SciPost Phys. 13, 083 (2022)

Superspin chains from superstring theory
Nafiz Ishtiaque1? , 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? nishtiaque@ias.edu

Abstract
We present a correspondence between two-dimensional N = (2, 2) supersymmetricgauge theories and rational integrable 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 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 gl(m|n) spin chains with spins valued in finite-dimensional representations.
Copyright N. Ishtiaque et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 18-02-2022
Accepted 23-08-2022
Check forupdatesPublished 05-10-2022
doi:10.21468/SciPostPhys.13.4.083

Contents
1 Introduction

1

2 Bethe/gauge correspondence for noncompact superspin chains
2.1 gl(m|n) and its Verma modules
2.1.1 Lie superalgebra gl(m|n)
2.1.2 Verma modules of gl(m|n)
2.1.3 Tensor products of Verma modules
2.2 Bethe side
2.2.1 Yangian
2.2.2 Representations of Y (gl(m|n))2.2.3 The spin chain
2.2.4 Bethe equations
2.3 Gauge side
2.3.1 The gauge theories
2.3.2 Vacuum equations
2.3.3 Twisted chiral ring
2.4 The correspondence

4
4
4
6
7
8
8
9
10
11
12
13
15
16
17

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SciPost Phys. 13, 083 (2022)

3 String theory realization of the Bethe/gauge correspondence
3.1 Brane construction of the gauge theory vacua
3.1.1 Semiclassical type IIA configuration
3.1.2 Lift to M-theory
3.1.3 Turning on ħh
3.2 Four-dimensional Chern–Simons theory
3.2.1 Case with m = 0 or n = 0
3.2.2 Case with nonzero m and n
3.3 Emergence of the spin chain
3.3.1 Line defects and spin chains
3.3.2 Line defects created by D3-branes
3.3.3 Line defects for parabolic Verma modules of scalar type
3.4 Fermionic Dualities
3.4.1 Odd reflections and fermionic dualities
3.4.2 Fermionic duality from string theory

17
17
17
19
20
20
21
23
26
26
27
30
30
31
33

4 Bethe/gauge correspondence for compact superspin chains
4.1 Covariant and contravariant representations of gl(m|n)
4.2 Line defects in covariant and contravariant representations
4.3 Brane construction of line defects
4.4 Two-dimensional N = (2, 2) supersymmetric gauge theories
4.5 Bethe/gauge correspondence for finite-dimensional representations

34
34
35
39
41
43

A Four-dimensional Chern–Simons theory with gauge supergroup from twisted
string theory
A.1 Topological strings
A.2 Topological open string field theory
A.3 Topological closed string field theory
A.4 Closed–open map
A.5 Four-dimensional Chern–Simons theory with gauge supergroup from the SU(3)invariant twist of type IIB string theory
References

1

46
47
48
49
50
52
55

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 gl(m) spin chain are identified with the vacua of a family ofgauge theories whose gauge group is the product of m − 1 unitary gauge groups. The gauge
theories have N = (4, 4) supersymmetry broken to 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 Am−1 as a subgraph.
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SciPost Phys. 13, 083 (2022)

In 2018, Nekrasov [3] presented a generalization of the above correspondence where therelevant spin chains carry superspins, namely rational gl(m|n) spin chains. The corresponding
gauge theories are essentially N = (2, 2) supersymmetric, as opposed to having softly broken
N = (4, 4) supersymmetry. One of the main results of this paper is an explanation of the originof this correspondence using superstring theory.
In fact, the goal of the present work is much more ambitious: we wish to place theBethe/gauge correspondence for superspin chains into a large web of dualities that relate
diverse phenomena in which the same superspin chains arise from different supersymmetricgauge 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 gl(m|0) = gl(m) are known and havebeen studied in recent years. Besides the Bethe/gauge correspondence already described,
the structures of rational gl(m) spin chains (and their trigonometric and elliptic generalizations) have appeared in quantization of the Seiberg–Witten geometries of four-dimensional
N = 2 supersymmetric gauge theories [4–6], the action of surface and line defects on supersymmetric indices of four-dimensional supersymmetric gauge theories [7–12], quantization of
the Coulomb branches of three-dimensional N = 4 supersymmetric gauge theories [13, 14],and correlation functions of local operators on interfaces in four-dimensional N = 4 superYang–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 tobe 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 [17–19] with gauge group 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 N = (1, 1) super Yang–Mills theory withgauge group U(m) in the presence of Ω-deformation [20–24]. The six-dimensional theory
describes the low-energy dynamics of a stack of m D5-branes, which comprise part of thebrane configurations.
Four-dimensional Chern–Simons theory placed on R2 × C is topological on the plane R2
and holomorphic on the complex plane C. Due to this holomorphic–topological property, line
defects extending along R2 automatically satisfy the Yang–Baxter relation. Moreover, eachline defect carries one complex parameter, its position in C. These facts imply that line defects
making up a square lattice in R2 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 ofline defects in four-dimensional Chern–Simons theory defines a one-dimensional quantum integrable spin chain. For Wilson lines, this spin chain is a rational gl(m) spin chain [17–19]. (Ifone replaces C in spacetime with C \ {0} or an elliptic curve, then one obtains trigonometricor elliptic 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 gl(m) spin chains in these setups.
Now, in view of the Bethe/gauge correspondence for rational 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 tothe superspin chain case, one would generalize, implicitly by string dualities, all of the gauge
theory phenomena mentioned above to their gl(m|n) versions. This is what we aim to achieve.
In this paper, we provide brane constructions of the gauge theories pertinent to theBethe/gauge correspondence for rational 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

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SciPost Phys. 13, 083 (2022)

GL(m|n). The latter theory is obtained from two copies of Ω-deformed six-dimensional
N = (1, 1) super Yang–Mills theory, with gauge groups U(m) and U(n), coupled by a fourdimensional hypermultiplet in the bifundamental representation of U(m) × U(n). In turn, thisgauge 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 arevalued in finite-dimensional or infinite-dimensional representations.
We introduce the Bethe/gauge correspondence for noncompact rational gl(m|n) spin chainsin 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 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 treatedin 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 foundthe Bethe/gauge correspondence for the rational gl(1|2) spin chain with spins taking values
in the natural (1|2)-dimensional representation C1|2 . Furthermore, they gave a string theory
argument to explain dualities between different families of gauge theories corresponding todifferent choices of Dynkin diagrams of 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
gl(m|n) spin chains from a similar point of view.
As we mentioned above, four-dimensional Chern–Simons theory with gauge supergroupcan be constructed from two copies of six-dimensional super Yang–Mills theory coupled byfour-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 thiswork on a rigorous mathematical footing. We describe the topological string construction inappendix 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 andother tools. We conclude this introduction by stating mathematical conjectures as examples
of such phenomena.
In section 2, we define a family of N = (2, 2) supersymmetric gauge theories labeled bym+n−1
the set of (m + n − 1)-tuples of nonnegative integers Z≥0, corresponding to a closed rational 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 M = (M1 , . . . , Mm+n−1 ) is a Calabi–Yau manifold M(M) with an action of
GL(L)m+n × GL(1). The topological A-twist of this sigma model, with mass parameters associated with the maximal torus T of GL(L)m+n × GL(1) turned on, is equivalent to the sectorof the spin chain in which there are M r magnons of type r. The highest weights of the Vermamodules are determined by the mass parameters.
Conjecture 1. The direct sum of equivariant cohomology groupsM
H T M(M)

(1)

m+n−1
M∈Z≥0

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

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SciPost Phys. 13, 083 (2022)

Conjecture 2. There is a homomorphism from the Bethe algebra of the Yangian Y (gl(m|n))to the direct sum of equivariant quantum cohomology ringsM
QH T M(M) .
(2)
m+n−1
M∈Z≥0

The first conjecture says that the Hilbert space of states of the A-model is the same as thatof the spin chain. For n = 0 and L = 1, the conjecture is proved in [26]. The second conjecturemeans 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 gl(m|n) spin chains. The brane configurations in the compact
case have been recently considered by Rimanyi and Rozansky [28] from the perspective ofgeometric construction of R-matrices [27], so we expect that the above conjectures also holdif the target spaces are varieties defined in [28].

2Bethe/gauge correspondence for noncompact superspin chains

The Bethe/gauge correspondence for superspin chains relates a closed spin chain with GL(m|n)
symmetry and two-dimensional gauge theories with 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 gl(m|n). After reviewing somebasic facts about 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 inwhat sense the two sides are equivalent.

2.1

gl(m|n) and its Verma modules

To begin with, let us review the structures of gl(m|n) and its Verma modules, with emphasis
on aspects that are important for the Bethe/gauge correspondence.
2.1.1

Lie superalgebra gl(m|n)
m|n

Let Cm|n be the vector space graded by Z2 = {0̄, 1̄} whose even subspace C0̄

= Cm and odd

m|nsubspace C1̄ = Cn . Let (b1 , . . . , bm ) and ( f1 , . . . , f n ) be the standard basis of Cm and thatof Cn , respectively. Throughout this section and the next section except section 3.4, we fix
an ordered basis (e1 , . . . , em+n ) of Cm|n that is a permutation of (b1 , . . . , bm , f1 , . . . , f n ). The
corresponding Z2 -grading [−]: {1, . . . , m + n} → Z2 is defined by¨
0̄ (ei ∈ Cm ) ;
(3)
[i] =
1̄ (ei ∈ Cn ) .
The space of endomorphisms End(Cm|n ) of Cm|n is also a Z2 -graded vector space, with theeven subspace
End(Cm|n )0̄ = Hom(Cm , Cm ) ⊕ Hom(Cn , Cn )(4)
and the odd subspace
End(Cm|n )1̄ = Hom(Cm , Cn ) ⊕ Hom(Cn , Cm ) .

5(5)

SciPost Phys. 13, 083 (2022)

The elementary matrix Ei j , which has 1 in the (i, j)th entry and 0 elsewhere, has grading[Ei j ] = [i] + [ j].
The Lie superalgebra gl(m|n) is the Z2 -graded vector space End(Cm|n ), endowed with thegraded commutator [−, −]: End(Cm|n ) ⊗ End(Cm|n ) → End(Cm|n ): for elements a, b withhomogeneous Z2 -grading,[a, b] = a b − (−1)[a][b] ba .
(6)
We will distinguish the elements of gl(m|n) from those of End(Cm|n ) by writing them as Ei jrather than Ei j . They satisfy the commutation relations[Ei j , Ekl ] = δ jk Eil − (−1)([i]+[ j])([k]+[l]) δli Ek j .

(7)

The Cartan subalgebra of gl(m|n) is generated by
H r = (−1)[r] E r r − (−1)[r+1] E r+1,r+1 ,r = 1,...,m + n − 1,(8)and one more diagonal matrix, say E11 . The elementary matrix Ei j has the root "i − " j , with"i = Eii∨(9)being the weight of ei in the natural (m + n)-dimensional representation. The positive rootsare "i − " j with i < j. The simple roots are
α r = " r − " r+1 ,r = 1,...,m + n − 1.

(10)

The elements having the roots α r and −α r are E r = E r,r+1 and F r = E r+1,r , respectively. They
satisfy[H r , Es ] = a rs Es ,(11)

[H r , Fs ] = −a rs Fs ,

(12)
[r]

[E r , Fs ] = δ rs (−1)

Hr ,

(13)

where

a rs = αs (H r ) = δ rs (−1)[r] + (−1)[r+1] − δ r+1,s (−1)[r+1] − δ r,s+1 (−1)[r](14)

is the (r, s)th entry of the Cartan matrix.
The structure of gl(m|n) can be encoded in a Dynkin diagram, in which a simple root α r
is represented by a blank node if a r r = ±2 and a crossed node if a r r = 0, and two nodes α r ,αs are connected by an edge if a rs 6= 0. As an example, consider the case with (m|n) = (3|2)
and (e1 , e2 , e3 , e4 , e5 ) = (b1 , b2 , f1 , f2 , b3 ). The associated Dynkin diagram isε1 − ε2ε2 − δ1

δ1 − δ2δ2 − ε3 ,(15)

where εi and δi are the weights of bi and f i , respectively.
Let us give a different presentation of the content of the Dynkin diagram in terms of aquiver diagram, which makes the connection to gauge theory transparent.
First, we represent "i by a vertical line, of one of two colors depending on its grading:¨([i] = 0̄) ;
"i =
(16)
([i] = 1̄) .

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SciPost Phys. 13, 083 (2022)

The ordered set ("1 , . . . , "m+n ) is then represented graphically as m vertical lines of one colorand n vertical lines of the other color, placed in the order specified by the choice of the Z2 grading:.

(17)

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

(18)

The rth node represents the simple root α r .
Finally, for each pair (r, s) with a rs 6= 0, we draw an arrow from the rth node to the sth
node and write the number a rs on the side. We can erase the vertical lines at this stage:
+2

−2
+1

−1

+1

+1

−1

.

(19)

+1

This quiver has the same content as the Dynkin diagram (15) modulo the action of the
Weyl group Sm × Sn which permutes the basis vectors (e1 , . . . , em+n ) without changing theZ2 -grading.
2.1.2

Verma modules of gl(m|n)

A representation of gl(m|n) in a Z2 -graded vector space V is a Lie superalgebra homomorphism
π: gl(m|n) → End(V ), where End(V ) is given the structure of a Lie superalgebra by the graded
commutator.
The Verma module M (λ): gl(m|n) → End(Vλ ), with highest weight
λ=

m+n
X

λi "i ,λi ∈ C ,

(20)

i=1

is a representation of gl(m|n) constructed from a highest-weight vector |Ωλ 〉 that is an eigenstate of the diagonal matrices:

Eii |Ωλ 〉 = λi |Ωλ 〉 ,(21)

Ei j |Ωλ 〉 = 0 ,(22)

i < j.

The other vectors in Vλ are created by the action of lowering operators {Ei j | i > j} on |Ωλ 〉,and two vectors are identified if they are related by the commutation relations (7).
More explicitly, the Fock space Vλ can be described as follows. Let us introduce an ordering
among all lowering operators and name them x 1 , . . . , x p . Then, by the Poincaré–Birkhoff–Witt(PBW) theorem, an element of Vλ is a linear combination of states of the form¨
Z≥0([ai ] = 0̄) ;np
n1
x 1 · · · x p |Ωλ 〉 , ni ∈(23){0, 1} ([ai ] = 1̄) .
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 Ei j changes the weight by "i − " j = −α j − α j+1 − · · · − αi−1 , a
state of M (λ) has a weight of the form
λ−

m+n−1
X

Mr αr ,

r=1

7

M r ∈ Z≥0 .

(24)

SciPost Phys. 13, 083 (2022)

We can also represent this weight graphically. To represent the highest weight λ, for eachvertical line we draw a diagonal line ending on it and write λi next to the ith diagonal line;
and to represent the weight (24), we draw M r horizontal line segments between the rth and(r + 1)st vertical lines. Here is an example for (M1 , M2 , M3 , M4 ) = (2, 3, 2, 1):

.
λ1

λ4

λ3

λ2

(25)

λ5

We convert this diagram into a quiver by replacing the diagonal lines with square nodes
and writing M r inside the rth circle node and 1 inside the square nodes:
+2

M1

1
λ1

−2
−1

M2

+1

−1

+1

1
λ2

1
λ3

M3

+1
+1

1
λ4

M4

.

(26)

1
λ5

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 V1 and V2 are Z2 -graded vector spaces, the tensor product V1 ⊗ V2 is naturally Z2 -graded.
Given two representations π1 : gl(m|n) → End(V1 ) and π2 : gl(m|n) → End(V2 ), the tensor
product representation π1 ⊗ π2 : gl(m|n) → End(V1 ⊗ V2 ) is defined by(π1 ⊗ π2 )(x)(v1 ⊗ v2 ) = π1 (x)v1 ⊗ v2 + (−1)[x][v1 ] v1 ⊗ π2 (x)v2 ,(27)

where v1 , v2 and x are homogeneous in Z2 -grading. The tensor products of more than two
representations can be defined recursively.
The tensor product of L Verma modules M (λ1 ), . . . , M (λ L ) has a highest-weight vector|Ωλ1 〉 ⊗ · · · ⊗ |Ωλ L 〉 with highest weight λ1 + · · · + λ L . A weight of M (λ1 ) ⊗ · · · ⊗ M (λ L ) takes
the form
L
m+n−1
X
X
λ` −
M r α r , M r ∈ Z≥0 .
(28)
`=1

r=1

Graphically, we represented it by a diagram similar to the diagram (25) for a weight of asingle Verma module, but with L diagonal lines ending on each vertical line. For example, thediagram
λ11
λ21
λ31

λ12
λ22
λ32

λ14
λ24
λ34

λ13
λ23
λ33

represents a weight with (M1 , M2 , M3 , M4 )M (λ1 ) ⊗ M (λ2 ) ⊗ M (λ3 ) of gl(3|2).

8

=

λ15
λ25
λ35

(29)

(2, 1, 0, 2) in the representation

SciPost Phys. 13, 083 (2022)

The corresponding quiver diagram is the same as before, except that the ith square node
~ i = (λ1 , . . . , λ L ):is now labeled L and accompanied by the L-tuple λ
i
i
+2

−2
−1

M1

L
~
λ1

M2

+1

−1

+1

L
~
λ2

L
~
λ3

M3

+1

M4

+1

.

L
~
λ4

(30)

L
~
λ5

The above quiver diagram will be identified with a quiver describing a two-dimensional
N = (2, 2) supersymmetric gauge theory that appears on the gauge theory side of theBethe/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 gl(m|n) spin chains, for which spins take values in representations of gl(m|n). More generally, spins in rational gl(m|n) spin chains are valued in representations of the Yangian Y (gl(m|n)) of gl(m|n).
2.2.1

YangianThe Yangian Y (gl(m|n)) is a Z2 -graded Hopf algebra, which in particular is a unital associativeZ2 -graded algebra. It is generated by elements(l)i, j = 1, . . . , m + n ,Ti j ,l ∈ Z>0 ,

(l)

(31)

(1)

with grading [Ti j ] = [i] + [ j]. The level-1 generators Ti j span a subalgebra isomorphic to(1)

gl(m|n), with the identification of generators being Ei j = (−1)[ j] T ji .
To describe the algebra relations for all generators in a compact form, let us introduce aformal variable σ and combine the generators into a single End(Cm|n ) ⊗ Y (gl(m|n))-valued
power series in σ−1 :T (σ) =m+n
X

Ei j ⊗ Ti j (σ) =i, j=1

m+n
∞
XX

ħ
hl
(l)
E ⊗ Ti j .
l ij
σi, j=1 l=0

(32)

(0)

Here Ti j = δi j and ħ
h is a complex parameter. We can think of T (σ) as an (m + n) × (m + n)matrix whose entries are elements of Y (gl(m|n))[[σ−1 ]]; it is called the monodromy matrix
and is a function of spectral parameter σ. In terms of the monodromy matrix, the algebra
relations for Y (gl(m|n)) are encoded in the RTT relation
R12 (σ1 − σ2 )T1 (σ1 )T2 (σ2 ) = T2 (σ2 )T1 (σ1 )R12 (σ1 − σ2 ) .

(33)

This is a relation between elements in End(Cm|n ) ⊗ End(Cm|n ) ⊗ Y (gl(m|n))[[σ−1 ]], and thesubscript(s) on an operator indicate which factor(s) of Cm|n the operator acts on.
The operator R12 (σ) ∈ End(Cm|n ) ⊗ End(Cm|n ) that appears in the RTT relation is therational gl(m|n) R-matrix. It is given by
R12 (σ) = σI ⊗ I + ħ
h P12 ,
9(34)

SciPost Phys. 13, 083 (2022)

where I is the identity matrix and
P12 =

m+n
X

(−1)[ j] Ei j ⊗ E ji .

(35)

i, j=1The permutation operator P12 swaps tensor factors as
P12 (ei ⊗ e j ) = (−1)[i][ j] e j ⊗ ei .

(36)

The R-matrix commutes with the automorphism group GL(m|n) of Cm|n :[g ⊗ g, R12 (σ)] = 0 ,g ∈ GL(m|n) .

(37)

The dynamics of a closed rational gl(m|n) spin chain is generated by the transfer matrixX m+nt(g, σ) = strCm|n g T (σ) =(−1)[i] g i j T ji (σ) ,g ∈ GL(m|n) .

(38)

i=1

The supertrace taken over Cm|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 ⊗ g ⊗1 from the left and R12 (σ1 −σ2 )−1from the right, then using the symmetry (37) of the R-matrix and taking the supertrace overCm|n ⊗ Cm|n , we see that transfer matrices for a fixed g commute with each other:[t(g, σ1 ), t(g, σ2 )] = 0 .

(39)Therefore, if we expand t(g, σ) in powers of σ−1 , the coefficients are mutually commuting
elements of Y (gl(m|n)). They generate a commutative subalgebra called the Bethe algebra (orthe Baxter algebra) of Y (gl(m|n)).
2.2.2

Representations of Y (gl(m|n))

While the Yangian Y (gl(m|n)) and its Bethe algebra are the algebraic structures underlying
rational gl(m|n) spin chains, to get a concrete physical realization of a spin chain we need to
specify a representation of Y (gl(m|n)).
A representation ρ : Y (gl(m|n)) → End(V ) of Y (gl(m|n)) maps the monodromy matrix toan End(V )-valued matrix
∞
XX
 m+n
ħ
hl
(l)
ρ T (σ) =E ⊗ ρ(Ti j ) .
l ij
σi, j=1 l=0(40)

Conversely, an End(V )-valued matrix satisfying the RTT relation determines a representationof Y (gl(m|n)).
Given a representation π: gl(m|n) → End(V ) of gl(m|n), we obtain a one-parameter familyof representations πζ : Y (gl(m|n)) → End(V ), ζ ∈ C, of Y (gl(m|n)) byπζ T (σ) = I ⊗ idV +

m+n
ħ
h X
(−1)[ j] Ei j ⊗ π(E ji ) .
σ − ζ i, j=1(41)

This is known as the evaluation module for π, and ζ is called the inhomogeneity parameter.
Note that we have(1)πζ (−1)[ j] T ji = π(Ei j ) .
(42)
10SciPost Phys. 13, 083 (2022)

For a representation π: gl(m|n) → End(V ) of gl(m|n), let us define a one-parameter familyof representations πc : gl(m|n) → End(V ), c ∈ C, by
πc (Ei j ) = π(Ei j ) + (−1)[i] cδi j idV .

(43)

In the associated Yangian representations, the parameter c is related to a shift in the inhomogeneity parameter. Suppose that ρ : Y (gl(m|n)) → End(V ) is a representation of Y (gl(m|n)).
Then, ρ(T (σ)) satisfies the RTT equation, and for any function f of σ, the RTT equation isstill satisfied when ρ(T (σ)) is replaced by f (σ)ρ(T (σ)). Therefore, if f (σ) can be expanded
in a power series in σ−1 starting from 1, then f (σ)ρ(T (σ)) defines a new representation of
Y (gl(m|n)). Sinceπζc T (σ) = πζ (T (σ)) +cħhσ − ζ + cħhI ⊗ idV =πζ−cħh T (σ) ,
σ−ζ
σ−ζ

(44)

we see that πζc and πζ−cħh are related in this manner.
To construct tensor product representations of Y (gl(m|n)), we use the coproduct∆: Y (gl(m|n)) → Y (gl(m|n)) ⊗ Y (gl(m|n)). The map ∆ is defined by the formulam+nX∆ T (σ) =(−1)([i]+[k])([k]+[ j]) Ei j ⊗ Tik (σ) ⊗ Tk j (σ) .

(45)i, j,k=1

Given two representations ρ1 : Y (gl(m|n)) → End(V1 ) and ρ2 : Y (gl(m|n)) → End(V2 ), the˙ 2 : Y (gl(m|n)) → End(V1 ⊗ V2 ) is defined bytensor product representation ρ1 ⊗ρ
˙ 2 = (ρ1 ⊗ ρ2 ) ◦ ∆ .
ρ1 ⊗ρ

(46)

˙ 2 )(T (σ)) satisfies the RTT relation.
A calculation shows that (ρ1 ⊗ρ
2.2.3

The spin chain

Now, fix a positive integer L, and choose L highest weights~ = (λ1 , . . . , λ L )λ(47)of gl(m|n) and L inhomogeneity parameters
ζ~ = (ζ1 , . . . , ζ L ) .

(48)

Also, choose a diagonal element1 of GL(m|n):g = diag(eφ1 , . . . , eφm+n ) ,φi ∈ C .

(49)

We consider the closed rational gl(m|n) spin chain of length L, with the spin at the `th site valued in the evaluation module M (λ` )ζ` for the Verma module M (λ` ) and the periodic boundary
condition twisted by g.
The Hilbert space of states of this spin chain is the tensor product
Vλ~ =

L
O
`=1

Vλ` ,

(50)

1
The spin chain can be defined for any choice of g, not necessarily diagonal ones. However, nondiagonalizablechoices of g do not appear to have a clear interpretation on the gauge theory side of the Bethe/gauge correspondence.

11

SciPost Phys. 13, 083 (2022)

and the L spins can be thought of as a single spin in the tensor product representation~ ~ = M (λ1 )ζ1 ⊗˙ · · · ⊗M˙ (λ L )ζ L .
M (λ)
ζ

(51)

The transfer matrix of the spin chain~ ~ t(g, σ)
M (λ)
ζ

(52)

generates commuting conserved charges acting on Vλ~ , 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 thecommuting conserved charges, or equivalently, diagonalize the transfer matrix (52) for all
values of the spectral parameter σ. These eigenvectors, referred to as Bethe vectors, are themain characters from the Bethe side of the Bethe/gauge correspondence.
The Bethe vectors for the rational gl(m|n) spin chain have been constructed by Bethe ansatz
methods [29, 30]. The construction starts with the highest-weight vector |Ωλ~ 〉 of the tensor
product representation
L
O
~ =
M (λ)
M (λ` ) .
(53)`=1

This state is called the pseudovacuum and satisfies
YLσ − ζ` + (−1)[i] λ`i ħh~ ~ Tii (σ) |Ω ~ 〉 =M (λ)|Ωλ~ 〉 ,ζλσ − ζ``=1

~ ~ Ti j (σ) |Ω ~ 〉 = 0 , i > j .
M (λ)
ζ
λ

(54)
(55)

According to our graphical notation, the pseudovacuum is represented by a diagram with nohorizontal segments:|Ωλ~ 〉 = λ1
1

λ21
λ31

λ12
λ22
λ32

λ14
λ24
λ34

λ13
λ23
λ33

λ15
λ25
λ35

.

(56)

Excited states are obtained from the pseudovacuum by the action of creation operators~~ ~ (Ti j (σ)) contains M (λ)(~ E ji ) and changes theM (λ)ζ~ (Ti j (σ)), i < j. The operator M (λ)ζ
gl(m|n) weight by " j − "i = −αi − αi+1 − · · · − α j−1 . Roughly speaking, we can interpret
this action as creating a single quasi-particle, or a magnon, of rapidity σ and type r for eachr = i, i + 1, . . . , j − 1. Graphically, we think of it as creating a horizontal line connecting theith and jth vertical lines:
σ1

σ2
σ3

T13 (σ1 )T25 (σ2 )T45 (σ3 )|Ωλ~ 〉 ∼

.

(57)

This is, however, not a precise correspondence because the left-hand side depends on theordering of creation operators.
The operator T ji (σ), i < j, changes the weight by αi + αi+1 + · · · + α j−1 , so annihilates
magnons of type m = i, i + 1, . . . , j − 1. It removes one horizontal line from each of the12

SciPost Phys. 13, 083 (2022)

intervals between the ith and jth vertical lines. If there is no horizontal line to remove, thenthe 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 gl(m|n) weight (28) is
specified by a Bethe root
M

M

M

m+n−1
({σ11 , . . . , σ1 1 }, {σ21 , . . . , σ2 2 }, . . . , {σ1m+n−1 , . . . , σm+n−1
}) ,

(58)

which is a solution of the Bethe equations
e

τr

Ms
m+n−1
Y Y
s=1

b

a

σ r r − σs s + 21 a rs ħ
h

bs
ar
1
h
bs =1 σ r − σs − 2 a rs ħ

= (−1)δ[r],[r+1]Here we have definedLY

σ r r − ζ` + (−1)[r] λ`r ħh − 21 c r ħ
h
a

1
r
`
[r+1] λ` ħh`=1 σ r − ζ + (−1)
r+1 h − 2 c r ħa,a r = 1, . . . , M r ,r = 1, . . . , m + n − 1 . (59)

τ r = (−1)[r+1] φ r+1 − (−1)[r] φ r

(60)

and
ci =

i
X
(−1)[ j] .

(61)

j=1

We represent this Bethe vector by a diagram with M r horizontal lines between the rth andM(r + 1)st vertical lines, with labels {σ1r , . . . , σ r r }. Therefore, the diagram
σ11

σ41

σ21

σ31

σ22

σ42

λ11

λ12

λ13

λ14

λ15

λ21

λ22

λ23

λ24

λ25

λ31

λ32

λ33

λ34

λ35

(62)

represents a Bethe vector of the closed gl(3|2) rational spin chain of length L = 3 that belongs to the magnon sector (M1 , M2 , M3 , M4 ) = (1, 2, 1, 2) and corresponds to the Bethe root({σ11 }, {σ21 , σ22 }, {σ31 }, {σ41 , σ42 }).
Note that the Bethe equations are invariant under the shiftζ` 7→ ζ` + c `ħh,λ`i 7→ λ`i + (−1)[i] c ` ,c` ∈ 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 gl(m|n)
spin chain discussed above corresponds to a family of two-dimensional 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 N = (2, 2) supersymmetric gauge theories, we refer thereader to [31].
13

SciPost Phys. 13, 083 (2022)

2.3.1

The gauge theories

The magnon sectors of the spin chain (the weight spaces of the Hilbert space (50)) are labeledby (m + n − 1)-tuples of nonnegative integers.
The sector with magnon numbers(M1 , . . . , Mm+n−1 ) corresponds to a theory with the product gauge group
U(M1 ) × · · · × U(Mm+n−1 ) .

(64)

Correspondingly, the theory has vector multiplets V r , r = 1, . . . , m+ n−1, one for each unitarygauge 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 U(M r ):2

φ r ∈ Hom(C Mr , C Mr ) ,[r] = [r + 1] ,r = 1, . . . , m + n − 1 .

(65)There are also chiral multiplets

Pi ∈ Hom(C Mi−1 , C Mi ) , i = 2, . . . , m + n − 1 ,(66)

e i ∈ Hom(C Mi , C Mi−1 ) , i = 2, . . . , m + n − 1 ,
P

(67)

L

Qi ∈ Hom(C , C

Mi−1

),

e i ∈ Hom(C Mi , C L ) ,Q

i = 2, . . . , m + n ,(68)

i = 1, . . . , m + n − 1 .

(69)

It is convenient to introduce the notationsφr = 0 ,[r] 6= [r + 1] ,

(70)

and
e m+n = Q1 = Q
e m+n = 0 .
e 1 = Pm+n = P
P1 = P(71)

These chiral multiplets are coupled by the superpotential
W=

m+n−1
X


e r+1 P r+1 − φ r P r P
e r + Pr Qr Q
er
trCMr φ r P

r=1



e r+1 P r+1 P r P
e r . (72)+ (−1)[r] − (−1)[r+1] P
The terms involving adjoint chiral multiplets are the cubic superpotentials required for
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].3
The field content of the theory can be encoded in a quiver diagram. This is the samequiver as the one that specifies a weight in the tensor product of L Verma modules of gl(m|n).
Here is the quiver for the now-familiar gl(3|2) example, with the arrows labeled with thecorresponding chiral multiplets:
φ1

M1

φ3
e2
P

M2

P2
e1
Q

L

Q2

e2
Q

L

e3
P

M3

P3
Q3

e3
Q

L

2

e4
P

M4

.

P4
Q4

e4
Q

L

(73)

Q5

L

Here and thereafter, statements about fields such as the one that follows only indicate the representations inwhich they are valued.
3
The quartic terms can be understood as follows. Suppose that [r] 6= [r + 1], say [r] = 0 and[r + 1] = 1, and introduce a pair of chiral multiplets φ±r in the adjoint representation of U(M r ). These
multiplets are massive and couple with bifundamental chiral multiplets via superpotential terms of the form
e r+1 P r+1 − φ+ P r P
e r + mφ+ φ− ). Integrating out φ± , we get the quartic term. From the point of view
trCMr (φ−r Prrrr
of the brane construction discussed in section 3.1, we imagine the situation in which NS5 r and NS5 r+1 are almost
orthogonal but not quite. The adjoint chiral multiplets φ±r correspond to the positions of D2-branes along R2±ħh .

14

SciPost Phys. 13, 083 (2022)

A circle node labeled M is a U(M ) gauge group. The theory has m + n copies of U(L) flavor
groups, denoted here by square nodes. We name them U(L)1 , U(L)2 , . . . , U(L)m+n from left
to right. The terms in the superpotential (72) correspond to closed paths of length three andfour in the quiver.
Apart from the U(L)m+n flavor symmetry, the theory has an important U(1) global symmetry preserved by the superpotential. We call it U(1)ħh . The charges of the chiral multiplets
under U(1)ħh are

φ r : 2(−1)[r] ,[i]

Pi : − (−1) ,[i]e i : − (−1) ,P121
e i : (−1)[i] .
Q
2Qi : (−1)[i] ,

(74)
(75)
(76)
(77)
(78)

e i , their charges coincide with the corresponding Cartan matrix elements.
For φ r and Pi , P
Most parameters of the spin chain correspond in the gauge theory to the twisted masses
with respect to the global symmetry U(L)m+n × U(1)ħh . We can turn on the twisted masses asfollows. First, we couple vector multiplets for U(L)m+n × U(1)ħh to the chiral multiplets andgauge 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
U(L)m+n × U(1)ħh 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 U(L)i , we give the vacuum expectation value
diag(µ1i , . . . , µiL ) .

(79)

e i . To the scalar field for U(1)ħh , we give theThis yields twisted masses −µ`i to Qi and +µ`i to Q
vacuum expectation value ħ
h/2. This yields a twisted mass qħ
h/2 to a chiral multiplet that has
charge q under U(1)ħh . To summarize, the twisted masses of the chiral multiplets are(φ r )ar br : (−1)[r]ħ
h,1(Pi )ai bi−1 : − (−1)[i]ħ
h,2e i )ai−1 b : − 1 (−1)[i]ħ
(P
h,
i
2
1
(Qi )ai−1 ` : − µ`i + (−1)[i]ħ
h,4e i )` a : + µ` + 1 (−1)[i]ħ
(Q
h.
i
i
4

(80)
(81)
(82)
(83)
(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
V r we construct an adjoint twisted chiral multiplet Σ r whose lowest component is the vector
multiplet scalar σ r . Then, we choose complexified FI parameters
t 1 , . . . , t m+n−1 ∈ C/2πiZ ,(85)

and turn on the twisted superpotential
f=−
W

m+n−1
X
r=1

15

t r tr Σ r .

(86)

SciPost Phys. 13, 083 (2022)

The real and imaginary parts of t r are related to the FI parameter r r and the theta angle θ r
for U(M r ) as t r = r r − iθ r .
2.3.2

Vacuum equations

We are interested in the vacua of this theory when it is defined on a periodic space. For
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].
aLet Σ r r , a r = 1, . . . , M r , be the diagonal components of Σ r , and let Σ and σ collectivelyaadenote {Σ r r } and their scalar components {σ r r }, respectively.
If the fields σ take generic large values and are slowly varying, the chiral multiplets and theoff-diagonal components of the vector multiplets (or equivalently the corresponding twistedchiral multiplets) can be considered as having large masses due to higgsing. Their masses are(Σ r )ar br : σ ar r − σ rbr ,(87)(φ r )ar br : σ ar r − σ rbr + (−1)[r]ħ
h,
1
bi−1
a
h,
− (−1)[i]ħ
(Pi )ai bi−1 : σi i − σi−12e i )ai−1 b : σ ai−1 − σ bi − 1 (−1)[i]ħ
h,
(P
i−1
i
i
2
1
ai−1
− µ`i + (−1)[i]ħ
(Qi )ai−1 ` : σi−1h,4e i )` a : µ` − σ ai + 1 (−1)[i]ħ
h,
(Q
i
i
i
4

(88)
(89)
(90)
(91)
(92)

where a r , b r are indices for the 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 Σ.
The effective theory is determined solely by a single holomorphic function of σ, the effective
feff , and it can be calculated exactly at one-loop order. Integrating out
twisted superpotential W
feff (σ) by the terma chiral multiplet whose mass due to the higgsing is m(σ) contributes to W−m(σ) log m(σ) − 1 .
(93)
An off-diagonal component of a vector multiplet also contributes in the same way [34]. Intefeff [31].
grating out high-energy modes of Σ does not alter the form of W
The vacua of the theory are the solutions of the vacuum equations
 f∂ Weff (σ)
exp
= 1,
a
∂ σr ra r = 1, . . . , M r ,r = 1, . . . , m + n − 1 .

(94)These equations are invariant under shifts of the exponent by integer multiples of 2πi, reflecting the fact that the imaginary part of the exponent is the effective theta angle. In the case athand, the vacuum equations readtre (−1)

M r +1×σ r r − σ r r + (−1)[r]ħ
hδ[r],[r+1] Y
L

br
ar
[r] h
b r =1 σ r − σ r + (−1) ħ

`=1

Y
Mr
M
r−1
Y

a

b

σ r r − µ`r+1 + 41 (−1)[r+1]ħ
haaµ`r − σ r r + 14 (−1)[r]ħ
h

b r+1
b r−1
a
a
M r+1
σ r r − σ r+1− 21 (−1)[r+1]ħ
hh Y
σ r r − σ r−1− 12 (−1)[r]ħ

b r−1
b r+1
ar
ar
1
1
[r] h
[r+1] ħhb r−1 =1 σ r−1 − σ r − 2 (−1) ħb r+1 =1 σ r+1 − σ r − 2 (−1)

a r = 1, . . . , M r ,
16

= 1,

r = 1, . . . , m + n − 1 , (95)

SciPost Phys. 13, 083 (2022)with M0 = Mm+n = 0. The factor e t r comes from the tree-level twisted superpotential (86),and the factor (−1) Mr +1 comes from the off-diagonal components of the vector multiplets.
Using the Cartan matrix (14), we can rewrite these equations as
e

τr

Ms
m+n−1
Y Y
s=1

L
h
σ r r − µ`r+1 + 14 (−1)[r+1]ħ
σ r r − σs s + 12 a rs ħ
hY
a

a

b

bs
ar
1
h `=1
bs =1 σ r − σs − 2 a rs ħa

σ r r − µ`r − 41 (−1)[r]ħ
h= (−1)δ[r],[r+1] ,

(96)

where

τ r = t r + iπ (1 − δ[r],[r+1] )(M r + 1) + M r−1 + M r+1 + L .
2.3.3

(97)

Twisted chiral ring

The above two-dimensional gauge theory has N = (2, 2) supersymmetry, generated by four
supercharges Q ± , Q± . Under a vector U(1) R-symmetry, Q± and Q ± have charges +1 and −1,
while under an axial U(1) R-symmetry, Q+ , Q − have charge +1 and Q− , Q + have charge −1.
The theory is unitary and the supercharges satisfy the reality conditions Q∗± = Q± . It turns outthat the theory has unbroken vector U(1) R-symmetries, and this fact implies the absence of
certain central charges Z, Z ∗ in the N = (2, 2) supersymmetry algebra.
The linear combination Q = Q+ + Q − satisfiese,Q2 = Z(98)e being another central charge. In the gauge theory that we are considering,with Ze =ħ
Zh Fħh +m+nLXXµ`i Fi` ,(99)

i=1 `=1

where Fħh is the generator of U(1)ħh and Fi` , ` = 1, . . . , L, are the generators of the maximal
e = 0, we can define the Qtorus of U(L)i . Since Q squares to zero in the sector in which Z
e -invariant states and in the algebra of Z
e -invariant operators. The
cohomology in the space of Z
subalgebra of the latter consisting of the elements represented by local operators is the twistedchiral ring of the theory.
The 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
e = 0. According to Hodge theory, the Q-cohomology
by Q and Q∗ . In particular, vacua have Zof states is isomorphic to the space of vacua.
Besides the Hamiltonian, the momentum P is also Q-exact: {Q, Q + − Q− } = 2P. It follows
that translations act trivially in the Q-cohomology. In particular, the twisted chiral ring iscommutative 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 theentire stress tensor is Q-exact. As a consequence, the Q-cohomology of states and the twistedchiral ring are topological, and there is a state–operator correspondence between them: thetwo are isomorphic as vector spaces.
Being topological, the twisted chiral ring can be computed in the effective theory. As aavector space, it is the space of polynomials in the scalar fields {σ r 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 chiralring acts by evaluation on the solution. Therefore, a vacuum is a simultaneous eigenstate of
the elements of the twisted chiral ring.

17

SciPost Phys. 13, 083 (2022)

2.4

The correspondence

The Bethe equations (59) and the vacuum equations (96) coincide under the identification


1
1
`
`
`
[i]
λi +
µi = ζ − (−1)ħ
h + ci ħ
h,
(100)
4
2
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 twistedchiral ring are identified with the commuting conserved charges of the spin chain. This is thestatement of the Bethe/gauge correspondence.
One conclusion we can immediately draw from the Bethe/gauge correspondence is thatthe gauge theory has no supersymmetric vacuum unless the assignment (M1 , . . . , Mm+n−1 ) of
the ranks of the unitary gauge groups corresponds to a weight of M (λ1 ) ⊗ · · · ⊗ M (λ L ). For
example, for (m|n) = (1|1), supersymmetry is broken if and only if
M1 > 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 thatsupersymmetry is broken in a two-dimensional N = (2, 2) supersymmetric gauge theory withgauge group U(M ), L f fundamental chiral multiplets and L a antifundamental chiral multiplets
if M > max(L f , L a ) [35].

3String theory realization of the Bethe/gauge correspondence

Although we have presented the Bethe/gauge correspondence for noncompact rational gl(m|n)
superspin chains, we have not yet explained why such a correspondence should exist. In thissection we provide an explanation using string theory.
We will discuss how to construct the vacua of the relevant gauge theories using branes, andhow to map these brane configurations to other ones that realize configurations of line defects
in four-dimensional Chern–Simons theory with gauge group G = GL(m|n). The emergence ofintegrable 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 instring theory. In fact, we have already represented the corresponding Bethe vectors graphicallyin a way that makes the connection to the brane construction transparent.
3.1.1

Semiclassical type IIA configuration

The construction uses NS5-branes
NS5i ,
D4-branes

D4`i ,

i = 1, . . . , m + n ,i = 1, . . . , m + n,
18

` = 1, . . . , L ,

(102)

(103)

SciPost Phys. 13, 083 (2022)

and D2-branes
D2ar r ,r = 1, . . . , m + n − 1,

a r = 1, . . . , M r ,(104)

in type IIA superstring theory. The indices i and r are Z2 -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:
Spacetime: R × Š 1 × C ×RX× RY × R2+ħh × R2−ħh ,NS5i ([i] = 0̄): R × Š 1 × C ×{X i }
× {Yi } × R2+ħh × {0} ,1
NS5i ([i] = 1̄): R × Š × C ×{X i }
× {Yi } × {0} × R2−ħh ,D4`i ([i] = 0̄): R × Š 1 × {µ`i } ×{X i }
× [Yi , ∞) × R2+ħh × {0} ,`1`
D4i ([i] = 1̄): R × Š × {µi } ×{X i }
× [Yi , ∞) × {0} × R2−ħh ,ara
D2 r : R × Š 1 × {σ r r } × [X r , X r+1 ] × {Yer } × {0} × {0} .

(105)

In the spacetime, Š 1 is a circle, RX and RY are lines, and R2+ħh and R2−ħh are planes. Corresponding to the vanishing FI parameters, we have
Yi = Yer = 0 ,(106)

for all i and r.
All of these branes wrap the cylinder R×Š 1 , which is the spacetime of the gauge theory. The
branes NS5i and D4`i extend over R2(−1)[i]ħh and are located at the origin of R2 [i]+1̄ . Moreover,
(−1)

ħ
h

NS5i extends over C, whereas D4`i extends along RX and ends on NS5i . Along RX , the NS5branes are ordered according to the ordered basis of Cm|n specifying the Dynkin diagram ofgl(m|n):
X 1 < X 2 < · · · < 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 C and the horizontal direction is RX ; the vertical lines are the NS5-branes. The diagonal lines ending on thevertical ones are the D4-branes. The horizontal line segments between the rth and (r + 1)stavertical lines are the D2-branes D2 r r , a r = 1, . . . , M r , suspended between the two NS5-branes:
C
RX

D214

D212

D211

D213

D222

RY

D224

λ11

λ12

λ13

λ14

λ15

λ21

λ22

λ23

λ24

λ25

λ31

λ32

λ33

λ34

λ35

a

b

.

(108)

Strings stretched between D2 r r and D2 r r produce the components (V r )ar br and (V r ) br arof the vector multiplet V r for the gauge group factor U(M r ). The D2-branes can move alongaaC, and the position of D2 r r in C determines the scalar field σ r r of the twisted chiral multipletaΣr r .
If [r] = [r + 1], the D2-branes suspended between NS5 r and NS5 r+1 can also move along2
R(−1)[r]ħh , over which the two NS5-branes extend. Accordingly, in this case strings with bothends attached on these D2-branes give rise to an additional chiral multiplet, namely the adjoint
chiral multiplet φ r . The positions of the D2-branes in R2(−1)[r]ħh are the diagonal componentsof the scalar field in φ r .
a r−1be r ) br a
Strings stretched between D2 r−1
and D2 r r yield the components (P r )ar−1 br and (Pr−1
aof the bifundamental chiral multiplets between U(M r−1 ) and U(M r ). Strings from D2i i to D4`ie i )` a and those from D4` to D2ai−1 give (Qi )ai−1 ` .
are responsible for (Q
i

i

19

i−1

SciPost Phys. 13, 083 (2022)

Various parametersof the gauge theory are identified as follows. The gauge couplingpfor U(M r ) is gs /ls (X r+1 − X r ), where gs is the string coupling and ls is the string length.
The position of D4`i in C determines the twisted mass µ`i . The FI parameter for U(M r ) is
(Yr+1 − Yr )/gs ls , while the theta angle is given by the difference in the periodic scalars on
NS5 r and NS5 r+1 (up to a shift by iπ which we will explain shortly). Since we are taking
Yi = 0 for all i, all FI parameters are zero. Introducing the twisted masses proportional to ħhrequires turning on a nontrivial B-field. For the moment we take ħ
h = 0.
The rotation symmetry of the directions orthogonal to the D2-brane worldvolumes becomes a global symmetry of the gauge theory. The rotation symmetry U(1)C of C is an axial
R-symmetry, under which the vector multiplet scalars have charge 2. The rotation symmetries
U(1)R2 of R2±ħh are vector R-symmetries. The adjoint chiral multiplet φ r has charge (2, 0) or±ħh(0, 2) under U(1)R2 × U(1)R2 , depending on whether [r] = 0̄ or 1̄.
+ħ
h
−ħ
h
Now, let us turn on FI parameters by displacing the NS5-branes along RY by different
amounts. As we vary their positions, the D2-branes suspended between them get rotated in
RX × RY by various angles. Such a configuration no longer preserves supersymmetry. If thetwisted masses are generic, the D2-branes cannot stretch between D4-branes without breaking
supersymmetry either. Moreover, if M r > L and [r] 6= [r + 1], suspending M r D2-branes
between NS5 r and the L D4-branes ending on NS5 r+1 (or between NS5 r+1 and the L D4branes ending on NS5 r ) breaks supersymmetry by the s-rule. It appears that there are nosupersymmetric vacua for generic FI parameters, twisted masses and magnon numbers.
This analysis is semiclassical, however. Quantum mechanically, D4-branes bend NS5branes 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
1
SM. Let ϑ be its coordinate with period 2π, and introduce a complex coordinate Y + iϑ for1
the cylinder RY × SM
. Further introducing w = e−(Y +iϑ) , we map the cylinder to the punctured
complex plane C× .
All NS5-branes and D4-branes are lifted to M5-branes in M-theory. For each i, NS5i and`
D4i , ` = 1, . . . , L, merge into a single M5-brane M5i , wrapping a Riemann surface Σi inaa
C × C× . The D2-brane D2 r r is lifted to an M2-brane M2 r r stretched between M5 r and M5 r+1 .
Hence, the brane configuration in M-theory is as follows:
Spacetime: R × Š 1 × C × C× ×RX
× R2+ħh × R2−ħh ,M5i ([i] = 0̄): R × Š 1 ×
Σi
×{X i }
× R2+ħh × {0} ,1
M5i ([i] = 1̄): R × Š ×
Σi
×
{X i }
× {0} × R2−ħh ,araa
M2 r : R × Š 1 × {(σ r r , w r r )} × [X r , X r+1 ] × {0} × {0} .

(109)

In terms of the coordinates (z, w) of C × C× , the Riemann surface Σi is defined by theequationLY
w = qi(z − µ`i ) ,
(110)
`=1

where qi is a constant. The zero of w at z = µ`i describes D4`i , which extends to +∞ in RY . Ifa
Σ r and Σ r+1 intersect in C × C× , then M2 r 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 preaaaserves supersymmetry if M2 r r is placed at z = σ r r and the coordinate σ r r , for each r and a r ,20

SciPost Phys. 13, 083 (2022)

satisfies the condition
qr

L
L
Y
Y
(σ ar r − µ`r ) = w r = q r+1(σ ar r − µ`r+1 ) .
`=1

(111)

`=1

Comparing these equations with the vacuum equations (96) for ħ
h = 0,eτraLY
σ r r − µ`

r+1

a

`=1

σ r r − µ`r= (−1)δ[r],[r+1] ,

(112)

q r+1
.
qr

(113)

we see that they coincide if we identify(−1)δ[r],[r+1] eτr =Since ϑi = (−1)[i] arg qi is the classical value of the periodic scalar field on NS5i , the difference(−1)[r+1] ϑ r+1 − (−1)[r] ϑ r is indeed equal to θ r , up to a shift by iπ.
3.1.3

Turning on ħhFinally, we explain how to make ħ
h 6= 0. The global symmetry U(1)ħh is the antidiagonal subgroup of U(1)R2 × U(1)R2 . To turn on the twisted masses for U(1)ħh , we follow the fluxtrap
+ħ
h
−ħ
h
procedure [36, 37]. This is done as follows.
First, we compactify C to a torus T 2 ∼= C/(R1 Z + iR2 Z) in the type IIA setup and apply
2
T-duality on both directions of T . The D2-branes become D4-branes wrapping the dual torusŤ 2 ∼= C/(Ř1 Z + iŘ2 Z). Next, we twist the product between Ť 2 and R2+ħh × R2−ħh by the action ofU(1)ħh . (More precisely, we replace Ť 2 × R2+ħh × R2−ħh with the quotient of C × R2+ħh × R2−ħh suchthat translations on C by Ř1 and iŘ2 are accompanied by the action of the elements exp(Re ħh)and exp(Im ħh) of U(1)ħh , respectively.) Last, we apply T-duality on Ť 2 and decompactify T 2 to
C by taking R1 , R2 → ∞. This last T-duality yields a certain B-field due to the twist in the
product between Ť 2 and R2+ħh × R2−ħh introduced earlier.
From the point of view of the gauge theory, the first step amounts to lifting the twodimensional theory on R × Š 1 to a four-dimensional theory on R × Š 1 × Ť 2 . Then, the second
step turns on a holonomy for the background gauge field for U(1)ħh . The last step dimensionally reduces the four-dimensional theory back to two dimensions. Since the components of afour-dimensional gauge field along Ť 2 become the complex scalar field for the corresponding
two-dimensional gauge field, this procedure induces the twisted masses for U(1)ħh .

3.2

Four-dimensional Chern–Simons theory

We can convert the brane configuration (105) to a configuration realizing line defects in fourdimensional Chern–Simons theory with gauge group GL(m|n). To do so, we apply T-duality
along Š 1 and then S-duality.
Under these dualities, Š 1 is mapped to the dual circle S 1 , the NS5-branes are mapped to D5branes, 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:
Spacetime: R × S 1 × C ×RX× RY × R2+ħh × R2−ħh ,1
D5i ([i] = 0̄): R × S × C ×{X i }
× {Yi } × R2+ħh × {0} ,D5i ([i] = 1̄): R × S 1 × C ×{X i }
× {Yi } × {0} × R2−ħh ,```D3i ([i] = 0̄): R × { yi } × {µi } ×{X i }
× [Yi , ∞) × R2+ħh × {0} ,D3`i ([i] = 1̄): R × { yi` } × {µ`i } ×{X i }
× [Yi , ∞) × {0} × R2−ħh ,ar
arar
F1 r : R × { ỹ r } × {σ r } × [X r , X r+1 ] × {Yer } × {0} × {0} .
21

(114)

SciPost Phys. 13, 083 (2022)

The positions of the D3-branes and the F1-branes on S 1 are given by the holonomies aroundŠ 1 of the gauge fields on their counterparts in the type IIA setup. The B-field inducing thetwisted masses for U(1)ħh 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 theD5-branes, with 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, isequivalent to four-dimensional Chern–Simons theory with gauge group GL(m|n).
Before demonstrating this equivalence, let us remark that related brane constructions haveappeared in the literature.4 In [41], Mikhaylov and Witten gave a brane construction of threedimensional Chern–Simons theory with gauge group GL(m|n), extending the construction forgauge group 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 GL(m) was proposed. In this construction, m D4branes 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
N = (1, 1) super Yang–Mills theory with gauge group U(m), placed on R × S 1 × C × R2+ħh . The
deformation is what is often called an Ω-deformation and controlled by ħ
h: we have
Q2 = ħh Fħh ,

(115)

where Fħh is the generator of U(1)ħh . (Here we are considering the situation in which there are
no D3-branes, hence no U(L) flavor symmetries.) In the six-dimensional theory, U(1)ħh is theantidiagonal subgroup of the rotation group U(1)R2 on R2+ħh and the subgroup U(1)R2 of the
+ħ
h

−ħ
h

R-symmetry group Spin(4) coming from the rotation symmetry of R2−ħh .
Six-dimensional (Euclidean) N = (1, 1) super Yang–Mills theory on R×S 1 ×C×R2+ħh reduces
to two-dimensional N = (8, 8) super Yang–Mills theory on R2+ħh by dimensional reduction. In
the undeformed case (when ħ
h = 0), the supercharge Q belongs to an N = (2, 2) subalgebraof the N = (8, 8) supersymmetry algebra. Accordingly, N = (1, 1) super Yang–Mills theory on
R×S 1 ×C×R2+ħh may be thought of as an N = (2, 2) supersymmetric gauge theory on R2+ħh , with
infinite-dimensional target space and gauge group. The Ω-deformation of the six-dimensional
theory induces an Ω-deformation of the two-dimensional theory.
In general, the Q-invariant sector of an Ω-deformed N = (2, 2) supersymmetric gauge theory on R2 is equivalent to a zero-dimensional theory [16, 23, 24]. Let G be the gauge group4

To relate these brane constructions to ours, we endow R2+ħh × R2−ħh 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 R2+ħh and R2−ħh .) If
we regard the Taub–NUT space as a circle fibration over R3 , then R2+ħh × {0} and {0} × R2−ħh are two semi-infinite
cigar-shaped subspaces extending in the opposite directions such that their tips touch at the origin of R3 ; see [38],appendix B. T-duality in the direction of the circle fibers produces an NS5-brane which sits at the origin of R3 andextends 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 theconstruction of [39]. The field theory counterpart of this T-duality was analyzed in [40]. From the D4–NS5 braneconfiguration we obtain the brane configuration of [41], roughly speaking, by further replacing C with a cylinder,
taking T-duality in the circumferential direction of the cylinder, and decompactifying the dual cylinder. (SuchT-duality was considered in [42].)

22

SciPost Phys. 13, 083 (2022)of the theory and GC be its complexification. By N = (2, 2) supersymmetry, the chiral multiplets take values in a Kähler manifold X with GC -action. The superpotential is a GC -invariant
holomorphic function W on X . The path integral with insertion of Q-invariant observables
localizes to a GC -invariant submanifold γ of X . This submanifold is essentially a Lefschetz
thimble: γ is the union of all gradient flows generated by the real part of W /ħh, terminating on
the GC -orbit of a chosen critical point of W . (For simplicity, we assume that the critical pointsof W are nondegenerate up to the GC -action.) The localized path integral takes the form
Z


2π
exp
W O,
(116)
ħ
h
γ/G
C

where O descends from the Q-invariant observables inserted in the path integral. This is thepath integral for a zero-dimensional gauge theory with gauge group GC and target space γ.
The action functional is −2πW /ħh.
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 GC down to G . Even though the action functional is holomorphic and its real part is not bounded from below, the integral (116)
can converge since Re(W /ħh) gets smaller and smaller along the backward gradient flows in γ.
For the six-dimensional N = (1, 1) super Yang–Mills theory on R × S 1 × C × R2+ħh , the gauge
group G is the space of maps from R × S 1 × C to U(m). In addition to the vector multiplet,the theory has three chiral multiplets in the adjoint representation of G . Their scalar fields are
Q-invariant and can be combined into a one-form on R × S 1 × C:

A = (A x + iX )dx + (A y + iY )d y + Az̄ dz̄ .

(117)Here, A x , A y are the components of the gauge field along R × S 1 , Az̄ is the antiholomorphic
component of the gauge field along C, and X , Y are two of the four scalar fields of the sixdimensional theory associated to motions along RX and RY , respectively. The superpotential
is given by
Z


i
2
W =− 2dz ∧ tr A ∧ dA + A ∧ A ∧ A ,(118)
e R×S 1 ×C
3
where e is the gauge coupling and tr is an invariant symmetric bilinear form on the Lie algebraof U(m), which we can take to be the trace in the defining representation.
According to the localization argument, the Ω-deformation of the six-dimensional theoryis equivalent to a zero-dimensional gauge theory. This zero-dimensional theory has infinitedimensional target space and gauge group, and can be more naturally interpreted as a fourdimensional gauge theory. Its action is
Z


2πi
2
(119)
− 2
dz ∧ trCm A00 ∧ dA00 + A00 ∧ A00 ∧ A00 .
ħ
he Σ×C
3
Here, we have written the partial gl(m) connection (117) as A00 to emphasize its place inthe Lie superalgebra 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 Ω-deformed sixdimensional N = (1, 1) super Yang–Mills theory on R×S 1 ×C×R2+ħh with gauge group U(m) is
equivalent to four-dimensional Chern–Simons theory on R × S 1 × C with gauge group GL(m)and coupling given by ħh.
Similarly, if we consider the case m = 0, the worldvolume theory on the D5-branes is an Ωdeformed six-dimensional N = (1, 1) super Yang–Mills theory on R × S 1 × C × R2−ħh with gauge

23

SciPost Phys. 13, 083 (2022)

group U(n). Its Q-invariant sector is equivalent to four-dimensional Chern–Simons theory on
R × S 1 × C with gauge group GL(n) and action
Z


2πi
2 11
11
11
11
11
+ 2
,
(120)
dz ∧ trCn A ∧ dA + A ∧ A ∧ A
ħhe R×S 1 ×C
3with the partial gl(n) connection A11 defined in the same way as A00 .
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 D5branes lead to two copies of four-dimensional Chern–Simons theory on R × S 1 × C, one withgauge group GL(m) and the other with gauge group GL(n), with opposite couplings. Arranging
A00 and A11 into a matrix


A00
0
0
A =
,
(121)
0
A11
we can write the sum of their action functionals as
Z


2 0
2πi
0
0
0
0
− 2
dz ∧ strCm|n A ∧ dA + A ∧ A ∧ A .
ħ
he R×S 1 ×C
3

(122)

The two copies are coupled through strings stretched between the two sets of D5-branes.
These strings produce a four-dimensional N = 2 hypermultiplet on R × S 1 × C in the bifundamental representation of GL(m) × GL(n). It consists of bosonic complex scalarsq ∈ Hom(Cm , Cn ) ,(123)q̃† ∈ Hom(Cm , Cn ) ,

(124)

†

n

m

(125)

n

m

q̃ ∈ Hom(C , C ) ,(126)

ψ ∈ Hom(Cm , Cn ) ,

(127)

†

ψ̃ ∈ Hom(C , C ) ,(128)ψ† ∈ Hom(Cn , Cm ) ,(129)

q ∈ Hom(C , C ) ,and fermionic Weyl spinors

m

n

n

m

ψ̃ ∈ Hom(C , C ) .

(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 R × S 1 andantiholomorphic translations on C are Q-exact. By redefining fields if necessary, we can takeQ to be the supercharge used in the holomorphic–topological twist studied in [44], with theparameter t = i.
It turns out that most of the action for the hypermultiplet is Q-exact. The remaining partof the action can be expressed in a suggestive form. Endow the cylinder R×S 1 with a complex
coordinate w, and define
1
A10 = −ψ̃†− dw + ψ− dw̄ − (ψ̃†+ − ψ+ )dz̄ ,2
1
A01 = ψ†− dw + ψ̃− dw̄ + (ψ̃†+ + ψ†+ )dz̄
2
24

(131)
(132)

SciPost Phys. 13, 083 (2022)

andc 10 = 4iq† ,(133)

c 01 = 4q̃† ,

(134)

10

= q̃ ,(135)

b01 = −iq ,(136)

B = i(ψ̃+ − ψ†+ ) ,B 01 = −i(ψ+ + ψ̃†+ ) .

(137)

b

10

We introduce a matrix



0
A01
A =
10
A
0
1

(138)


(139)

and matrices c 1 , b1 , B 1 defined likewise. On these matrices Q acts by
Q · A1 = −d0 c 1 ,

(140)

1

Q · c = 0,(141)

Q · b1 = B 1 ,

(142)

1

(143)

Q · B = 0,
whered0 = d − dz ∂z = dw ∂w + dw̄ ∂w̄ + dz̄ ∂z̄ .
The non-Q-exact part of the action isZ
2πidz ∧ strCm|n (A1 ∧ dA1 ) .
− 2
ħhe R×S 1 ×C

(144)

(145)

(Since this is quadratic in fermions, the prefactor is inessential.)
To describe the intersecting D5-branes, we couple this bifundamental hypermultiplet to thetwo copies of four-dimensional Chern–Simons theory by identifying the flavor groups GL(m)and GL(n) with the gauge groups of the latter. Concretely, we replace the de Rham differentialthat appears in the above formulas with the gauge-covariant differential
dA0 = d + A0 .

(146)

Thus, the action of Q on the fields is modified to
Q · A1 = −d0A0 c 1 ,(147)

Q · c1 = 0 ,

(148)

1

1

Q·b =B ,(149)

Q · B1 = 0 ,(150)

and the action functional for the bifundamental hypermultiplet becomes
Z
2πi
− 2
dz ∧ strCm|n (A1 ∧ dA0 A1 ) .
ħ
he R×S 1 ×C

(151)

Combining A0 and A1 into a single matrix
A00 A01
A=A +A =,A10 A11
0

1

25

(152)

SciPost Phys. 13, 083 (2022)

we can write the total action, which is the sum of the actions (122) and (151), as
Z


2πi
2
− 2
dz ∧ strCm|n A ∧ dA + A ∧ A ∧ A .
ħ
he R×S 1 ×C
3

(153)

This is the action for four-dimensional Chern–Simons theory with gauge group 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 GL(m|n) gauge transformation, thegauge group of the theory is still GL(m) × GL(n), not GL(m|n). Second, the gauge-invariant
action (151) for the bifundamental hypermultiplet is not Q-invariant due to the coupling to
A0 . Its Q-variation gives
Z
4πi
− 2
dz ∧ strCm|n (A1 ∧ [F 0 , c 1 ]) ,(154)
ħhe R×S 1 ×C
where F 0 is the curvature of A0 .
The two problems are solved simultaneously if we correct the Q-action on A0 and B 1 to
Q · A0 = {A1 , c 1 } ,1Q · B 1 = [{c 1 , c 1 }, b1 ] .
2

(155)
(156)

With this modification, the Q-variation of the bosonic action (122) cancels that of the fermionicaction (151). At the same time, c 1 , b1 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 GL(m|n) down to GL(m) × GL(n) [45].
To make the last point more explicit, let us introduce a ghost c 0 , an antighost b0 and an
auxiliary field B 0 for gauge fixing of GL(m) × GL(n). The BRST charge Q B acts on the fields by
Q B · A0 = −d0A c 0 ,0

1 0 0{c , c } ,2Q B · b0 = B 0 ,QB · c0 =
0

(157)
(158)
(159)

QB · B = 0(160)

Q B · A1 = {c 0 , A1 } ,(161)

Q B · c 1 = [c 0 , c 1 ] ,

(162)

and

1

0

1

Q B · b = [c , b ] ,(163)

Q B · B 1 = {c 0 , B 1 } .

(164)

Let us postulate that1
Q · c 0 = − {c 1 , c 1 } ,2Q · b0 = 0 ,Q · B0 = 0 .

(165)
(166)
(167)

Then, the modified BRST chargeb = QB + Q
Q
26

(168)

SciPost Phys. 13, 083 (2022)

b 2 = 0 andsatisfies Q
b · A = −d0 c ,
Q
A
1 0 0
1
b · c = {c , c } − {c 1 , c 1 } + [c 0 , c 1 ] ,Q
2
2b · b = B,
Q

(170)

b · B = 0,
Q

(172)

(169)

(171)

where
c = c0 + c1 ,(173)

b = b0 + b1 ,
0

1

(174)
0

1

B = B + B + [c , b ] .

(175)

b
The 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 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 alonglines in R × S 1 × C. As such, they create line defects in the four-dimensional Chern–Simons
theory on R × S 1 × C, extending along R and supported at points in C. Such a configuration ofline defects in four-dimensional Chern–Simons theory is naturally identified with an integrable
spin chain [17–19]. 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 R × R × 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

A = A x dx + A y d y + Az̄ dz̄ .
We insert line defectsL` ,` = 1, . . . , L ,

(176)

(177)

extending in the x-direction, which we regard as the time direction. Along the y-axis, wearrange L1 , . . . , L L in the ascending order. They are supported at points ζ1 , . . . , ζ L in C.
Solutions of the equation of motion for four-dimensional Chern–Simons theory, away fromthe line defects, are flat connections. Away from the line defects, flat connections can begauged 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 thespaces attached to the line defects:
V = V1 ⊗ ··· ⊗ V L .

(178)This is identified with the Hilbert space of an open spin chain with L sites. The space V `supported on L` is the state space for the `th spin.
The four-dimensional Chern–Simons theory is topological on R × R and holomorphic on
C. Due to the topological invariance on R × R, the Hamiltonian is zero. To change the state,
27

SciPost Phys. 13, 083 (2022)

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 amonodromy matrix T (σ) in the spin chain. The spectral parameter σ is the position of the
Wilson line in C, and the holomorphy on C implies that T (σ) is holomorphic in σ.
If we introduce two Wilson lines and make them intersect in R×R, we get an R-matrix at theintersection. The two sides of the RTT relation (33) correspond to two different configurationsof two open Wilson lines crossing each other and the line defects L1 , . . . , L L . The topologicalinvariance on R × R and the existence of the extra dimensions C imply the equivalence of thetwo configurations.
This R-matrix can be computed by perturbation theory [19], and it was found to be theR-matrix for the rational spin chain with G symmetry. Therefore, this setup produces an open
rational spin chain. For G = GL(m|n), the R-matrix is the one given in (34), up to someequivalence relations.
To obtain a closed spin chain, we simply compactify the y-axis 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
L L and L1 , 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 matrixt(g, σ) 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 R×S 1 , the positions of the D3-branes on S 1 do not matter.
For each `, we gather the m + n D3-branes D3`i , i = 1, . . . , m + n, to the same position y ` on
S1:`
= y` .
(179)y1` = · · · = ym+n
Since they are also located at the same point ζ` in C up to first order in ħh, we can regard them```as creating a single line defect L supported on the line R × { y } × {ζ } in R × S 1 × C, treating
the differences µ`i − ζ` , i = 1, . . . , m + n, in the positions in C as parameters of the line defect.
The D3-branes thus create L line defects L1 , . . . , L L .
From the Bethe/gauge correspondence we know what the Hilbert space V ` of L` must be:
it is the evaluation module of the Yangian Y (gl(m|n)) with spectral parameter ζ` , obtained
from the Verma module Vλ` of 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 fourdimensional Chern–Simons theory on R × S 1 × C, the positions of the D5-branes in RX and RY
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 situationin which all D5-branes are coincident, say X i = Yi = 0 for all i.
The Q-invariant sector of the theory living on the D3-branes that create L` is the BF theory
with gauge group G = GL(m|n), defined on R × [0, ∞). This theory can be obtained fromfour-dimensional Chern–Simons theory on R × [0, ∞) × C by dimensional reduction on C. It
has the actionZ
1
σ(dη + η ∧ η) ,(180)
ħh R×[0,∞)where η is the gauge field and σ is a scalar field valued in g∗ , the dual of the Lie algebrag = gl(m|n). The field σ comes from the reduction of the antiholomorphic component Az̄ of28

SciPost Phys. 13, 083 (2022)

the four-dimensional gauge field on C.
One way to see that this is the right theory is to note that upon exchanging the directionsof S 1 and RY , the D3-branes are T-dual to D5-branes on R × S 1 × C × R2±ħh (after compactifying
C to a torus). Since the D5-branes are described by four-dimensional Chern–Simons theory
with gauge group GL(m|n), the D3-branes are described by the BF theory. The exchange oftwo 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 g∗ . The spaceof functions on g∗ is the symmetric algebra S(g) of g. Linear functions are elements of g, and
the Poisson bracket between them is given by the Lie bracket. Extending the Poisson bracket
to S(g) by the Leibniz rule, we endow 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 R × [0, ∞) ⊂ R × RY , one at Y = 0 and theother at Y = ∞. It is more convenient to think of the spacetime of the theory as the limit of
R × [0, r] ,(181)

as r → ∞. Physically, we can realize this setup by making the D3-brane D3`i end on an
NS5-brane NS5`i with worldvolume
R × { y ` } × {µ`i } × RX × {r} × R2+ħh × R2−ħh ⊂ R × S 1 × C × RX × RY × R2+ħh × R2−ħh .

(182)

Since the BF theory is topological, the value of r does not matter. In particular, we can takethe limit r → 0. In this limit the theory reduces to a one-dimensional quantum mechanicalsystem. This quantum mechanical system describes the line defect after coupling to the fourdimensional Chern–Simons theory.
The Hilbert space of the BF theory depends on the boundary conditions imposed on thetwo boundaries R × {0} and R × {r}. On each boundary, we impose a boundary condition thatdefines what is known as a coisotropic brane in the context of Poisson sigma models [49].
The boundary condition on R × {0} is simple. The imaginary part of the component η xof the gauge field along R parametrizes the positions of the D3-branes in RX . 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 haveη|R×{0} = 0 .
(183)
The field σ 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 thefour-dimensional Chern–Simons theory by gauging.
We propose that the boundary condition on R × {r} is determined by the positions of theNS5-branes in C as follows. The diagonal part of σ parametrizes these positions. Let

g = n− ⊕ h ⊕ n+(184)

be the triangular decomposition of g with respect to the chosen basis; thus h is spanned bydiagonal matrices, and n+ and n− are spanned by strictly upper triangular matrices and strictly
lower triangular matrices, respectively. Then, the boundary condition is
η|R×{r} ∈ b ,(185)

σ|R×{r} ∈ n∗− + λ` ,

(186)

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SciPost Phys. 13, 083 (2022)

where b = h ⊕ n+ is the Borel subalgebra and λ` is an element of h∗ . This condition breaks thegauge group on the boundary to the Borel subgroup B whose Lie algebra is b.5
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.
dim n
dim g
Let {Tα }α=1 − be a basis of n− and extend it a basis {Ta }a=1 of g. Let σa = 〈Ta , σ〉.
Classically, the algebra of local observables is the algebra of gauge-invariant polynomials in{σa }.
On the boundary R × {0}, local observables are simply polynomials in {σa } since there
is no gauge symmetry there. The algebra of local observables on the boundary is therefore
S(g) at the classical level. Quantum corrections lead to a noncommutative deformation ofS(g). At the quantum level, the algebra is isomorphic to the universal enveloping algebra ofPdim gPdim gg with bracket ħh[−, −]; if [Ta , Tb ] = c=1 f ab c Tc , then [σa , σ b ] = ħ
h c=1 f ab c σc quantum
mechanically [50, 51]. In our setup ħh is a complex parameter rather than a formal parameter,
so the algebra is isomorphic to U(g), with σa mapped to ħ
h Ta . (The quantization map from
S(g) to U(g) is complicated for polynomials of higher degree.)On the boundary R×{r}, the algebra of local observables is trivial. The boundary condition
says that local operators are constructed entirely from {σα }. Nontrivial polynomials cannot
commute with 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 {σα }. As a vector space, the space of observables is generated from the “highest-weightvector” 1 by the action of “creation operators” {σα }. On this vector space the algebra U(g)acts. Thus, the Hilbert space is a Verma module of U(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 R × {r} implies that thehighest weight of the module is λ` . There is, however, a quantum correction which shifts thehighest weight.6 The highest weight is actuallyλ` = λ` − ρ ,(187)

where ρ is the Weyl vector defined in terms of the character of the b-module g/b as1
ρ(−) = − strg/b ad(−) .
(188)
2Since ρ(n+ ) = ρ([b, b]) = 0, ρ can be regarded as an element of h∗ . This is the graded half
sum of positive roots. When g is an ordinary Lie algebra, ρ is the ordinary Weyl vector andthe above shift was derived in [52] based on results from [53, 54].
For g = gl(m|n), we have
ρ=

m+n
X

m+n
X1
1
(−1)[k]+[l] ("k − "l ) =(−1)[i] (m − n + (−1)[i] − 2ci )"i .
2
2
i=1
k,l=1

(189)

k<l
5
The choice of Borel subgroup is determined by the positions of the D5-branes in RX , which are in turn givenby the vacuum expectation value of the time componentA x = A x + iX of the gauge field; the u(m|n)-valued
Pm+nfield X has the vacuum expectation value 〈X 〉 = i=1 iX i Eii . In this background, a state evolving for duration
Pm+n
T is scaled by the factor exp(T 〈A x 〉) = exp(−T i=1 X i Eii ). (We have set 〈A x 〉 = 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 latticemodel equivalent to the spin chain, then the periodic boundary condition is twisted
Pm+nby the action of exp(− i=1 X i Eii ). In the magnon sector (M1 , . . . , Mm+n−1 ), this action is multiplication by
P L Pm+nPm+n−1exp(− `=1 i=1 λ`i X i + r=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.
6
This shift can be understood as originating from normal ordering of creation and annihilation operators, whichcorrespond to the positive and negative roots. We will see a similar shift in section 4.1.

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Comparing the relations (187) and (306), we find


1
1
1
`
[i]
λi −
= (ζ` − µ`i ) + (m − n − ci ) .
(−1)
4
ħ
h
2
3.3.3

(190)

Line defects for parabolic Verma modules of scalar type

There is a generalization of the above brane construction which produces line defects inparabolic Verma modules of scalar type.
Ps
Let (l1 , . . . , ls ) be an ordered partition of m + n: α=1 lα = m + n. The partition specifies aparabolic subalgebra p of gl(m|n), namely the subalgebra consisting of upper-triangular blockdiagonal matrices with diagonal blocks of orders l1 , . . . , ls . A character χ of p is determinedby an s-tuple of complex numbers (χ1 , . . . , χs ) asχ(−) = str(χ ∨ −) ,(191)

χ ∨ = diag(χ1 , . . . , χ1 , . . . , χα , . . . , χα , . . . , χs , . . . , χs ) .
| {z }(192)

where the matrix χ ∨ is given bylα times

For each α, we take lα 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 GL(m|n). The boundary conditionon the D5-brane side is the same as before. On the NS5-brane side, the boundary condition is
η|R×{r} ∈ p ,
⊥

(193)
`

σ|R×{r} ∈ p + λ ,(194)

where p⊥ is the annihilator of p in g∗ and λ` is a character of p. Classically (that is to say, whenλ` is of order ħ
h−1 and quantities of order ħ
h0 are ignored), the value of λ`α is the position of theαth NS5-brane in C.
We expect that the Hilbert space of the BF theory with these boundary conditions is aparabolic Verma module of scalar typeU(g) ⊗U(p) Cλ` −ρ .

(195)

1
ρ = − strg/p adp
2

(196)

The character ρ of p is defined by

and Cλ` −ρ is the one-dimensional U(p)-module determined by the character λ` − ρ.

3.4

Fermionic Dualities

As we have emphasized in our discussions, the Lie superalgebra gl(m|n) does not possess aunique Dynkin diagram. A Dynkin diagram is specified by a choice of ordered basis of Cm|n(or a choice of Z2 -grading if we identify Dynkin diagrams related by the action of the Weylgroup), 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 highestweight representation, but the highest weight is not preserved because the definition of raisingand lowering operators is altered.
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SciPost Phys. 13, 083 (2022)

Odd reflections change how we describe representations of gl(m|n), and the descriptionof 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 [55–60],
with a notable exception of the work by Orlando and Reffert [25] where they employed thepoint of view of string theory to discuss the fermionic dualities for the supersymmetric t-J
model, which is the rational gl(1|2) spin chain with spins valued in the natural representation
C1|2 . Here we offer a string theory explanation for important aspects of fermionic dualities forthe rational 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 choiceof ordered basis (e1 , . . . , em+n ) of Cm|n , which is a permutation of (b1 , . . . , bm , f1 , . . . , f n ) suchthat (b1 , . . . , bm ) and ( f1 , . . . , f n ) are the standard bases of Cm and Cn , respectively. There is anatural identification between these basis vectors and their weights:(e1 , . . . , em+n ) ↔ ("1 , . . . , "m+n ) ,(197)(b1 , . . . , bm ) ↔ (ε1 , . . . , εm ) ,(198)( f1 , . . . , f n ) ↔ (δ1 , . . . , δn ) .

(199)

In the following discussion we will consider permutations of ("1 , . . . , "m+n ) induced by thoseof (e1 , . . . , em+n ).
For a given choice of ordered basis ("1 , . . . , "m+n ) of the dual of the Cartan subalgebra ofgl(m|n), the set of positive roots is
Φ+ = {"i − " j | i < j}(200)

Π = {" r − " r+1 | r = 1, . . . , m + n − 1} .

(201)

and the set of simple roots is

A root "i − " j is said to be even if [i] = [ j] and odd if [i] 6= [ j].
Pick an odd simple root αs = "s − "s+1 and apply to the ordered basis the adjacent transposition σs : {1, . . . , m + n} → {1, . . . , m + n} interchanging "s and "s+1 :("σs (1) , . . . , "σs (s−1) , "σs (s) , "σs (s+1) , "σs (s+2) , . . . , "σs (m+n) )= ("1 , . . . , "s−1 , "s+1 , "s , "s+2 , . . . , "m+n ) . (202)The adjacent transposition alters the notion of positive and simple roots. For the new orderedbasis ("σs (1) , . . . , "σs (m+n) ), the set of positive roots is+Φ+
α = {"σs (i) − "σs ( j) | i < j} = {−αs } ∪ Φ \ {αs }
s

(203)

and the set of simple roots isΠαs = {"σs (r) − "σs (r+1) | r = 1, . . . , m + n − 1}
= {"s−1 − "s+1 , "s+1 − "s , "s − "s+2 } ∪ Π \ {αs−1 , αs , αs+1 } .

(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 αs .
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As an example, take (m|n) = (3|2) and ("1 , "2 , "3 , "4 , "5 ) = (ε1 , ε2 , δ1 , δ2 , ε3 ). This choiceof ordered basis gives the Dynkin diagram (15). There are two odd simple roots, α2 = ε2 − δ1and α4 = δ2 −ε3 , represented by the crossed nodes. Reflection with respect to α2 swaps ε2 andδ1 , leading to the new ordered basis (ε1 , δ1 , ε2 , δ2 , ε3 ). The Dynkin diagram corresponding
to the reflected simple roots is
ε1 − δ1

δ1 − ε2ε2 − δ2δ2 − ε3 .

(205)

We see that all simple roots are now odd. Reflection of the original ordered basis with respect
to α4 results in the ordered basis (ε1 , ε2 , δ1 , ε3 , δ2 ) and the Dynkin diagram
ε1 − ε2ε2 − δ1

δ1 − ε3

ε3 − δ2 .

(206)

Odd reflections change the characterization of highest weights. Let us see how Vermamodules are transformed. Fix an ordered basis of Cm|n and consider the Verma module M (λ),
with the highest-weight vector |Ωλ 〉. Let αs be an odd root. After the odd reflection about αs ,the roles of the raising operator Es,s+1 and the lowering operator Es+1,s are exchanged, while
all other lowering operators remain unchanged. Consequently, the state|Ωλ0 〉 = Es+1,s |Ωλ 〉(207)

is annihilated by all elements of the new set of raising operators, that is, it is a highest-weightstate with respect to the new ordered basis. According to the PBW theorem (23), the states ofthe form
n p−1 n p
n p−1 1−n pnnx 1 1 · · · x p−1 Es+1,s |Ωλ 〉 = x 1 1 · · · x p−1 Es,s+1 |Ωλ0 〉 ,(208)
form a basis of the Fock space Vλ for M (λ), where (x 1 , . . . , x p−1 , Es+1,s ) is an ordered set oflowering operators in the original ordered basis. (Note that n p is either 0 or 1.) By the PBWtheorem, we see that M (λ) is the Verma module M (λ0 ) with respect to the new ordered basis,with
λ0 = λ − αs .
(209)
In the spin chain, the highest weights of the Verma modules placed at the spin sites aretransformed by an odd reflection. The weight of each state of the spin chain remains the same,so the magnon numbers must be transformed as
L
m+n−1
L
m+n−1
X
X
X
X
` 0
0 0
`
(λ ) −
Mr αr =
λ −Mr αr ,
`=1

`=1

r=1

α0r = "σs (r) − "σs (r+1) ,

(210)

r=1

or more explicitly,¨
M r0 =

L + Ms−1 + Ms+1 − Ms(r = s) ;Mr(r 6= s) .

(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 thesolutions 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.

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3.4.2

Fermionic duality from string theory

In the D1–D3–NS5 duality frame that is S-dual to (114), the choice of ordered basis of Cm|n isreflected in the ordering of NS5-branes along RX ; they are ordered as NS51 , NS52 , . . . , NS5m+nfrom left to right. Therefore, the string theory interpretation of the reflection about an odd
root αs is clear: it swaps the positions of NS5s and NS5s+1 . We wish to understand the effectof 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 L` is specified by the parameters λ`i , i = 1, . . . , m + n. The position ofD3`i in C is given byζ` − (−1)[i] λ`i .
(212)
h−1 , by the relation (190) this quantity coincides to order ħh0Indeed, if we take λ`i to be of order ħ``with the twisted mass µi , which is identified with the classical location of D3i . Swapping thepositions of NS5s and NS5s+1 also exchanges D3s and NS5s+1 while keeping their locations
in C fixed. Thus, {λ`i } are transformed to new values {(λ`i )0 } such that (λ`s )0 = λ`s+1 and(λ`s+1 )0 = λ`s in the new ordered basis. This simply means that we have(λ` )0 =m+nX(λ` )0i "σs (i) = λ` .

(213)

i=1

Although λ` is invariant under the odd reflection, the Weyl vector ρ is transformed to a
new Weyl vector ρ 0 . Since ρ is the half sum of even positive roots minus the half sum of odd
positive roots, from the relation (203) between Φ+ and Φ+
α we seesρ 0 = ρ + αs .

(214)Then, the relation (187) between λ` and the highest weight λ` shows that λ` is transformedto (λ` )0 according to the formula (209).
Exchanging the pairs (NS5s , D3s ) and (NS5s+1 , D3s+1 ) does not only transform the highest
weights, but also change the magnon numbers. In the brane picture, we can understand thisphenomenon as creation and annihilation of D1-branes due to the Hanany–Witten effect [61].7
In order to exchange the positions of the brane pairs, we first need to move each D1-brane
between NS5s and NS5s+1 so that one of its end is attached to one of the NS5-branes, say
NS5s+1 . Then, we displace NS5s+1 into the page and start moving it to the left. At one point
the D3-branes ending on NS5s+1 pass through NS5s . As a result, the D1-branes ending onthese D3-branes are annihilated and a new D1-brane is created on each of those D3-branes
that did not have D1-brane ending on it. In the case in which [s] = 0̄, [s + 1] = 1̄ and(Ms−1 , Ms , Ms+1 ) = (1, 2, 1), the process looks as follows:

→

→

→

. (215)The dotted lines indicate the annihilation of D1-branes. We see that the numbers of D1-branes
between NS5-branes transform as in the formula (211).
In general, Hanany–Witten processes for type IIA brane configurations for two-dimensional 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.
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In the D2–D4–NS5 duality frame, the above manipulation is expected to lead to an infrared duality of N = (2, 2) supersymmetric gauge theories. Indeed, there is a known dualitytransformation that sends a theory with U(Nc ) gauge group, N f fundamental chiral multiplets
and Na antifundamental chiral multiplets to a theory with U(Nc0 ) gauge group, Na fundamental chiral multiplets and N f antifundamental chiral multiplets, plus mesons transforming in
the bifundamental representation of the flavor group U(Na ) × U(N f ) [62]. The rank of thedual gauge group is Nc0 = max(N f , Na ) − Nc . This is consistent with what we have found sinceNc = Ms and N f = Na = L + Ms−1 + Ms+1 in our case. However, it appears that the mesons areabsent from our final brane configuration. Fortunately, the mesons, being neutral under thegauge 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 (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 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 gl(m|n)

Finite-dimensional representations of gl(m|n) are most easily discussed in the distinguishedgrading, in which¨
0̄ (i ≤ m) ;
(216)
[i] =
1̄ (i > m) .
For this reason,in this section we exclusively use the distinguished grading. We will write aPm+n
weight λ = i=1 λi "i as (λ1 , . . . , λm |λm+1 , . . . , λm+n ).
The Verma module M (λ) with highest weight λ contains a unique maximal submodule. In the distinguished grading, the corresponding simple quotient module L(λ) is finitedimensional if and only if
λi − λi+1 ∈ Z≥0 ,i = 1, . . . , . . . , m + n − 1 ,i 6= m ,(217)

in other words, if and only if (λ1 , . . . , λm ) and (λm+1 , . . . , λm+n ) are highest weights of finitedimensional irreducible representations of gl(m) and gl(n), respectively. Any finite-dimensional
irreducible representation of gl(m|n) is isomorphic to L(λ) for some λ.
We will consider two classes of finite-dimensional irreducible representations of gl(m|n),called covariant representations and contravariant representations. Covariant representations
appear in tensor products of copies of the natural representation

Cm|n = L (1, 0, . . . , 0|0, . . . , 0) ,(218)
whereas contravariant representations arise from tensor products of copies of the dual repre-

35

SciPost Phys. 13, 083 (2022)

sentation8
(Cm|n )∗ = L (0, . . . , 0|0, . . . , 0, −1) .

(219)

Both covariant and contravariant representations are indexed by the so-called (m|n)-hook partitions.
A partition Y = (Y1 , . . . , Yl(Y ) ) of size |Y | and length l(Y ) is an l(Y )-tuple of positive integers
such that Y1 ≥ · · · ≥ Yl(Y ) and Y1 + · · · + Yl(Y ) = |Y |. It can be represented by a Young diagramwith l(Y ) rows, with the αth row consisting of Yα boxes. The conjugate partition Y 0 has theYoung diagram that is the transpose of the Young diagram for Y .
A partition Y is said to be (m|n)-hook if Ym+1 ≤ n. If Y is an (m|n)-hook partition, then Y 00
is an (n|m)-hook partition, Yn+1
≤ m. We let Hm|n denote the set of all (m|n)-hook partitions.
For an (m|n)-hook partition Y , we define the integral weightY \ = (Y1 , . . . , Ym |〈Y10 − m〉, . . . , 〈Yn0 − m〉) ,(220)

where 〈a〉 = max(0, a). The even part of Y \ is represented by the Young diagram formed by the
first m rows of Y . The Young diagram for the odd part of Y \ is the transpose of the remainderof 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 highestweight representation L(Y \ ). The contravariant representation labeled by Y is the dual representation L(Y \ )∗ = L(Ye \ ). Its highest weight Ye \ equals the minus of the lowest weight of
L(Y \ ) and is given by(221)
Ye \ = (−〈Ym − n〉, . . . , −〈Y1 − n〉| − Yn0 , . . . , −Y10 .

4.2

Line defects in covariant and contravariant representations

Now we construct quantum mechanical systems whose Hilbert spaces are covariant and contravariant representations of gl(m|n). Coupled to four-dimensional Chern–Simons theory withgauge group GL(m|n), they describe line defects valued in these finite-dimensional irreducible
representations.
Let K, K be nonnegative integers and consider a pair of fields
ϕ ∈ Hom(CK|K , Cm|n ) ,(222)

ϕ̃ ∈ Hom(Cm|n , CK|K )(223)

transforming in the bifundamental representations of GL(m|n) × GL(K|K). Their components
are Z2 -graded, with the grading given by[ϕαi ] = [ϕ̃iα ] = [i] + [α] ,(224)

where i and α are indices for Cm|n and CK|K , respectively. The even components are bosonicand the odd ones are fermionic. The action of the theory isZ
1
strCm|n (ϕ dϕ̃) .
(225)
ħ
h R
8
The dual π∗ of a representation π is given by π∗ = τ ◦ π, where τ(X ) = −X st is the Chevalley automorphism.
The supertranspose X st of X is defined by X istj = (−1)([i]+[ j])[ j] X ji . Our definition of supertranspose differs from acommonly used definition by a factor of (−1)[i]+[ j] . With this definition, the quantum mechanical action (225) isinvariant under the natural action of GL(m|n) × GL(K|K).

36

SciPost Phys. 13, 083 (2022)

We will find it convenient to define
i i
ϕ ,
ħ
h α
χiα = (−1)[α] ϕ̃iα ,
ηαi =

(226)
(227)

and
χ̃αi = (−1)[i][α]+[α] ηαi ,i
η̃αi = (−1)[i][α]+[α] χiα .
ħ
h

(228)
(229)

Then, the canonical commutation relations readββ[ηαi , χ j ] = [χ̃αi , η̃ j ] = δαβ δ ij .

(230)

The theory has a GL(m|n) × GL(K|K) global symmetry. The associated conserved charges
areqi j =K+K
X

χiα ηαj + (−1)[i] cδi j ,

α=1
m+n
X

Q αβ = −(231)

β(−1)([α]+[β])[α] χi ηαi + (−1)[α] Cδαβ

(232)

i=1

and satisfy the gl(m|n) ⊕ gl(K|K) commutation relations:[qi j , qkl ] = δ jk qil − (−1)([i]+[ j])([k]+[l]) δli qk j ,(233)[Q αβ , Q γδ ] = δβγQ αδ − (−1)([α]+[β])([γ]+[δ]) δδαQ γβ ,(234)

[qi j , Q αβ ] = 0 .

(235)

The constants c and C account for the ambiguity in operator ordering and will be fixed in a
moment. Under GL(m|n)×GL(K|K), the sets of fields {χiα } and {χ̃αi } transform as the standard
basis vectors for Cm|n ⊗ (CK|K )∗ and (Cm|n )∗ ⊗ CK|K , respectively:[qi j , χkα ] = δ jk χiα =m+nX(Ei j )lk χlα ,

(236)

l=1
γ

β

[Q αβ , χi ] = −(−1)([α]+[β])[α] δαγ χi =[qi j , χ̃αk ] = −(−1)([i]+[ j])[i] δki χ̃αj =

K+K
X

δ=1
m+n
X

st
(−Eαβ
)δγ χiδ .

(−Eistj )lk χ̃αl ,

(237)

(238)

l=1

[Q αβ , χ̃γi ] = δβγ χ̃αi =

K+K
X
δ=1

(Eαβ )δγ χ̃δi .

(239)

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

B=

K+K
X

B αβ Eαβ

α,β=1
α≤β

37

(240)

SciPost Phys. 13, 083 (2022)

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

B = diag(b1 , . . . , bK+K ) .

(241)

In this background, the action becomes
1
ħ
h

Z m+n
X K+K
X
R i=1 α=1(−1)[i] (ϕαi dϕ̃iα + bα ϕαi ϕ̃iα ) ,(242)

and the GL(K|K) symmetry is broken to the stabilizer of the gauge field, which is genericallythe 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 = iħ
hK+K
X

bαQ αα .

(243)

α=1

The Hilbert space of the theory is a Z2 -graded Fock space constructed from a vacuum state |0〉by the action of the creation operators. The action of ϕαi changes H by iħh bα while ϕ̃iα changes
H by −iħh bα . Those component fields that increase Re(iH/ħh) are creation operators, and those
9that decrease it are annihilation operators. We can think of Re(iH/ħh) as energy.
Suppose that we give the background value such that
0 < Re bK+K < Re bK+K−1 < · · · < Re b1 .

(244)

Then, χiα is a creation operator and ηαi is an annihilation operator. Requiring the vacuum
to be invariant under (the maximal torus of) GL(K|K), we find c = C = 0. Let F be thecorresponding Fock space.
The Fock space F decomposes into tensor products of covariant representations of gl(m|n)
and contravariant representations of gl(K|K) [63, 64]:
M\\
F=
L(Ym|n ) ⊗ L(Y )∗ .
(245)
K|K

Y ∈Hm|n ∩HK|K(We use subscripts to distinguish weights for gl(m|n) and gl(K|K).) For example, the firstexcited states take the form
m+n
X K+K
X
cαi χiα |0〉(246)
i=1 α=1

and span a subspace isomorphic to
Cm|n ⊗ (CK|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 represen\tation L(Ym|n ) of gl(m|n). To do so, we impose constraints that singles out the summand\

L(Ym|n ) ⊗ L(Y\K|K)∗ and further projects it to the subspace of lowest-energy states. Since the9

If the time axis is compactified to a circle of radius 1, then the partition function involves trace twisted byexp(−iH/ħh). Creation operators should make this factor smaller.

38

SciPost Phys. 13, 083 (2022)

raising operator Q αβ , α < β, changes the energy Re(iH/ħh) by − Re bα + Re bβ < 0, the lowestenergy states have the highest weight with respect to gl(K|K).
We implement this projection by making B dynamical. The vacuum expectation value of
B is given by the diagonal matrix (241). Let us add to the action the Chern–Simons termZ\−i
Ye (B ) .
(248)
K|K

R

Then, the equations of motion for B are
Q αβ = 0 ,
and10

α<β

(249)

m−n
.
(250)
2
The former equations restrict the Fock space to the subspace of states that contains highestweight vectors of covariant representations of gl(K|K):
M\
L(Ym|n ) ⊗ |ΩYe \ 〉 .
(251)\
Q αα = (Ye K|K )α +K|KY ∈Hm|n ∩HK|K

With the choice
\
Ye

K|K

K+K

\

= Ye K|K −

m−n X
"α ,
2 α=1

the second equation selects the highest weight Ye

\

K|K

\

(252)

, thereby reducing the Hilbert space to thecovariant representation L(Ym|n ) ⊗ |ΩYe \ 〉 of gl(m|n).
K|K

In order to construct line defects in contravariant representations, we take
Re bK+K < Re bK+K−1 < · · · < Re b1 < 0 .

(253)

In this case, χ̃iα is a creation operator and η̃αi is an annihilation operator. The corresponding
Fock space Fe decomposes as
M
\
\
Fe =
L(Ym|n )∗ ⊗ L(Y ) .
(254)
K|K

Y ∈Hm|n ∩HK|K

Making B dynamical and adding the Chern–Simons term
Z
−i

Y

\

K|K

(B )

(255)

R

to the action, we can reduce the Hilbert space to the contravariant representation L(Y \ )∗ , with
Y

\

K|K

10

=Y

\
K|K

K+K

−

m−n X
"α .
2 α=1

The Weyl quantization of the classical expression of Q αα equals Q αα + (−1)[α] (m − n)/2.

39

(256)

SciPost Phys. 13, 083 (2022)

4.3

Brane construction of line defects

The quantum mechanical system discussed above can be constructed with D3-branes and D5branes. Let us remove from the brane system (114) the semi-infinite D3-branes and the F1branes stretched between the D5-branes, and instead introduce infinite D3-branes D3α , α = 1,
. . . , K + K:
Spacetime: R × S 1 × C ×RX× RY × R2+ħh × R2−ħh ,1
D5i (i ≤ m): R × S × C × {X i } × {Yi } × R2+ħh × {0} ,D5i (i > m): R × S 1 × C × {X i } × {Yi } × {0} × R2−ħh ,D3α (α ≤ K): R × { y} × {ζ} × {− Re bα } × RY × R2+ħh × {0} ,D3α (α > K): R × { y} × {ζ} × {− Re bα } × RY × {0} × R2−ħh .

(257)

We claim that strings stretched between the D3-branes and the D5-branes give rise to thequantum mechanical system in question.
The K D3-branes D3α , α ≤ K, and the m D5-branes D5i , i ≤ m, share the three-dimensional
spacetime R × R2+ħh . Strings stretched between them produce an N = 4 hypermultiplet in thebifundamental representation of U(K) × U(m). Let
ϕ 00 ∈ Hom(CK , Cm ) ,
ϕ̃

00

m

K

∈ Hom(C , C )

(258)
(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 Ω-deformationinduced by the background RR two-form, and it has the effect of localizing the hypermultiplet
to the quantum mechanical model with action [23]
Z
1
(260)
trCm (ϕ 00 dϕ̃ 00 ) .
ħ
h RHere we are using ϕ 00 , ϕ̃ 00 to denote the one-dimensional fields that descend from the threedimensional scalar fields.
Similarly, from strings stretched between the K D3-branes D3α , α > K, and the n D5branes D5i , i > m, we get an N = 4 hypermultiplet in the bifundamental representation ofU(K) × U(n) on the three-dimensional spacetime R × R2−ħh . By an Ω-deformation, the theory
localizes to a quantum mechanical system with actionZ
1−
trCn (ϕ 11 dϕ̃ 11 ) ,
(261)
ħ
h R
where
ϕ 11 ∈ Hom(CK , Cn ) ,(262)

ϕ̃ 11 ∈ Hom(Cn , CK ) .

(263)

The branes D3α , α ≤ K, and D5i , i > m, intersect along the time axis R, and from strings
stretched between them we get fermions
ϕ 10 ∈ Hom(CK , Cn ) ,(264)

ϕ̃ 10 ∈ Hom(Cn , CK ) .

(265)

They are described by the action
1
ħ
h

Z

trCn (ϕ 10 dϕ̃ 10 ) .

R

40

(266)

SciPost Phys. 13, 083 (2022)

This is the dimensional reduction of the two-dimensional chiral fermions that arise from anintersection of D4-branes and D6-branes [65, 66].
In the same way, from strings stretched between D3α , α > K, and D5i , i ≤ m, we get
fermionic fields
ϕ 01 ∈ Hom(CK , Cm ) ,(267)

ϕ̃ 01 ∈ Hom(Cm , CK ) ,(268)

described by the action
1
−
ħ
h

Z

trCm (ϕ 01 dϕ̃ 01 ) .

(269)

R

These four quantum mechanical systems can be combined into the single quantum mechanical system described by the action (225), with the fields
ϕ 00 ϕ 01
ϕ=
,
(270)
ϕ 10 ϕ 11
ϕ̃ 00 ϕ̃ 01
ϕ̃ =
.
(271)
ϕ̃ 10 ϕ̃ 11
The creation operator χiα adds a string stretched between D3α and D5i . The annihilation
operator ηαi 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 theconstruction 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 theGL(K|K) gauge symmetry to a Borel subgroup. Which Borel subgroup is selected is determinedby the ordering of the D3-branes on RX . For the ordering (244) for {Re bα }, it is the standard
Borel subgroup.
The situation in which there are no strings stretched between the D3-branes and the D5branes corresponds to the vacuum of the Fock space F . Here is how the vacuum looks like for(m|n) = (1|2) and (K|K) = (2|1):

|0〉 =

.

(272)

D51 D52 D53

D31 D32 D33\

To project to a covariant representation L(Ym|n ) of gl(m|n), we fix the number of strings
ending on each D3-brane. Let us illustrate how this works with an example in which\
Y = (5, 1, 1). For this choice of Y , we have Y 0 = (3, 1, 1, 1, 1), Ym|n = (5|2, 0) and
\
Ye

K|K

\

= (0, −4| − 3). The brane configuration for the highest-weight state of L(Ym|n ) ⊗ L(Ye

\

K|K

)

is the following:|ΩY \ 〉 ⊗ |ΩYe \ 〉 =
m|n

.

(273)

K|K

D31 D32 D33

F1s

D51 D52 D53

A string ending on D5i from the left contributes "i to the gl(m|n) weight, and a string endingon D3α from the right contributes −"α to the gl(K|K) weight. This configuration is the tensor
41

SciPost Phys. 13, 083 (2022)

product of two highest-weight vectors because we cannot shorten any of the strings and stretch
it between another pair of D3-brane and D5-brane; by doing so we would get more than one
strings between D33 and D51 , but that is prohibited as such strings have necessarily coincident
worldsheets and are fermionic. This brane diagram shows|ΩY \ 〉 ⊗ |ΩYe \ 〉 = (χ23 )2 χ13 (χ12 )4 |0〉 .
m|n

(274)

K|K

\

The other vectors in L(Ym|n ) ⊗ |ΩYe \ 〉 can also be represented by brane configurations. For
K|K

example,q21 |ΩY \ 〉 ⊗ |ΩYe \ 〉 =
m|n

K|K

3
X
α=1

χ2α η1α (χ23 )2 χ13 (χ12 )4 |0〉(275)= −4(χ23 )2 χ13 χ22 (χ12 )3 |0〉 + (χ23 )3 (χ12 )4 |0〉
is a linear combination of two states, which one obtains from the highest-weight vector byextending one of the strings to the right:

q21 |ΩY \ 〉 ⊗ |ΩYe \ 〉 = −4
m|n

K|K

D51 D52 D53

D31 D32 D33+. (276)
D51 D52 D53

D31 D32 D33

The construction of a line defect in a contravariant representation of gl(m|n) is analogous.
For a contravariant representation, the D3-branes are placed to the right of the D5-branes.
For example, for the same choice (m|n) = (1|2), (K|K) = (2|1) and Y = (5, 1, 1), we have\\
Yem|n = (−3| − 1, −3) and Y= (5, 1|1), and the highest-weight vector is represented by the
K|Kconfiguration
.

(277)

D51 D52 D53 D31 D32 D33

4.4

Two-dimensional N = (2, 2) supersymmetric gauge theories

Applying S-duality and T-duality on S 1 to the brane configurations for a line defect, we obtain
D2–D4–NS5 brane configurations which describe two-dimensional N = (2, 2) supersymmetricfield theories. For a general choice of (K|K) and Y , the resulting theory does not seem to admit
a simple gauge theory description.
If we restrict to covariant representations with K = 0 and Ym+1 = 0 and contravariant
0
representations with K = 0 and Yn+1
= 0, the two-dimensional theories are particularly nice.
Let us consider these cases.
For a covariant representation with K = 0 and Ym+1 = 0, the relevant highest weights\\
are Ym|n = (Y1 , . . . , Ym |0, . . . , 0) and Ye= (−Y 0 , . . . , −Y10 ). The brane configuration for the
K
K|K
\
\
e
highest-weight vector of L(Ym|n )⊗ L(Y ) has one string stretched between NS5i and D3K−α+1
K|K

42

SciPost Phys. 13, 083 (2022)

for each α = 1, . . . , Yi . The following diagram depicts the highest-weight vector for\(m|n) = (3|2), (K|K) = (0|4) and Y = (4, 2, 1), for which Y 0 = (3, 2, 1, 1), Ym|n = (4, 2, 1|0, 0)
and Ye\K|K= (−1, −1, −2, −3):

.

(278)

D51 D52 D53 D54 D55

D31 D32 D33 D34

By moving the D3-branes past D5-branes, we can bring this configuration to another configuration in which the D3-branes are located between D5-branes and have no strings attached:11
D31 D32

D34

D33

.
D51

D52

D53

D54

(279)

D55

The strings that were initially present get annihilated by the Hanany–Witten transition. The
number of D3-branes between D5i and D5i+1 is equal to Yi − Yi+1 .
If we stretch strings between D5-branes in this configuration, then by the reverse Hanany–\
Witten moves we get a configuration for excited states in L(Ym|n ) ⊗ |ΩYe \ 〉. For example, theK|Kconfiguration(280)
represents an excited state with weight (4, 2, 1|0, 0) − 3α1 − 2α2 − 2α3 − 2α4 = (1, 3, 1|0, 2).
One such state is represented by the configuration

.

(281)
\

0
Similarly, for a contravariant representation with K = 0 and Yn+1
= 0, we have Yem|n= (0, . . . , 0|− Yn0 , . . . , −Y10 ) and Y\

K|K= (Y1 , . . . , YK ), and the brane configuration for the highest-\\
weight vector of L(Yem|n ) ⊗ L(Y ) can be brought to a configuration without any strings. Take
K|K\
an example with (K|K) = (3|0) and Y = (2, 2, 1), for which Y 0 = (3, 2), Yem|n = (0, 0, 0| − 2, −3)

and Y

\

K|K

= (2, 2, 1):
.

(282)

This configuration represents an excited state with weight (0, 0, 0| − 2, −3) − α1 − α2 − α3 − α4
= (−1, 0, 0| − 2, 0). The number of D3-branes between an adjacent pair of D5-branes can beread off from Y 0 .
11
An obstruction to generalize the present argument to more general covariant and contravariant representations
is that we do not understand what happens when a D3-brane passes through a D5-brane of the same color in thepresence of the RR two-form for Ω-deformation.

43

SciPost Phys. 13, 083 (2022)

The dual D2–D4–NS5 brane configurations are of the type studied by Hanany and Hori [33]and realize quiver gauge theories. The quiver for the configuration (280) is
3

2

2

2

(283)
1

2

1

1

2

1

and the quiver for the configuration (281) is
1

.
1

1

1

(284)

1

The ranks of the gauge nodes are given by the numbers of F1-branes, and the ranks of the flavor
nodes are given by the numbers of D3-branes. There are N = (4, 4) cubic superpotential terms
involving adjoint chiral multiplets.
The flavor symmetry for the chiral multiplets charged under the mth gauge node is doubled
due to the lack of cubic superpotential term. From the brane point of view, this is because aD4-brane between NS5m and NS5m+1 can be broken into half on one of the NS5-branes which
has the same color as the D4-brane:

→

→

.

(285)

a

Strings stretched between D2mm , am = 1, . . . , Mm , and one half of the D4-brane produce afundamental chiral multiplet for U(Mm ), while strings between those D2-branes and the otherhalf of the D4-brane produce an antifundamental chiral multiplet.

4.5

Bethe/gauge correspondence for finite-dimensional representations

Generalizing the above brane construction, we can obtain the Bethe/gauge correspondence
for spins valued in arbitrary finite-dimensional representations of gl(m|n). This is essentiallythe correspondence proposed by Nekrasov [3].
Consider the rational gl(m|n) spin chain of length L, with the `th spin takes values infinite-dimensional highest-weight representations L(λ` ). For the moment, let us assume thatthe highest weights are all integral and satisfy
λ`1 ≥ · · · λ`m ≥ 0 ≥ λ`m+1 ≥ · · · ≥ λ`m+n .

(286)This is the case if all of the representations are of the type studied in section 4.4. We define
nonnegative integers
K r` = λ`r − λ`r+1 ,`Km= λ`m ,`
K r = λ`r − λ`r+1 ,`K m = −λ`m+1 .

r = 1, . . . , m − 1,

(287)
(288)

r = m + 1, . . . , m + n − 1,

(289)
(290)

We look at a sector of fix magnon numbers (M1 , . . . , Mm+n−1 ). The Bethe equations depend
only on the highest weights and the magnon numbers, so their form remain the same as in thecase of Verma modules.
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SciPost Phys. 13, 083 (2022)

The gauge theory corresponding to this magnon sector is similar to the theory discussed
in section 2.3 and has the same gauge symmetry. The difference is that the chiral multiplets
e i , i = 1, . . . , m + n, are replaced by chiral multiplets
Qi , Q`R`r ∈ Hom(CKr , C Mr ) , r = 1, . . . , m ,

(291)

K r`

(292)

e ` ∈ Hom(C Mr , C ) , r = 1, . . . , m ,
R
r
K r`

S`r ∈ Hom(C , C Mr ) , r = m + 1, . . . , m + n − 1 ,K r`e` ∈ Hom(C Mr , C ) , r = m + 1, . . . , m + n − 1 .
S
r

(293)
(294)

Letting
Kr =
Kr =

L
X
`=1
L
X

K r` ,

(295)

`

Kr ,

(296)

`=1

e r ∈ Hom(C Mr , CKr ), r = 1,
we can combine them into chiral multiplets R r ∈ Hom(CKr , C Mr ), R
Kr
Mr
Mr
Kr
e
. . . , m − 1, and S r ∈ Hom(C , C ), S r ∈ Hom(C , C ), r = m + 1, . . . , m + n − 1.
e r φ r R r ), and a flavor
For r < m, the theory has the N = (4, 4) superpotential term trCKr (Resymmetry U(K r ) act on R r and R r . For r = m, the cubic superpotential is absent and two
e m . Similar statements hold for S r and S
e r . Undercopies of U(Km ) act separately on Rm and ReeU(1)ħh , R r and R r have charge −1 and S r and S r have charge +1.
The gauge and matter contents of the theory can be encoded in a quiver. For (m|n) = (3, 2),
the quiver is
φ1
e2
P

M1
R1

e1
R

K1

K3

e3
S

S3
e4
P

e3
P

M2

P2

M3

P3
R2

e2
R

K4

K3
φ2

e4
S

M4

P4

e3
R

R3

K3

K3

K2

S4

.

(297)

φ4

Furthermore, the theory admits a brane construction. For example, for (M1 , M2 , M3 , M4 ) =(3, 2, 2, 2) and (K1 , K2 , K3 |K 3 , K 4 ) = (2, 1, 1|1, 1), the brane configuration for the above quiver
is
.

(298)

Note that in order to realize the flavor symmetry U(Km )2 × U(K m )2 , each D4-brane between
NS5m and NS5m+1 needs to be brought to the NS5-brane of the same color and broken intohalf.
We turn on mass parameters for the global symmetries in such a way that higgsing give

45

SciPost Phys. 13, 083 (2022)

the following masses:
1(R`r )ar l : σ ar r − µ`r,l − ħh,2e ` )l a : µ̃` − σ ar − 1 ħ
(R
h,
r
r
r,l
r
2
1
h,
(S`r )ar l : σ ar r − ν`r,l + ħ2e` )l a : ν̃` − σ ar + 1 ħ
(S
h.
r
r
r,l
r
2

(299)
(300)
(301)
(302)

We necessarily have µ`r,l = µ̃`r,l and ν`r,l = ν̃`r,l for r 6= m.
e ` ) contributes
From these expressions for the masses, we see that for r ≤ m, the pair (R`r , Rrto the vacuum equations the factor
`

a
Kr
Y
σ r r − µ`r,l − 21 ħ
h

,
ar
1
`
h
l=1 σ r − µ̃ r,l + 2 ħ(303)e` ) contributes the factor
and for r ≥ m, the pair (S`r , S
r
`

a
Kr
Y
σ r r − ν`r,l + 21 ħ
h

.
ar
1
`
h
l=1 σ r − ν̃ r,l − 2 ħ(304)

For the Bethe/gauge correspondence to exist, the above factors should reproduce the factor
σ r r − ζ` + (−1)[r+1] λ`r+1ħ
h − 21 c r ħhaa

σ r r − ζ` + (−1)[r] λ`r ħh − 21 c r ħ
h

,

(305)

in the Bethe equations. This is indeed possible if we identify the parameters as
11µ`r,l = µ̃`r,l = ζ` − (λ`r + c r + )ħh + lħ
h,22
11ν`r,l = ν̃`r,l = ζ` + (λ`r+1 + c r − )ħh + lħ
h.
2
2

(306)
(307)

We have`aKrYhσ r r − µ`r,l − 12 ħar1`hl=1 σ r − µ̃ r,l + 2 ħ

σ r r − µ`r,K ` − 12 ħ
h
a

=

r

σ r r − µ`r,1 + 12 ħha

σ r r − ζ` + (λ`r − K r` )ħ
h − 12 c r ħ
h
a

=

a

σ r r − ζ` + λ`r ħh − 12 c r ħ
h

,

(308)

for r ≤ m and
`

a
Kr
Y
σ r r − ν`r,l + 21 ħ
h
1
r
`
h
l=1 σ r − ν̃ r,l − 2 ħah
σ r r − ν`r,1 + 12 ħa=

σ r r − ν` ` − 12 ħha

σ r r − ζ` − λ`r+1ħ
hh − 12 c r ħ
a

=

r,K r

`

σ r r − ζ` − (λ`r+1 + K r )ħh − 12 c r ħ
h
a

,

(309)

`

for r ≥ m, so we obtain the factor (305) using the definitions of K r` and K r .
Now, let us consider the case in which the representations of the spin variables are arbitrary
finite-dimensional ones. Even in this general case, most of the above argument actually goes`

through, with the same definitions of K r` and K r for r 6= m and the identifications (306)46

SciPost Phys. 13, 083 (2022)

and (307). The only place that fails is where we set r = m: if we choose nonnegative integers``
Km, K m and write down the product of the factors (303) and (304) for r = m, we get

σmm − ζ` − λ`m+1ħ
hh − 21 cmħaa

σmm − ζ` + λ`mħh − 21 cmħh`
σmm − ζ` + (λ`m − Km)ħh − 12 cmħ
h
a

`

a

σmm − ζ` − (λ`m+1 + K m )ħh − 12 cmħ
h



,

(310)

whereas the Bethe equations do not contain the second fraction in the parenthesis.
We can cancel this unwanted factor if we introduce additional chiral multiplets`e ` ∈ Hom(C Mm , CKm ) ,R
mS`m ∈ Hom(C`Km, C Mm )

(311)
(312)

and give them appropriate masses. These chiral multiplets are produced by semi-infinite D4branes ending on NS5m and NS5m+1 .

Acknowledgments
We would like to thank Kevin Costello, Mykola Dedushenko and Philsang Yoo for helpful discussions. NI gratefully acknowledges support from the Institute for Advanced Study and the
National Science Foundation under Grant No. PHY-1911298. The work of SFM is fundedby the Natural Sciences and Engineering Research Council of Canada (NSERC). Research at
Perimeter Institute is supported in part by the Government of Canada through the Department
of Innovation, Science and Economic Development Canada and by the Province of Ontario
through the Ministry of Colleges and Universities. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not necessarilyreflect the views of the funding agencies.

A

Four-dimensional Chern–Simons theory with gauge supergroupfrom twisted string theory

In this appendix we present an alternative construction of four-dimensional Chern–Simons
theory with gauge group GL(m|n), using the framework of twists of superstring theory as
developed in [67]. Twisted superstring theory refers to superstring theory in a particular RR
background where the bosonic ghost for local supersymmetries may take a nonzero nilpotent
vacuum expectation value Q. When one considers D-branes in such backgrounds, the coupling
between D-branes and the bosonic ghost dictates that Q is added to the BRST differential of theworldvolume theory [67]. Therefore, the field content of worldvolume theories of branes intwisted superstrings are naturally Q-cohomology of that of the supersymmetric gauge theories
one would find in the absence of the additional RR background. As such, twisted superstrings
affords a useful framework for studying protected sectors of supersymmetric gauge theories.
Costello and Li [67] give conjectural descriptions of such twists of superstrings in termsof topological strings. These conjectures have passed several consistency checks [67–69] andhave been proven at the level of the free limit of the supergravity approximation [70]. Taking
these conjectures as a starting point, one can derive simple descriptions of twists of worldvolume theories of D-branes using mathematical tools from the study of topological strings.
Though such calculations require machinery from homological algebra, they have the benefitof calculational ease. Tractable models of twisted worldvolume theories can be determinedfrom an Ext-algebra computation, and the action functional can be read off from an algebraic
47

SciPost Phys. 13, 083 (2022)

structure and trace on the Ext-algebra; no term-matching arguments involving the Dirac–Born–
Infeld action are required.
In this appendix, we work with field theory in the Batalin–Vilkovisky (BV) formalism as
articulated by [71, 72]. In particular, we freely make use of the language of L∞ -algebras.
Much of the below is exposited elsewhere in the literature. The construction of twisted supergravity and the conjectural descriptions of twists of superstrings in terms of topological strings
are given in [67]. Many of the examples below are worked out in [73] where more formal
aspects of the framework are articulated and some mathematical applications are discussed.
We hope the exposition of this appendix will have the simultaneous benefit of illustrating the
calculational utility of twisted superstrings, and making our constructions parseable to moremathematically minded readers.

A.1

Topological strings

We begin with some recollections on topological strings. The worldsheet theory of a topologicalstring theory is a two-dimensional oriented topological quantum field theory. Treating suchtheories via the language of functorial field theory, the results of [74, 75] tell us that suchtheories are determined by the data of a Calabi–Yau category. Physically, we think of objectsof this category as D-branes in our topological string theory, and the space of homomorphisms
between two objects as the complex computing BRST cohomology of the states of open strings
stretched between the branes. It is known that spaces of open string states have an algebra
structure, with respect to which the action for open string field theory takes a simple form [76].
The data of a Calabi–Yau category is exactly what is needed to make precise this algebraic
structure; this will be elaborated more on subsection A.2 below.
Example A.1. Let M be a symplectic four-manifold and X a Calabi–Yau three-fold. The SU(3)invariant twist of type IIB string theory is given by the Calabi–Yau 5-category Fuk(M )⊗Coh(X ).
Here, Fuk(M ) refers to the Fukaya category of M and Coh(X ) refers to the category of coherent
sheaves on X . This describes a topological string theory that looks like a combination of the
A-model into M and the B-model into X .
Here the terminology is meant to indicate that the above mixed A-B model conjecturally
arises from type IIB string theory in an RR background in which the bosonic ghost takes a vacuum expectation value given by an SU(3)-invariant nilpotent element of the ten-dimensional
N = (2, 0) supersymmetry algebra.
Remark A.2. Let us elaborate on our description of the A-model directions. For us the main relevant example will be when M = R2N . In this case, we will use a version of the Fukaya categorythat we will denote Fuk0 where we discard counts of pseudo-holomorphic discs with nonzero
area. Explicitly, the objects in the category will consist of Lagrangians in M , and for two Lagrangians L1 , L2 ⊂ R2N with clean intersection, we have that HomFuk0 (L1 , L2 ) = Ω• (L1 ∩ L2 ).
This will suffice for our purposes as we will primarily care about perturbative phenomena on
worldvolume theories of branes, so we may neglect worldsheet instantons.
In addition to this restriction on the space of homomorphisms, this category does not include as objects, coisotropic A-branes. To the authors’ knowledge, it is an open mathematicalproblem to construct a version of the Fukaya category that includes as objects such branes.
Fortunately, we will not need to consider such branes in our analysis.
To a topological string theory, we may associate two field theories which are versions ofopen string field theory and closed string field theory respectively. The former recovers twistsof worldvolume theories of branes in the physical string while the latter contains twists ofsupergravity.
48

SciPost Phys. 13, 083 (2022)

A.2

Topological open string field theory

Let C be an A∞ -category and let F ∈ C be an object. Then HomC (F , F ) is an A∞ -algebra, andskew-symmetrizing the A∞ -operations yields an L∞ -algebra. Now suppose our C is in fact
a Calabi–Yau N -category, and as such can be thought of as determining a topological string
theory. Then for any object F ∈ C , we have an invariant pairing tr: HomC (F , F ) → C[N ].
In examples of interest, where C is attached to a 2N -manifold thought of the target spacetime of our topological string, HomC (F , F ) will arise as sections of a natural graded vector
bundle over the support of F , the L∞ -structure maps will be given by polydifferential operators, and the trace map will factor through integration over the support of F . In such instances,the data of this L∞ -algebra and the trace pairing then determine the data of a perturbativeZ2 -graded BV theory – the space of fields of the theory is Π HomC (F , F ) and the action is givenbyX
1
S(α) =
tr α ⊗ `k (α⊗k ) ,(A.1)(k + 1)!
k≥1where `k : HomC (F , F )⊗k → HomC (F , F ) are the L∞ -structure maps. This theory is theworldvolume theory of the D-brane F in the topological string theory determined by C . The
conjectural descriptions of twists of superstrings in terms of topological strings imply that for
C coming from a twist of a superstring theory, the worldvolume theory of F is a twist of theworldvolume theory of the corresponding brane in the physical string theory.
Example A.3. Consider the SU(3)-invariant twist of type IIB string theory on R4 × C3 fromexample A.1, and consider a stack of n D5-branes wrapping R2 × C2 . As explained in theabove example, this twist of type IIB string theory is described by the Calabi–Yau category
C = Fuk0 (R4 ) ⊗ Coh(C3 ). The object describing our stack of branes is given by (R2 , OCn 2 ). We
have that
ExtC (R2 , OCn 2 ), (R2 , OCn 2 ) = HomFuk0 (R2 , R2 ) ⊗ ExtCoh(C3 ) (OCn 2 , OCn 2 )= Ω• (R2 ) ⊗ ExtCoh(C3 ) (OC 2 , OC2 ) ⊗ gl(n)•2= Ω (R ) ⊗ Ω

0,•

(A.2)

2

(C )["] ⊗ gl(n).

In the last step, we have used the following general result:Lemma A.4. Let X be a Calabi–Yau manifold and let Y ⊂ X be holomorphic.
ExtCoh(X ) (OY , OY ) ∼= Ω0,• (Y, ∧• NX /Y ).
We can describe the L∞ -structure as follows.
Ω• (R2 ) ⊗ Ω0,• (C2 ) ⊗ gl(n) given by

Then

There is an L∞ -structure on`1 = d ⊗ 1Ω0,• (C2 ) ⊗ 1gl(n) + 1Ω• (R2 ) ⊗ ∂¯ ⊗ 1gl(n) ,(A.3)`2 = ∧ ⊗ ∧ ⊗ [−, −]gl(n) ,(A.4)`k = 0 ,(A.5)

k ≥ 3.

The L∞ -structure on Ω• (R4 ) ⊗ Ω0,• (C2 )["] ⊗ gl(n) is given by the semidirect productΩ• (R4 ) ⊗ Ω0,• (C2 ) ⊗ gl(n) n " Ω• (R4 ) ⊗ Ω0,• (C2 ) ⊗ gl(n) .
(A.6)
The trace pairing induced from the Calabi–Yau structure on C is givenZ
tr: α 7→Tr(α) ∧ Ω ,R2 ×C2|1

49

(A.7)

SciPost Phys. 13, 083 (2022)

where Ω denotes the holomorphic volume form on C2 and Tr is the Killing form on gl(n). Thus,
we find that the action of the theory is exactly
Z


1
1
¯
S(α, β) =Tr β(d + ∂ )α + β ∧ [α, α] ∧ Ω ,
(A.8)
2
6
R2 ×C2|1forα ∈ Ω• (R2 ) ⊗ Ω0,• (C2 ) ⊗ gl(n) ,•2β ∈ Ω (R ) ⊗ Ω

0,•2(C )" ⊗ gl(n) .

(A.9)(A.10)

This is exactly the holomorphic–topological twist of six-dimensional N = (1, 1) super Yang–
Mills theory, dubbed the rank (1, 1) partially holomorphic topological twist in [77].

A.3

Topological closed string field theory

Let Z be the worldsheet theory determined by the Calabi–Yau category C . Naively, the closed
string states of the theory should be given by the local operators of the worldsheet theory,
Z(S 1 ). However, the worldsheet theory in the physical string is coupled to two-dimensional
gravity – closed string states should be those local operators invariant under reparametrizations of the worldsheet. Since the worldsheet theory is topological, Cartan’s magic formula
tells us that small reparametrizations will act homotopically trivially on the space of local operators. In the setting of topological strings, there is a natural homotopy action of S 1 on Z(S 1 )1– the closed string states will be the invariants Z(S 1 )S . In terms of categorical data, this iscomputed by the cyclic cochains of the category C , HC• (C ). There is a natural way to equipa shift of HC• (C ) with an odd Poisson tensor and an L∞ -structure. In examples in which thegraded vector space underlying HC• (C ) arises as the space of sections of some graded vector
bundle, this gives HC• (C ) the structure of a Z2 -graded Poisson BV theory. The constructionsof the L∞ - and shifted Poisson structures in this generality are extraneous for our purposes –we will be focused on the following examples.
Example A.5. Suppose C = Coh(X ) with X Calabi–Yau. Then the cyclic formality theorem [78]tells us that there is an equivalence of L∞ -algebrasHC• (C ) ∼(A.11)= PV•,• (X )[[t]] , `1 = ∂¯ + t∂ , `2 = {−, −} ,where t is a parameter of degree 2, ∂ denotes the divergence operator, and {−, −} denotes
the Schouten bracket of polyvector fields. The Poisson tensor has Poisson kernel (∂ ⊗ 1)δ∆(X ) ,where ∆(X ) ⊂ X × X denotes the diagonal. This theory is Kodaira–Spencer gravity articulated
as Bershadsky–Cecotti–Ooguri–Vafa theory studied by [79–81].
Example A.6. Suppose C = Fuk(M ) with M being a symplectic manifold. The Hochschild(co)chains admit a description in terms of the quantum cohomology of the target. Together
with the abstract L∞ -structure and the Z2 -graded Poisson structure, we expect that the result
will be a version of the Kähler gravity [82]. We will discard worldsheet instantons coming fromthe A-model directions of the twists of string theory we consider. Therefore, our ansatz will bethat the closed string field theory for the A-model directions is described by the L∞ -algebraΩ• (M ) with L∞ -structure given by `1 = d, `2 = ∧ and Poisson structure given by the wedge
and integrate pairing. We will abusively continue to denote the closed string field theory in
the A-model sans worldsheet instantons by HC• (Fuk(M )).
Example A.7. Putting the above two examples together, we can describe the closed string
field theories for the twists of type IIB string theory we are interested in. The closed string
50

SciPost Phys. 13, 083 (2022)

field theory for the SU(3)-invariant twist of type IIB string theory on R4 × C3 is given by the
L∞ -algebra Ω• (R4 ) ⊗ PV•,• (C3 )[[t]] with
`1 = d ⊗ 1PV•,• (C3 )[[t]] + 1Ω• (R4 ) ⊗ (∂¯ + t∂ ) ,(A.12)`2 = ∧ ⊗ {−, −} ,(A.13)`k = 0 ,(A.14)k ≥ 3.

The Poisson tensor is given by the Poisson kernel (∂ ⊗ 1)δ∆(C3 ) δ∆(R4 ) .

A.4

Closed–open map

Given a Calabi–Yau category C and an object F ∈ C , there is always an L∞ -mapHC• (C ) → CE• HomC (F , F ) .

(A.15)

Here, the target denotes Chevalley–Eilenberg cochains on the L∞ -algebra HomC (F , F ); thisis a model for Hamiltonian vector fields on the formal moduli space describing fluctuations of
the brane F . This map takes a closed string field and produces a single trace-operator on theworldvolume theory of F , which describes how the closed string field couples to the worldvolume theory of F . We will wish to apply this to examples where C = Fuk0 (R10−2N ) ⊗ Coh(CN ),and F = (R5−N , OCn k ) for k ≤ N . Then we have thatHC• (C ) = Ω• (R10−2N ) ⊗ PV0,• (CN )[[t]] ,•HomC (F , F ) = Ω (R

5−N

)⊗Ω

0,•

(A.16)k(C )["k+1 , . . . , "N ] ⊗ gl(n) .

(A.17)

This map should be thought of as given by a sum of disk amplitudes with boundaryon the brane F and with an arbitrary number of marked points on the interior labelingclosed string insertions. We will only consider single closed string insertions of the form
1 ⊗ µ ∈ Ω• (R10−2n ) ⊗ PV0,• (CN ). In particular the field does not depend on the A-twisted
directions of spacetime, or the parameter t. For such fields we have the following explicit
formula for the linear component of the closed–open mapaa

1 ⊗ w11 · · · w NN ∂wb1 · · · ∂wbN 7→ I(α) ,1(A.18)

N

where
1
I(α) =(n + 1)!

ZaR5−N ×Ck|N −k

a

b

b

k+1
· · · "NNTr(w11 · · · w NN "k+1

× ∂"bk+1 α ∧ · · · ∧ ∂"bN α ∧ ∂wb1 α ∧ · · · ∧ ∂wbk α) ∧ Ω . (A.19)
k+1

N

1

k

A version of this result including formulas for the deformation to all orders in open string
insertions is proved in [81]. It is worth emphasizing that deriving formulas for this map at allorders is an extremely nontrivial problem – for µ ∈ PV2,0 this is the content of the holomorphic
analogue of Kontsevich’s theorem on deformation quantization.
Example A.8. Consider the SU(3)-invariant twist of type IIB string theory on R4 × C3 . We
fix once and for all coordinates z, w1 , w2 on C3 . We saw in the example above that astack of n D5-branes wrapping R2 × C2z,w gives rise to a holomorphic–topological twist of1
six-dimensional N = (1, 1) super Yang–Mills theory with gauge group U(n). Let us now consider what happens when we turn on a field 1 ⊗ w1 w2 ∈ Ω• (R4 ) ⊗ PV0,• (C3 ). Recall that thefields of the relevant twist of six-dimensional N = (1, 1) super Yang–Mills theory were givenby Ω• (R2 ) ⊗ Ω0,• (C2z,w )["] ⊗ gl(n).
151SciPost Phys. 13, 083 (2022)

The image of the closed string field w1 w2 under the closed open map becomes the functionalZ
Tr(αw1 ∂" α ∧ Ω) .

I(α) =(A.20)

R2 ×C2|1

Equivalently, this deforms the L∞ -structure on Ω• (R2 ) ⊗ Ω0,• (C2 )["] ⊗ gl(n) so that
`1 = d ⊗ 1Ω0,• (C2 ["]) ⊗ 1gl(n) + 1Ω• (R2 ) ⊗ (∂¯ + w1 ∂" ) ⊗ 1gl(n) . The differential w1 ∂" has the effectof deforming the complex of fields of the theory into


w1 ∂"
Ω• (R2 ) ⊗ Ω0,• (C2 )" −−→ Ω0,• (C2 ) ⊗ gl(n) .

(A.21)

This is the Koszul resolution of the locus w1 = 0, so is quasi-isomorphic toΩ• (R2 )⊗Ω0,• (C)⊗ gl(n). This is exactly four-dimensional Chern–Simons theory as a Z2 -graded
BV theory.
Remark A.9. Note that the above construction differs slightly from the construction of fourdimensional Chern–Simons theory via Ω-deformation in [16]. Conjecturally, the quadratic superpotential we have introduced should describe those components of the RR two-form usedin [16] that are not exact for the twist we are performing. However, checking this explicitly is adifficult task. Moreover, at the level of field theory, the construction in [16] came from subjecting the holomorphic–topological twist of six-dimensional N = (1, 1) super Yang–Mills theory
with BV fields given by Ω• (R4 )⊗Ω0,• (C)⊗ gl(n) to a B-type Ω-background along R2 ⊂ R4 . Sucha construction involves replacing a factor of Ω• (R2 ) with the Cartan model for S 1 -equivariant
1
cohomology of R2 , which is given by the abelian dg Lie algebra (Ω• (R2 )[u]S , d + u ι∂θ ), where
u is an equivariant parameter, and ∂θ is the infinitesimal action of rotations. The localization theorem for equivariant cohomology tells us that for generic values of the equivariant
parameter, the complex of fields of our theory is quasi-isomorphic to Ω• (R2 ) ⊗ Ω0,• (C) ⊗ gl(n).
Note that the relation uL∂θ = [d, u ι∂θ ] coming from Cartan’s magic formula tells us that theinfinitesimal action of rotations on the fields of our theory is homotopically trivial.
However, in the above construction we instead work with a more minimal twist of sixdimensional N = (1, 1) super Yang–Mills theory; the twist of the previous paragraph is gotten
from deforming the differential on Ω• (R2 )⊗Ω0,• (C2 )["]⊗gl(n) so that `1 = d⊗1Ω0,• (C)["] ⊗1gl(n)+ 1Ω• (R2 ) ⊗ (∂¯ + "∂w1 ) ⊗ 1gl(n) . Instead of taking this further twist and working equivariantly
along the topological plane that the w1 -plane becomes, we turned on a deformation comingfrom a quadratic superpotential. It is worth noting that there is a map from a twist of the fourdimensional N = 4 superconformal algebra to the closed string sector of the SU(3)-invariant
twist of type IIB string theory – the quadratic superpotential deformation considered abovelies in the image of this map. Moreover, note that we have that Lw1 ∂w = ["∂w1 , w1 ∂" ]; we see1that the superconformal deformation also makes the complexified action of rotations exact for
the B-twist supercharge. This appears to be part of a general pattern where a superconformal
deformation of a holomorphic theory is equivalent to an Ω-deformation of a further topologicaltwist [73, 83, 84].
Remark A.10. It is also interesting to consider the superpotential w1 w2 as a deformation of theentire topological string theory, that is, as a deformation of the category of branes. Morally, it
should deform the category of coherent sheaves on C3 to the category of matrix factorizations
for the superpotential w1 w2 ; the B-model directions of the topological string are turned into a
Landau–Ginzburg B-model. The category of matrix factorizations in this case can be describedas the category of modules for the Jacobi algebra of the superpotential w1 w2 , which in thiscase is just the algebra C[z]. Thus, we see that the SU(3)-invariant twist of type IIB string
theory localizes to a six-dimensional topological string theory on R4 × C; this makes contact
with the work of [85].
52SciPost Phys. 13, 083 (2022)

A.5

Four-dimensional Chern–Simons theory with gauge supergroup from theSU(3)-invariant twist of type IIB string theory

In this section we will arrive at four-dimensional Chern–Simons theory with gauge supergroupusing the formalism developed in the previous subsections. The calculation is essentially aneasy corollary of the examples therein.
We consider the SU(3)-invariant twist of type IIB string theory on R4 × C3 with a configuration of D-branes as in the following table:

n D5
m D5

R2
◦
◦

R2
×
×

Cz
×
×

Cw1
×
◦

Cw2
◦ .
×

(A.22)A cross mark means that the D5-brane extends in that direction. We also turn on a closed
string field given by the quadratic superpotential w1 w2 . We arrive at a field theory description
for this system by first computing the open string field theory using the techniques above, andthen applying the closed–open map.
Let F1 = (R2 , OCm2 ), F2 = (R2 , OCn 2 ) denote the objects in the categories of D-branes
z,w1

z,w2

corresponding to the stacks of n and m D5-branes, respectively. We first wish to compute
HomC (F1 ⊕ F2 , F1 ⊕ F2 ). Since Hom commutes with direct sums, we have four summands:• From example A.8, we have that
HomC (F1 , F1 ) = Ω• (R2 ) ⊗ Ω0,• (C2z,w )["2 ] ⊗ gl(m) ,(A.23)

HomC (F2 , F2 ) = Ω• (R2 ) ⊗ Ω0,• (C2z,w )["1 ] ⊗ gl(n) .

(A.24)

1
2

Our convention above is that "i denotes a section of the normal bundle of C2z,w ⊂ C3 ,jwhere i 6= j.
The trace pairing is given bytr(α) =

Zdz dw j tr(α) .

(A.25)

R2 ×C2|1• We can compute, using free resolutions of the structure sheaves of the w i -planes, that
HomC (F1 , F2 ) = Ω• (R2 ) ⊗ Ω0,• (Cz ) ⊗ Hom(Cm , Cn )[−1] ,(A.26)

HomC (F2 , F1 ) = Ω• (R2 ) ⊗ Ω0,• (Cz ) ⊗ Hom(Cn , Cm )[−1] .

(A.27)

Each of these are abelian L∞ -algebras, with `1 = d ⊗1Ω0,• (Cz ) ⊗1Hom +1Ω• (R2 ) ⊗ ∂¯ ⊗1Hom .
There is a natural trace pairing on the direct sumΩ• (R2 ) ⊗ Ω0,• (Cz ) ⊗ T ∗ Hom(Cm , Cn )[−1] .

(A.28)

Letting X s denote fields valued in Hom(Cm , Cn ) and Y s fields valued in the cotangent
direction, the pairing is given byZtr(X 1 + Y1 , X 2 + Y2 ) = dw TrCn (X 1 Y2 ) − TrCm (Y1 X 2 ) .
(A.29)
The action functional induced by this pairing and abelian L∞ -structure is exactly the
BV action for a free hypermultiplet in the Kapustin twist. Restricted to fields of ghost
number 1, this recovers exactly the action (145).
53

SciPost Phys. 13, 083 (2022)

Thus we see that the entire space of open string states is given by 0,•
Ω (Cw1 )["2 ] ⊗ gl(n)


⊕




•
2
0,•
0,•

E = Ω (R ) ⊗ Ω (Cz ) ⊗ Ω (Cw2 )["1 ] ⊗ gl(m)
.


⊕



(A.30)T ∗ Hom(Cm , Cn )[−1]

We now determine the L∞ -structure. This is as usual gotten by skew-symmetrizing thenatural A∞ -structure on HomC (F1 ⊕ F2 , F1 ⊕ F2 ). In terms of the above direct summands,
the A∞ -structure is given in terms of the following operations:• HomC (Fi , Fi ) ⊗ HomC (Fi , Fi ) → HomC (Fi , Fi ) ,A ⊗ B 7→ AB ;• HomC (Fi , F j ) ⊗ HomC (F j , Fi ) → HomC (F j , Fi ) ,Y ⊗ X 7→ Y X ;• HomC (Fi , Fi ) ⊗ HomC (F j , Fi ) → HomC (F j , Fi ) ,A ⊗ X 7→ AX ,where i, j = 1, 2, i 6= j.
This induces the following L∞ -structure:• The first kind of A∞ -operation above gives L∞ -structures on HomC (Fi , Fi ) with
`1 = d ⊗ 1Ω0,• (C2 )[" j ] ⊗ 1g + 1Ω• (R2 ) ⊗ ∂¯ ⊗ 1g ,(A.31)`2 = ∧ ⊗ [−, −]g ,(A.32)

where g = gl(m) for i = 1 and g = gl(n) for i = 2. As explained in example A.3 theseare precisely holomorphic–topological twists of the six-dimensional N = (1, 1) vector
multiplets for gl(m) and gl(n).
• There is a bracketΩ• (R2 ) ⊗ Ω0,• (Cz ) ⊗ T ∗ Hom(Cm , Cn )[−1]

⊗2

 0,•

Ω (Cw1 )["2 ] ⊗ gl(m)⊕→ Ω• (R2 ) ⊗ Ω0,• (Cz ) ⊗ 
 , (A.33)Ω0,• (Cw2 )["1 ] ⊗ gl(n)explicitly given by wedging the form factors and taking the commutator of the matrices
in T ∗ Hom(Cm , Cn ).
• Finally, there is a bracketΩ0,• (Cw1 )["2 ] ⊗ gl(m)⊕Ω• (R2 ) ⊗ Ω0,• (Cz ) ⊗ Ω0,• (Cw2 )["1 ] ⊗ gl(n)⊗•2Ω (R ) ⊗ Ω

0,•(Cz ) ⊗ T ∗ Hom(Cm , Cn )→ Ω• (R2 ) ⊗ Ω0,• (Cz ) ⊗ T ∗ Hom(Cm , Cn ) , (A.34)

explicitly given by wedge product of forms and the natural action of gl(m) ⊕ gl(n) on
T ∗ Hom(Cm , Cn ).
54

SciPost Phys. 13, 083 (2022)

The last of these brackets encodes the coupling between the hypermultiplets in the Kapustin twist and the twist of the six-dimensional N = (1, 1) vector multiplet. The second
bracket, which breaks the Z-grading down to a Z2 -grading, encodes an extra gauge symmetry.
The open string field theory we have found can formally be regarded as four-dimensional
Chern–Simons theory on R2 × Cz for a dg Lie superalgebra. We may schematically encode theabove brackets by writing the above dg Lie algebra as
 0,•Ω (Cw1 )["2 ] ⊗ gl(m)Hom(Cm , Cn )[−1].
(A.35)
Hom(Cn , Cm )[−1]Ω0,• (Cw2 )["1 ] ⊗ gl(n)Let us now analyze the effect of the closed string field w1 w2 . As we saw before, the imageof this closed string field under the closed open map only affects the differential on the above
dg Lie superalgebra. Explicitly, the deformation looks like


 Ω

0,•

w1 ∂"2(Cw1 )"2 −−−→ Ω0,•
(Cw1 ) ⊗ gl(m)w2 ∂"1nΩ (Cw2 )"1 −−−→ Ω (Cw2 ) ⊗ gl(n)


gl(m)
Hom(Cm , Cn )[−1]∼. (A.36)=
Hom(Cn , Cm )[−1]gl(n)Hom(C , C )[−1]nHom(C , C )[−1]
m

0,•

m

0,•

The remaining Lie brackets equip the above with with the structure of the Lie superalgebra gl(m|n). Thus, we have found exactly four-dimensional Chern–Simons theory for the Liealgebra gl(m|n) as claimed.
We note that the BRST transformations induced by the above Lie brackets are slightly
different from those identified in the main body of the paper. This is an artifact of working
with a particular model for the underlying L∞ algebra. For comparison, we explicate the BRST
transformations below.
Note that the cochain complex underlying our L∞ algebra arises naturally as
the totalization of a Z × Z/2-graded cochain complex, where the fields valued in
Hom(Cm , Cn ) ⊕ Hom(Cn ⊕ Cm ) are placed in bidegree (•, 1). Though the lie brackets arisingfrom the coupling of the hypermultiplets to the vector multiplets broke the Z-grading down toa Z/2-grading, these brackets are easily seen to preserve the above grading Z × Z/2 grading.
We fix the following notation for components of our fields
αi j ∈ Ωi (R) ⊗ Ω0, j ⊗ (gl(m) ⊕ gl(n)),βi j ∈ Ωi (R) ⊗ Ω0, j ⊗ (Hom(Cm , Cn ) ⊕ Hom(Cn , Cm ))(A.37)and denote the corresponding linear operators the same way. The BRST variations determinedby the then take the form
Qαi j = dα(i−1) j + ∂¯ αi( j−1) +X

X[αab , αcd ] +a+c=i,b+d= j

Qβi j = dβ(i−1) j + ∂¯ βi( j−1) +[βab , βcd ] ,(A.38)

a+c=i,b+d= jX[αab , βcd ] .

(A.39)

a+c=i,b+d= j

The brackets in these equations are the relevant brackets on the L∞ algebra we’ve identified. It would be interesting to construct an explicit L∞ equivalence between the BV complex
we have identified and the L∞ algebra consisting of the fields A, c, b, B and BRST transformations from section 169.

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SciPost Phys. 13, 083 (2022)

Remark A.11. As in remark A.10, we can consider the effect of the quadratic superpotential
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