\pgfsys@color@rgb@stroke{0.05859375}{0.125}{0.5}\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{42.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{28.0pt}{12.60013pt}\pgfsys@lineto{56.0pt}{33.60013pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.21875,0.12890625,0.09765625}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.87253pt}{9.70662pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.9609375}{0.47265625}{0.2265625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.9609375,0.47265625,0.2265625}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{84.0pt}{12.60013pt}\pgfsys@lineto{112.0pt}{33.60013pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.21875,0.12890625,0.09765625}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{72.87253pt}{9.70662pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\definecolor[named]{pgffillcolor}{rgb}{0.9609375,0.47265625,0.2265625}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{140.0pt}{12.60013pt}\pgfsys@lineto{168.0pt}{33.60013pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.21875,0.12890625,0.09765625}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.05859375}{0.125}{0.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.05859375,0.125,0.5}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{196.0pt}{12.60013pt}\pgfsys@lineto{224.0pt}{33.60013pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.21875,0.12890625,0.09765625}{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.21875,0.12890625,0.09765625}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{184.87253pt}{9.70662pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\color[rgb]{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}$\lambda^{1}_{5}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (56)

Excited states are obtained from the pseudovacuum by the action of creation
operators $M(\vec{\lambda})_{\vec{\zeta}}(T_{ij}(\sigma))$, $i<j$. The
operator $M(\vec{\lambda})_{\vec{\zeta}}(T_{ij}(\sigma))$ contains
$M(\vec{\lambda})({\mathcal{E}}_{ji})$ and changes the $\mathfrak{gl}(m|n)$
weight by
$\varepsilon_{j}-\varepsilon_{i}=-\alpha_{i}-\alpha_{i+1}-\dotsb-\alpha_{j-1}$.
Roughly speaking, we can interpret this action as creating a single quasi-
particle, or a _magnon_ , of rapidity $\sigma$ and type $r$ for each $r=i$,
$i+1$, $\dotsc$, $j-1$. Graphically, we think of it as creating a horizontal
line connecting the $i$th and $j$th vertical lines:

$T_{13}(\sigma_{1})T_{25}(\sigma_{2})T_{45}(\sigma_{3})|\Omega_{\vec{\lambda}}\rangle\sim\leavevmode\hbox
to191.1pt{\vbox to55.8pt{\pgfpicture\makeatletter\hbox{\hskip
21.59999pt\lower-13.20012pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}
\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}{}{{{}{}}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.35013pt}\pgfsys@lineto{84.0pt}{28.35013pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{18.45241pt}{31.78683pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\color[rgb]{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}$\sigma_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\par\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.0pt}{23.10013pt}\pgfsys@lineto{168.0pt}{23.10013pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{101.45244pt}{26.53683pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\color[rgb]{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}$\sigma_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\par\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}{}{{{}{}}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{126.0pt}{9.45026pt}\pgfsys@lineto{168.0pt}{9.45026pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{135.97186pt}{12.88696pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}\pgfsys@color@rgb@stroke{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0.21875}{0.12890625}{0.09765625}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0.21875,0.12890625,0.09765625}\color[rgb]{0.21875,0.12890625,0.09765625}\definecolor[named]{pgfstrokecolor}{rgb}{0.21875,0.12890625,0.09765625}$\sigma_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \par{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{3.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.0pt}{27.29974pt}\pgfsys@lineto{42.0pt}{29.39987pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{3.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{84.0pt}{22.25995pt}\pgfsys@lineto{84.0pt}{23.9403pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
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}{}\pgfsys@moveto{105.0pt}{12.60013pt}\pgfsys@lineto{126.0pt}{33.60013pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.05859375,0.125,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0.05859375,0.125,0.5}\pgfsys@color@rgb@stroke{0.05859375}{0.125}{0.5}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0.05859375,0.125,0.5}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{147.0pt}{12.60013pt}\pgfsys@lineto{168.0pt}{33.60013pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (57)

This is, however, not a precise correspondence because the left-hand side
depends on the ordering of creation operators.

The operator $T_{ji}(\sigma)$, $i<j$, changes the weight by
$\alpha_{i}+\alpha_{i+1}+\dotsb+\alpha_{j-1}$, so annihilates magnons of type
$m=i$, $i+1$, $\dotsc$, $j-1$. It removes one horizontal line from each of the
intervals between the $i$th and $j$th vertical lines. If there is no
horizontal line to remove, then the state is annihilated.

Eigenvectors of the transfer matrix are excited states constructed by certain
linear combinations of creation operators. It turns out that a Bethe vector
with the $\mathfrak{gl}(m|n)$ weight (28) is specified by a _Bethe root_

$(\\{\sigma_{1}^{1},\dotsc,\sigma_{1}^{M_{1}}\\},\\{\sigma_{2}^{1},\dotsc,\sigma_{2}^{M_{2}}\\},\dotsc,\\{\sigma_{m+n-1}^{1},\dotsc,\sigma_{m+n-1}^{M_{m+n-1}}\\})\,,$
(58)

which is a solution of the _Bethe equations_

$e^{\tau_{r}}\prod_{s=1}^{m+n-1}\prod_{b_{s}=1}^{M_{s}}\frac{\sigma_{r}^{a_{r}}-\sigma_{s}^{b_{s}}+\frac{1}{2}a_{rs}\hbar}{\sigma_{r}^{a_{r}}-\sigma_{s}^{b_{s}}-\frac{1}{2}a_{rs}\hbar}\\\
=(-1)^{\delta_{[r],[r+1]}}\prod_{\ell=1}^{L}\frac{\sigma_{r}^{a_{r}}-\zeta^{\ell}+(-1)^{[r]}\lambda_{r}^{\ell}\hbar-\frac{1}{2}c_{r}\hbar}{\sigma_{r}^{a_{r}}-\zeta^{\ell}+(-1)^{[r+1]}\lambda_{r+1}^{\ell}\hbar-\frac{1}{2}c_{r}\hbar}\,,\\\
a_{r}=1,\dotsc,M_{r}\,,\quad r=1,\dotsc,m+n-1\,.$ (59)

Here we have defined

$\tau_{r}=(-1)^{[r+1]}\phi_{r+1}-(-1)^{[r]}\phi_{r}$ (60)

and

$c_{i}=\sum_{j=1}^{i}(-1)^{[j]}\,.$ (61)

We represent this Bethe vector by a diagram with $M_{r}$ horizontal lines
between the $r$th and $(r+1)$st vertical lines, with labels
$\\{\sigma_{r}^{1},\dotsc,\sigma_{r}^{M_{r}}\\}$. Therefore, the diagram

$\sigma_{1}^{1}$$\sigma_{3}^{1}$$\sigma_{4}^{2}$$\sigma_{4}^{1}$$\sigma_{2}^{1}$$\sigma_{2}^{2}$$\lambda^{3}_{1}$$\lambda^{3}_{2}$$\lambda^{3}_{3}$$\lambda^{3}_{4}$$\lambda^{3}_{5}$$\lambda^{2}_{1}$$\lambda^{2}_{2}$$\lambda^{2}_{3}$$\lambda^{2}_{4}$$\lambda^{2}_{5}$$\lambda^{1}_{1}$$\lambda^{1}_{2}$$\lambda^{1}_{3}$$\lambda^{1}_{4}$$\lambda^{1}_{5}$
(62)

represents a Bethe vector of the closed $\mathfrak{gl}(3|2)$ rational spin
chain of length $L=3$ that belongs to the magnon sector
$(M_{1},M_{2},M_{3},M_{4})=(1,2,1,2)$ and corresponds to the Bethe root
$(\\{\sigma_{1}^{1}\\},\\{\sigma^{1}_{2},\sigma^{2}_{2}\\},\\{\sigma_{3}^{1}\\},\\{\sigma_{4}^{1},\sigma_{4}^{2}\\})$.

Note that the Bethe equations are invariant under the shift

$\zeta^{\ell}\mapsto\zeta^{\ell}+c^{\ell}\hbar\,,\quad\lambda^{\ell}_{i}\mapsto\lambda^{\ell}_{i}+(-1)^{[i]}c^{\ell}\,,\quad
c^{\ell}\in\mathbb{C}\,.$ (63)

This is a consequence of the relation (44) between a representation with
shifted highest weight and a representation with shifted inhomogeneity
parameter. Since multiplying the transfer matrix by a function of the spectral
parameter does not change its eigenvectors, the Bethe equations remain the
same if we shift the highest weights and inhomogeneity parameters as above.

###  2.3 Gauge side

Now we turn to the gauge side of the Bethe/gauge correspondence. The closed
rational $\mathfrak{gl}(m|n)$ spin chain discussed above corresponds to a
family of two-dimensional ${\mathcal{N}}=(2,2)$ supersymmetric gauge theories
whose field contents are described by quivers. These theories have
supersymmetric vacua that are in one-to-one correspondence with the Bethe
vectors of the spin chain. For background knowledge on ${\mathcal{N}}=(2,2)$
supersymmetric gauge theories, we refer the reader to [31].

####  2.3.1 The gauge theories

The magnon sectors of the spin chain (the weight spaces of the Hilbert space
(50)) are labeled by $(m+n-1)$-tuples of nonnegative integers. The sector with
magnon numbers $(M_{1},\dotsc,M_{m+n-1})$ corresponds to a theory with the
product gauge group

$\mathrm{U}(M_{1})\times\dotsb\times\mathrm{U}(M_{m+n-1})\,.$ (64)

Correspondingly, the theory has vector multiplets $\mathsf{V}_{r}$, $r=1$,
$\dotsc$, $m+n-1$, one for each unitary gauge group factor.

In addition, the theory has various chiral multiplets. If $[r]=[r+1]$, then
there is one chiral multiplet transforming in the adjoint representation of
$\mathrm{U}(M_{r})$:222Here and thereafter, statements about fields such as
the one that follows only indicate the representations in which they are
valued.

$\upphi_{r}\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{M_{r}},\mathbb{C}^{M_{r}})\,,\quad[r]=[r+1]\,,\quad
r=1,\dotsc,m+n-1\,.$ (65)

There are also chiral multiplets

$\displaystyle\mathsf{P}_{i}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{M_{i-1}},\mathbb{C}^{M_{i}})\,,$
$\displaystyle i=2,\dotsc,m+n-1\,,$ (66)
$\displaystyle\widetilde{\mathsf{P}}_{i}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{M_{i}},\mathbb{C}^{M_{i-1}})\,,$
$\displaystyle i=2,\dotsc,m+n-1\,,$ (67) $\displaystyle\mathsf{Q}_{i}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{L},\mathbb{C}^{M_{i-1}})\,,$
$\displaystyle i=2,\dotsc,m+n\,,$ (68)
$\displaystyle\widetilde{\mathsf{Q}}_{i}$
$\displaystyle\in\mathop{\mathrm{Hom}}\nolimits(\mathbb{C}^{M_{i}},\mathbb{C}^{L})\,,$
$\displaystyle i=1,\dotsc,m+n-1\,.$ (69)

It is convenient to introduce the notations

$\upphi_{r}=0\,,\quad[r]\neq[r+1]\,,$ (70)

and

$\mathsf{P}_{1}=\widetilde{\mathsf{P}}_{1}=\mathsf{P}_{m+n}=\widetilde{\mathsf{P}}_{m+n}=\mathsf{Q}_{1}=\widetilde{\mathsf{Q}}_{m+n}=0\,.$
(71)

These chiral multiplets are coupled by the superpotential

$W=\sum_{r=1}^{m+n-1}\mathop{\mathrm{tr}}\nolimits_{\mathbb{C}^{M_{r}}}\Bigl{(}\upphi_{r}\widetilde{\mathsf{P}}_{r+1}\mathsf{P}_{r+1}-\upphi_{r}\mathsf{P}_{r}\widetilde{\mathsf{P}}_{r}+\mathsf{P}_{r}\mathsf{Q}_{r}\widetilde{\mathsf{Q}}_{r}\\\
+\bigl{(}(-1)^{[r]}-(-1)^{[r+1]}\bigr{)}\widetilde{\mathsf{P}}_{r+1}\mathsf{P}_{r+1}\mathsf{P}_{r}\widetilde{\mathsf{P}}_{r}\Bigr{)}\,.$
(72)

The terms involving adjoint chiral multiplets are the cubic superpotentials
required for ${\mathcal{N}}=(4,4)$ supersymmetry, which the theory possesses
if either $m=0$ or $n=0$. The last quartic terms are present only for the
gauge group factors without adjoint chiral multiplets [32].333The quartic
terms can be understood as follows. Suppose that $[r]\neq[r+1]$, say $[r]=0$
and $[r+1]=1$, and introduce a pair of chiral multiplets $\upphi_{r}^{\pm}$ in
the adjoint representation of $\mathrm{U}(M_{r})$. These multiplets are
massive and couple with bifundamental chiral multiplets via superpotential
terms of the form
$\mathop{\mathrm{tr}}\nolimits_{\mathbb{C}^{M_{r}}}(\upphi_{r}^{-}\widetilde{\mathsf{P}}_{r+1}\mathsf{P}_{r+1}-\upphi_{r}^{+}\mathsf{P}_{r}\widetilde{\mathsf{P}}_{r}+m\upphi_{r}^{+}\upphi_{r}^{-})$.
Integrating out $\upphi_{r}^{\pm}$, we get the quartic term. From the point of
view of the brane construction discussed in section 3.1, we imagine the
situation in which $\mathrm{NS5}_{r}$ and $\mathrm{NS5}_{r+1}$ are almost
orthogonal but not quite. The adjoint chiral multiplets $\upphi_{r}^{\pm}$
correspond to the positions of D2-branes along $\mathbb{R}^{2}_{\pm\hbar}$.

The field content of the theory can be encoded in a quiver diagram. This is
the same quiver as the one that specifies a weight in the tensor product of
$L$ Verma modules of $\mathfrak{gl}(m|n)$. Here is the quiver for the now-
familiar $\mathfrak{gl}(3|2)$ example, with the arrows labeled with the
corresponding chiral multiplets:

$\widetilde{\mathsf{P}}_{2}$$\mathsf{P}_{2}$$\widetilde{\mathsf{P}}_{3}$$\mathsf{P}_{3}$$\widetilde{\mathsf{P}}_{4}$$\mathsf{P}_{4}$$\upphi_{1}$$\upphi_{3}$$\widetilde{\mathsf{Q}}_{1}$$\mathsf{Q}_{2}$$\widetilde{\mathsf{Q}}_{2}$$\mathsf{Q}_{3}$$\widetilde{\mathsf{Q}}_{3}$$\mathsf{Q}_{4}$$\widetilde{\mathsf{Q}}_{4}$$\mathsf{Q}_{5}$$M_{1}$$M_{2}$$M_{3}$$M_{4}$$L$$L$$L$$L$$L$
(73)

A circle node labeled $M$ is a $\mathrm{U}(M)$ gauge group. The theory has
$m+n$ copies of $\mathrm{U}(L)$ flavor groups, denoted here by square nodes.
We name them $\mathrm{U}(L)_{1}$, $\mathrm{U}(L)_{2}$, $\dotsc$,
$\mathrm{U}(L)_{m+n}$ from left to right. The terms in the superpotential (72)
correspond to closed paths of length three and four in the quiver.

Apart from the $\mathrm{U}(L)^{m+n}$ flavor symmetry, the theory has an
important $\mathrm{U}(1)$ global symmetry preserved by the superpotential. We
call it $\mathrm{U}(1)_{\hbar}$. The charges of the chiral multiplets under
$\mathrm{U}(1)_{\hbar}$ are

$\displaystyle\upphi_{r}$ $\displaystyle\colon 2(-1)^{[r]}\,,$ (74)
$\displaystyle\mathsf{P}_{i}$ $\displaystyle\colon-(-1)^{[i]}\,,$ (75)
$\displaystyle\widetilde{\mathsf{P}}_{i}$ $\displaystyle\colon-(-1)^{[i]}\,,$
(76) $\displaystyle\mathsf{Q}_{i}$
$\displaystyle\colon\frac{1}{2}(-1)^{[i]}\,,$ (77)
$\displaystyle\widetilde{\mathsf{Q}}_{i}$
$\displaystyle\colon\frac{1}{2}(-1)^{[i]}\,.$ (78)

For $\upphi_{r}$ and $\mathsf{P}_{i}$, $\widetilde{\mathsf{P}}_{i}$, their
charges coincide with the corresponding Cartan matrix elements.

Most parameters of the spin chain correspond in the gauge theory to the
twisted masses with respect to the global symmetry
$\mathrm{U}(L)^{m+n}\times\mathrm{U}(1)_{\hbar}$. We can turn on the twisted
masses as follows. First, we couple vector multiplets for
$\mathrm{U}(L)^{m+n}\times\mathrm{U}(1)_{\hbar}$ to the chiral multiplets and
gauge the global symmetry. Then, we give vacuum expectation values to the
adjoint scalar fields in these vector multiplets. Finally, we take the limit
in which the gauge couplings for
$\mathrm{U}(L)^{m+n}\times\mathrm{U}(1)_{\hbar}$ go to zero, thereby freezing
the vector multiplets just added. The vacuum expectation values of the scalar
fields appear as complex mass parameters for the chiral multiplets, which are
the twisted masses in question.

To the scalar field for $\mathrm{U}(L)_{i}$, we give the vacuum expectation
value

$\mathop{\mathrm{diag}}\nolimits(\mu_{i}^{1},\dotsc,\mu_{i}^{L})\,.$ (79)

This yields twisted masses $-\mu_{i}^{\ell}$ to $\mathsf{Q}_{i}$ and
$+\mu_{i}^{\ell}$ to $\widetilde{\mathsf{Q}}_{i}$. To the scalar field for
$\mathrm{U}(1)_{\hbar}$, we give the vacuum expectation value $\hbar/2$. This
yields a twisted mass $q\hbar/2$ to a chiral multiplet that has charge $q$
under $\mathrm{U}(1)_{\hbar}$. To summarize, the twisted masses of the chiral
multiplets are

$\displaystyle(\upphi_{r})^{a_{r}}{}_{b_{r}}$
$\displaystyle\colon(-1)^{[r]}\hbar\,,$ (80)
$\displaystyle(\mathsf{P}_{i})^{a_{i}}{}_{b_{i-1}}$
$\displaystyle\colon-\frac{1}{2}(-1)^{[i]}\hbar\,,$ (81)
$\displaystyle(\widetilde{\mathsf{P}}_{i})^{a_{i-1}}{}_{b_{i}}$
$\displaystyle\colon-\frac{1}{2}(-1)^{[i]}\hbar\,,$ (82)
$\displaystyle(\mathsf{Q}_{i})^{a_{i-1}}{}_{\ell}$
$\displaystyle\colon-\mu_{i}^{\ell}+\frac{1}{4}(-1)^{[i]}\hbar\,,$ (83)
$\displaystyle(\widetilde{\mathsf{Q}}_{i})^{\ell}{}_{a_{i}}$
$\displaystyle\colon+\mu_{i}^{\ell}+\frac{1}{4}(-1)^{[i]}\hbar\,.$ (84)

Lastly, we need FI parameters and theta angles in order to account for the
twist parameters for the periodic boundary condition of the spin chain. To do
so, from each vector multiplet $\mathsf{V}_{r}$ we construct an adjoint
twisted chiral multiplet $\Upsigma_{r}$ whose lowest component is the vector
multiplet scalar $\sigma_{r}$. Then, we choose complexified FI parameters

$t_{1},\dotsc,t_{m+n-1}\in\mathbb{C}/2\pi\mathrm{i}\mathbb{Z}\,,$ (85)

and turn on the twisted superpotential

${\widetilde{W}}=-\sum_{r=1}^{m+n-1}t_{r}\mathop{\mathrm{tr}}\nolimits\Upsigma_{r}\,.$
(86)

The real and imaginary parts of $t_{r}$ are related to the FI parameter
$r_{r}$ and the theta angle $\theta_{r}$ for $\mathrm{U}(M_{r})$ as
$t_{r}=r_{r}-\mathrm{i}\theta_{r}$.

####  2.3.2 Vacuum equations

We are interested in the vacua of this theory when it is defined on a periodic
space. For ${\mathcal{N}}=(2,2)$ supersymmetric gauge theories, the exact low-
energy effective descriptions are known and we can use these descriptions to
determine their vacua. See [31] for detailed discussions in the abelian case.
Nonabelian examples are treated in [33].

Let $\Upsigma_{r}^{a_{r}}$, $a_{r}=1$, $\dotsc$, $M_{r}$, be the diagonal
components of $\Upsigma_{r}$, and let $\Upsigma$ and $\sigma$ collectively
denote $\\{\Upsigma_{r}^{a_{r}}\\}$ and their scalar components
$\\{\sigma_{r}^{a_{r}}\\}$, respectively.

If the fields $\sigma$ take generic large values and are slowly varying, the
chiral multiplets and the off-diagonal components of the vector multiplets (or
equivalently the corresponding twisted chiral multiplets) can be considered as
having large masses due to higgsing. Their masses are

$\displaystyle(\Upsigma_{r})^{a_{r}}{}_{b_{r}}$
$\displaystyle\colon\sigma_{r}^{a_{r}}-\sigma_{r}^{b_{r}}\,,$ (87)
$\displaystyle(\upphi_{r})^{a_{r}}{}_{b_{r}}$
$\displaystyle\colon\sigma_{r}^{a_{r}}-\sigma_{r}^{b_{r}}+(-1)^{[r]}\hbar\,,$
(88) $\displaystyle(\mathsf{P}_{i})^{a_{i}}{}_{b_{i-1}}$
$\displaystyle\colon\sigma_{i}^{a_{i}}-\sigma_{i-1}^{b_{i-1}}-\frac{1}{2}(-1)^{[i]}\hbar\,,$
(89) $\displaystyle(\widetilde{\mathsf{P}}_{i})^{a_{i-1}}{}_{b_{i}}$
$\displaystyle\colon\sigma_{i-1}^{a_{i-1}}-\sigma_{i}^{b_{i}}-\frac{1}{2}(-1)^{[i]}\hbar\,,$
(90) $\displaystyle(\mathsf{Q}_{i})^{a_{i-1}}{}_{\ell}$
$\displaystyle\colon\sigma_{i-1}^{a_{i-1}}-\mu_{i}^{\ell}+\frac{1}{4}(-1)^{[i]}\hbar\,,$
(91) $\displaystyle(\widetilde{\mathsf{Q}}_{i})^{\ell}{}_{a_{i}}$
$\displaystyle\colon\mu_{i}^{\ell}-\sigma_{i}^{a_{i}}+\frac{1}{4}(-1)^{[i]}\hbar\,,$
(92)

where $a_{r}$, $b_{r}$ are indices for the $\mathrm{U}(M_{r})$ gauge group
factor. After integrating out these heavy fields, we are left with an
effective description of the theory that involves only $\Upsigma$.

The effective theory is determined solely by a single holomorphic function of
$\sigma$, the _effective twisted superpotential_
${\widetilde{W}}_{\mathrm{eff}}$, and it can be calculated exactly at one-loop
order. Integrating out a chiral multiplet whose mass due to the higgsing is
$m(\sigma)$ contributes to ${\widetilde{W}}_{\mathrm{eff}}(\sigma)$ by the
term

$-m(\sigma)\bigl{(}\log m(\sigma)-1\bigr{)}\,.$ (93)

An off-diagonal component of a vector multiplet also contributes in the same
way [34]. Integrating out high-energy modes of $\Upsigma$ does not alter the
form of ${\widetilde{W}}_{\mathrm{eff}}$ [31].

The vacua of the theory are the solutions of the vacuum equations

$\exp\biggl{(}\frac{\partial{\widetilde{W}}_{\mathrm{eff}}(\sigma)}{\partial\sigma_{r}^{a_{r}}}\biggr{)}=1\,,\quad
a_{r}=1,\dotsc,M_{r}\,,\quad r=1,\dotsc,m+n-1\,.$ (94)

These equations are invariant under shifts of the exponent by integer
multiples of $2\pi\mathrm{i}$, reflecting the fact that the imaginary part of
the exponent is the effective theta angle. In the case at hand, the vacuum
equations read

$e^{t_{r}}(-1)^{M_{r}+1}\Biggl{(}\prod_{b_{r}=1}^{M_{r}}\frac{\sigma_{r}^{a_{r}}-\sigma_{r}^{b_{r}}+(-1)^{[r]}\hbar}{\sigma_{r}^{b_{r}}-\sigma_{r}^{a_{r}}+(-1)^{[r]}\hbar}\Biggr{)}^{\delta_{[r],[r+1]}}\prod_{\ell=1}^{L}\frac{\sigma_{r}^{a_{r}}-\mu_{r+1}^{\ell}+\frac{1}{4}(-1)^{[r+1]}\hbar}{\mu_{r}^{\ell}-\sigma_{r}^{a_{r}}+\frac{1}{4}(-1)^{[r]}\hbar}\\\
\times\prod_{b_{r-1}=1}^{M_{r-1}}\frac{\sigma_{r}^{a_{r}}-\sigma_{r-1}^{b_{r-1}}-\frac{1}{2}(-1)^{[r]}\hbar}{\sigma_{r-1}^{b_{r-1}}-\sigma_{r}^{a_{r}}-\frac{1}{2}(-1)^{[r]}\hbar}\prod_{b_{r+1}=1}^{M_{r+1}}\frac{\sigma_{r}^{a_{r}}-\sigma_{r+1}^{b_{r+1}}-\frac{1}{2}(-1)^{[r+1]}\hbar}{\sigma_{r+1}^{b_{r+1}}-\sigma_{r}^{a_{r}}-\frac{1}{2}(-1)^{[r+1]}\hbar}=1\,,\\\
a_{r}=1,\dotsc,M_{r}\,,\quad r=1,\dotsc,m+n-1\,,$ (95)

with $M_{0}=M_{m+n}=0$. The factor $e^{t_{r}}$ comes from the tree-level
twisted superpotential (86), and the factor $(-1)^{M_{r}+1}$ comes from the
off-diagonal components of the vector multiplets. Using the Cartan matrix
(14), we can rewrite these equations as

$e^{\tau_{r}}\prod_{s=1}^{m+n-1}\prod_{b_{s}=1}^{M_{s}}\frac{\sigma_{r}^{a_{r}}-\sigma_{s}^{b_{s}}+\frac{1}{2}a_{rs}\hbar}{\sigma_{r}^{a_{r}}-\sigma_{s}^{b_{s}}-\frac{1}{2}a_{rs}\hbar}\prod_{\ell=1}^{L}\frac{\sigma_{r}^{a_{r}}-\mu_{r+1}^{\ell}+\frac{1}{4}(-1)^{[r+1]}\hbar}{\sigma_{r}^{a_{r}}-\mu_{r}^{\ell}-\frac{1}{4}(-1)^{[r]}\hbar}=(-1)^{\delta_{[r],[r+1]}}\,,$
(96)

where

$\tau_{r}=t_{r}+\mathrm{i}\pi\bigl{(}(1-\delta_{[r],[r+1]})(M_{r}+1)+M_{r-1}+M_{r+1}+L\bigr{)}\,.$
(97)

####  2.3.3 Twisted chiral ring

The above two-dimensional gauge theory has ${\mathcal{N}}=(2,2)$
supersymmetry, generated by four supercharges $Q_{\pm}$,
${\overline{Q}}_{\pm}$. Under a vector $\mathrm{U}(1)$ R-symmetry,
${\overline{Q}}_{\pm}$ and $Q_{\pm}$ have charges $+1$ and $-1$, while under
an axial $\mathrm{U}(1)$ R-symmetry, ${\overline{Q}}_{+}$, $Q_{-}$ have charge
$+1$ and ${\overline{Q}}_{-}$, $Q_{+}$ have charge $-1$. The theory is unitary
and the supercharges satisfy the reality conditions
$Q_{\pm}^{*}={\overline{Q}}_{\pm}$. It turns out that the theory has unbroken
vector $\mathrm{U}(1)$ R-symmetries, and this fact implies the absence of
certain central charges $Z$, $Z^{*}$ in the ${\mathcal{N}}=(2,2)$
supersymmetry algebra.

The linear combination $Q={\overline{Q}}_{+}+Q_{-}$ satisfies

$Q^{2}={\widetilde{Z}}\,,$ (98)

with ${\widetilde{Z}}$ being another central charge. In the gauge theory that
we are considering,

${\widetilde{Z}}=\hbar
F_{\hbar}+\sum_{i=1}^{m+n}\sum_{\ell=1}^{L}\mu_{i}^{\ell}F_{i}^{\ell}\,,$ (99)

where $F_{\hbar}$ is the generator of $\mathrm{U}(1)_{\hbar}$ and
$F_{i}^{\ell}$, $\ell=1$, $\dotsc$, $L$, are the generators of the maximal
torus of $\mathrm{U}(L)_{i}$. Since $Q$ squares to zero in the sector in which
${\widetilde{Z}}=0$, we can define the $Q$-cohomology in the space of
${\widetilde{Z}}$-invariant states and in the algebra of
${\widetilde{Z}}$-invariant operators. The subalgebra of the latter consisting
of the elements represented by local operators is the _twisted chiral ring_ of
the theory.

The ${\mathcal{N}}=(2,2)$ supersymmetry algebra with $Z=Z^{*}=0$ says that the
Hamiltonian $H$ satisfies $\\{Q,Q^{*}\\}=2H$. Therefore, $H$ is positive
semidefinite and vacuum states are annihilated by $Q$ and $Q^{*}$. In
particular, vacua have ${\widetilde{Z}}=0$. According to Hodge theory, the
$Q$-cohomology of states is isomorphic to the space of vacua.

Besides the Hamiltonian, the momentum $P$ is also $Q$-exact:
$\\{Q,Q_{+}-{\overline{Q}}_{-}\\}=2P$. It follows that translations act
trivially in the $Q$-cohomology. In particular, the twisted chiral ring is
commutative since we can switch the order of two local $Q$-cohomology classes
along the time axis by moving them around continuously inside the two-
dimensional spacetime.

In fact, for the theory considered here, not just the Hamiltonian and the
momentum but the entire stress tensor is $Q$-exact. As a consequence, the
$Q$-cohomology of states and the twisted chiral ring are topological, and
there is a state–operator correspondence between them: the two are isomorphic
as vector spaces.

Being topological, the twisted chiral ring can be computed in the effective
theory. As a vector space, it is the space of polynomials in the scalar fields
$\\{\sigma_{r}^{a_{r}}\\}$ modulo the action of the Weyl group of the gauge
group and the relations imposed by the vacuum equations (96). On a vacuum
state, specified by a solution of the vacuum equations, an element of the
twisted chiral ring acts by evaluation on the solution. Therefore, a vacuum is
a simultaneous eigenstate of the elements of the twisted chiral ring.

###  2.4 The correspondence

The Bethe equations (59) and the vacuum equations (96) coincide under the
identification

$\mu_{i}^{\ell}=\zeta^{\ell}-(-1)^{[i]}\biggl{(}\lambda^{\ell}_{i}+\frac{1}{4}\biggr{)}\hbar+\frac{1}{2}c_{i}\hbar\,,$
(100)

together with the obvious identification between parameters for which we have
been using the same symbols. (We are measuring twisted masses in an
appropriate unit so that they are numbers here.)

Thus, the vacua of the gauge theory are identified with the Bethe vectors of
the corresponding magnon sector of the spin chain. Under this identification,
the elements of the twisted chiral ring are identified with the commuting
conserved charges of the spin chain. This is the statement of the Bethe/gauge
correspondence.

One conclusion we can immediately draw from the Bethe/gauge correspondence is
that the gauge theory has no supersymmetric vacuum unless the assignment
$(M_{1},\dotsc,M_{m+n-1})$ of the ranks of the unitary gauge groups
corresponds to a weight of $M(\lambda^{1})\otimes\dotsb\otimes
M(\lambda^{L})$. For example, for $(m|n)=(1|1)$, supersymmetry is broken if
and only if

$M_{1}>L$ (101)

because the fermionic lowering operator can be applied at most $L$ times, at
which point all spin sites are occupied by fermionic excitations. This is
consistent with the known result that supersymmetry is broken in a two-
dimensional ${\mathcal{N}}=(2,2)$ supersymmetric gauge theory with gauge group
$\mathrm{U}(M)$, $L_{f}$ fundamental chiral multiplets and $L_{a}$
antifundamental chiral multiplets if $M>\max(L_{f},L_{a})$ [35].

##  3 String theory realization of the Bethe/gauge correspondence

Although we have presented the Bethe/gauge correspondence for noncompact
rational $\mathfrak{gl}(m|n)$ superspin chains, we have not yet explained why
such a correspondence should exist. In this section we provide an explanation
using string theory.

We will discuss how to construct the vacua of the relevant gauge theories
using branes, and how to map these brane configurations to other ones that
realize configurations of line defects in four-dimensional Chern–Simons theory
with gauge group $G=\mathrm{GL}(m|n)$. The emergence of integrable spin chains
is understood naturally in the latter setup.

Moreover, we will give an explanation of fermionic dualities known in the
literature of integrable superspin chains.

###  3.1 Brane construction of the gauge theory vacua

The gauge theory and its vacua described in sections 2.3 can be constructed
with branes in string theory. In fact, we have already represented the
corresponding Bethe vectors graphically in a way that makes the connection to
the brane construction transparent.

####  3.1.1 Semiclassical type IIA configuration

The construction uses NS5-branes

$\mathrm{NS5}_{i}\,,\quad i=1,\dotsc,m+n\,,$ (102)

D4-branes

$\mathrm{D4}_{i}^{\ell}\,,\quad i=1,\dotsc,m+n,\quad\ell=1,\dotsc,L\,,$ (103)

and D2-branes

$\mathrm{D2}_{r}^{a_{r}}\,,\quad r=1,\dotsc,m+n-1,\quad
a_{r}=1,\dotsc,M_{r}\,,$ (104)

in type IIA superstring theory. The indices $i$ and $r$ are
$\mathbb{Z}_{2}$-graded as before.

First, let us consider the case in which all FI parameters are zero. In this
case, a semiclassical brane configuration for a vacuum state of the gauge
theory is summarized as follows:

$\begin{array}[]{r@{\colon\ }c@{\ \times\ }c@{\ \times\ }c@{\ \times\ }c@{\
\times\ }c@{\ \times\ }c@{\ \times\
}c}\mathrm{Spacetime}&\mathbb{R}&\check{S}^{1}&\mathbb{C}&\mathbb{R}_{X}&\mathbb{R}_{Y}&\mathbb{R}^{2}_{+\hbar}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{NS5}_{i}\
([i]=\bar{0})&\mathbb{R}&\check{S}^{1}&\mathbb{C}&\\{X_{i}\\}&\\{Y_{i}\\}&\mathbb{R}^{2}_{+\hbar}&\\{0\\}\\\
\mathrm{NS5}_{i}\
([i]=\bar{1})&\mathbb{R}&\check{S}^{1}&\mathbb{C}&\\{X_{i}\\}&\\{Y_{i}\\}&\\{0\\}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{D4}_{i}^{\ell}\
([i]=\bar{0})&\mathbb{R}&\check{S}^{1}&\\{\mu_{i}^{\ell}\\}&\\{X_{i}\\}&[Y_{i},\infty)&\mathbb{R}^{2}_{+\hbar}&\\{0\\}\\\
\mathrm{D4}_{i}^{\ell}\
([i]=\bar{1})&\mathbb{R}&\check{S}^{1}&\\{\mu_{i}^{\ell}\\}&\\{X_{i}\\}&[Y_{i},\infty)&\\{0\\}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{D2}_{r}^{a_{r}}&\mathbb{R}&\check{S}^{1}&\\{\sigma_{r}^{a_{r}}\\}&[X_{r},X_{r+1}]&\\{{\widetilde{Y}}_{r}\\}&\\{0\\}&\\{0\\}\end{array}$
(105)

In the spacetime, $\check{S}^{1}$ is a circle, $\mathbb{R}_{X}$ and
$\mathbb{R}_{Y}$ are lines, and $\mathbb{R}^{2}_{+\hbar}$ and
$\mathbb{R}^{2}_{-\hbar}$ are planes. Corresponding to the vanishing FI
parameters, we have

$Y_{i}={\widetilde{Y}}_{r}=0$ (106)

for all $i$ and $r$.

All of these branes wrap the cylinder $\mathbb{R}\times\check{S}^{1}$, which
is the spacetime of the gauge theory. The branes $\mathrm{NS5}_{i}$ and
$\mathrm{D4}_{i}^{\ell}$ extend over $\mathbb{R}^{2}_{(-1)^{[i]}\hbar}$ and
are located at the origin of $\mathbb{R}^{2}_{(-1)^{[i]+\bar{1}}\hbar}$.
Moreover, $\mathrm{NS5}_{i}$ extends over $\mathbb{C}$, whereas
$\mathrm{D4}_{i}^{\ell}$ extends along $\mathbb{R}_{X}$ and ends on
$\mathrm{NS5}_{i}$. Along $\mathbb{R}_{X}$, the NS5-branes are ordered
according to the ordered basis of $\mathbb{C}^{m|n}$ specifying the Dynkin
diagram of $\mathfrak{gl}(m|n)$:

$X_{1}<X_{2}<\dotsb<X_{m+n}\,.$ (107)

The graphical representation (62) for a Bethe vector can be reinterpreted as
the above brane configuration. In that picture, the vertical direction is the
direction of $\mathbb{C}$ and the horizontal direction is $\mathbb{R}_{X}$;
the vertical lines are the NS5-branes. The diagonal lines ending on the
vertical ones are the D4-branes. The horizontal line segments between the
$r$th and $(r+1)$st vertical lines are the D2-branes
$\mathrm{D2}_{r}^{a_{r}}$, $a_{r}=1$, $\dotsc$, $M_{r}$, suspended between the
two NS5-branes:

$\mathbb{R}_{X}$$\mathbb{C}$$\mathbb{R}_{Y}$$\mathrm{D2}_{1}^{1}$$\mathrm{D2}_{3}^{1}$$\mathrm{D2}_{4}^{2}$$\mathrm{D2}_{4}^{1}$$\mathrm{D2}_{2}^{1}$$\mathrm{D2}_{2}^{2}$$\lambda^{3}_{1}$$\lambda^{3}_{2}$$\lambda^{3}_{3}$$\lambda^{3}_{4}$$\lambda^{3}_{5}$$\lambda^{2}_{1}$$\lambda^{2}_{2}$$\lambda^{2}_{3}$$\lambda^{2}_{4}$$\lambda^{2}_{5}$$\lambda^{1}_{1}$$\lambda^{1}_{2}$$\lambda^{1}_{3}$$\lambda^{1}_{4}$$\lambda^{1}_{5}$
(108)

Strings stretched between $\mathrm{D2}_{r}^{a_{r}}$ and
$\mathrm{D2}_{r}^{b_{r}}$ produce the components
$(\mathsf{V}_{r})^{a_{r}}{}_{b_{r}}$ and $(\mathsf{V}_{r})^{b_{r}}{}_{a_{r}}$
of the vector multiplet $\mathsf{V}_{r}$ for the gauge group factor
$\mathrm{U}(M_{r})$. The D2-branes can move along $\mathbb{C}$, and the
position of $\mathrm{D2}_{r}^{a_{r}}$ in $\mathbb{C}$ determines the scalar
field $\sigma_{r}^{a_{r}}$ of the twisted chiral multiplet
$\Upsigma_{r}^{a_{r}}$.

If $[r]=[r+1]$, the D2-branes suspended between $\mathrm{NS5}_{r}$ and
$\mathrm{NS5}_{r+1}$ can also move along $\mathbb{R}^{2}_{(-1)^{[r]}\hbar}$,
over which the two NS5-branes extend. Accordingly, in this case strings with
both ends attached on these D2-branes give rise to an additional chiral
multiplet, namely the adjoint chiral multiplet $\upphi_{r}$. The positions of
the D2-branes in $\mathbb{R}^{2}_{(-1)^{[r]}\hbar}$ are the diagonal
components of the scalar field in $\upphi_{r}$.

Strings stretched between $\mathrm{D2}_{r-1}^{a_{r-1}}$ and
$\mathrm{D2}_{r}^{b_{r}}$ yield the components
$(\mathsf{P}_{r})_{a_{r-1}}{}^{b_{r}}$ and
$(\widetilde{\mathsf{P}}_{r})^{b_{r}}{}_{a_{r-1}}$ of the bifundamental chiral
multiplets between $\mathrm{U}(M_{r-1})$ and $\mathrm{U}(M_{r})$. Strings from
$\mathrm{D2}_{i}^{a_{i}}$ to $\mathrm{D4}_{i}^{\ell}$ are responsible for
$(\widetilde{\mathsf{Q}}_{i})^{\ell}{}_{a_{i}}$ and those from
$\mathrm{D4}_{i}^{\ell}$ to $\mathrm{D2}_{i-1}^{a_{i-1}}$ give
$(\mathsf{Q}_{i})^{a_{i-1}}{}_{\ell}$.

Various parameters of the gauge theory are identified as follows. The gauge
coupling for $\mathrm{U}(M_{r})$ is $\sqrt{g_{s}/l_{s}(X_{r+1}-X_{r})}$, where
$g_{s}$ is the string coupling and $l_{s}$ is the string length. The position
of $\mathrm{D4}_{i}^{\ell}$ in $\mathbb{C}$ determines the twisted mass
$\mu_{i}^{\ell}$. The FI parameter for $\mathrm{U}(M_{r})$ is
$(Y_{r+1}-Y_{r})/g_{s}l_{s}$, while the theta angle is given by the difference
in the periodic scalars on $\mathrm{NS5}_{r}$ and $\mathrm{NS5}_{r+1}$ (up to
a shift by $\mathrm{i}\pi$ which we will explain shortly). Since we are taking
$Y_{i}=0$ for all $i$, all FI parameters are zero. Introducing the twisted
masses proportional to $\hbar$ requires turning on a nontrivial B-field. For
the moment we take $\hbar=0$.

The rotation symmetry of the directions orthogonal to the D2-brane
worldvolumes becomes a global symmetry of the gauge theory. The rotation
symmetry $\mathrm{U}(1)_{\mathbb{C}}$ of $\mathbb{C}$ is an axial R-symmetry,
under which the vector multiplet scalars have charge $2$. The rotation
symmetries $\mathrm{U}(1)_{\mathbb{R}^{2}_{\pm\hbar}}$ of
$\mathbb{R}^{2}_{\pm\hbar}$ are vector R-symmetries. The adjoint chiral
multiplet $\upphi_{r}$ has charge $(2,0)$ or $(0,2)$ under
$\mathrm{U}(1)_{\mathbb{R}^{2}_{+\hbar}}\times\mathrm{U}(1)_{\mathbb{R}^{2}_{-\hbar}}$,
depending on whether $[r]=\bar{0}$ or $\bar{1}$.

Now, let us turn on FI parameters by displacing the NS5-branes along
$\mathbb{R}_{Y}$ by different amounts. As we vary their positions, the
D2-branes suspended between them get rotated in
$\mathbb{R}_{X}\times\mathbb{R}_{Y}$ by various angles. Such a configuration
no longer preserves supersymmetry. If the twisted masses are generic, the
D2-branes cannot stretch between D4-branes without breaking supersymmetry
either. Moreover, if $M_{r}>L$ and $[r]\neq[r+1]$, suspending $M_{r}$
D2-branes between $\mathrm{NS5}_{r}$ and the $L$ D4-branes ending on
$\mathrm{NS5}_{r+1}$ (or between $\mathrm{NS5}_{r+1}$ and the $L$ D4-branes
ending on $\mathrm{NS5}_{r}$) breaks supersymmetry by the s-rule. It appears
that there are no supersymmetric vacua for generic FI parameters, twisted
masses and magnon numbers.

This analysis is semiclassical, however. Quantum mechanically, D4-branes bend
NS5-branes on which they end and the conclusion is altered.

####  3.1.2 Lift to M-theory

Important aspects of the quantum corrections to the above brane configuration
can be understood by uplift to M-theory. The M-theory spacetime contains an
additional compact direction $S^{1}_{\mathrm{M}}$. Let $\vartheta$ be its
coordinate with period $2\pi$, and introduce a complex coordinate
$Y+\mathrm{i}\vartheta$ for the cylinder $\mathbb{R}_{Y}\times
S^{1}_{\mathrm{M}}$. Further introducing $w=e^{-(Y+\mathrm{i}\vartheta)}$, we
map the cylinder to the punctured complex plane $\mathbb{C}^{\times}$.

All NS5-branes and D4-branes are lifted to M5-branes in M-theory. For each
$i$, $\mathrm{NS5}_{i}$ and $\mathrm{D4}_{i}^{\ell}$, $\ell=1$, $\dotsc$, $L$,
merge into a single M5-brane $\mathrm{M5}_{i}$, wrapping a Riemann surface
$\Sigma_{i}$ in $\mathbb{C}\times\mathbb{C}^{\times}$. The D2-brane
$\mathrm{D2}_{r}^{a_{r}}$ is lifted to an M2-brane $\mathrm{M2}_{r}^{a_{r}}$
stretched between $\mathrm{M5}_{r}$ and $\mathrm{M5}_{r+1}$. Hence, the brane
configuration in M-theory is as follows:

$\begin{array}[]{r@{\colon\ }c@{\ \times\ }c@{\ \times\ }c@{\ \times\ }c@{\
\times\ }c@{\ \times\
}c}\mathrm{Spacetime}&\mathbb{R}&\check{S}^{1}&\mathbb{C}\times\mathbb{C}^{\times}&\mathbb{R}_{X}&\mathbb{R}^{2}_{+\hbar}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{M5}_{i}\
([i]=\bar{0})&\mathbb{R}&\check{S}^{1}&\Sigma_{i}&\\{X_{i}\\}&\mathbb{R}^{2}_{+\hbar}&\\{0\\}\\\
\mathrm{M5}_{i}\
([i]=\bar{1})&\mathbb{R}&\check{S}^{1}&\Sigma_{i}&\\{X_{i}\\}&\\{0\\}&\mathbb{R}^{2}_{-\hbar}\\\
\mathrm{M2}_{r}^{a_{r}}&\mathbb{R}&\check{S}^{1}&\\{(\sigma_{r}^{a_{r}},w_{r}^{a_{r}})\\}&[X_{r},X_{r+1}]&\\{0\\}&\\{0\\}\end{array}$
(109)

In terms of the coordinates $(z,w)$ of $\mathbb{C}\times\mathbb{C}^{\times}$,
the Riemann surface $\Sigma_{i}$ is defined by the equation

$w=q_{i}\prod_{\ell=1}^{L}(z-\mu_{i}^{\ell})\,,$ (110)

where $q_{i}$ is a constant. The zero of $w$ at $z=\mu_{i}^{\ell}$ describes
$\mathrm{D4}_{i}^{\ell}$, which extends to $+\infty$ in $\mathbb{R}_{Y}$. If
$\Sigma_{r}$ and $\Sigma_{r+1}$ intersect in
$\mathbb{C}\times\mathbb{C}^{\times}$, then $\mathrm{M2}_{r}^{a_{r}}$ can be
placed at an intersection point so that its worldvolume is orthogonal to the
M5-branes.

Therefore, the M2-branes can be suspended between the M5-branes in a manner
that preserves supersymmetry if $\mathrm{M2}_{r}^{a_{r}}$ is placed at
$z=\sigma_{r}^{a_{r}}$ and the coordinate $\sigma_{r}^{a_{r}}$, for each $r$
and $a_{r}$, satisfies the condition

$q_{r}\prod_{\ell=1}^{L}(\sigma_{r}^{a_{r}}-\mu_{r}^{\ell})=w_{r}=q_{r+1}\prod_{\ell=1}^{L}(\sigma_{r}^{a_{r}}-\mu_{r+1}^{\ell})\,.$
(111)

Comparing these equations with the vacuum equations (96) for $\hbar=0$,

$e^{\tau_{r}}\prod_{\ell=1}^{L}\frac{\sigma_{r}^{a_{r}}-\mu_{r+1}^{\ell}}{\sigma_{r}^{a_{r}}-\mu_{r}^{\ell}}=(-1)^{\delta_{[r],[r+1]}}\,,$
(112)

we see that they coincide if we identify

$(-1)^{\delta_{[r],[r+1]}}e^{\tau_{r}}=\frac{q_{r+1}}{q_{r}}\,.$ (113)

Since $\vartheta_{i}=(-1)^{[i]}\arg q_{i}$ is the classical value of the
periodic scalar field on $\mathrm{NS5}_{i}$, the difference
$(-1)^{[r+1]}\vartheta_{r+1}-(-1)^{[r]}\vartheta_{r}$ is indeed equal to
$\theta_{r}$, up to a shift by $\mathrm{i}\pi$.

####  3.1.3 Turning on $\hbar$

Finally, we explain how to make $\hbar\neq 0$. The global symmetry
$\mathrm{U}(1)_{\hbar}$ is the antidiagonal subgroup of
$\mathrm{U}(1)_{\mathbb{R}^{2}_{+\hbar}}\times\mathrm{U}(1)_{\mathbb{R}^{2}_{-\hbar}}$.
To turn on the twisted masses for $\mathrm{U}(1)_{\hbar}$, we follow the
fluxtrap procedure [36, 37]. This is done as follows.

First, we compactify $\mathbb{C}$ to a torus
$T^{2}\cong\mathbb{C}/(R_{1}\mathbb{Z}+\mathrm{i}R_{2}\mathbb{Z})$ in the type
IIA setup and apply T-duality on both directions of $T^{2}$. The D2-branes
become D4-branes wrapping the dual torus
$\check{T}^{2}\cong\mathbb{C}/(\check{R}_{1}\mathbb{Z}+\mathrm{i}\check{R}_{2}\mathbb{Z})$.
Next, we twist the product between $\check{T}^{2}$ and
$\mathbb{R}^{2}_{+\hbar}\times\mathbb{R}^{2}_{-\hbar}$ by the action of
$\mathrm{U}(1)_{\hbar}$. (More precisely, we replace
$\check{T}^{2}\times\mathbb{R}^{2}_{+\hbar}\times\mathbb{R}^{2}_{-\hbar}$ with