SciPost Phys. 16, 052 (2024)

Line operators in 4d Chern-Simons theory and Cherkis bows
Nafiz Ishtiaque1 and Yehao Zhou2
1 IHES, 91440 Bures-sur-Yvette, France
2 Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

Abstract
We show that the phase spaces of a large family of line operators in 4d Chern-Simons theory with GLn gauge group are given by Cherkis bow varieties with n crosses. These line
operators are characterized by Hanany-Witten type brane constructions involving D3,
D5, and NS5 branes in an Ω-background. Linking numbers of the five-branes and mass
parameters for the D3 brane theories determine the phase spaces and in special cases
they correspond to vacuum moduli spaces of 3d N = 4 quiver theories. Examples include
line operators that conjecturally create T, Q, and L-operators in integrable spin chains.
Copyright N. Ishtiaque and Y. Zhou.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 31-01-2023
Accepted 06-02-2024
Check forupdatesPublished 22-02-2024
doi:10.21468/SciPostPhys.16.2.052

Contents
1 Introduction

2

2 Brane Construction of Line Operators
2.1 General Considerations (Arbitrary Linking Numbers)
2.2 Special Cases (with Quiver Descriptions)55
8

3 Boundary Conditions and Topological Twists in 4d N = 4 SYM
3.1 1/2-BPS Boundary Conditions in 4d N = 4 SYM
3.1.1 D5 Boundary
3.1.2 NS5 Boundary
3.2 Topological Twists Preserved by 1/2-BPS Boundary Conditions

9
10
11
12
13

4 Ω-deformed Kapustin-Witten (KW) Theory and BF Phase Spaces
4.1 KW Theory as a 2d B-model
4.1.1 2d Multiplets and Supersymmetry
4.1.2 Actions
4.1.3 Ω-deforming the B-model
4.2 Boundary Conditions in BF Theory
4.2.1 D5 Boundary
4.2.2 NS5 Boundary
4.3 Phase Spaces of BF Theories with Boundaries
4.3.1 BF Theories and Cherkis Bows
4.3.2 BF Theories and Branches of Vacua of 3d N = 4 Theories

14
14
16
17
20
20
21
26
28
28
30

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SciPost Phys. 16, 052 (2024)

5 Line Operators in 4d Chern-Simons Theory
5.1 Example: The T-Operators (Wilson Lines)
5.2 Example: The Q-Operators (Minuscule ’t Hooft Lines)
5.3 Example: The L-Operators (Wilson-’t Hooft)
5.4 Open Bow Diagrams

31
32
36
37
38

6 Conclusion

39

A Ω-deformed Supersymmetry

40

References40

1

Introduction

4d Chern-Simons (CS) theory [1,2] is a holomorphic-topological quantum field theory on a 4manifold Σ×C where Σ is a topological surface and C is a Riemann surface with a holomorphic
volume form ω. It is a gauge theory with a complex gauge group GC . The only dynamical
field in this theory is the Lie algebra valued connection A x dx + A y d y + Az dz where x, y are
local coordinates on Σ and z, z are local holomorphic and anti-holomorphic coordinates on C.
The action for the theory is:
Z


1
2
ω ∧ tr A ∧ (d + ∂ )A + A ∧ A ∧ A .
(1.1)
ħ
h Σ×C
3
The theory is topological on Σ and holomorphic on C. Here we only focus on the case Σ = R2 ,C = C, ω = dz and the gauge group is GC = GLn = GL(n, C) with Lie algebra gln = gl(n, C).
In this paper we study phase spaces of line operators in 4d CS theory that are local in theholomorphic direction C. A line operator in a gauge theory with gauge group G can be definedby taking a quantum mechanics with H-symmetry, a homomorphism G ! H by which G actson its phase space, and coupling it to the gauge theory. By the phase space of the line operator
we refer to the phase of this quantum mechanics. The line operators we study are motivatedby certain brane constructions. We label these line operators in the GLn theory bya spectral parameter, z
an n-tuple of integers,

K = (K1 , · · · , Kn )a p-tuple of integers,

L = (L1 , · · · , L p )and, a (p − 1)-tuple of complex numbers,
satisfying
N :=

n
X

Ki =

i=1

p
X

Li .

(1.2)

C
ϱ = (ϱ1C , · · · , ϱ p−1
)

(1.3)

i=1

Our main result is that the line operator labeled by z, ϱ, K , and L – which we denote byLzϱ (K , L) – can be described classically in terms of a complex symplectic space called a Cherkis

bow variety, denoted in this paper by Mbow
ϱ (K , L). Our result is similar to how flag varieties
correspond to classical phase spaces of Wilson lines in 3d CS theory [3]. The bow varieties
were introduced by Cherkis as certain moduli spaces of instantons [4–6] and further describedby Nakajima and Takayama as quiver varieties [7] who also showed that for certain choicesof K and L these varieties coincide with the Coulomb branches of 3d N = 4 quiver gauge
2

SciPost Phys. 16, 052 (2024)

theories. Note that the phase spaces are independent of the spectral parameter and we will
omit references to this parameter for the most part.
To each line operator Lzϱ (K , L) we assign two 3d N = 4 theories which we denote byTϱ [U(N )]LK and Tϱ∨ [U(N )]LK . These two theories are mirrors [8] of each other. The bow variety

L
∨
L
Mbow
ϱ (K , L) is the Coulomb branch of Tϱ [U(N )] K and the Higgs branch of Tϱ [U(N )] K . WhenK and L satisfy certain constraints, the 3d theory Tϱ [U(N )]LK admits a linear quiver description.
Coulomb branches of these quiver theories are known to be slices in the affine Grassmannian
whose deformation quantization results in shifted truncated Yangians [9–12]. Line operators
in 4d CS theory form integrable spin chains which carry natural Yangian actions [13, 14]. Ourconstruction then suggests that not only the (deformation) quantization of slices in the affine
Grassmannian but also that of Bow varieties should result in algebras with RTT representations(e.g. shifted truncated Yangian). From this point of view, our results are closely related to theBethe/Gauge correspondence of Nekrasov and Shatashvili [15–17], we will not discuss thisconnection in depth in this paper.
It was shown in [18] that bow varieties also arise as certain moduli spaces of vacua in
4d N = 4 theories with impurity walls. These defect 4d theories are effectively describedby 3d N = 4 theories and the relevant solutions to the vacuum equations become the Higgs
branches of these 3d theories. This establishes an equivalence between bow varieties and
Higgs branches as hyper Kähler spaces. Using Ω-deformation, we localize these defect 4dtheories to 2d holomorphic BF theories with line defects. We then identify the complex phasespaces of these BF theories with the complex bow varieties described in [7]. Combinatorial
data associated with Hanany-Witten brane configurations [19] involving D3-D5-NS5 branes
were also used in the mathematical literature [20] to define bow varieties, which is precisely
the association between bow varieties and Higgs branches of 3d N = 4 theories made in [18]and in our paper. These authors define stable envelopes for torus equivariant cohomology ofbow varieties and being motivated by S-duality in string theory they study symplectic duality
between stable envelopes of bow varieties defined by mirror Hanany-Witten configurations.
The description of 3d N = 4 vacuum branches as bow varieties derived in [18] and this paper
puts the results of [20] in a more physical context.
A similar characterization of some spins in terms of algebras appears in [21] in the context
of studying solutions to the RTT relations in rational gln spin chains. Given an R-matrix, avariety of T-operators, or more generally referred to as L-operators in [21] are characterized
as follows. A module Vq of U(glq )⊗ Weyl⊗q(n−q) , where Weyl is the Weyl algebra or the Heisenberg algebra with q(n − q) oscillators, becomes an induced module for the Yangian Y (gln ) via
a homomorphism Y (gln ) ! U(glq ) ⊗ Weyl⊗q(n−q) . Given Vq , the authors of [21] construct solutions Lq (z) ∈ End Cn ⊗ Vq ⊗ C[z] to the RTT relation that are linear in z. The operators
L n and L1 are called T and Q-operators respectively. [22] shows that the Q-operator can becomputed from 4d CS theory by computing the expectation value of crossing Wilson and ’t
Hooft lines. Here Wilson and ’t Hooft lines can be constructed as topological line defects labeled by the module Cn of gln and the Fock module for Weyl⊗(n−1) respectively. We shall showthat the family of line operators that we study includes operators with phase spaces whose
deformation quantization leads to the algebras U(glq ) ⊗ Weyl⊗q(n−q) for all q = 1, · · · , n. It istherefore natural to conjecture that crossing these line operators with Wilson lines shall leadto the L-operators of [21], though we leave the computation of such expectation values for
future work.
This paper is structured as follows. In §2 we discuss the brane construction of 4d CS
theories with line operators. We follow the construction of 4d CS as the Ω-deformed worldvolume theory of a stack of D5 branes from [23]. A family of line operators in this theory can beconstructed by introducing NS5 branes and suspending D3 branes between the five-branes. Welabel a line operator by the linking numbers of the five-branes – the linking numbers of the D5

3

SciPost Phys. 16, 052 (2024)

and the NS5 branes constitute the tuples K and L respectively. The remaining parameters z andϱ will denote locations of the NS5 branes in certain directions. Supersymmetric configurationsof the D3 branes parameterize the classical phase spaces that we assign to the line operators
created by the D3 branes. These supersymmetric configurations can be seen as the spaces ofvacua of the world-volume theory of the D3 branes. In general we can describe this theory
as a 4d N = 4 theory on an interval, which in principle can effectively be described as a 3d
N = 4 theory at low energy. This defines the 3d theory Tϱ∨ [U(N )]LK and we define the theory
Tϱ [U(N )]LK as its mirror. The phase space assigned to the line operator is the Higgs branch ofTϱ∨ [U(N )]LK which is also the Coulomb branch of the mirror. For some restricted set of linking
numbers we can rearrange the five and the three-branes using Hanany-Witten transitions ina way that leads to a standard description of the world-volume theory of the D3 branes as
3d N = 4 quiver gauge theories [19]. In these quiver cases the ϱiC s will denote the complex
Fayet–Iliopoulos (FI) parameters in Tϱ∨ [U(N )]LK and complex twisted masses in Tϱ [U(N )]LK . 3d
N = 4 theories admit FI parameters and twisted masses that are triplets of real scalars rotatedby SU(2) × SU(2) R-symmetry acting on the hyper-Kähler vacuum branches [8]. However, we
will land on a complex description of the vacuum branches as opposed to the hyper-Kähler
description, and as such only a complex linear combination of two of the components of a real
triplet will be prominent in our discussion.
In §3 and §4 we take a more field theoretic route to arrive at the description of the phasespaces as bow varieties. Instead of looking at the 3d N = 4 descriptions, we look directly at
the 4d world-volume theory of the D3 branes with boundaries and domain walls provided by
the five-branes. The 4d theory is N = 4 super Yang-Mills (SYM) and the five-branes impose
1/2-BPS boundary conditions on the fields of this theory [24, 25]. In §3 we check that these
1/2-BPS boundaries preserve the Kapustin-Witten twist [26] of the 4d N = 4 SYM labeled by
the twist parameter t = i (as defined in §3.1 of [26], see also (3.26)). In §4 we show thatturning on Ω-deformation reduces this twisted 4d theory to 2d BF theory on an interval. In thissetup the phase space attached to the line operator is the phase space of the BF theory, whichis the moduli space of solutions to complex Nahm’s equation with boundary conditions. We
determine the boundary conditions in the BF theory descended from boundary conditions of
the original 4d theory and we show that the phase spaces coincide with Cherkis bow varietiesL
Mbow
ϱ (K , L). When the linking numbers are restricted such that Tϱ [U(N )] K admits a quiverL
description we check that Mbow
ϱ (K , L) is indeed the coulomb branch of Tϱ [U(N )] K using results from [7]. In the final section §5 we discuss some examples of line operators, their phasespaces, and the quantized algebras. As special cases we find line operators whose quantization
give algebras related to the T, Q, and L-operators as described in [21].
The main contributions of the paper are as follows. We show that Bow varieties provide
a geometric characterization of spins in rational gln spin chains. We do this by showing thatthese varieties are phase spaces of line operators in 4d CS theory and then appealing to the
known relation between these line operators and integrable spin chains. The result suggeststhat bow varieties can be endowed with the structure of classical integrable systems whichquantize to rational gln spin chains. We identify these phase spaces with supersymmetricvacua of 4d N = 4 SYM theories on intervals with domain walls, as in [18]. These vacua
correspond to the Higgs branch vacua of some effective 3d theories, or equivalently, to the
Coulomb branch vacua of the mirror theories. Applying Ω-deformation to the 4d setup weshow that a protected subsector of these 4d theories can be described as 2d BF theories withboundaries and line defects. We map boundary conditions and domain walls of the 4d theory
to those of the BF theory. The bow varieties become the complex phase spaces of these defect
BF theories.

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SciPost Phys. 16, 052 (2024)

2

Brane Construction of Line Operators

2.1

General Considerations (Arbitrary Linking Numbers)

4d CS theory with GLn gauge group can be constructed from a stack of n D5 branes. Let Σ be aa 2d surface, C a Riemann surface, and TN the 4d Taub-NUT space. We start in type IIB string
theory with the 10d space-time being T ∗ Σ × C × TN. The TN can be described very concretely
in terms of a coordinate x⃗ of R3 and a circle with coordinate θ – in terms of which the TN
metric can be written as:
2
dsTN
= Ud⃗
x · d⃗x+1(dθ + ω⃗ · d⃗x )2 ,
U(2.1)⃗ is a vector field on R3 \R satisfying dU = ⋆R3 d(ω⃗ · d⃗
x ). We see
where U = 1r + λ12 and ωthat the TN circle collapses at the center of R3 andpat infinity its radius asymptotes to λ. R3
can be parameterized by a radial coordinate ρ := x⃗ · x⃗ and two angular coordinates. A 2d
surface inside TN located at fixed values of these two angular coordinates and parameterized
by r and θ has the shape of a “cigar”. TN can therefore be described as a family of cigars
parameterized by an S 2 , all the cigars sharing a single point – the tip – located at the centerof the TN. We need to fix a specific supergravity background, the defining characteristic of
the background is that it preserves a supercharge Q which induces a B-type Ω-deformation of
the world-volume theory of any brane wrapping a cigar in TN. This requires turning on – inaddition to a nontrivial metric – a dilaton and a RR 2-form. For details about this particular
background we refer to [23].1 Here we simply note that if any D-brane is placed in thisbackground wrapping a cigar inside TN then from the point of view of the cigar the D-brane
theory looks like an Ω-deformed B-model. The supercharge Q squares to a rotation of theTN circle which we schematically write as Q2 ∼ Lħh∂θ , here ∂θ is the vector field generatingrotation of the TN circle and ħ
h acts as a deformation parameter.
To find 4d CS theory we introduce a stack of n D5 branes wrapping Σ × C × Cig where Cig
is some chosen cigar inside TN. The world-volume theory of the D5 branes is 6d N = (1, 1)
U(n) SYM, which upon Ω-deformation along Cig reduces to 4d GLn CS theory on Σ × C at thelevel of BRST cohomology [23]. To get line operators in this theory we introduce NS5 branes
that share 3 directions with the D5s, wrap the entire TN, and do not wrap any direction in
C. We further introduce D3 branes suspended between the D5s and the NS5s. The D3 branes
have finite extent in one of the directions and at low energy the corresponding world-volume
theory is a 3d N = 4 theory, which upon Ω-deformation becomes a 1d topological quantum
mechanics2 (TQM) coupled to the 4d CS theory [28, 29]. Different configurations of D3 andNS5 branes lead to different TQMs and each TQM defines a line operator in the 4d CS theory.
We shall simplify our discussions slightly by taking Σ = R2 and C = C. For notational
simplicity we also write TN as R2+ħh × R2−ħh . The notation is meant to imply that the supercharge
Q squares to a bosonic symmetry that rotates R2+ħh and R2−ħh in opposite angular directions.
These two planes can be thought of as two antipodal cigars inside TN. We choose coordinateswith indices running from 0 to 9 to label directions in our 10d space-time and we summarize
the directions wrapped by the D3, D5, and NS5 branes in Table 1.
The NS5 branes and consequently the D3 branes have fixed positions in the C direction. To
create a single line operator these branes must be coincident in the C direction, the location ofthese branes in this direction will be the spectral parameter associated with the line operator.
Different numbers of NS5s and D3s lead to different line operators. Since we want to create
4d CS theory with GLn gauge group, the number of D5 branes is fixed to be n. Suppose
Some T-dual version of this background was introduced in [27] with the name Taub-trap background.
Topological means that the action for the quantum mechanics depends only on the symplectic form of thetarget, there is no Hamiltonian.
1

2

5

SciPost Phys. 16, 052 (2024)

Table 1: Directions wrapped by various branes in the construction of 4d CS theory
with line operators. The 10d space-time is T ∗ Σ × C × T N . We have replaced TN with
R2+ħh × R2−ħh and we have chosen indices for 10d coordinates parameterizing various
components as follows: Σ = R207 , T ∗ Σ = R40734 , C = C = R289 , R2+ħh = R212 , andR2−ħh = R256 . We write Rni ···i to refer to the Rn space parameterized by coordinates1

x i1 , · · · , x i n .

D5
D3
NS5

n

R2+ħh
R2−ħh
C
0 1 2 3 4 5 6 7 8 9
× × ×
× × ×
× × × ×
× × ×
× × ×

there are p NS5 branes. We choose an ordering of the five-branes in the x 3 direction andreading from left to right, we label the NS5 branes as NS51 , ... , NS5 p and the D5 branes as
D51 , ... , D5n . A particular configuration can now bePdescribedP
by an n-tuple K = (K1 , · · · , Kn )npand a p-tuple L = (L1 , · · · , L p ) of integers satisfying i=1 Ki = i=1 L p where Ki and L i are thelinking numbers of D5i and NS5i respectively (concept introduced in [19] but our terminology
is from [25]):3
Ki := No. of (D3s to the left – D3s to the right + NS5s to the right) of D5i(2.2)

L i := No. of (D3s to the right – D3s to the left + D5s to the left) of NS5i

Here left and right refers to the x 3 direction. If we put all the NS5 branes to the left of all the D5
branes then the linking numbers are simply the numbers of D3 branes ending on the D5 (resp.
NS5) branes from the left (resp. right). We depict the corresponding configuration pictoriallyin Fig. 1. We only impose the constraints on the linking numbers that our diagrams do notnP

Ki D3si=2NS5 p−1

NS51

..
.

···

..
.

..
.

NS5 p

..
.

..
.

L1 D3s

C

···

..
.

Kn D3s

..
.

N D3s
p−2
P
j=1

L j D3sp−1
P
j=1

L j D3sx3(Kn−1 + Kn )D3s
D51

D5n−1

D5n

Figure 1: Brane configuration for the line operator Lϱ (K , L) in GLn 4d CS theory
linking numbers K = (K1 , · · · , Kn ) and L = (L1 , · · · , L p ) satisfying
Pn labeledPbypChϱ = ħ
h(ϱ1C , · · · , ϱ p−1) are complex FI parameters, they are
i=1 Ki =j=1 L j = N . ħ
not visible in the classical brane picture. We denote the 3d N = 4 theory describing
the low energy dynamics of the D3 branes by Tϱ∨ [U(N )]LK .
become disconnected, i.e., there are nonzero number of D3 branes between any two adjacentfive-branes and that the brane configurations remain supersymmetric, i.e., the s-rule [19] is
3

To be precise, L i – what we are calling the linking number of the ith NS5 brane – is really n minus the linking
number of the NS5 brane where n is the total number of D5 branes. However we shall keep referring to L i as thelinking number of NS5i simply for convenience. Our notion of linking number coincides precisely with that of the“charge of a five-brane” from [20].

6SciPost Phys. 16, 052 (2024)

Pn
Pj
not violated. These require j=i K j > 0 for all 1 ≤ i ≤ n, i=1 L i > 0 for all 1 ≤ j ≤ p, andN ≤ np for example, though these conditions are not sufficient in general.
The D3 branes have finite length in the x 3 direction and at low energy their world-volume
theory is a 3d N = 4 theory on R3012 . We denote the 3d N = 4 theory describing the D3 branes
at low energy by T ∨ [U(N )]LK . This theory can be deformed by both FI parameters and twistedmasses. Turning on FI parameters deforms the Higgs branch and turning on generic twistedmasses lifts most of the Higgs branch leaving only isolated vacua [30–32]. We only turn on FI
parameters, keeping the twisted masses zero, so that we have smooth Higgs branches. In termsof the branes, FI parameters correspond to differences in locations of the NS5 branes in theR3789 direction [19]. We have already mentioned that we take the NS5 branes to be coincident
in these directions to have a single line operator. In practice we will assume that the differencesin their locations in these direction are of order O(ħh) so that classically they are still coincident
but these differences will generate FI deformations of order O(ħh) in the theory. Thus the 3dtheory in general can be deformed by turning on the FI parameters ħhϱ⃗j = ħ
h(ϱ1, j , ϱ2, j , ϱ3, j )where ħhϱ
⃗ j can schematically be interpreted as the difference between the locations of NS5 j
and NS5 j+1 in the R3789 direction. In the Ω-background we get a holomorphic description of
the Higgs branch which is deformed by the parameters associated with the locations of theNS5 branes in the C direction. We therefore define the complex FI parametersħhϱ Cj := ħh(ϱ j,2 − iϱ j,3 ) .

(2.3)

The remaining FI parameter, also called the real FI parameter, is traded for stability conditions
during the construction of Higgs branches as invariant quotients [33]. We therefore label
C). Forour theory using only the complex parameters: Tϱ∨ [U(N )]LK where ϱ := (ϱ1C , · · · , ϱ p−1
arbitrary linking numbers K and L a more concrete description of this theory is not immediate.
To create a line operator we do not assume any constraints on the linking numbers other thanpreservation of supersymmetry.
We shall denote the Higgs branch of a theory T by MH (T ). A 3d N = 4 theory on R3012
with B-type Ω-deformation on the R212 plane localizes to a TQM on R0 whose target is the
Higgs branch of the 3d theory [28, 29]:4Ω-deformation on R2

12
T on R3012 −−−−−−−−−−−!
TQM on R0 with the target MH (T ) .

(2.4)

Before the localization by Ω-background, the 3d theory couples to the 6d N = (1, 1) U(n) SYM
theory of the D5 branes as a 3d defect. After localization we find the aforementioned TQM
coupled to the 4d GLn CS theory. The 4d CS connection couples to the flavor current of the
TQM. The flavor symmetry of the TQM is the complexification of the flavor symmetry of the
Higgs branch of the 3d N = 4 theory, which depends on the exact brane configuration and isdifficult to describe concisely in general. Let us schematically denote this flavor symmetry of
the 3d theory by F (K , L):
F (K , L) := Higgs branch flavor symmetry group of Tϱ∨ [U(N )]LK ,(2.5)

and the flavor symmetry of the TQM by FC (K , L). The flavor symmetry of the 3d theory is a
subgroup of the U(n) that rotates the n D5 branes. After complexification by Ω-background
we therefore find FC to be a subgroupof GLn and the gln -valued connection A of the 4d CSRtheory couples to the TQM via R tr(AµMH ) where µMH is the (dual of the) Lie(FC ) valued
4This quantum mechanics can also be studied as a protected sub-sector of the 3d N = 4 superconformal algebra[34–37]. The relation between certain protected sub-sectors of a superconformal algebras and Ω-deformations
persists in 4d N = 2 (and possibly more generally) and in fact the 3d version can be derived as a dimensional
reduction thereof [38, 39].

7

SciPost Phys. 16, 052 (2024)

moment map on MH . This TQM defines a line operator in the 4d CS theory which we denoteby Lϱ (K , L). The phase space of the line operator is the target of the TQM:
Phase space of Lϱ (K , L) = MH (Tϱ∨ [U(N )]LK ) ,(2.6)

and the TQM quantizes (the holomorphic functions on) this phase space into an operator
algebra which we denote by Aϱ (K , L):Aϱ (K , L) := Operator algebra that couples to Lϱ (K , L) .

2.2

(2.7)

Special Cases (with Quiver Descriptions)

In the remainder of this section, let us look at the cases where the 3d theory Tϱ∨ [U(N )]LK admits
a quiver description as examples.5
In order to have a quiver description, we impose the following constraints on the linking
numbers:
0 < Ki < p ,1 ≤ i ≤ n,ev j :=jX

j−1
Xei ≥ 0 ,Li −( j − i)w

i=1

0 < j < p,

(2.8)

i=1

where,

e j := #{Ki | Ki = p − j} .
w

With these constraints we can bring the D5 branes between the NS5 branes using HananyWitten transitions [19] such that there are equal number of D3 branes on both sides of any
D5 brane. The brane configuration after the Hanany-Witten transitions can be depicted as inFig. 2. By standard arguments [19], the configuration of Fig. 2 leads to the 3d gauge theorye p−1 D5sw

e 1 D5swe
v1 D3s

···

···

NS51

..
.

···

e p D5sw

NS5 p−1

···

..
.

C
NS5 p

..
.

e
vp−1 D3se
vp D3sx3

Figure 2: A brane configuration found after applying some Hanany-Witten transitions to the configuration in Fig. 1, assuming the constraints (2.8). The two braneconfigurations lead to the same IR description of the D3 brane world-volume theory
in terms of the same 3d N = 4 gauge theory.
defined by the quiver in Fig. 3. This 3d theory lives on R3012 with B-type Ω-deformation turned
on with respect to the rotation of the R212 plane. In other words, this theory can be viewed
as an Ω-deformed 2d B-model on R212 with matter fields valued in the infinite dimensional
hyper-Kähler target space Map (R0 , X ) where
X :=

p−1
M

T ∗ Hom(Cevi , Cwe i ) ⊕p−2
MT ∗ Hom(Cevi , Cevi+1 ) ,

(2.9)

i=1

i=1

For comparison, the Higgs branches of these theories would be bow varieties given by what [7] calls cobalanced
dimension vectors.
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SciPost Phys. 16, 052 (2024)

e
v1

e
v2

···

e
vp−1

e1
w

e2
w

···

e p−1
w

Pn
Pp
Figure 3: Quiver for the 3d N = 4 theory Tϱ∨ [U(N )]LK with N = i=1 Ki = j=1 L j .
The ranks of the gauge and flavor groups are defined form K and L via (2.8). The
complex FI parameter associated to the abelian factor of the jth gauge node is ħ
hϱ Cj
C
).
and ϱ = (ϱ1C , · · · , ϱ p−1

and the gauge group Map (R0 , G) where
G :=

p−1
Y

U(e
vi ) .

(2.10)

i=1

The Ω-deformation localizes the 2d theory to a 1d TQM supported on R0 with the target
being the Higgs branch of the 3d theory, as a complex symplectic manifold. Note that the
Higgs and Coulomb branches of 3d N = 4 theories are naturally hyper-Kähler spaces, but
the Ω-background breaks the SU(2) symmetry rotating the complex and symplectic structures
down to a U(1) fixing some particular structures. The Higgs branch as a complex
Qmanifoldp−1
is simply the symplectic reduction of X by the complexified gauge group GC =vi ,i=1 GLesubject to stability conditions:

MH (Tϱ∨ [U(N )]LK ) = X

ϱ GC .

(2.11)

This TQM couples to the 4d CS theory and creates the line operator Lϱ (K , L). The Higgs
Q p−1
e i ) which complexifies after the
branch flavor symmetry in the 3d theory is F := i=1 SU(wQ p−1Ω-deformation to the flavor symmetry FC = i=1 SLwe i of the TQM. This TQM defines the line
operator in the 4d CS theory whose phase space is the Higgs branch (2.11).

3

Boundary Conditions and Topological Twists in 4d N = 4 SYM

For more general linking numbers we will not have nice quiver descriptions of the 3d theories
associated to line operators as we did in the special cases of §2.2. Instead, we will analyze
the D3 brane world-volume theory as a 4d N = 4 theory on a finite interval with boundaries
and domain walls. Moreover, we need to look at the Ω-deformation of this theory. In thissection we study the supersymmetry preserved by boundaries and domain walls to identify
the topological twist of the 4d theory that will be relevant in the context of Ω-deformation.
Type IIB string theory in flat R1,9 background has 32 real supercharges parameterized by
two chiral spinors of the same chirality. We can take them to be in the 16 representation ofSO(1, 9). A stack of D3 branes breaks half of the supersymmetry by relating the two chiralspinors. Let us denote the remaining independent spinor by ϵ. The chirality constraint on ϵ is:Γ0···9 ϵ = ϵ ,(3.1)

where Γ I for I ∈ {0, · · · , 9} are the SO(1, 9) gamma matrices and Γ I1 ···I k refers to the comP1
sgn(σ)
pletely anti-symmetrized product Γ I1 ···I k := k!
Γ Iσ(1) · · · Γ Iσ (k) where the sum isσ∈Sk (−1)
9SciPost Phys. 16, 052 (2024)

over the symmetric group of order k! and sgn(σ) is the signature of the permutation σ. The
world-volume theory of the D3 branes is the 4d SYM with N = 4 supersymmetry which preserves 16 supercharges. The D3-D5, as well as the D3-NS5, boundary preserves half of thesesupercharges. In fact, when placed as in Table 1, both boundaries preserve exactly the same
8 supercharges [24]. We briefly recap the boundary conditions on the fields of the 4d N = 4
SYM at the two boundaries, following [24].

3.1

1/2-BPS Boundary Conditions in 4d N = 4 SYM

Bosonic fields of the 4d N = 4 SYM theory with a compact gauge group G consist of a connection A0 dx 0 + · · · + A3 dx 3 and six adjoint scalars φ4 , · · · , φ9 which correspond to fluctuations of the D3 branes in the transverse R64···9 space. The bosonic symmetry of the theory isSO(1, 3) × SO(6) where SO(1, 3) is the isometry of the space-time and SO(6) rotates the six
scalars. A 1/2-BPS boundary containing the temporal direction breaks this symmetry down to
I := SO(1, 2) × SO(3) × SO(3) ,(3.2)

where SO(1, 2) is the isometry of the 3d boundary and the two factors of SO(3) each rotates
three of the six scalars.6 Looking at the brane configuration of Table 1 it is apparent that
1,2
the D3-D5-NS5 configuration preserves the isometry of R012 , an SO(3) R-symmetry rotating
R3456 , and another SO(3) R-symmetry rotating R3789 . According to this symmetry breaking, we
rename the six scalars as:X⃗ = (X 1 , X 2 , X 3 ) := (φ7 , φ8 , φ9 ) ,Y⃗ = (Y1 , Y2 , Y3 ) := (φ4 , φ5 , φ6 ) ,(3.3)

and label the respective R-symmetries by SO(3)X and SO(3)Y . Under the symmetrySO(1, 3) × SO(6) of the 4d theory without boundaries, the spinors parameterizing the supersymmetry transform as:Sℓ ⊕ S r := (2, 1, 4) ⊕ (1, 2, 4) .

(3.4)

Here (2, 1) and (1, 2) refer to the left and the right handed representations of SO(1, 3)
whereas 4 and 4 refer to the two four dimensional spinor representations of SO(6). Under theSO(1, 2) ⊆ I isometry of the boundary both left and right handed spinors of SO(1, 3) transform
as 2. Under the remaining R-symmetry SO(3) × SO(3) ⊆ I, both 4 and 4 transform as (2, 2).
Therefore, under the symmetry I preserved by the boundaries the two representations Sℓ andS r become isomorphic:Sℓ = S r = V8 := (2, 2, 2) .

(3.5)

As a representation of I we thus have Sℓ ⊕ S r = V8 ⊗ V2 where V2 = R2 . A key result from [24]is that for any choice of ϵ0 ∈ V2 , there is a 1/2-BPS boundary condition of 4d N = 4 SYM
preserving all the supercharges parameterized by spinors of the form v ⊗ ϵ0 ∈ V8 ⊗ V2 .
There is an SL(2, R) group acting on V2 generated by the even elements of the 10d Clifford
algebra that commute with I, namely:
B0 = Γ456789 ,B1 = Γ3456 ,B2 = Γ3789 .

(3.6)

They satisfy −B02 = B12 = B22 = 1, B0 B1 = −B1 B0 = B2 , etc. On V2 these can be represented as:



0 1
B0 =
,
−1 0
6



0 1
B1 =
,
1 0

The fermions transform under spin representations of I.

10



1 0
B2 =
.
0 −1

(3.7)

SciPost Phys. 16, 052 (2024)

In order for a symmetry to be preserved by a boundary, the normal component of theassociated current must vanish at that boundary. We shall impose this constraint on the supersymmetry generated by a spinor u ⊗ ϵ0 . The 4d N = 4 vector multiplet contains fermions inthe same spinor representation as the spinors parameterizing the supersymmetry. Therefore,
at the boundary the fermion is valued in V8 ⊗ V2 , just like the supersymmetry generators. Assuming that at the boundary the fermion takes the form v ⊗ ϑ, [24] spells out the boundary
conditions for the vector multiplet fields needed to preserve the supersymmetry generated byspinors of the form u ⊗ ϵ0 :7
ϵ 0 (Fµν − εµνλ F 3λ B0 )ϑ = 0 ,(3.8a)Dµ X a (ϵ 0 B1 ϑ) = 0 ,(3.8b)

Dµ Ya (ϵ 0 B2 ϑ) = 0 ,(3.8c)[X a , Yb ](ϵ 0 B0 ϑ) = 0 ,(3.8d)ϵ 0 ([X b , X c ] − εabc D3 X a B2 )ϑ = 0 ,(3.8e)ϵ 0 ([Ya , Yb ] − εabc D3 Yc B1 )ϑ = 0 .

(3.8f)


Here µ, ν, · · · ∈ {0, 1, 2} and a, b, · · · ∈ {1, 2, 3}. ϵ 0 is the row vector t −s for any column
 s. F is the curvature of A. Let us now focus separately on the D5 and the NS5
vector ϵ0 =tboundaries.
3.1.1

D5 Boundary

At the D3-D5 boundary the Y⃗ fields will satisfy Dirichlet boundary conditions since the D5
branes have fixed locations in the R3456 space. For now, let us set the constant value of the Y⃗
fields to zero:Y⃗ |D5 = 0 .
(3.9)
The constant value depends on the locations of the D5 branes in the R3456 directions. Here weare not being specific about the number of D5 branes and we assume all the D5 branes to belocated at the origin. A few comments about these locations are made in Remark 4.1.
(3.9) satisfies (3.8c) and (3.8d). The remaining equations from (3.8) are satisfied if we
impose the following constraints on Aµ and X at the D5 boundary:ελµν F 3λ + γFµν = 0 ,u
D3 X a + εabc [X b , X c ] = 0 ,2(3.10a)(3.10b)for some numbers γ and u along with the following constraints on the vectors ϵ0 and ϑ:
ϵ 0 (1 + γB0 )ϑ = 0 ,(3.11a)ϵ 0 B1 ϑ = 0 ,(3.11b)ϵ 0 (1 + uB2 )ϑ = 0 .

(3.11c)By an overall scaling, which leaves (3.11) invariant, we can choose ϵ0 to be of the form:
 
a
ϵ0 =
,
a ∈ R ∪ {∞} .
(3.12)
1
Note that compared to the equations presented in §2.1 of [24] our equations have B1 and B2 swapped. The
reason is that, we use the same definitions for B1 and B2 as in [24] and just like [24] we use X⃗ and Y⃗ to refer
to coordinates parallel and normal to D5 branes, but our D5 branes wrap R3789 whereas the D5 branes in [24]wrap R3456 .
711

SciPost Phys. 16, 052 (2024)

The equations (3.11) now completely fix γ, u, and ϑ (up to scaling) in terms of a: 
a2 − 1a2 − 1
1
.
,
u= 2
,
ϑ=
γ=
a
2a
a +1

(3.13)

Thus, for any value of a there is a set of D5-type 1/2-BPS boundary conditions on thebosonic fields given by (3.9) and (3.10) where γ and u are determined by (3.13). The standard
Dirichlet boundary condition on the gauge field sets the components of the curvature parallelto the boundary to zero at the boundary. We achieve this by choosing:a = ∞.

(3.14)In this limit γ ! ∞ and u ! 1, turning (3.10) into the ordinary Dirichlet condition for thegauge field and Nahm’s equation for X⃗ :



3.1.2

Fµν |D5 = 0 ,
1
D3 X a + εabc [X b , X c ]
= 0.
2
D5

(3.15a)(3.15b)NS5 Boundary

Similar to (3.9), at the NS5 boundary the X⃗ fields must satisfy Dirichlet boundary condition:
X⃗ |NS5 = 0 .

(3.16)

The constant value 0 corresponds to locations of NS5 branes in R3789 directions. For now weassume the NS5 branes to be at the origin, but later we shall separate them in the R3789 direction
shifting the above boundary condition. (3.16) satisfies (3.8b) and (3.8d). The remaining
equations are satisfied if we impose at the NS5 boundary:ελµν F 3λ + γ′ Fµν = 0 ,D3 Ya +

(3.17a)′uεabc [Yb , Yc ] = 0 ,2(3.17b)for some numbers γ′ and u′ along with:ϵ 0 (1 + γ′ B0 )ϑ′ = 0 ,(3.18a)′ϵ 0 B2 ϑ = 0 ,(3.18b)′′(3.18c)ϵ 0 (1 + u B1 )ϑ = 0 .

By assuming the same form for ϵ0 as in (3.12) we find the following expressions for γ′ , u′ , andϑ′ from (3.18):
 
2a
2a
−a
′
′
′
γ =
,
u =
,
ϑ =
.
(3.19)
1
1 − a2
1 + a2In the D5 case, the choice a = ∞ leads to the standard Dirichlet equations (3.15) for thebosonic fields. This special value of a leads to γ′ , u′ ! 0 and thus the following boundary
conditions for the gauge field and Y⃗ at the NS5 boundary preserve the same supersymmetryas (3.15):
F3λ |NS5 = 0 ,(3.20a)D3 Ya |NS5 = 0 .

(3.20b)We recognize them as the standard Neumann boundary conditions for the gauge field and thescalar fields.
12

SciPost Phys. 16, 052 (2024)

3.2

Topological Twists Preserved by 1/2-BPS Boundary Conditions

In order to get 4d CS theory from the D5 branes of Table 1, we first need to perform a supersymmetric twist of the type IIB string theory background which induces the holomorphictopological twist of the 6d N = (1, 1) SYM describing the D5 brane dynamics [23, §6.1]. Whatis the effect of this twisted string theory background on the 4d N = 4 theory describing theD3 branes? In [26], Kapustin and Witten defined a family of topological twists of 4d N = 4
SYM parameterized by a complex number t ∈ CP1 . The aforementioned twisted string theory
induces precisely the Kapustin-Witten (KW) twist corresponding to t = i [40, §6.2]. However,
the analysis of [40] only looked at the bulk 4d theory and not at the boundary conditions.
So, for consistency, we must check that this particular twist is preserved by the boundary conditions (3.9), (3.15), (3.16), and (3.20) given by the choice a = ∞. The question of KW
twists preserved by 1/2-BPS boundary conditions was addressed in [41]. We recall the setup
to establish notation.
The topological twist is most naturally performed in Euclidean signature. We therefore
Wick rotate R1,9 to R10 . Essentially the only important change from our earlier discussion isthe chirality constraint on an SO(10) spinor, which changes from (3.1) to:Γ0···9 ϵ = −iϵ ,(3.21)

where Γ I now refers to generators of the Clifford algebra associated to SO(10).
We start by viewing R80···7 as the cotangent bundle of R40123 . The choice of the subspace
R44567 of R44···9 as the cotangent directions explicitly breaks the SO(6) R-symmetry of 4d N = 4
supersymmetry down to SO(4) × SO(2) where SO(4) and SO(2) acts on R44567 and R289 respectively. Let us denote the rotation groups of R40123 and R44567 by SO(4) and SO(4)R respectively.
∼
We then pick an isomorphism c : SO(4) −! SO(4)R and declare the image of the diagonalembedding (1 × c) : SO(4) ! SO(4) × SO(4)R to be the new space-time isometry. We denote
this new isometry by SO(4)′ . Up to an inconsequential relabeling of coordinates, we can concretely write down the action of c on the generators of SO(4) in terms of the following evenelements of the 10d Clifford algebra:∼c∗ : so(4) −! so(4)R ,c∗ : Γµν 7! Γµ+4,ν+4 ,µ, ν ∈ {0, 1, 2, 3} .

(3.22)

This redefinition of space-time isometry reduces the bosonic symmetry from SO(4) × SO(6) toJ := SO(4)′ × SO(2) .

(3.23)The spin representation 4 of SO(6)R transforms as (2, 1)1 ⊕ (1, 2)−1 under SO(4)R × SO(2).8
Similarly, 4 transforms as the complex conjugate (2, 1)−1 ⊕ (1, 2)1 . Therefore, taking into
account the identification of SO(4) isometry and SO(4)R , the spin representations Sℓ and S rfrom (3.4) transform under J as:
Sℓ = (1, 1)−1 ⊕ (3, 1)−1 ⊕ (2, 2)1 ,S r = (1, 1)−1 ⊕ (1, 3)−1 ⊕ (2, 2)1 .

(3.24)We see that there are exactly two supercharges that transform as scalars under the twisted
isometry SO(4)′ . One of these two supercharges was left handed in the untwisted theory andthe other right handed. Let us denote the spinors parameterizing these supercharges by ϵℓ andϵ r respectively. Define an arbitrary complex linear combination of ϵℓ and ϵ r :ϵe := uϵℓ + vϵ r .

(3.25)The scalar supercharge parameterized by ϵe squares to zero modulo gauge transformation.
Twisting now simply means passing to the cohomology of this scalar nilpotent supercharge.
8

The superscript refers to the SO(2) charge.

13

SciPost Phys. 16, 052 (2024)

The choice of ϵe is defined up to an irrelevant scaling. Therefore, the true parameter for the
choice of a scalar supercharge is the ratiot :=v∈ CP1 .
u(3.26)

Here we allow the value ∞ for the possibility that ϵe = ϵ r . This t is the complex parameter for
the family of KW twists.
The question we want answered is: does the choice a = ∞ (from (3.12)) preserve the KW
twist t = i? Witten showed in [41] that t and a are nicely related:9
a=i

t +i
.
t −i

(3.28)

In particular, we find that the choice a = ∞, which gives us the standard Dirichlet and Neumanboundary conditions, preserves precisely the topological twist labeled by t = i:a=∞⇔t = i.

(3.29)

This twist is of the B-type [26], i.e., we can view the twisted 4d theory as a 2d B-model on R212
with fields valued in some appropriate infinite dimensional space. This of course is the twist
which, once Ω-deformation is turned on, will reduce to 2d BF theory on R203 – as we show
below.

Ω-deformed Kapustin-Witten (KW) Theory and BF Phase Spaces4

Now that we know precisely which twist to use, we can look at the specific twisted theory andperform Ω-deformation [42–44]. We shall find that the KW theory in a B-type Ω-background
localizes to 2d BF theory. While this is certainly known in literature [40, 45, 46], some details
have yet to appear, so for completion we go through the intermediate steps in this section. First
we look at the bulk theory, disregarding the boundaries, and then we look at the boundary
conditions in terms of the fields of the twisted theory.

4.1

KW Theory as a 2d B-model

In the untwisted 4d N = 4 SYM, the six adjoint scalars φ4 , · · · , φ9 transform as the vector ofSO(6). Under the twisted space-time symmetry SO(4)′ the four scalars φ4 , · · · , φ7 transform
as a 1-form:
φ := φ4 dx 0 + φ5 dx 1 + φ6 dx 2 + φ7 dx 3 .
(4.1)
This 1-form can be combined with the connection to form a complexified connection:

Aµ := Aµ + iφµ+4 ,Aµ := Aµ − iφµ+4µ ∈ {0, 1, 2, 3} .

(4.2)

9

This relation is a consequence of comparing the constraints on the supersymmetries preserved by the boundary
conditions and the twisting procedure. The constraints that ϵℓ and ϵ r are scalars under SO(4)′ can be written as(1 + c∗ )(Γµν )(ϵ• ) = (Γµν + Γµ+4,ν+4 )ϵ• = 0 for • ∈ {ℓ, r}. Combining these with the 10d chirality constraint (3.21)
and the 4d chirality constraints Γ0123 ϵℓ = −ϵℓ , Γ0123 ϵ r = ϵ r we can get:

1 − t22t1+iB+B(ϵℓ + tϵ r ) = 0 .
(3.27)
0
1
1 + t2
1 + t2
It can be shown that ϵ0 , as defined in (3.12), satisfies the same equation iff a and t are related by (3.28). The
overall sign of a is different in (3.28) relative to [41] because our definition of a in (3.13) differs by a sign.

14

SciPost Phys. 16, 052 (2024)

The remaining two scalars φ8 and φ9 transform under the vector representation of SO(2), we
repackage them into a complex scalar and its conjugate with definite SO(2) charges:
σ := φ8 − iφ9 ,σ := φ8 + iφ9 .

(4.3)

Let us denote by Q ℓ and Q r the supercharges parameterized by ϵℓ and ϵ r . The superchargeparameterized by uϵℓ + vϵ r is then:Q(t) = uQ ℓ + vQ r .

(4.4)

As mentioned earlier, the supersymmetry generated by Q(t) depends only on the ratio t = v/u.
We denote the supersymmetry variation of a field Φ generated by Q(i) as:δ T Φ := [Q(i), Φ} .

(4.5)

The bracket is anti-symmetric/symmetric if Φ is bosonic/fermionic.
The vector multiplet in the KW theory contains the following fields:
Bosonic
FermionicΩ0 (R4 , gC ) Ω1 (R4 , gC ) Ω2 (R4 , gC )P, σ, σ
A, A
η, η
ψ, ψ
χ

(4.6)

where R4 = R40123 is the space-time of the 4d theory. With the addition of the auxiliary field P
and for t = i the supersymmetry transformation δ T is nilpotent off-shell:
δ2T = 0 .

(4.7)

This auxiliary field can be integrated out by adding an irrelevant δ T -exact term to the action(see §3.4 of [26]). The fermionic fields all come from the fermions of the 4d N = 4 vector
multiplet. The untwisted vector multiplet contains fermions in the representation Sℓ ⊕S r whichis the same as in (3.4). In the twisted theory this representation breaks down as representationof the twisted space-time symmetry SO(4)′ as in (3.24). We see that there are two scalars, aself-dual 2-form, an anti self-dual 2-form, and two 1-forms. The self-dual and anti self-dual
forms combine to make up a complete 2-form. We see these fields in the above field content.
Variations of the fields under the twisting supercharge are:
δT A = 0 ,δT ψ = Dσ ,

δT A = ψ ,δT ψ = 0 ,δT σ = 0 ,δT η = P ,δT σ = η ,δT η = 0

δT P = 0 ,δT χ = F .

(4.8)

Here the covariant derivative is defined with respect to the complex connection:

D = d + A∧ ,(4.9)

and F = dA + 21 A ∧ A is the curvature of A.10 δ T clearly satisfies (4.7).
10

We need to define the wedge product involving Lie algebra valued forms as our convention differs frommany standard literature. Let g be any Lie algebra and M a g-module with the Lie algebra homomorphismρ : g ! End(M ). Let a = a I1 ···I p dx I1 ∧ · · · ∧ dx I p be a g-valued p-form and v = vJ1 ···Jq dx J1 ∧ · · · ∧ dx Jq an M valued q-form. Then we define the wedge product between them as:a ∧ v := ρ(a I1 ···I p )(vJ1 ···Jq )dx I1 ∧ · · · ∧ dx I p ∧ dx J1 ∧ · · · ∧ dx Jq .
µ(4.10)

In particular, for a Lie algebra valued connection Aµ dx – thinking of g as the adjoint module of g – we get fromthe above definition A ∧ A = [Aµ , Aν ]dx µ ∧ dx ν . This differs from many literature where A ∧ A only involves theassociative product in the universal enveloping algebra of the Lie algebra which then needs to be anti-symmetrized
to get the Lie bracket. This is the reason for the factor of half in our expression for the curvature which may seem
unfamiliar to some.

15

SciPost Phys. 16, 052 (2024)

We shall write down the bosonic part of the action of the KW theory (in (4.44)) after we
introduce a 2d formulation of the theory.
4.1.1

2d Multiplets and Supersymmetry

The 4d KW theory on R40123 can be viewed as a 2d B-model on R212 with an infinite dimensional
gauge group and fields valued in certain infinite dimensional target spaces. More specifically,
the 2d vector multiplet contains the connections

A(2) := Ai dx i ,A(2):= Ai dx i .

(4.11)In this section the indices i, j, · · · run over 1 and 2. Viewed as 1-forms on R212 , they are valued
in the infinite dimensional Lie algebra:

GC := Ω0 (R203 , gC ) .

(4.12)The Lie bracket of GC consists of point-wise multiplication on R203 and the ordinary Lie bracket
on gC . The full 2d B-model vector multiplet contains the following fields:Ω0 (R212 , GC )Bosonic
FermionicΩ1 (R212 , GC )

Ω2 (R212 , GC )(2)

A(2) , A(2)ψ := ψi dx i χ (2) := 21 χi j dx i ∧ dx jP
η

(4.13)

B-model supersymmetry transformations of these fields [47] coincide with the KW supersymmetry (4.8) restricted to these fields:(2)= ψ(2) ,δ T A(2) = 0 ,δT Aδ T ψ(2) = 0 ,δT η = P ,(4.14)

δ T χ (2) = F (2) .

δT P = 0 ,Here F (2) = d(2) A(2) + 21 A(2) ∧ A(2) is the curvature of A(2) .
The remaining two components of the 4d connections, namely:mA(c) := A(c)m dx ,

A

(c)

(c)

:= Am dx m ,(4.15)(c)belong to 2d chiral multiplets. Here m, n, · · · run over 0 and 3. The fields A(c) and A are
scalars on R212 – the world-volume of the B-model – and they are gC valued 1-forms on R203 .
So the chiral multiplet is valued in the infinite dimensional space:

X := Ω1 (R203 , gC ) .

(4.16)

The full content of this chiral multiplet is as follows:Ω0 (R212 , X)
Bosonic
FermionicΩ1 (R212 , X)

(c)A(c) , Aψ(c) := ψm dx m χ (c) := χim dx i ∧ dx m

Ω2 (R212 , X)

F, FM

(4.17)We have introduced the auxiliary fields F, F, M ∈ Ω2 (R212 , X) ⊂ Ω3 (R40123 , gC ). To compare the
B-model with the KW theory we will need to integrate out the bosonic auxiliary fields F and
F. After integrating out these two fields we will find that

M12 = −ψm dx m .
16(4.18)

SciPost Phys. 16, 052 (2024)

B-model supersymmetry transformations of these chiral multiplet fields are:(c)

δ T A(c) = 0 ,δT Aδ T χ (c) = Fim dx i ∧ dx m ,δ T ψ(c) = 0 ,δ T F = D(2) χ (c) + D(c) χ (2)= ψ(c) ,(4.19)

δT M = F ,δT F = 0 .

The differentials are defined as:D(2) := dx i ∂ x i + A(2) ∧ ,D(c) := dx m ∂ x m + A(c) ∧ .

(4.20)

These transformations coincide with the restriction of the KW supersymmetry to the above
fields, excluding the newly introduced auxiliary fields.
The remaining bosonic fields of the KW theory, namely σ and σ fit into another GC valued
chiral multiplet with the following field content:

BosonicΩ0 (R212 , GC )σ, σ
ηFermionicΩ1 (R212 , GC )ψ(c):= ψm dx

Ω2 (R212 , GC )G, Gm

(4.21)

N

Similar to the previous chiral multiplet, we have introduced auxiliary fields G, G and N. We
integrate out G and G, after which we are lead to the identification

N12 = χ03 dx 0 ∧ dx 3 .

(4.22)The supersymmetry transformations of these fields are:
δT σ = 0 ,
δT ψ

(c)

δT σ = η ,= D(2) σ ,δ T G = D(2) ψ(c)δT η = 0 ,+ σ ∧ χ (2) ,(4.23)

δT N = G ,δT G = 0 .

The transformations (4.14), (4.19), and (4.23) satisfy the topological supersymmetry algebra:
δ2T = 0 ,(4.24)

the same as (4.7).
Taking into account the on-shell identifications (4.18) and (4.22), which we will justifyin (4.39) and (4.43) respectively, all the fields of the 4d KW theory have been encoded into
various 2d B-model multiplets. Let us now look at the actions for these multiplets.
4.1.2

Actions

We define the following inner product on the Lie algebra GC : for a, b ∈ GC = Ω0 (R203 , gC )Z
Z

tr(a b) =dx 0 ∧ dx 3 tr(a b) ,⋆03 tr(a b) =R203

(4.25)

R203

here ⋆03 is the Hodge star operator on R203 . The 2d vector multiplet action can be written as:Zδ T tr (− ⋆12 P + 2iD(2) ⋆12 φ (2) )η − χ (2) ⋆12 F

SV =

R212

17

(2)

SciPost Phys. 16, 052 (2024)

=

Z



ij
1
i
+ ···
d x tr (−P + 2iD φi )P − Fi j F
2
4

R40123(4.26)The ellipses hide terms involving fermions. We shall often ignore such terms for the sake ofsimplicity. Note that the above action, when viewed as a 4d action, contains kinetic terms for
only A1 and A2 . It is missing the kinetic terms for A0 and A3 . These components are part of a
2d chiral multiplet (4.17) and therefore their kinetic term will come from 2d chiral multiplet
actions, as we shall now see.
In addition to the inner product (4.25) on GC , we also need an inner product on X (4.16) towrite down an action for the chiral multiplets. For a, b ∈ Ω p (R203 , gC ) define the inner product:Z
d2 x tr(a ∧ ⋆03 b) .

〈a, b〉 :=(4.27)

R203

We can then write down the action for the first chiral multiplet (4.17) as follows:
Z

¬
¶
SC1 =
δ T −F im dx i ∧ dx m , ⋆12 χim dx i ∧ dx m + 2iD(c) ⋆03 φ (c) , ⋆12 η − M, ⋆12 F
R2

=

Z 12
R40123
imd4 x tr −F Fim + 2iDm φ m P − Fm12 Fm12 + · · ·and that of the second chiral multiplet (4.21) as:Z DE
S C2 ==

δT − D(2)

σ, ⋆12 ψ(c)(4.28)¬¶+ 〈[σ, σ], ⋆12 η〉 − N, ⋆12 G

R2

Z 12
R40123
id4 x tr −D σDi σ + [σ, σ]P − G12 G12 + · · ·(4.29)As before, we have omitted terms containing fermions in the final expressions.
To complete the 2d theory we also need a superpotential. We shall find that we recover
the KW theory precisely with the following superpotential:
Z
 

1 (c)
(c)
(c)
(c) (c)(c)
W (σ, A ) := σ, ⋆03 F=
tr σ d A + A ∧ A.
(4.30)
2
R2
03This leads to the superpotential action:
Z ­
 ­

· 
· 
δW
δW
δW
δW
SW =
F,
− F,
+ G,
− G,
+ ···
(c)
δσ
δσ
δA(c)
R212
δA
Z
m

=

R4012303

d4 x tr(−Fm12 Dm σ + F 12 D m σ + G12 F 03 − G12 F ) + · · ·(4.31)The total B-model action is the sum of (4.26), (4.28), (4.29), and (4.31):
SB2d = SV + SC1 + SC2 + SW .

(4.32)The matching of the 2d B-model action and the 4d KW action will occur on shell. We needto integrate out the bosonic auxiliary fields. Terms in (4.32) containing P are:
Z
 


1
4
2
i
m
d x tr −P + 2 iD φi + iDm φ + [σ, σ] P .
(4.33)
2
R4
0123

18

SciPost Phys. 16, 052 (2024)

Therefore, P can be integrated out by the following equation of motion:
1
P = iDµ φµ + [σ, σ] .
2

(4.34)The effect of integrating P out is to replace the terms (4.33) in the action involving P with thefollowing term:
Z

2
1
4
µ
d x tr iD φµ + [σ, σ] .
(4.35)
2
R4
0123

Terms in the action involving F, F are:Zmd4 x tr −Fm12 Fm12 − Fm12 Dm σ + F 12 D m σ .

(4.36)

R40123

Their equations of motion are:Fm12 = −Dm σ ,Fm12 = D m σ .

By integrating out F, F we replace the action (4.36) by:
Z
 m

−
d4 x tr D σDm σ .

(4.37)(4.38)

R40123

Also note that, after integrating out these fields the supersymmetry variation of M (4.19)
becomes:δ T M12 = F12 = −Dm σ .
(4.39)
Comparing this with the supersymmetry transformation of ψ in the KW theory (4.8) leads tothe identification (4.18).
Terms in the action involving G, G are:
Z

03 
.
(4.40)
d4 x tr −G12 G12 + G12 F 03 − G12 F
R40123

Their equations of motion:

G12 = −FG12 = F03 ,03.

(4.41)

By integrating out G, G we replace the action (4.40) by:Z−
R40123d4 x F03 F

03.

(4.42)

Integrating out G makes the on shell supersymmetry of N (4.23):
δ T N12 = G12 = F03 .

(4.43)

Comparing this with the supersymmetry of the KW theory (4.8) we find the identification(4.22).
Thus, after integrating out all the auxiliary fields, the B-model action becomes:

Z


2
µν
µ
1
1
µ
2d on shell4
SB −−−!
d x tr − Fµν F − D σDµ σ + iD φµ + [σ, σ] + · · · . (4.44)
2
2
R4
0123

This is the fermion free part of the 4d KW action [26], or equivalently, the dimensional reduction of the holomorphic-topological twist of 6d N = (1, 1) SYM – reduced along the holomorphic directions [23].
19SciPost Phys. 16, 052 (2024)

4.1.3 Ω-deforming the B-model
A B-model on R2 with a U(1) symmetry generated by rotation of the space-time admits an Ωdeformation [48]. Let V = ħh∂θ be the generator of rotation where θ is the angular coordinate
on R2 and ħ
h is a deformation parameter. The B-model supersymmetry variation δ T can be
deformed in a way such that instead of squaring to zero, it squares to the action of V :δ T ⇝ δV

such that on fields

δ2V = LV (modulo gauge transformations) ,(4.45)

where LV is the Lie derivative with respect to the vector field V . The precise deformationsof δ T acting on the vector multiplet (4.14) and the chiral multiplets, (4.19) and (4.23), aregiven in (A.1), (A.2), and (A.3) respectively. Suppose that the B-model has chiral multiplets
valued in some complex symplectic target Υ , a compact gauge group G, and a GC -invariant
holomorphic superpotential W : Υ ! C. Then the Ω-deformed theory localizes to a matrix
model with the action ħ1h W and the complexified gauge group GC . Path integrals in this matrix
model are performed over a Lagrangian subspace of Υ – the choice of the Lagrangian depends
on the choice of boundary conditions for the B-model fields at infinity of the space-time R2 .
Applying this process to the B-model description of the KW theory with compact gauge group
G and superpotential W (4.30) from the last section we find that once Ω-deformation is turned
on with respect to the rotation of the R212 plane, the theory localizes to a 2d BF theory on R203
with gauge group GC and the following action [28, 29]:
Z
1
1
SBF = W =dx 0 ∧ dx 3 tr σF03 .
(4.46)
ħ
h
ħ
h R2
03

Varying this action we get the following equations of motion:

F03 = 0 and D(c) σ = 0 ,(4.47)

where the covariant derivative D(c) only has differentials and connections in the directions of
R203 , as defined in (4.20).

4.2

Boundary Conditions in BF Theory

Instead of considering the BF theory on the infinite plane as in (4.46) we can put it on a strip
R × I where we consider the x 3 direction to be a finite interval I. We can choose a gauge
where A0 = 0. In this gauge the second equation of motion from (4.47) says that σ and A3
are independent of x 0 and they further satisfy:∂3 σ + [A3 , σ] = 0 ,(4.48)

which is GC -invariant. Appropriate boundary conditions must be imposed on σ and gaugetransformations at the boundary which we shall discuss in this section. For now we note thatthe moduli space of solutions to (4.48) modulo complex gauge transformations is the same,as a complex symplectic manifold, as the moduli space of solutions to Nahm’s equation for the
triple (Im A3 , Re σ, −Im σ) = (φ7 , φ8 , φ9 ) modulo real gauge transformations [49].
Fields of the BF theory are the complex connection A0 dx 0 +A3 dx 3 and the complex adjoint
scalar σ. These fields are written in terms of the fields of 4d N = 4 SYM in (4.2) and (4.3).
In the presence of a boundary, it is better to use the notation for the scalar fields suited to thesymmetry of the boundary (3.3). We have:

A0 = A0 + iY1 ,A3 = A3 + iX 1 ,20

σ = X 2 − iX 3 .

(4.49)

SciPost Phys. 16, 052 (2024)

We can now translate the 1/2-BPS boundary conditions of the 4d theory into boundary conditions of the BF theory. We can readily use the analysis of the moduli space of solutions toNahm’s equation as a complex space from [24]. In this reference the choice of complex structure was arbitrary and breaks the hyper-Kähler symmetry of the moduli space. This extended
symmetry is broken in our case in the process of twisting when we broke the SO(6) R-symmetryof 4d N = 4 SYM down to SO(4) × SO(2). This SO(2) acts on the moduli space of solutions
to Nahm’s equation and making the action holomorphic is equivalent to choosing a complex
structure. The choice is made by declaring σ to be a holomorphic coordinate.
We create boundaries for the BF theory by starting with 4d N = 4 SYM on an interval andthen turning on Ω-deformation. In terms of branes, We start with a configuration where D3
branes are suspended between five-branes. We consider two types of boundaries, one createdby D5 branes, corresponding to the boundary conditions of §3.1.1 and the other created by
NS5 branes corresponding to the boundary conditions of §3.1.2. Similar setups also appearedin [50] where the author shows using supersymmetric localization that interfaces in 4d N = 4
SYM created by five-branes can be reduced to a 2d/1d setup involving interfaces in 2d YangMills (YM) theory. This setup was further used in [51] to compute 1-point correlation functionsin 2d YM in the presence of a D5-type interface. 2d BF theory is a zero coupling limit of 2d
YM and these results should be relevant for defining traces of algebras coupled to the line
operators of 4d CS that these defects create in our setup.
4.2.1

D5 Boundary

The most general D5 type boundary we consider comes from the brane configuration in Fig. 4.
There are n ordered D5 branes such that the ith D5 brane has linking number Ki . Since there
n
P

···

..
.

..
.

Ki D3s

i=2

C

···

Kn D3s

..
.

..
.

N D3s

x3

(Kn−1 + Kn )D3s
D51

D5n−1

D5n

Figure 4: A generic D5 type boundary determined by K = (K1 , · · · , Kn ) and Ki is thelinking
Pnnumber of the ith D5 brane. We denote the total number of D3 branes by
N = i=1 Ki .
are no NS5 branes to the right of any D5 brane, Ki coincides with the number of D3 branes
ending on the ith D5 brane from the left according to the definition of linking numbers (2.2).
In some other brane configuration related by Hanany-Witten transitions the number of D3
branes ending on the five-branes will generally be different but linking numbers are invariant
under such transformations. The brane configuration of Fig. 4 can be tabulated as in Table 2.
We have specified the positions of the branes in some of the directions. The ith D5 brane islocated at x 3 = t i , x 4 = mi , x 5 = x 6 = 0. We need these branes to be located at the centerof the R256 plane in order to preserve the U(1) symmetry that acts by rotating the R212 and theR256 planes and which we use to turn on the Ω-deformation. Our brane configuration must
be invariant under this rotation. All D3 branes attached to the ith D5 brane must be locatedat (mi , 0, 0) in R3456 . In order to have a gauge theory on the D5 branes with the gauge group
U(n) we must ultimately take the limit where for all i we have t i ! t and mi ! m for somereal numbers t and m. Having a nonzero m would modify the D5 boundary condition on Y⃗
21

SciPost Phys. 16, 052 (2024)

Table 2: Directions wrapped by the D3 and D5 branes from Fig. 4. We denote the D3
branes attached to the ith D5 brane by D3αi for different values of α. We have also
noted that fluctuations of the D3 branes in R3456 and R3789 direction corresponds tothe triples of adjoint scalars Y⃗ and X⃗ respectively in the 4d N = 4 U(N ) SYM theory
on the D3 branes. Here N is the total number of D3 branes.

0

Scalar Fields
D5i
D3αi

R2+ħh
1 2

3

4
Y1
× × × t i mi× × × × mi

R2−ħh
5 6
Y2 Y3
0 0
0 0

C
7
X1
×

8
X2
×

9
X3
×

(3.9) by putting the triple (m, 0, 0) ⊗ 1N ∈ R3 ⊗ Z(glN ) of central elements on the right handside instead of zero. We can set m = t = 0 by choosing the origin of our coordinate systemappropriately. We have left the locations of the D3 branes in the R3789 directions unspecified
because they are not fixed by this brane configuration and are not part of the D5 type boundary
conditions.
Remark 4.1 (Further Comments on Twisted Masses). We will ultimately let the D3 branes inFig. 4 end on NS5 branes located to the left of all the D5 branes. Then, with finite extension
in the x 3 direction, the low energy dynamics of the D3 branes will be captured by a 3d N = 4
theory, the one we denoted as T ∨ [U(N )]LK in §2. In terms of this 3d theory, the mi ’s are real
twisted masses. As we discussed in §2, we expect to find the Higgs branch of this 3d theory
as the phase space for the line operators under consideration. Nonzero twisted masses would
lift parts of the Higgs branch. Note furthermore that in 3d N = 4 mass parameters usually
come in real triplets, in terms of our brane construction such a triplet of twisted masses would
correspond to the locations of the D3 branes in the R3456 directions. Since we want to preserve
the rotational symmetry of the Ω-background, we could only turn on one component of thesemasses.
The world-volume theory of the D3 branes is 4d N = 4 SYM with U(N ) gauge symmetry.
After twisting and turning on Ω-deformation this becomes 2d BF theory with gauge group GLN
where
n
X
N=
Ki .
(4.50)
i=1

The 1/2-BPS boundary conditions for the 4d SYM fields are the D5 type conditions mentionedin (3.9), (3.15). The real scalars X⃗ satisfy Nahm’s equation (3.15b). We can combine twoof the three equations to write down the following equation for the complexified BF fields σ
and A3 :D3 σ := ∂3 σ + [A3 , σ] = 0 .
(4.51)
And instead of imposing the remaining real equation and then considering solutions to theequations of motion with boundary conditions modulo real gauge transformations, in the BF
theory we only impose the complex equation and quotient by complex gauge transformations.
In the special cases when the integers Ki satisfy the following inequality:
Ki ≥ Ki+1 ≥ 0

for all1 ≤ i < n,(4.52)

the boundary of Fig. 4 corresponds to a simple polar boundary condition on the real adjoint
scalars X i of the 4d theory. Assuming that the boundary is located at x 3 = 0, the fields X i

22

SciPost Phys. 16, 052 (2024)

behave near the boundary as (see §3.5 of [24]):⃗t+ O(1) .
x3

X⃗ (x 3 ) =(4.53)

where ⃗t = (t 1 , t 2 , t 3 ) are images of a standard su(2) basis under a homomorphismρ K : su(2) ! u(N ) satisfying [t i , t j ] = iεi jk t k . As the notation suggests, the homomorphism
depends on the linking numbers K . The form (4.53) of the 4d fields becomes the following
boundary behavior of the BF field σ:t−+ O(1) ,(4.54)
x3
where t − := t 2 − it 3 is a nilpotent element of glN . The homomorphism ρ K is such that t − hasthe following Jordan normal form:


νK1
..

,
(4.55).
σ(x 3 ) =

ν Kn
where νK is a regular nilpotent matrix of size K × K in the normal form.


0 1
0 1




.
.
.

.
.
νK = 
.
.


0 1

(4.56)

0
For arbitrary linking numbers that do not satisfy the inequality (4.52), such a simple boundary condition does not exist. Instead, we keep the D5 branes separated along the x 3 -direction
and treat them as interfaces between theories with possibly different gauge groups. The boundary conditions on the fields of the 4d theory at these interfaces were described in [24] and we
can easily translate them into boundary conditions for the BF fields.
From Fig. 4, between the D5i−1 and the D5i branes there are
Ni :=

n
X

Kj ,

(4.57)

j=i

D3 branes which lead to a 4d theory with U(Ni ) gauge group and consequently to a 2d BF
theory with GLNi gauge group on an interval. To the left of the D51 brane there are N1 = N
D3 branes leading to the GLN BF theory. Thus the D5i brane creates an interface between twotheories such that the rank of the gauge group jumps from Ni to Ni+1 from left to right. The
last, i.e. the nth D5 brane is just a special case of this such that it interfaces between a GLNn
BF theory, and a trivial/empty theory.
To make contact with the diagrammatic notations for bow varieties from [6] and [7] wedraw the stack of D3 brane segments between two adjacent five-branes (of any kind) as a wavyNline and label the line by the number of D3 branes in the stack, such as. We mark each
D5 brane by a . So we represent the boundary from Fig. 4 diagrammatically as in Fig. 5.
To discuss the boundary conditions at an interface, let us focus on the ith D5 brane between two BF theories with gauge groups GLNi and GLNi+1 , i.e., we are focusing on the portion···

Ni Ni+1· · · of a bow diagram. We always choose a gauge with A0 = 0. The BF equa-tions of motion (4.47) then imply, in particular, that A3 and σ are independent of the x 0
23

SciPost Phys. 16, 052 (2024)

···

N1

N3

N2

···

Nn

Figure 5: The bow diagram for the D5 type boundary whose brane diagram is given
in Fig. 4. Here K = (K1 , · · · , Kn ) is an n-tuple of integers satisfying (4.57).
coordinate and therefore solving the equations of motion becomes a problem in one dimension. Thus, for a wavy line with label Ni we have a glNi connection D3 := ∂3 + A3 ∧ and acovariantly constant (according to (4.48)) adjoint scalar σ. We assume that the D5 brane interface is located at x 3 = 0. We denote the fields to the left and to the right of the interface±by − and + respectively, we thus denote the BF fields by A±
3 , σ and the 4d SYM fields by
A±3 , X 1± , X 2± , X 3± . We consider two separate cases, Ni ̸= Ni+1 and Ni = Ni+1 .
• Ni ̸= Ni+1 . For concreteness let us assume Ni > Ni+1 . The inequality Ni > Ni+1 means
Ni+1 D3 branes pass through the D5 brane and the remaining (Ni − Ni+1 ) D3 branes end
on the D5 brane from the left. We learn from [24] (see §3.4.4) that whenever D3 branes
end on D5 branes, the adjoint scalars X⃗ from the 4d theory picks up poles compatible
with the real Nahm’s equations, and the scalars continue through the interface smoothly
if the D3 branes extend past the D5 branes. More specifically, the real connection A+3
and the adjoint scalars X⃗ + cross over smoothly from x 3 > 0 to x 3 < 0. But X⃗ − picks up
some poles at x 3 = 0 – near the interface it takes the form:11 +
⃗X⃗ (0) + O(x 3 )B−3X⃗ (x ) =.
(4.58)⃗tC⃗+ O(1)
x3⃗ and C⃗ are matrices of size Ni+1 ×(Ni −Ni+1 ) and (Ni −Ni+1 )×Ni+1 respectively that
Here Bare regular at x 3 = 0. The upper left corner of size Ni+1 × Ni+1 is regular and matches
the value of X⃗ + at the interface. In the lower right corner ⃗t = (t 1 , t 2 , t 3 ) are the imagesof a standard su(2) basis under an irreducible homomorphism su(2) ! u(Ni − Ni+1 ). As
usual, in the BF theory we only keep the constraints on σ = X 2 −iX 3 . The aforementioned
boundary condition then says that σ+ is regular at the interface and σ− behaves near
the interface as: +
σ (0) + O(x 3 )B2 − iB3− 3
σ (x ) =,(4.59)
t 2 −it 3
C2 − iC3+ O(1)
x3
where t 2 − it 3 ∈ glNi −Ni+1 is a regular nilpotent element.
Boundary conditions for Ni < Ni+1 follows easily by swapping the superscript + and −on fields and changing Ni − Ni+1 to Ni+1 − Ni .
• Ni = Ni+1 . In this case we have the same gauge group on both sides of the D5 brane.
Thus the D5 brane can be interpreted as a domain wall in a single gauge theory. In the
4d picture, we have N = 4 U(Ni ) SYM and the open strings stretched between the stackof D3 branes and the D5 brane introduce a hypermultiplet, localized at the domain wall,transforming under the quaternionic representation Z := T ∗ CNi of the gauge group.
The full 4d theory on R40123 can be viewed as a 3d theory with N = 4 supersymmetry
supported on R3012 with fields valued in the infinite dimensional space of functions of x 3 .
The 4d fields X⃗ and A3 , as functions of x 3 , become coordinates on an infinite dimensional
hyper-Kähler target space. The scalars of the domain wall hypermultiplet parameterize
11

D3 branes ending on D5s appear as monopoles in the three dimensional space transverse to the D3s and parallelto the D5s (parameterized by X⃗ ). It was shown in [49, 52] that solutions to Nahm’s equations with the following
boundary conditions coincide with such monopoles.

24

SciPost Phys. 16, 052 (2024)

the finite dimensional hyper-Kähler target Z. From this point of view the gauge group isalso infinite dimensional, consisting of U(Ni ) valued functions of x 3 . This gauge group
acts on the 4d fields, as well as on the fields localized at the domain wall. Only the
restrictions of the group valued functions at the domain wall – which we take to be at
x 3 = 0 – act on the localized fields. So we get a hyper-Kähler action of U(Ni ) on Z. The
hyper-Kähler moment map of the infinite dimensional gauge group on the full target
space is the following:12µ⃗ (x 3 ) = D3 X⃗ (x 3 ) + X⃗ (x 3 ) × X⃗ (x 3 ) + δ(x 3 )(X⃗ + (0) − X⃗ − (0) + µ⃗Z ) ,(4.60)

where µ⃗ Z is the moment map of the U(Ni ) action on Z. Supersymmetric field configurations in the 4d theory are those on which the moment map vanishes. Away fromthe domain wall this simply means satisfying the standard Nahm’s equation. The delta
function supported at the domain wall implies that X⃗ is discontinuous at x 3 = 0 andjumps according to:X⃗ + (0) − X⃗ − (0) + µ⃗Z = 0 .
(4.61)Let us choose holomorphic coordinates on Z:

such that:(I, J) ∈ Hom(C, CNi ) ⊕ Hom(CNi , C) = T ∗ CNi ,(4.62)

1 †(I I − J † J), Re I J, −Im I Jµ⃗ =2(4.63)Z.

After twisting and turning on omega deformation, the domain wall hypermultiplets localize to analytically continued topological quantum mechanics with target Z coupled
to the restriction of the gauge field of the BF theory at the domain wall [28, 29]. The
kinetic term of the quantum mechanics simply takes the form J∂0 I. The coupling of
the BF theory to this quantum mechanics via the term J A0 I causes the BF field σ to bediscontinuous at the location of the defect and from (4.61) we get the following form of
the discontinuity:
σ+ (0) − σ− (0) + I J = 0 .
(4.64)
This can also be seen as a Gauss law constraint in the coupled theory after gauging away
A0 .
Gauge symmetry is broken at a D5 interface for the gauge fields that satisfy Dirichlet boundNi Ni+1
· · · with Ni ≥ Ni+1 the u(Ni+1 ) part ofary condition. For example, in case of · · ·the connection crosses over smoothly from one side to the other, but the components of theu(Ni ) connection orthogonal to the u(Ni+1 ) directions are set to zero at the domain wall by
Dirichlet boundary condition. At the location of the D5 brane, gauge symmetry is reduced
from U(Ni ) to U(Ni+1 ) going from left to right. Suppose g − (x 3 ) and g + (x 3 ) are gauge transformation matrices to the left and to the right of the D5 brane, then at x 3 = 0 they are required
to satisfy:
 +

g (0)
0
−
g (0) =.
(4.65)
0
1Ni −Ni+1
Away from the interface the A3 -covariant derivative D3 and the field σ transform by the adjoint
action as usual:(D3 , σ) 7! (g D3 g −1 , gσg −1 ) .
(4.66)
12
If there are boundaries of the 4d theory away from the domain wall in either direction, there can be contribution
to the moment map localized at those boundaries. We ignore this possibility here since we are now only interested
in the boundary conditions coming from the domain wall.

25

SciPost Phys. 16, 052 (2024)

Table 3: Directions wrapped by the D3 and NS5 branes from Fig. 6. We denote theD3 branes attached to the jth NS5 brane by D3αj for different values of α. The fields
X⃗ and Y⃗ of the 4d N = 4 theory living on the D3 branes consist of fluctuations of theD3 branes in R3456 and R3789 respectively, as in Table 2.
0
Scalar Fields
D3αj
NS5 j

R2+ħh
1 2

3

× × × ×× × × τj

4
Y1
×

R2−ħh
5 6
Y2 Y3
0 0
× ×

C
7
X1
r1, j
r1, j

8
X2
r2, j
r2, j

9
X3
r3, j
r3, j

This makes sense at the interface as well. In case Ni = Ni+1 and we have the additional fields
I and J localized at the interface, they transform under the gauge transformation as:(I, J) 7! (g(0)I, J g(0)−1 ) .
4.2.2(4.67)

NS5 Boundary

The most general NS5 type boundaries result from brane configurations such as Fig. 6. There
NS5 p−1

NS51

..
.

···

..
.

NS5 p

..
.

..
.

L1 D3s

C

···

N D3s
p−2
P
j=1

L j D3s

p−1
P

L j D3sx3

j=1

Figure 6: A generic NS5 type boundary determined by L = (L1 , · · · , L p ) where L iis the P
linking number of the ith NS5 brane. The total number of D3 branes isp
M := j=1 L j .
are p NS5 branes labeled NS51 , ..., NS5 p such that the linking number of NS5i is L i . When
there are no D5 branes to the left of any NS5 brane, as in Fig. 6, L i coincides with the numberof D3 branes ending on NS5i from the right according to (2.2). The NS5 and the D3 braneworld-volumes are as in Table 3. The jth NS5 brane is located at x 3 = τ j , x 7 = r1, j , x 8 = r2, j ,x 9 = r3, j . All the D3 branes are located at the center of the R2−ħh plane to preserve the U(1)ħh
symmetry. We get a single line operator in 4d CS if all the D3 branes are coincident, in otherwords by taking the limit ⃗r j := (r1, j , r2, j , r3, j ) ! ⃗r := (r1 , r2 , r3 ) for some j-independent point⃗r ∈ R3789 . In practice, we shall impose this limit only classically, adding O(ħh) modifications asfollows:⃗r := ⃗r p ,⃗r j ! ⃗r j+1 + ħhϱ⃗j ,j ∈ {1, · · · , p − 1} .
(4.68)
Here ϱ
⃗ i = (ϱi,1 , ϱi,2 , ϱi,3 ) ∈ R3789 are some arbitrary points. The terms proportional to ħh haveno effect as far as classical brane geometry is concerned. These terms will affect our computations by deforming the moment map constraints coming from boundary conditions at NS5
domain walls. When we can describe the low energy theory of the D3 branes in terms of a 3d
N = 4 quiver as in Fig. (3), the parameters ϱ⃗ j correspond to the triplets of real FI parameterassociated to the jth gauge node. In such cases deformations of the Higgs branch by FI parameters is a standard phenomenon. The locations of the D3 branes in the x 4 direction, i.e.
the field Y1 , is not fixed at this boundary, rather it satisfies the Neumann boundary condition(3.20b).
26

SciPost Phys. 16, 052 (2024)

As in the previous section, we treat this boundary by keeping the NS5 branes separated
along the x 3 -direction, assuming τ1 < τ2 < · · · < τ p . Then each NS5 brane can be seen as adomain wall between two theories with possibly different gauge groups. In a bow diagram,⃗r

we represent an NS5 brane by a hollow circle ⃝ with its location in the R3789 direction on top.
As before, we draw the stack of D3 branes between two consecutive five-branes as a wavy linewith a label showing the number of D3 branes. From Fig. 6 we see that for i = 1, · · · , p − 1
the number of D3 branes between NS5 j and NS5 j+1 is:
Mj =

j
X

Li .

(4.69)

i=1

Pp
To the right of of the last NS5 brane there are M p := M = i=1 L i D3 branes. The bow diagram
corresponding to the brane configuration of Fig. 6 is in Fig. 7.
⃗r1

M1

⃗r2 M

2

···

M p−1 ⃗r p M p···Figure 7: The bow diagram for the NS5 type boundary whose brane construction
appears in Fig. 6. Here L = (L1 , · · · , L p ) is a p-tuple of integers such that the L j s are
related to the M j ’s by (4.69). ⃗r j is the location of NS5 j in R3789 with the constraint(4.68) that ⃗r j+1 − ⃗r j = ħhϱ⃗ j for j ∈ {1, · · · , p − 1}.
M j−1 ⃗r j M j· · · of a bow diaLet us focus on the jth NS5 brane, i.e., the portion · · ·gram. Before twisting and turning on Ω-deformation the jth NS5 brane creates a domain wall
between two 4d N = 4 SYM theories with gauge groups U(M j−1 ) and U(M j ). Let us assume
that the NS5 brane is located at x 3 = 0. We can write the half of the 4d space R40123 withx 3 < 0 as R3012 × L− where L− is the negative half line. Similarly denote the other half of
R40123 by R3012 × L+ where L+ is the positive half line. We can think of the 4d N = 4 SYM on
R40123 as a 3d N = 4 theory supported at x 3 = 0. The 3d theory has the infinite dimensional
gauge group G − × G + := Map(L− , U(M j−1 )) × Map(L+ , U(M j )). The fields X⃗ and A3 , as functions of L± , form two adjoint hypermultiplets of the 3d theory. The two hypermultiplets areacted on by the two factors of the gauge group respectively. Unlike a D5 brane, open strings
can stretch across an NS5 brane, the open strings connecting the stacks of D3 branes to theleft and to the right of NS5 j provide a bifundamental hypermultiplet to the 3d theory. This
hypermultiplet transforms by the action of the restriction of the infinite dimensional gauge
group to x 3 = 0. The restriction is simply U(M j−1 ) × U(M j ) and the hypermultiplet transe := T ∗ Hom(C M j−1 , C M j ). The hyper-Kähler moment maps of the G± actions on theforms as Z
hypermultiplets are:e
⃗ (x 3 ) = D X⃗ ± (x 3 ) + X⃗ ± (x 3 ) × X⃗ ± (x 3 ) + δ(x 3 ) ±X⃗ ± (0) + µeµ⃗ ±Z .
(4.70)±
3
Here X⃗ ± are functions on L± respectively, and µ−Z , µ+Z are the moment maps for the actions ofe . By introducing holomorphic coordinatesU(M j−1 ) and U(M j ) on Zeee,
(I, J) ∈ Hom(C M j−1 , C M j ) ⊕ Hom(C M j , C M j−1 ) = Z
we can write these moment maps as:


1 †
e
†
Z
(J J − I I ) , −Re I J , Im I J ,
µ
⃗+ =
2


1
e
Z
†
†
µ
⃗− =
(J J − I I) , Re J I , −Im J I .
227(4.71)(4.72)

SciPost Phys. 16, 052 (2024)

The boundary condition coming from the NS5 domain wall is simply that the terms in the
moment maps localized at the domain wall vanish, with shifts by central elements dependingon the location of NS5 j in R3789 :eX⃗ + (0) + µ⃗ +Z = ⃗r j 1 M j ,eX⃗ − (0) − µ⃗ −Z = ⃗r j 1 M j−1 .

(4.73)

This becomes the following constraints on the BF fields:
σ+ (0) + I J = r C
j 1M j ,σ− (0) + J I = r C
j 1 M j−1 ,(4.74)

where we have defined:rC
j := r2, j − ir3, j .

(4.75)

rC
is the location of NS5 j in the holomorphic C direction. The real variable r1, j labeling thejlocation of NS5 j in the x 7 direction does not appear in the BF theory.
There is some ambiguity in the exact value of the central element in the equations (4.74).
The field σ+ appears in two equations, coming from the two NS5 interfaces on two sides of
the wavy segment:
rC
j
···

Mj

rC
j+1
··· .

(4.76)

At each interface we have some localized bi-fundamental hypermultiplets. To distinguish thehypermultiplets from the left and the right interfaces, let us put a label on the fields as follows:(I k , Jk ) ∈ Hom(C Mk−1 , C Mk ) ⊕ Hom(C Mk , C Mk−1 ) .

(4.77)

Then (I j , J j ) and (I j+1 , J j+1 ) are the hypermultiplet fields at the left and the right interfaces
respectively. If NS5 j and NS5 j+1 are located at x 3 = x L and x 3 = x R respectively, then we get
the following two constraints on σ+ :σ+ (x L ) + I j J j = r C
j 1M j ,σ+ (x R ) + J j+1 I j+1 = r C
j+1 1 M j .

(4.78)

We can shift σ+ by a constant amount: σ+ ! σ+ +r C
1 , which changes these two equations
j+1 M jto:
σ+ (x L ) + I j J j = ħhϱ Cj 1 M j ,σ+ (x R ) + J j+1 I j+1 = 0 .
(4.79)
Here ħ
hϱ Cj = r C− rC
is the complex FI parameter for the abelian factor of the U(M j ) gaugejj+1
group associated with the M j D3 branes between the jth and the ( j + 1)st NS5 branes. We fix
this ambiguity by adopting the convention that as in the second equation in (4.79), at a rightNS5 interface there is no deformation of the moment map constraint, whereas as in the firstequation in (4.79), at a left NS5 interface the moment map constraint is deformed by a central
element given by the complex FI parameter.
Gauge symmetry is not broken at an NS5 interface. If g − and g + are the gauge transformation matrices on the left and the right of an NS5 interface located at x 3 = 0, then thebifundamental fields I and J at the interface transform under the gauge transformation as:(I, J) 7! (g + (0)I g − (0)−1 , g − (0)J g + (0)−1 ) .

4.3

Phase Spaces of BF Theories with Boundaries

4.3.1

BF Theories and Cherkis Bows

(4.80)

Putting these all together, we create a 2d BF theory with GLN gauge group as follows.
We suspend N D3 branes between n D5 and p NS5 branes. The boundary configurations
28

SciPost Phys. 16, 052 (2024)

arePn describedP p by the linking numbers K = (K1 , · · · , Kn ) and L = (L1 , · · · , L p ) satisfying
i=1 Ki =i= j L j = N . When all the NS5s are positioned to the left of all the D5s, Ki andL j are the net numbers of D3 branes ending on D5i and NS5 j from the left and from the rightrespectively (cf. (2.2)). Additionally, we need to keep track of the locations r Cof the NS5jbranes in the C direction. The theories will depend only on the differences between positions,
namely (cf. (4.68), (2.3), and (4.75)):Cħhϱ Cj = r C
j − r j+1 ,j ∈ {1, · · · , p − 1} .

(4.81)

Also define:z := r pC .

(4.82)

This is the spectral parameter associated to the line operator. Since this is coordinate dependent, the field theories, or the phase spaces associated to the line operators are independent of
it. The BF theory constructed from such brane configurations is denoted by TϱBF (K , L) whereCϱ := (ϱ1C , · · · , ϱ p−1).
nP

Ki D3si=2NS5 p−1

NS51

..
.

..
.

···

..
.

NS5 p

..
.

..
.

L1 D3s

C

..
.

···

Kn D3s

..
.

N D3s
p−2
P

p−1
P

L j D3sj=1

x3(Kn−1 + Kn )D3s

L j D3s

D51

j=1

D5n−1

D5n

Figure 8: Brane configuration for the GLN BF theory TϱBF (K , L) on an interval labeledby the linking numbers K and L. This figure is a duplicate of Fig. 1.
The worldvolume theory of the D3 branes is the 4d N = 4 U(N ) SYM theory with two 1/2BPS boundaries. Turning on Ω-deformation localizes the 4d theory to the 2d BF theory withan adjoint scalar σ and a gauge field A. We associate the following bow diagram – denotedby Bowϱ (K , L) – to this BF theory:
r1C

M1

r2C

M2

···

M p−1

r pC

N

N2

···

Nn−1

Nn

The bow diagram Bowϱ (K , L) defined by K = (K1 , · · · Kn ) andPn
Pi
L = (L1 , · · · L p ) satisfying Ni =
K
and
M
=
j
i
j=i
j=1 L j with M p = N1 = N .
CCϱ = (ϱ1 , · · · , ϱ p−1 ) contains deformation parameters where ħhϱ Cj = r C
− rC
.
j
j+1

Figure 9:

A wavy line segment with the label m corresponds to a GLm BF theory between two interfaces.
We gauge away the component of the gauge field parallel to the interfaces. Then each line
segment represents the equation of motion D3 σ = 0 where D3 is the gauge covariant derivativein the direction normal to the interfaces. At the D5 and NS5 interfaces boundary conditions are
imposed as described in the previous sections. The space of solutions to the equation of motion
subjected to the boundary conditions modulo gauge transformations is called the Cherkis Bow
Variety as defined in [6, 7], which we denote by Mbow
ϱ (K , L).

PϱBF (K , L) := Phase space of TϱBF (K , L) = Mbow
ϱ (K , L) .
29

(4.83)

SciPost Phys. 16, 052 (2024)

4.3.2

BF Theories and Branches of Vacua of 3d N = 4 Theories

We have constructed the BF theory on an interval by applying Ω-deformation to a 4d theory
on an interval. By the topological symmetry of the 4d theory, we can in principle consider thelimit where the interval shrinks to zero and we have an Ω-deformed 3d theory which localizes
to a TQM. Of course, the phase space remains invariant under this scaling and the phase spaceof the TQM is the same as the phase space of the BF theory. The problem in implementing
this in general is that we do not have a concrete description of the effective 3d theory forarbitrary linking numbers K and L of the five-branes in the brane construction such as inFig. 8. Regardless, in §2 we gave the name T ∨ [U(N )]LK to this 3d theory which has N = 4
supersymmetry. We know on general grounds [28, 29] that a 3d N = 4 theory reduces, upon
the B-type Ω-deformation, to a TQM whose phase space is the Higgs branch of the 3d theory.
In other words,
PϱBF (K , L) = MH (Tϱ∨ [U(N )]LK ) .
(4.84)
Using mirror symmetry, and denoting the mirror of Tϱ∨ [U(N )]LK by Tϱ [U(N )]LK we get the same
BF phase space as a Coulomb branch:PϱBF (K , L) = MC (Tϱ [U(N )]LK ) := Coulomb branch of Tϱ [U(N )]LK .

(4.85)

Note that mirror symmetry exchanges FI parameters with twisted masses [8] and so in the
mirror theory Tϱ [U(N )]LK , the parameters ħhϱ are twisted masses. Mirror symmetry is realized
in type IIB string theory as S-duality [19] which changes D5 branes to NS5s, NS5s to D5s, andleaves the D3 branes as D3s. Once again, for generic linking numbers of the five-branes, neitherTϱ∨ [U(N )]LK nor Tϱ [U(N )]LK has a quiver description and we can not give a more concrete
description of their branches of vacua than saying that they are the Cherkis bow varieties wehave computed in the last subsection.
To make contact with existing results about vacuum branches, we now specialize to braneconfigurations where Tϱ [U(N )]LK has a quiver description. The requirements are similar to(2.8) of §2.2. In that section we looked at example where Tϱ∨ [U(N )]LK has a quiver description,
now we need the “mirror” constraints on the linking numbers, namely:13
0 < Lj < n,vi := −

i
X
j=1

Kj +

n−1
X

1 ≤ j ≤ p,min(i, j)w j ≥ 0 ,1 ≤ j < n,(4.86)

j=1

where,

w i := #{L j | L j = i} .

With these constraints, we can bring the NS5 branes between D5 branes such that there are
equal numbers of D3 branes on both sides of every NS5 brane. Then by applying S-duality
we find the 3d N = 4 theory Tϱ [U(N )]LK defined by the quiver in Fig. 10. This is a gauge
Qn−1
Qn−1
theory with the gauge group i=1 U(vi ) and flavor group i=1 SU(w i ). An edge between two
nodes corresponding to two groups U(v) and U(w) represents a bifundamental hypermultiplet
transforming under the representation T ∗ Hom(C v , Cw ) of U(v) × U(w). It was proved in [7]that the Coulomb branch of this theory is a bow variety:

MC (Tϱ [U(N )]LK ) = Mbow
ϱ (K , L) .

(4.87)

Here Mbow
ϱ (K , L) is precisely the bow variety associated to the bow diagram of Fig. 9. The
relation (4.85) between BF phase spaces and Coulomb branches now implies our earlier claim(4.83). Of course, our claim is that (4.83) holds true for more general linking numbers, evenwhen the 3d theories involved have no quiver descriptions.
13

The bow varieties constructed with these constraints are given by what are called balanced dimension vectorsin [7].

30

SciPost Phys. 16, 052 (2024)

v1

v2

···

vn−1

w1

w2

···

w n−1

Pn
Pp
Figure 10: Quiver for the 3d N = 4 theory Tϱ [U(N )]LK with N = i=1 Ki = j=1 L j .
The ranks of the gauge and flavor groups are defined form K and L via (4.86).
Cϱ = (ϱ1C , · · · , ϱ p−1) where ħhϱ Cj s are complex twisted masses for the flavor symmetry. The mirror of this theory is denoted by Tϱ∨ [U(N )]LK , if this also admits a quiverdescription then the quiver is given by Fig. 3.

5Line Operators in 4d Chern-Simons Theory

The line operator Lϱ (K , L), created by the brane configuration of Fig. 8, appears in 4d CS
theory as a coupled TQM whose target space is the phase space of the BF theory TϱBF (K , L).
Thus,
(5.1)
Phase space of Lϱ (K , L) = PϱBF (K , L) = Mbow
ϱ (K , L) .
The equality between BF phase spaces and bow varieties is from (4.83). The TQM quantizes
the algebra of functions on this phase space. So this algebra Aϱ (K , L), which couples to the line
operator, can be characterized as the deformation quantization of functions on Bow varieties:Aϱ (K , L) = Deformation quantization of C[Mbowϱ (K , L)] .

(5.2)

This algebra will show up in the study of integrable spin chains as follows. Consider a gln
spin chain of length L consisting of spins R1 (z1 ) , · · · , R L (z L ). Here Ri (zi ) is the evaluation
module of the Yangian of gln associated with the gln module Ri and spectral parameter zi .
NL
The Hilbert space of the spin chain is the tensor product H := i=1
Ri (zi ). Following [13],this spin chain can be created in 4d GLn CS theory by putting L parallel Wilson lines carrying
the representations Ri . The Wilson lines are along the topological plane of 4d CS and have
fixed locations zi in the holomorphic C direction. Viewing these Wilson lines as vertical wez
can introduce Lϱ0 (K , L) as a horizontal line operator, as in Fig. 11. Here z0 is simply thelocation of the operator in the C direction. This horizontal line operator creates a monodromyzLϱ0 (K , L)···

R1 (z1 )R2 (z2 ) · · ·R L (z L )Figure 11:
Schematic diagram of a gln spin chain with Hilbert space
NL
H = i=1 Ri (zi ), realized in 4d GLn CS theory as an arrangement of vertical Wilz
son lines. Lϱ0 (K , L) has been introduced as a horizontal line operator to create thezmonodromy matrix Lϱ0 (K , L|H).

31

SciPost Phys. 16, 052 (2024)zmatrix which we denote by Lϱ0 (K , L|H). This monodromy matrix acts on an extended space
V ⊗ H where the auxiliary space V is some module for Aϱ (K , L). Without studying geometricquantization of Mbow
ϱ (K , L) we can not say exactly which module to assign to this line operator.
What we can say at this stage is which operator algebra the monodromy matrix belongs to:Lzϱ0 (K , L|H) ∈ Aϱ (K , L) ⊗ End(H) .

(5.3)

In other words, the monodromy matrix can be seen as a matrix that acts on H with entries in thealgebra Aϱ (K , L). Taking a partial trace of the monodromy matrix over the Aϱ (K , L)-module
produces a transfer matrix. Some closely related traces were studied recently in [45,53]. Notethat while the monodromy matrix itself will generally depend on the spectral parameter z0 of
the horizontal line, the algebra Aϱ (K , L) is independent of it.14
Integrability of the spin chain means that the monodromy matrices satisfy the RTT relations given the R-matrix. In [21] the authors explicitly constructed a class of monodromy
matrices that are linear in z0 for the Hilbert space H = R1 (z1 ) = Cn being the fundamental representation of gln . Examples of these monodromy matrices include the T, Q, andthe L-operators corresponding to elements of U(gln ) ⊗ End(Cn ), Weyl⊗k(n−k) ⊗ End(Cn ), andU(glk )⊗Weyl⊗k(n−k) ⊗End(Cn ) respectively. Below we describe the bow varieties and the quivers defining the line operators that create monodromy matrices associated with the algebrasU(gln ), Weyl⊗k(n−k) , and U(glk ) ⊗ Weyl⊗k(n−k) . We conjecture that they are in fact the T, Q,
and L-operators from the literature.
By some abuse of notation, in the following we use the terms T, Q, and L-operators to referto the horizontal line operator from Fig. 11 for certain K and L and not the corresponding
element of Aϱ (K , L) ⊗ End(H).

5.1

Example: The T-Operators (Wilson Lines)

A basic example of line operators in 4d GLn CS theory is the T-operator. In our notation (1.2)
a T-operator Lzϱ (K , L) is characterized by havingnz }| {
K = L = (1, · · · , 1) .

(5.4)

C
Given the mass parameters15 ϱ = (ϱ1C , · · · , ϱn−1), the phase space of this operator is the(1,··· ,1)

Coulomb branch of the 3d N = 4 theory Tϱ [U(N )](1,··· ,1) . This is a much studied theory
in the 3d N = 4 literature and it is often denoted simply as Tϱ [U(n)], which we adopt in thissection. We shall prove at the end of this section that
Proposition 5.1. The T-operator, i.e., Lzϱ (K , L) for K and L given by (5.4) carries a gln Vermamodule. Up to the action of the Weyl group, the Verma module has the highest weightCλ − ρ. The weight λ is determined by its Dynkin labels (ϱ1C , · · · , ϱn−1), and the Weyl vectorPn−1 n−3n−11
ρ = 2 α∈∆+ α = ( 2 , 2 , · · · , − 2 ) is the half-sum of the positive roots.
To prove this proposition we shall compute the values of the Casimirs of gln take in thesemodules. This gives us a Verma module assuming the representation is of the highest weighttype. More general representations are possible but not considered in this paper.
14
If we treat the spectral parameter z as a formal variable, as opposed to a complex number, then there may be
a factor of C[z] in the algebra, for example see §5.1.
15
Mass parameters can be either twisted masses or FI parameters depending on the duality frame.

32

SciPost Phys. 16, 052 (2024)

The theory Tϱ [U(n)] can be described by the following quiver.
n−1

···

n−2

1

n

Figure 12: The quiver for the 3d N = 4 theory Tϱ [U(n)], whose Coulomb branch isthe phase space for the T-operator.
It is well-known that the Coulomb branch of Tϱ [U(n)] is a deformation of the nilpotent
cone Ngln [11]. If we turn on all complex twisted masses and treat them as undeterminedparameters, then the Coulomb branch is gl∗n ,16 and it quantizes to the universal enveloping
algebra Uħh (gln ) [56, Example 6.2]. Incorporate the mass parameters ϱ and the spectral parameter z into the following complex numbers:
ri := z + ħ
h

n−1
X

ϱ Cj ,

1 ≤ i ≤ n.

(5.5)

j=i

Then the Coulomb branch can be equivalently described by the following bow diagram.
r1

1

r2

2

···

rn

n−1

n

n−1

···

2

1

Figure 13: Bow diagram whose associated bow variety is the Coulomb branch ofTϱ [U(n)] from Fig. 12.
The theory Tϱ [U(n)] is self mirror and so the dual theory Tϱ∨ [U(n)] is also described by
the same quiver as in Fig. 12. Let An be the quantized Higgs branch algebra of Tϱ∨ [U(n)],N⊗k(k+1)then An is the quantum Hamiltonian reduction of C[rn ] ⊗ n−1
with respect to
k=1 WeylħhLn−1
the action of k=1 glk . Let us introduce the variables:
I k ∈ Hom(Ck−1 , Ck ) ,Jk ∈ Hom(Ck , Ck−1 ) ,1 ≤ k ≤ n − 1,

(5.6)

satisfying the Weyl algebra commutation relations:αi[Jk,hδβα δ ij ,j , I k,β ] = ħ

1 ≤ i, j ≤ k,1 ≤ α, β ≤ k − 1 .

(5.7)

We shall use Weyl ordering to promote functions of classical variables into functions of operators:
1(5.8)(PQ)W := (PQ + QP) .
2
Now we can write down the quantum moment map equations for the Hamiltonian reduction:
i
α
a
i
i
α
a
i
(I k,αJk,hϱkC δ ij = I k,αJk,hδ ij + ħhϱkC δ ij = 0 .
j )W − (I k+1, j Jk+1,a )W + ħ
j − I k+1, j Jk+1,a − ħ
16Using [54, Theorem 7.6.1] and [55, Theorem 2.11].

33

(5.9)

SciPost Phys. 16, 052 (2024)

Lemma 5.2. Treat the ri ’s from (5.5) as formal variables, then An is isomorphic to the centrally
extended universal enveloping algebraUħh (gln ) ⊗Uħh (gln )GLn C[r1 , · · · , rn , ħh] ,(5.10)

where the map Uħh (gln )GLn ! C[r1 , · · · , rn , ħh] is given by

n 
Y
n−1
C(X , u) =u − rk −
ħ
h .
2
k=1

(5.11)

Here C(X , u) is the Capelli determinant17 of the generators X ij of Uħh (gln ) with standard comliimutation relations [X ij , X m]=ħhδ lj X m−ħhδm
X lj . Moreover, the explicit isomorphism is given
by

ħ
h
i
α
i
i
α
i
X ij 7! (I n,αJn,j )W + rn δ j = I n,α Jn, j + (n − 1) + rn δ j .
2(5.12)

Proof. We prove the lemma by induction on n.
The case n = 1 is obvious,
and we prove the induction step as follows.
By induction, An−1 is isomorphic toUħh (gln−1 ) ⊗Uħh (gln−1 )GLn−1 C[r1 , · · · , rn−1 , ħh], and Uħh (gln−1 )GLn−1 ! C[r1 , · · · , rn−1 , ħh] is given by

Qn−1
n−2
C(Y, u) = k=1 u − rk − 2 ħ
h , where C(Y, u) is the Capelli determinant of generators Yba ofaαUħh (gln−1 ), and the explicit isomorphism is given by Yba 7! I n−1,αJn−1,b+ ħ2h (n − 2) + rn−1 δ ab .
⊗n(n−1)

Then An is the quantum Hamiltonian reduction of C[rn ] ⊗ Weylħh
the gln−1 action and moment map equation⊗ An−1 with respect to


n  a
i
a
− rn + ħh δb = 0 .
Yba − I n,bJn,i2(5.13)

Now
recallthenotation of section 6.1 of [56], we can set B+ := Y − (rn + n−2
ħ
h
)
1
,
ψ
=
J
,
ψ
=
I
and
n−1
2
then quotient by the right ideal generated by B− (which turns out to be a two-sided ideal),
and then we obtain an algebra homomorphism Cħh [M(n − 1, n)] ! An . Under this map, wehaveqdet 1n − I






1
Xe
ħ
h
J !
7 qdet 1n −[we defined, Xe := X − rn + (n − 1) 1n ]u − B−
u
2

 

σ(1)
X
e
e σ(n)
X
X
σ(1)
n
1
=
sgn(σ) δ1 −· · · δσ(n)
−
n
n−1
n−1
u
−
ħ
h
u
+
h
σ∈Sn
2
2 ħ
=

C(X , u + rn + (n − 1)ħ
h)h) · · · (u − n−1h)(u + n−1
2 ħ
2 ħ

.

(5.14)On the other hand, [56, Proposition 6.2] implies that18
qdet 1n − IC(B+ , u + n−1
h)
1
2 ħ
J =
n−3
n−5
u − B−(u + 2 ħh)(u + 2 ħh) · · · (u − n−1
h)
2 ħ

l
i
i
The Capelli determinant C(B, u) of operators B ij satisfying gln commutation relations [B ij , Bm]=ħhδ lj Bm
−ħ
hδ m
B lj
P
σ(1)
is defined as C(B, u) = σ∈Sn sgn(σ)(u − (n − 1)ħ
h − B)1 · · · (u − B)σ(n)
.
n




1
1
18
We use the identity qdet 1n − I u−B− J qdet 1n + I u−B+ J = 1 [57] to derive the first equality from [56, Proposition 6.2].
17

34

SciPost Phys. 16, 052 (2024)=C(Y, u + rn + (n − 23 )ħ
h)h) · · · (u − n−1h)(u + n−3
2 ħ
2 ħ

.

(5.15)

Combining the above two equations we see that

 

n−1
ħ
h
C(X , u) = u − rn −ħh C Y, u −,22
 Qn−1ħh, thusand by the induction hypothesis C Y, u − ħ2h = k=1 u − rk − n−1
2

n 
Y
n−1
u − rk −C(X , u) =
ħ
h .
2
k=1

(5.16)

Since An is generated by X ij and r1 , · · · , rn as a C[ħh] algebra, we obtain a surjective mapUħh (gln ) ⊗Uħh (gln )GLn C[r1 , · · · , rn , ħh] ↠ An .
Since modulo ħh, the former algebra becomes the ring of functions on the variety gl∗n ×gl∗n /GLn a∗nwhere an is the Cartan subalgebra, and the map a∗n ! gl∗n /GLn is identified with the quotientby the Weyl group a∗n ! a∗n /Sn ∼= gl∗n /GLn . It is known that gl∗n ×gl∗n /GLn a∗n is a normal variety
of dimension n2 , and An /(ħh) is the ring of function on a (deformation of) Nakajima quiver
2 19
variety of dimension n , thus the surjective map Uħh (gln ) ⊗Uħh (gln )GLn C[r1 , · · · , rn , ħh] ↠ An is
an isomorphism modulo ħ
h. Since both sides of the map are flat over C[ħh], the map must be
injective as well, thus An is isomorphic to Uħh (gln ) ⊗Uħh (gln )GLn C[r1 , · · · , rn , ħh].
Remark 5.3 (Comparing the Higgs and Coulomb Branches). The reader might notice the difference between the Coulomb branch algebra of Tϱ [U(n)], which is Uħh (gln ), and the Higgs
branch algebra of Tϱ∨ [U(n)] which is Uħh (gln ) ⊗Uħh (gln )GLn C[r1 , · · · , rn , ħh]. This difference is actually superfluous, since mass parameters in our definition of Coulomb branch are symmetrized,in the sense that we only take symmetric polynomials in the mass parameters, and they form
the GLn -invariant part Uħh (gln )GLn ∼h]Sn of the Coulomb branch algebra, where= C[r1 , · · · , rn , ħSn permutes r1 , · · · , rn . On the other hand, FI parameters in our definition of Higgs branch are
not symmetrized. If we extend the the Coulomb side by adding non-symmetric polynomials ofmass parameters, we get Uħh (gln ) ⊗Uħh (gln )GLn C[r1 , · · · , rn , ħh], which is isomorphic to the Higgs
branch algebra of Tϱ∨ [U(n)].
Proof of Proposition 5.1. In practice, the ri ’s from (5.5) are locations of the NS5 branes in theholomorphic C direction and so we should treat them as complex numbers instead of formal
variables. If we evaluate the algebra An from Lemmah, λk ∈ C, then we get
Qn 5.2 at rk = λk ħ
the central quotient algebra Uħh (gln )/ C(X , u) − k=1 (u − (λk + n−1)ħ2 h) , which acts on theVerma module with the highest weight (with some choice of ordering for the gln fundamental
weights)(λ1 , · · · , λn ) − ρ .
(5.17)
This follows from the fact that the Capelli determinant acts on the ground state |λ〉 as
C(X , u)|λ〉 =nYu − (n − k)ħ
h − X kk |λ〉 ,(5.18)

k=1

Let h be the Cartan subalgebra of the Lie algebra g of a reductive Lie group G, then the quotient g∗ /G isisomorphic to h∗ /W where W is the Weyl group of G, and moreover there exists an open subset g∗reg ⊂ g∗ such thatits complement in g∗ has codimension 2 and that the natural map g∗reg ! h∗ /W is smooth, see [58, Section 3.1].
Then it follows from the aforementioned facts that g∗ ×h∗ /W h∗ is Cohen-Macaulay, and it contains an smooth open
subset g∗reg ×h∗ /W h∗ whose complement in g∗ ×h∗ /W h∗ has codimension 2, thus g∗ ×h∗ /W h∗ is normal [59, Theorem39]. Since the projection g∗ ×h∗ /W h∗ ! g∗ is finite and surjective, we see that dim g∗ ×h∗ /W h∗ = dim g∗ .
19

35

SciPost Phys. 16, 052 (2024)

whereas, according to Lemma 5.2:

 
n 
Y
n − 2k + 1
C(X , u)|λ〉 =u − (n − k)ħ
h − λk −ħh |λ〉 .
2
k=1

(5.19)We recognize n−2k+1
as the k-th component of the Weyl vector ρ. Comparing the above two2
equations give us the highest weight (5.17). The Dynkin labels of the weight λ = (λ1 , · · · , λn )are λi − λi+1 = ħ
h−1 (rn − rn+1 ) = ϱiC for 1 ≤ i ≤ n − 1.
The brane construction of the T-operator involves n D5, n NS5, and n D3 branes – one D3
brane suspended between each pair of five-branes. This setup was also used in [46] to create
Wilson lines valued in Verma modules for 4d GLm|n CS theory with highest weights given by
locations of NS5 branes in the holomorphic direction shifted by the Weyl vector.20 It was
shown in [60] that the action of quantized Coulomb branch algebras on their Verma modules
can be interpreted in terms of monopole operators acting on vortex configurations.

5.2

Example: The Q-Operators (Minuscule ’t Hooft Lines)

One class of important examples of line operators in 4d GLn Chern-Simons theory are the
Q-operators, also known as the minuscule ’t Hooft operators [22]. They are labeled by
Q 0 , Q 1 , · · · , Q n−1 where Q k in our notation is Tϱ (K , L) for the tuples:n−kkz }| { z }| {K = (0, · · · , 0, 1, · · · , 1) ,L = (k) .

(5.20)

Q 0 is the trivial line operator whose phase space is just a point, and for 0 < k ≤ n/2, the phasespace of Q k is the Coulomb branch of the 3d N = 4 theory of the quiver Fig. 14a. It is known
···

1

k−1

···

k

k−1

k

···

1

1
(a) Quiver for the theory Tϱ [U(n)]LK where K , L are given by (5.20). There are n − 1 circles
in the quiver. The Coulomb branch of this theory is the phase space of the Q-operator Q k .

z

k

k

···

k−1

k

···

2

1

(b) The Coulomb branch can also be described as the bow variety for this bow diagram withn crosses.

Figure 14: 3d N = 4 quiver and bow diagram associated with the Q-operator Q k .
that the Coulomb branch of this quiver is the affine space A2k(n−k) [11], and it quantizes tothe Weyl algebra⊗k(n−k)Weylħh= C〈x i j , yl m | 1 ≤ i, l ≤ k < j, m ≤ n〉/([x i j , ylm ] = ħhδil δ jm ) .

20(5.21)

The choice of Weyl ordering (5.8) in the quantum moment map (5.9) and the definition of the generators ofgln (5.12) is important to get the shift by the Weyl vector. It is not clear to us if there is some canonical/physicalreason that singles out the Weyl ordering over other ordering schemes.

36

SciPost Phys. 16, 052 (2024)

There is only one mass parameter ϱ = (z) which can be adjoined as a formal variable or evaluated at some complex value to get the algebra (5.21). The phase space can be equivalently
described by the bow variety of Fig. 14b.
All the Q-operators are created using a single NS5 brane. The operator Q k is created byconnecting k D5 branes to the only NS5 brane using k D3 branes.
Phase spaces of Q-operators were computed in [22] by directly solving the equations of
motion of 4d CS theory in the presence of minuscule ’t Hooft lines. The authors further described the quantization of ’t Hooft operator in terms of quantization of 3d N = 4 Coulomb
branches. The n distinct choices of minuscle coweights of gln correspond in our example tothe choice of k.
Remark 5.4 (All the Q-operators from [21]). In the work of Bazhanov et. al. [21], the Q
operators are labeled by subsets of {1, · · · , n}. The operators Q k in our paper are denoted
by Q {1,··· ,k} in [21], and other Q operators Q I with |I| = k are obtained by permutation of
coordinates. From the phase space perspective, the phase space of Q k is identified with thecotangent bundle of a big cell of Gr(k, n) [22], and a matrix g ∈ GLn transforms this phasespace to the cotangent bundle of other open cells. If g is taken to be permutation matrix,nthen we get the Weyl conjugations of the standard big cell, and in total there are k such
Weyl conjugations, which are in one-to-one correspondence with the operators Q I with |I| = k
in [21].

5.3

Example: The L-Operators (Wilson-’t Hooft)

A class of more complicated examples of line operators in 4d Chern-Simons theory are Loperators. They are labeled by L0 , L1 , · · · , L n , where L0 is the trivial line operator whose phasespace is just a point, L n is the T-operator from §5.1, and for 0 < k < n, the phase space of L k
is the Coulomb branch of the 3d N = 4 theory of the quiver Fig. 15a. We claim that the
···

k

···

k−1

k

1

k
(a) Quiver for the 3d N = 4 theory whose Coulomb branch corresponds to the phase spaceof the L-operator L k . There are n − 1 circles in the quiver. The Coulomb branch quantizes tothe algebra (5.22)

r1

1

r2

2

···

k−1

rk

k

k

···

k

k−1

···

2

1

(b) Bow diagram with n crosses whose associated variety is the Coulomb branch of the above
3d N = 4 quiver.

Figure 15: 3d N = 4 quiver and bow diagram associated with the L-operator L k .
quantized Coulomb branch algebra of this quiver is⊗k(n−k)Uħh (glk ) ⊗ Weylħh

.

(5.22)

In fact, using [56, Lemma 6.3] we see that the Coulomb branch algebra of this quiver is
the GLk invariant subalgebra of Weylħ⊗kn
h , where GLk embeds into GLn diagonally and actson the second tensor component of Ckn = Ck ⊗ Cn . Thus, the Coulomb branch algebra is37

SciPost Phys. 16, 052 (2024)22⊗k(n−k)GLk
GLk ∼(Weylħ⊗k⊗ Weylħh
, and it is easy to see that (Weylħ⊗k= Uħh (glk ). The phase spaceh )h )of the L-operator can be equivalently described by the bow variety associated to the diagram
Fig. 15b.
In 4d CS theory the L-operator is a dyonic line whose electric charge corresponds to aC
) and a spectral parepresentation of Uħh (glk ) determined by mass parameters (ϱ1C , · · · , ϱk−1
rameter z that are related to the ri ’s from Fig. 15b by a relation similar to (5.5). The magnetic
charge is determined by a minuscle coweight of gln (cf. §5.2).

5.4

Open Bow Diagrams

We define an open bow diagram to be a linear diagram with circles and crosses such that it
is allowed to have nothing on one or both ends, i.e. there could be semi-infinite D3 branes
which do not end on five-branes. And we define the open bow variety associated to an open
bow diagram to be similar to a bow variety, except that the symmetries at the open ends
are not being gauged or quotient out. For example, Fig. 16 is an open bow diagram whose
corresponding open bow variety is well-known to be T ∗ GLk . It is a D5 type boundary providing
GLk

k−1

···

2

1

Figure 16: The open bow variety corresponding to pure Dirichlet boundary condition. The GLk over the open edge, as opposed to just k, is meant to imply that thecorresponding GLk symmetry has not been gauged.
the pure Dirichlet boundary condition.
Note that for any Poisson variety X with a Hamiltonian GLk action, the Hamiltonian reduction (X × T ∗ GLk ) GLk is isomorphic to X as a Poisson variety. This implies that gluing
the above open bow diagram to some other open bow diagram does not change the variety,therefore we see that an open bow variety of a given open bow diagram is actually isomorphic
to the bow variety associated to the new bow diagram formed by gluing the open ends of theabove bow diagram and the given bow diagram. In terms of branes, we then get an open bow
variety by allowing the D3 branes to end on D5 branes such that all of these D5 branes acquire
linking number 1.
The identification between open bow variety and bow variety also holds at the quantum
level. In fact, it is known that T ∗ GLk quantizes to the ring Dħh (GLk ) of differential operatorson GLk , and for any C[ħh]-algebra A equipped with a Hamiltonian GLk -action, the quantum
Hamiltonian reduction (A ⊗ Dħh (GLk )) glk is isomorphic to A.
We can glue any two open bow varieties that have the same ranks at their open edges.
Let A L and AR be two algebras associated to two open bow diagrams with rank k at theiropen edges. The same rank k at the open edges means both A L and AR are equipped with
Hamiltonian GLk actions. Then the glued variety quantizes to the quantum Hamiltonian reduction (A L ⊗ AR ) glk . This leads to the notion of boundary algebras, such that we get thealgebra associated to a bow variety by gluing two boundary algebras. Similar boundary algebras were studied by Dedushenko-Gaiotto [45, 53] where they not only constructed boundary
algebras but also their traces by computing correlation functions in a hemisphere background(as opposed to our Ω-background).
In terms of branes, the gluing corresponds to identifying D5 branes at the Dirichlet boundaries, joining D3s from opposite sides of the same D5, and removing the D5s from the configuration. We illustrate the process for k = 3 in Fig. 17.

38

SciPost Phys. 16, 052 (2024)

AR

AR

AL

AL

(a) Two different brane configurations with
Dirichlet boundary conditions at one end, corresponding to the algebras A L and AR .

AL(b) Identifying the D5 branes from the two different Dirichlet boundaries.

AR

AL

AR

(d) The D5 branes are moved away and weare left with the configuration corresponding
to the algebra (A L ⊗ AR ) gl3 .

(c) Joining the D3 branes from opposite sidesof the D5 branes.

Figure 17: Gluing two boundary algebras A L and AR with Hamiltonian GL3 actions –the brane picture. The gray squares with A L and AR hide whatever configurations ofbranes are needed to get the algebras A L and AR . The vertical and horizontal lines
are D5 and D3 branes respectively.

6

Conclusion

Let us end with a few remarks on open threads. Instead of putting the line operators Lzϱ (K , L)horizontally just to create a monodromy operator as in Fig. 11, we can put a collection ofthese operators vertically to create the spin chain itself. We then get direct products of bow
varieties as phase spaces of these spin chains.21 It will be interesting to consider geometricand brane quantization of these phase spaces following [61–63], especially considering thatquantization of these bow varieties should result in quantum integrable systems according tothis construction.We are further exploring operator relations in 4d CS theory corresponding
to the famous TQ and QQ-relations [64–67] and their relations to fusions of bow varieties.
When the phase spaces can be identified with vacuum moduli spaces of 3d N = 4 quivergauge theories, these results are along the lines of many known relations between integrable
spin chains and such 3d theories [15,16,68–72]. Known connections between spin chains and3d gauge theories with Hanany-Witten type brane construction was used in [69] to establish
bispectral dualities of integrable spin chains. Brane constructions similar to that of §5.1 wereused in [46] to create spin chains with Verma modules and to illustrate fermionic dualities ofsuperspin chains. We hope that our brane constructions and characterization of line operators
using Cherkis bow varieties will further aid the study of non-trivial dualities of integrable spin
chains and related gauge theories.

Acknowledgments
Funding information Majority of the work done by NI for this paper took place at the Institute for Advanced Study in Princeton, NJ 08540, USA with support from the National Sci21

Assuming that the 4d CS theory itself has a trivial phase space (a single point) without the lines.

39

SciPost Phys. 16, 052 (2024)

ence Foundation under Grant No. PHY-1911298. At IHÉS NI is supported by the Huawei
Young Talents Program Fellowship. Kavli IPMU is supported by World Premier International
Research Center Initiative (WPI), MEXT, Japan. YZ also thanks Perimeter Institute for Theoretical Physics in Waterloo, ON N2L 2Y5, Canada, where part of his work was done as a graduate
student.

AΩ-deformed Supersymmetry

Let V be a U(1) space-time symmetry acting on R212 – the world-volume of the 2d B-modelfrom §4.1. Turning on Ω-deformation using V deforms the B-model supersymmetry (4.14),(4.19), and (4.23). We denote the deformed supersymmetry variation by δV . It acts on the
B-model multiplets as follows.
On vector multiplet (deformation of (4.14)):
δV A(2) = ιV χ (2) ,δV A

δV ψ(2) = 2ιV F (2) − 2iD(2) ιV φ (2) ,δV P = ιV D

(2)

(2)

= ψ(2) − ιV χ (2) ,δV η = P ,

η,

δV χ

(2)

=F

(2)

(A.1)
.

On the first chiral multiplet (deformation of (4.19)):
δV A

δV χ (c) = d(c) A(2) + D(2) A(c) + ιV F ,(c)= ψ(c) − ιV χ (c) ,



(c)
ψ(c) − ιV χ (c) = ιV d(c) A(2) + D(2) A,

δV A(c) = ιV χ (c) ,δVδV F = D(2) χ (c) + D(c) χ (2)δV M = F ,δV F = D(2) ιV M .
(A.2)

On the second chiral multiplet (deformation of (4.23)):
δV σ = ιV ψ
δV ψ

(c)

(2)

δV σ = η ,,= D(2) σ + ιV G ,δV G = D(2) ψ(c)δV η = ιV D(2) σ ,+ σ ∧ χ (2) ,(A.3)

δV N = G ,δV G = D(2) ιV N .

The deformed supersymmetry satisfies the algebra (deformation of (4.24)):
δ2V = D(2) ιV + ιV D(2) = LV + δι A(2) .
gauge

(A.4)

V

gauge

Here LV is Lie derivative with respect to V and δι A(2) denotes gauge transformation generatedV

by ιV A(2) . The Ω-deformed theory localizes to the fixed point of V with LV -invariant fields
and action given by the LV -invariant superpotential of the B-model.

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