A domain wall in twisted M-theory

Jihwan Oh1*, Yehao Zhou2,

1 Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2
6GG, United Kingdom

2 Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo,
ON N2L 2Y5, Canada

*<EMAIL_ADDRESS>
## Abstract

We study a four-dimensional domain wall in twisted M-theory. The domain wall
is engineered by intersecting D6 branes in the type IIA frame. We identify the
classical algebra of operators on the domain wall in terms of a higher vertex
operator algebra, which describes the holomorphic subsector of a 4d
$\mathcal{N}=1$ supersymmetric field theory. We conjecture the quantum
deformation of the classical algebra is isomorphic to the bulk algebra of
operators from which we establish twisted holography of the domain wall.

###### Contents

  1. 1 Introduction
  2. 2 Twisted M-theory
    1. 2.1 Background
    2. 2.2 Compatibility of M2 and M5 branes in twisted M-theory
    3. 2.3 KK monopole, intersecting D6 branes and a domain wall
  3. 3 4d VOA in the 5d Chern-Simons theory
    1. 3.1 Holomorphic twist of 4d $\mathcal{N}=1$ theory
    2. 3.2 Higher vertex operator algebra
    3. 3.3 The boundary condition of the 5d gauge field
    4. 3.4 Physical observables on the domain wall
    5. 3.5 5d/4d system and anomaly cancellation
  4. 4 Algebra of operators on the domain wall
    1. 4.1 The commutation relation
    2. 4.2 The large N limit
    3. 4.3 Conjecture
  5. 5 Conclusion and open questions
  6. A 4d $\mathcal{N}=1$ SUSY transformation of a chiral multiplet
  7. B Mode algebra computation

##  1 Introduction

Topological twists[1, 2] have been a standard tool to study a supersymmetric
quantum field theory along with $\Omega-$deformation [3, 4]. Since the twists
make the translation generator $Q$-exact with respect to the scalar
supercharge $Q$ of the twisted theory, the Q-cohomology class of a Q-closed
local operator is independent of its position. One can slightly generalize the
notion of the topological twist and discuss the holomorphic version. In this
case, the Q-cohomology class only retains the holomorphic dependence. By
construction, the resulting holomorphic field theory needs to be even
dimensional. Outstanding examples include the 2d $(0,2)$ theory [5, 6, 7] and
4d $\mathcal{N}=1$ theory [8, 9, 10]. The protected subsector of the 2d
example naturally forms a familiar vertex operator algebra. On the other hand,
the holomorphic sector of 4d $\mathcal{N}=1$ theory reorganizes itself into a
higher vertex operator algebra(higher VOA) [11, 12, 13], which will be the
main player of this paper. Higher VOA naturally appears in twisted M-theory
[14, 15], where there are 4 holomorphic directions and 7 topological
directions.

Twisted M-theory [15, 16] is eleven-dimensional supergravity background
specified by a product geometry of a hyperKahler 4-manifold and a 7-manifold
with $G_{2}$-holonomy, equipped with the above mentioned twist and
$\Omega-$deformation. The hyperKahler 4-manifold is equipped with the
holomorphic twist and the $G_{2}$-manifold is equipped with the topological
twist, along with the $\Omega-$deformation. Due to the $\Omega$-deformation,
the M-theory degree of freedom localizes to the 5d Chern-Simons theory, which
is topological in one direction and holomorphic in four directions; it comes
from Taub-NUT geometry inside the $G_{2}-$manifold, or a KK-monopole. Once we
introduce M2 and M5 branes, the worldvolume theories of M2 and M5 branes are
twisted and deformed(and localized to the subspace of the spacetime). To
preserve supersymmetry and to be compatible with the twisted background, the
M2 branes should wrap a three-cycle in the $G_{2}$ manifold and the M5 branes
should wrap four-cycle in the $G_{2}$ manifold. The supersymmetric coupling
between the membranes and the supergravity localizes to the coupling between
defects and the 5d Chern-Simons theory; the M2 branes and M5 branes give rise
to line and surface defects.

Twisted holography in the context of M-theory is an exact isomorphism between
the algebra of operators of the 5d Chern-Simons theory and the algebra of
operators on membranes. The relevant algebras are affine $gl(1)$ Yangian[17,
18] and its shifted and truncated version [19, 20, 21, 22, 23]. The
isomorphism is induced by Koszul duality111For another nice set of physical
examples of Koszul duality not in the context of twisted holography, see [24,
25]., which is a common name for a duality between a certain pair of algebras.
Conjecturally, twisted holography describes the protected subsectors of well-
established M2 and M5 brane holography [26, 27]. We will discuss the relevant
background for twisted M-theory in §2.1, §2.2.

One can also consider a network of M2 and M5 branes [28, 23] and identify the
relevant algebra [29]. Following the idea of [16], where the author
beautifully derived the M2 brane algebra from the perturbative computation in
the 5d Chern-Simons theory, one can reproduce the algebra associated to the
network of M2 and M5 branes [30].

Twisting supergravity is not necessarily restricted to the eleven-dimensional
example. Indeed, the most complete example for twisted holography is [31] the
twisted type IIB theory and a protected subsector of D3 brane holography. See
also [32] as an example of a D2-D4 system and [33] as an example of a D1-D5
system. Literatures that employ direct localization in both supergravity and
field theory include [34, 35].

In this paper, we study the seemingly last available BPS defect in the twisted
M-theory that has not been not studied: an extra KK-monopole, which is
different from the KK monopole that descends to the 5d Chern-Simons theory.
The new KK-monopole creates a domain wall-like defect wrapping the 4
holomorphic directions in the 5d Chern-Simons theory. Different from the line
and surface defects engineered by membranes, the domain wall-like defect is
engineered by changing the $G_{2}$-holonomy 7-manifold [36, 37]. It passes
tight constraints for the allowed BPS objects in twisted M-theory. (1) It
preserves some amount of supersymmetry. (2) It is compatible with the
topological holomorphic background. (3) It preserves the $G_{2}$-holonomy
structure. (4) It preserves the $U(1)$ isometry of the $G_{2}$ manifold.

For a practical computation and convenient visualization, it is better to work
in the type IIA frame by reducing a circle in the $G_{2}$ manifold. In type
IIA, we obtain two sets of intersecting D6 branes [38], one set from the
original Taub-NUTK geometry and the other set from the change we made. The
original $K$ D6 brane worldvolume theory gives the 5d $G=U(K)$ Chern-Simons
theory and the new $N$ D6’ brane intersects with the 5d Chern-Simons theory as
a domain wall. On the wall, D6-D6’ strings are localized and realize the 4d
$\mathcal{N}=1$ chiral multiplet [38]. Due to the holomorphic twist on the
wall, we get a higher VOA. We will give more details about the intersecting D6
brane configuration in §2.3.

The major difference between the higher VOA coming from the holomorphic twist
and the familiar 2d VOA is that in the former case the descent operators play
the key role to generate meromorphicity, which is the essential ingredient for
the algebraic structure under the vertex operator algebra. Like the usual
cohomological field theory, the set of Q-cohomology classes in the higher VOA
includes a certain integrated version of the descent operators and in our case
they are holomorphic descents of local Q-cohomology classes. The holomorphic
descents themselves are not Q-closed, but combining with the holomorphic top
form of the 4d spacetime manifold $\mathbb{C}^{2}$ and integrating over
$S^{3}\subset\mathbb{C}^{2}$, it becomes a Q-cohomology class. The descent
operator can avoid Hartog’s theorem that prohibits a meromorphic OPE between
two holomorphic functions on $\mathbb{C}^{d\geq 2}$, since the descent
operator is not a holomorphic function, but a differential form. We will
explain in more detail about this set of operators and how it generates the
higher VOA in §3.1, §3.2. Since the domain wall forms a boundary of the 5d
Chern-Simons theory, we need to account for the boundary condition of the 5d
Chern-Simons gauge field, which we did in §3.3. Collecting these ingredients,
we propose the physical observables living at the domain wall in §3.4. The
5d/4d system that we have discussed is anomalous due to the chiral fermion at
the domain wall. It can be canceled by tuning a parameter of the 5d Chern-
Simons theory on one side of the wall, as we will discuss in §3.5.

In §4, we will compute the commutation relation of the physical observables
that we have constructed in the previous section. We prove that the resulting
algebra is the universal enveloping algebra of
Diff${}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K}$. We conjecture
that the quantum deformation of our classical computation is isomorphic to the
Koszul dual algebra of operators in the 5d Chern-Simons theory. A priori, this
conjecture must be true due to the powerful theorem by Costello [16]: there is
a unique deformation of
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$ and it is
the algebra of operators in the 5d Chern-Simons theory. In this sense, the
proof for the conjecture is already given. However, we prefer to call it the
conjecture since we do not know how to systematically incorporate the quantum
corrections only using our machinery but not relying on the Costello’s
theorem.

We conclude in §5 by summarizing the discussion and provide open questions
that we want to answer in the future. In Appendix, we provide details of some
computations that we have abbreviated in the main text. We collect relevant 4d
$\mathcal{N}=1$ supersymmetry transformation of a chiral multiplet in Appendix
A. Lastly, we provide details of the key commutation relation computation in
Appendix B.

##  2 Twisted M-theory

In §2.1, we will review how to define a twisted supergravity and its
deformation under the $\Omega-$background. We will focus on the 11-dimensional
example, which we call the twisted M-theory. Due to the Omega background, the
bulk dynamics drastically simplifies to a non-commutative version of the 5d
Chern-Simons theory, which is topological in 1 direction and holomorphic in 4
directions. We will briefly explain the algebra of operators in the 5d Chern-
Simons theory. In §2.2, we will explore various BPS objects in the twisted
M-theory and study their compatibility with the twisted background. Finally,
in §2.3, we identify the BPS object in M-theory that realizes the domain wall
in the 5d Chern-Simons theory.

###  2.1 Background

Twisted supergravity [14] is defined as the supergravity in a background where
the bosonic ghost $\Psi$ of the local supersymmetry takes a nonzero value. To
satisfy the supergravity equation of motion, $\Psi$ needs to square to zero,
i.e., it is nilpotent. In the presence of $U(1)$ isometry in the background
geometry, we may turn on the $\Omega$-background for the twisted supergravity
background $X$ and deform it to $X_{\epsilon}$. $X_{\epsilon}$ is equipped
with $\Psi_{\epsilon}$, which squares not to zero but ${\epsilon}V$, where $V$
is the vector field generating the $S^{1}$ action.

In [15] Costello defines the twisted and $\Omega-$deformed 11-dimensional
supergravity and proves that it satisfies the equation of motion of the
11-dimensional supergravity. We will call this background twisted M-theory.
The twisted M-theory background222Appendx A of [15] contains the exact
expression of the background and the proof for it to be a consistent
supergravity background. is specified by the triple $(\Psi_{\epsilon},g,C)$,
which are a bosonic ghost, a metric, and an M-theory 3-form. The
11-dimensional manifold is a product of a 7-manifold $\mathcal{M}^{T}_{7}$
with $G_{2}$ holonomy and a hyper-Kahler 4-manifold $\mathcal{M}^{H}_{4}$
parametrized by $z$ and $w$. The background induced by $\Psi_{\epsilon}$ makes
$\mathcal{M}_{7}^{T}$ topological and $\mathcal{M}_{4}^{H}$ holomorphic.

The background geometry is given by

$\displaystyle(\mathbb{C}_{z}\times\mathbb{C}_{w})^{\text{H}}\times(\mathbb{R}_{t}\times\mathbb{C}\times\text{TN}_{K})^{\text{T}}.$
(1)

TNK is the Taub-NUTK manifold, which can be thought of as a
circle$(S^{1}_{TN})$ fibration over the base $\mathbb{R}^{3}$. There is an
$S^{1}_{{\epsilon}_{1}}\times S^{1}_{{\epsilon}_{2}}$ action on
$\mathbb{C}\times\text{TN}_{K}$. The first $S^{1}_{{\epsilon}_{1}}$ action,
parametrized by ${\epsilon}_{1}$, acts on $\mathbb{C}$ and the base of the
Taub-NUT manifold simultaneously. We will therefore put the subscript
${\epsilon}_{1}$ on the $\mathbb{C}$ and denote it as
$\mathbb{C}_{{\epsilon}_{1}}$. Also, we will denote the Taub-NUT base
$\mathbb{R}^{3}$ as $\mathbb{R}^{3}_{TNB}$. The second
$S^{1}_{{\epsilon}_{2}}$ action, parametrized by ${\epsilon}_{2}$, acts on the
Taub-NUT circle. Finally, the $S^{1}\times S^{1}$ action preserves the
holomorphic volume form of $\mathbb{C}_{{\epsilon}_{1}}\times\text{TN}_{K}$.

For a practical computation, it is helpful to reduce $S^{1}_{TN}$ to go to
type IIA theory. Note that we do not lose essential information by the circle
reduction, since it is a Q-exact operation, as proven in [15]. Under the
reduction, the M-theory geometry produces $K$ D6 branes. Furthermore, due to
the localization effect of the Omega background on
$\mathbb{C}_{{\epsilon}_{1}}$, the 7d maximally SYM of D6 brane worldvolume
reduces to the 5d Chern-Simons theory on
$\mathbb{R}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}$:

$\displaystyle\frac{1}{{\epsilon}_{1}}\int_{\mathbb{R}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}dz\wedge
dw\wedge\left(AdA+\frac{2}{3}A\wedge_{{\epsilon}_{2}}A\wedge_{{\epsilon}_{2}}A\right),$
(2)

where $\wedge_{{\epsilon}_{2}}$ is a combination of a wedge product and a
Moyal product $\star_{{\epsilon}_{2}}$. The Moyal product between two
holomorphic functions is defined as

$\displaystyle
f\star_{{\epsilon}_{2}}g=fg+{\epsilon}_{2}\frac{1}{2}{\epsilon}_{ij}\frac{\partial}{\partial_{z_{i}}}f\frac{\partial}{\partial_{z_{j}}}g+\ldots.$
(3)

Due to the holomorphic top form in the action, the 5d gauge field has only
three components.

$\displaystyle A=A_{t}dt+A_{\bar{z}}d\bar{z}+A_{\bar{w}}d\bar{w}.$ (4)

The origin of the non-commutativity that induces the Moyal product is the
B-field that descends from the M-theory 3-form C-field

$\displaystyle C=dz\wedge dw\wedge V^{\flat},$ (5)

where $V$ is a vector field that generates the rotation on $S^{1}_{TN}$ and
$V^{\flat}$ is the corresponding dual 1-form, whose component is given by
$V^{\flat}_{\mu}=g^{TN}_{\mu\nu}V^{\nu}$, where $g^{TN}_{\mu\nu}$ is the Taub-
NUT metric. We reduce $S^{1}_{TN}$ when we pass to the type IIA frame.

The reason that we may entirely focus on the open string part(the D6 brane
worldvolume theory) ignoring the closed strings from the geometry is that the
B-field combines with the B-model background(holomorphic) on
$\mathbb{C}_{z}\times\mathbb{C}_{w}$ and effectively produces the A-model
background(topological). As we already have A-model background in the other 6
directions,
$\mathbb{R}_{t}\times\mathbb{C}_{{\epsilon}_{1}}\times\mathbb{R}^{3}_{TNB}$,
all 10 directions in type IIA are topological. In other words, physical closed
string observables in the relevant Q-cohomology have a trivial dependence on
spacetime coordinates. Hence, they can be ignored [15].

Let us discuss the algebra of operators in the 5d Chern-Simons theory:
Obs${}^{5d}_{{\epsilon}_{1},{\epsilon}_{2}}$. We will first describe the case
with ${\epsilon}_{1}=0$, Obs${}^{5d}_{0,{\epsilon}_{2}}$ and turn on the
deformation parameter ${\epsilon}_{1}$ later. Since the equation of motion of
the 5d Chern-Simons theory is $F=0$, all operators have a positive ghost
number. The ghosts have a trivial dependence on $t$ and holomorphic dependence
on $z$ and $w$. Due to the non-commutative background on
$\mathbb{C}_{z}\times\mathbb{C}_{w}$, $[z,w]={\epsilon}_{2}$, the operators
form a non-commutative algebra of holomorphic functions on
$\mathbb{C}_{z}\times\mathbb{C}_{w}.$ Together with the BRST differential
${\delta}$, the algebra of operators of the 5d Chern-Simons theory forms a
graded associative algebra isomorphic to

$\displaystyle\wedge^{*}\mathbb{C}[z,w]_{{\epsilon}_{2}}\otimes\mathfrak{gl}_{K}\cong\wedge^{*}\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K},$
(6)

where Diff${}_{{\epsilon}_{2}}\mathbb{C}$ is the algebra of differential
operators on $\mathbb{C}_{z}$ with $[z,\partial_{z}]={\epsilon}_{2}$. In other
words, the classical algebra Obs${}^{5d}_{0,{\epsilon}_{2}}$ is the Chevalley-
Eilenberg algebra $C^{*}(\mathfrak{g})$ of cochains on the Lie algebra
$\mathfrak{g}=\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K}$.
A Koszul dual algebra of the Lie algebra cochain $C^{*}(\mathfrak{g})$ is the
universal enveloping algebra $U(\mathfrak{g})$.

When ${\epsilon}_{1}\neq 0$, Obs${}^{5d}_{0,{\epsilon}_{2}}$ receives quantum
corrections and deforms into an $A_{\infty}$ algebra333For a physical
description of $A_{\infty}$ algebra, see [28].
Obs${}^{5d}_{{\epsilon}_{1},{\epsilon}_{2}}$ and we denote the Koszul dual of
it as $U_{{\epsilon}_{1}}(\mathfrak{g})$. Costello showed that
$U(\mathfrak{g})$ has a non-trivial(section 9 of [16]) and unique(section 15
of [16]) deformation $U_{{\epsilon}_{1}}(\mathfrak{g})$ and identified it with
the algebra of operators on $N$ M2 branes in the large $N$ limit.

As long as we do not break the supersymmetry and respect the topological
holomorphic twist of the twisted supergravity, we can introduce M2 and M5
branes that inherit all twists in the background. They span the following
directions in the twisted M-theory.

M2: $\displaystyle\mathbb{R}_{t}\times\mathbb{C}_{{\epsilon}_{1}},$ (7) M5:
$\displaystyle\\{0\\}\times\text{TN}_{K}\times\mathbb{C}_{z}.$

Each of the membranes supports an interesting quantum field theory [15, 16],
which is a twisted subsector of 3d $\mathcal{N}=4$ ADHM gauge theory and 6d
$\mathcal{N}=(2,0)$ SCFT, respectively. Due to the Omega background applied to
each of the worldvolume, the field theories that encode the protected
subsector are localized to $\mathbb{R}_{t}$ and $\mathbb{C}_{z}$:

$\displaystyle\text{M2: Topological quantum mechanics on }\mathbb{R}_{t},$ (8)
$\displaystyle\text{M5: Free fermion vertex operator algebra on
}\mathbb{C}_{z}.$

In the limit of a large number of membranes, the algebra of operators on the
M2 branes is 1-shifted affine $gl(1)$ Yangian with one central element
eliminated and the algebra of operators on the M5 branes is $W_{\infty}$
algebra, or affine $gl(1)$ Yangian.

Note that in some literature, $\mathbb{C}_{{\epsilon}_{1}}\times\text{TN}_{K}$
part of the twisted M-theory background is replaced with

$\displaystyle\mathbb{C}_{{\epsilon}_{1}}\times\frac{\mathbb{C}_{{\epsilon}_{2}}\times\mathbb{C}_{{\epsilon}_{3}}}{\mathbb{Z}_{K}},\quad\text{where
}{\epsilon}_{1}+{\epsilon}_{2}+{\epsilon}_{3}=0.$ (9)

This presentation is useful to explain a triality automorphism of the algebra
associated with the M2 and M5 branes(see [28], for instance). However, we will
follow the original presentation [15], which did not show ${\epsilon}_{2}$,
${\epsilon}_{3}$ subscripts explicitly, since the triality does not play an
important role in our discussion.

###  2.2 Compatibility of M2 and M5 branes in twisted M-theory

Let us comment on the compatibility of the M2 and M5 branes and the 11d
topological holomorphic background and explain why these two BPS defects can
not engineer the domain wall-like defect, which is the main object of this
paper.

M2 branes are entirely embedded in $\mathcal{M}^{T}_{7}$. It preserves 3d
$\mathcal{N}=4$ supersymmetry in the presence of the Taub-NUT geometry. There
are two types of topological twists available in 3d: Rozansky-Witten twist[40]
and the mirror version of it. Both twists are compatible with the twisted
supergravity background and it is useful to work in either of the two twists
to give a complete picture of the algebra of operators on the M2 branes.
Clearly, the three-dimensional M2 branes can not engineer the four-dimensional
domain wall.

Next, out of the 6 directions of M5 brane worldvolume, 4 directions lie in
$\mathcal{M}_{7}^{T}$ and 2 directions are in $\mathcal{M}_{4}^{H}$. There are
two possible twists available to the 6d $(2,0)$ theory444We are grateful to
Kevin Costello for the discussion on this point.. First, a topological
holomorphic twist that gives 4 topological directions and 2 holomorphic
directions. Second, a holomorphic twist that gives all 6 directions
holomorphic. Since we only have 4 holomorphic directions in the twisted
M-theory background, only the first option is feasible and this background
exactly applies to the M5 branes that appear in (LABEL:M2M5). In other words,
since the 6d (2,0) theory does not admit a twist generating 2 topological and
4 holomorphic directions, we are not allowed to use M5 branes to engineer the
domain wall-like defects in the 5d Chern-Simons theory, which require the M5
branes to wrap $\mathcal{M}_{4}^{H}$.

There are remaining BPS objects in M-theory: the M9 brane and the KK monopole.
The KK monopole wraps different directions to those of the already existing KK
monopole, which is equivalent to the Taub-NUT geometry.

###  2.3 KK monopole, intersecting D6 branes and a domain wall

We would like to engineer a codimension-1 defect in the 5d Chern-Simons theory
from the remaining BPS objects. However, the M9 brane only admits a
holomorphic twist(on the entire 10 directions) on its worldvolume, so we
exclude the M9 brane since it is not compatible with the 7 topological
directions. Next, let us consider

New KK monopole:
$\displaystyle\mathbb{C}_{z}\times\mathbb{C}_{w}\times\mathbb{R}^{3},$ (10)

where $\mathbb{R}^{3}\subset\mathcal{M}^{T}_{7}$(we will specify the
$\mathbb{R}^{3}$ more precisely later in the IIA frame.) Since the 7d SYM on
the worldvolume of the KK monopole admits a topological holomorphic twist that
induces 3 topological and 4 holomorphic directions, like the original KK
monopole associated with $TN_{K}$, the new KK monopole is compatible with the
twisted background.

Let us now argue that the new KK monopole is compatible with the other
necessary conditions for twisted M-theory: 1. $G_{2}$-holonomy, 2.
$S^{1}_{{\epsilon}_{1}}\times S^{1}_{{\epsilon}_{2}}$ action.

First, the multi(original plus new) KK monopole configuration corresponds to a
singular $G_{2}$-holonomy manifold [37, 36], if we tune $\mathbb{R}^{3}$ to
intersect with $\mathbb{R}_{t}\times\mathbb{C}_{{\epsilon}_{1}}$ at one point.
If there are $N$ new KK monopoles, the singularity of the $G_{2}$-holonomy
manifold represents the unfolding of an $A_{N+K}$ singularity into $A_{N}$ and
$A_{K}$ singularities [41]. More precisely, it is a cone on a weighted
projective space WCP${}^{3}_{N,N,K,K}$ [37, 36]. Let us call this new geometry
as $\widetilde{\mathcal{M}}^{T}_{7}$.

Second, we will argue that $\widetilde{\mathcal{M}}^{T}_{7}$ admits
$S^{1}_{{\epsilon}_{1}}\times S^{1}_{{\epsilon}_{2}}$ isometry. The original
Taub-NUT circle $S^{1}_{TN}$ is intact by the introduction of new KK
monopoles. $\widetilde{\mathcal{M}}^{T}_{7}$ has an $S^{1}_{{\epsilon}_{2}}$
isometry that rotates $S^{1}_{TN}$. Indeed, when we go to the type IIA frame,
we take this circle as the M-theory circle to reduce555Historically, the brane
configuration, which we get after the circle reduction in type IIA, was first
discovered [38] and M-theory lift was introduced later [37, 36].. Next, it is
helpful to go to the type IIA frame to see the presence of
$S^{1}_{{\epsilon}_{1}}$ action. Reducing $S^{1}_{TN}$, we map two types of KK
monopoles in the M-theory to $K$ D6 and $N$ D6’ branes in the type IIA frame
that wrap the 4 holomorphic directions and 3 topological directions.

D6:
$\displaystyle\mathbb{C}_{z}\times\mathbb{C}_{w}\times\mathbb{R}_{t}\times\mathbb{C}_{{\epsilon}_{1}},$
(11) D6’:
$\displaystyle\mathbb{C}_{z}\times\mathbb{C}_{w}\times\mathbb{R}^{3},$

In general, such a D6-D6’ configuration does not preserve supersymmetry.
However, if we align them to make certain angles, we can preserve 4d
$\mathcal{N}=1$ supersymmetry on $\mathbb{C}_{z}\times\mathbb{C}_{w}$ [38].
More precisely, we may split the 6 topological
directions($\widetilde{\mathcal{M}}^{T}_{7}/S^{1}_{TN}$) into three 2-planes
$\mathcal{P}_{t}$, $\mathcal{P}_{1}$, $\mathcal{P}_{2}$. Then, we embed each
$\mathbb{R}_{i}$ of
$\mathbb{R}_{t}\times\mathbb{C}_{{\epsilon}_{1}}=\mathbb{R}_{t}\times\mathbb{R}_{1}\times\mathbb{R}_{2}$
of D6 branes in $\mathcal{P}_{i}$, respectively. We parametrize
$\mathbb{R}_{i}$ with $x_{i}$ and the orthogonal axis in $\mathcal{P}_{i}$
with $y_{i}$. Next, we orient each $\mathbb{R}^{\prime}_{i}$ of
$\mathbb{R}^{3}=\mathbb{R}^{\prime}_{t}\times\mathbb{R}^{\prime}_{1}\times\mathbb{R}^{\prime}_{2}$
of the D6’ branes in $\mathcal{P}_{i}$ to make an angle $\varphi_{i}$ with
$\mathbb{R}_{i}$ such that [38]

$\displaystyle\varphi_{t}\pm\varphi_{1}\pm\varphi_{2}=0~{}\text{mod}~{}2\pi$
(12)

for some choice of signs.

Figure 1: $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$. Each of them is
parametrized by $(x_{1},y_{1})$ and $(x_{2},y_{2})$ respectively.

Given the supersymmetry, let us check the $S^{1}_{{\epsilon}_{1}}$ isometry in
$\widetilde{\mathcal{M}}^{T}_{7}/S^{1}\sim\mathbb{R}_{t}\times\mathbb{C}_{{\epsilon}_{1}}\times\mathbb{R}^{3}_{TNB}$.
Recall that we took $TN_{K}\approx S^{1}_{TN}\times\mathbb{R}^{3}_{TNB}$. In
the present case, it suffices to show that the metric on both D6 and D6’
branes is invariant under $S^{1}_{{\epsilon}_{1}}$ action that simultaneously
rotates $\mathbb{C}_{{\epsilon}_{1}}\times\mathbb{R}^{2}_{tnb}$, where
$\mathbb{R}^{2}_{tnb}$ is a 2-plane embedded in $\mathbb{R}^{3}_{TNB}$.

Let us first consider two independent rotations, parametrized by $\theta_{1}$,
$\theta_{2}$, respectively. They act on the coordinates $(x_{1},x_{2})$ of
$\mathbb{C}_{{\epsilon}_{1}}$ and $(y_{1},y_{2})$ of $\mathbb{R}^{2}_{tnb}$ as
follows

$\displaystyle\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}\rightarrow\begin{pmatrix}\cos\theta_{1}x_{1}+\sin\theta_{1}x_{2}\\\
\sin\theta_{1}x_{1}-\cos\theta_{1}x_{2}\end{pmatrix},\quad\begin{pmatrix}y_{1}\\\
y_{2}\end{pmatrix}\rightarrow\begin{pmatrix}\cos\theta_{2}y_{1}+\sin\theta_{2}y_{2}\\\
\sin\theta_{2}y_{1}-\cos\theta_{2}y_{2}\end{pmatrix}.$ (13)

The relevant part of the metric on the original D6 branes is preserved
trivially under the rotation.

$\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}\rightarrow dx_{1}^{2}+dx_{2}^{2}$
(14)

If we denote $a=\tan\varphi_{1}$, $b=\tan\varphi_{2}$, we may write down the
relevant part of the metric on the new D6’ branes as

$\displaystyle ds^{2}$
$\displaystyle=d(x_{1}+ay_{1})d(x_{1}+ay_{1})+d(x_{2}+by_{2})d(x_{2}+by_{2})$
(15)
$\displaystyle=dx_{1}^{2}+dx_{2}^{2}+a^{2}dy_{1}^{2}+b^{2}dy_{2}^{2}+a(dx_{1}dy_{1}+dy_{1}dx_{1})+b(dx_{2}dy_{2}+dy_{2}dx_{2}).$

For (15) to be invariant under (13), $a$, $b$, $\theta_{1}$, $\theta_{2}$ need
to satisfy the following set of equations.

$\displaystyle a^{2}\sin^{2}\theta_{2}+b^{2}\cos^{2}\theta_{2}$
$\displaystyle=b^{2},$ (16) $\displaystyle
a^{2}\cos^{2}\theta_{2}+b^{2}\sin^{2}\theta_{2}$ $\displaystyle=a^{2},$
$\displaystyle(a^{2}-b^{2})\cos\theta_{2}\sin\theta_{2}$ $\displaystyle=0,$
$\displaystyle a\cos\theta_{1}\cos\theta_{2}+b\sin\theta_{1}\sin\theta_{2}$
$\displaystyle=a,$ $\displaystyle
a\sin\theta_{1}\sin\theta_{2}+b\cos\theta_{1}\cos\theta_{2}$
$\displaystyle=b,$ $\displaystyle
a\cos\theta_{1}\sin\theta_{2}-b\sin\theta_{1}\cos\theta_{2}$
$\displaystyle=0,$ $\displaystyle
a\sin\theta_{1}\cos\theta_{2}-b\cos\theta_{1}\sin\theta_{2}$
$\displaystyle=0.$

There is an obvious solution:

$\displaystyle a=b\text{ and }\theta_{1}=\theta_{2}.$ (17)

In principle, $a=b$ can be anything, but to be consistent with Figure 4, let
us choose them to be $1$. As this condition fixes
$\varphi_{1}=\varphi_{2}=\pi/4$, by (12), we can also fix $\varphi_{t}$. Since
we want to preserve exactly $\mathcal{N}=1$ supersymmetry on the domain wall,
$\varphi_{t}$ should not be 0666In this case, the D6 and D6’ branes intersect
in 5 directions, which is not of our interest. In fact, configuration with
$\varphi_{t}=0$ and its M-theory uplift was discussed in [37, 36], but another
set of KK monopoles other than the two sets of KK monopoles should be
considered in that case., but any value, which satisfies (12). The existence
of the solution (17) guarantees the $U(1)_{{\epsilon}_{1}}$ isometry, which
simultaneously($\theta_{1}=\theta_{2}$) rotates
$\mathbb{C}_{{\epsilon}_{1}}\times\mathbb{R}^{2}_{tnb}$.

Now that we have shown the compatibility of the new KK monopoles with the
twisted M-theory, let us discuss the degree of freedom that they introduce. It
is helpful to stay in the type IIA frame to do so. There are massless D6-D6’
strings, which are localized at the 4d intersection
$\mathbb{C}_{z}\times\mathbb{C}_{w}$, and they give rise to the 4d
$\mathcal{N}=1$ chiral multiplet [38], charged under both Chan-Paton
symmetries associated with the D6 and D6’ branes. We can also understand this
matter content as off-diagonal components in the decomposition of $A_{N+K}$
singularity into $A_{N}$ and $A_{K}$ singularities in the M-theory [41, 37,
36]. Since the worldvolume theory of D-branes probing the twisted supergravity
background inherits the twist, the D6-D6’ string or the 4d $\mathcal{N}=1$
chiral multiplet on $\mathcal{M}^{H}_{4}=\mathbb{C}_{z}\times\mathbb{C}_{w}$
is holomorphically twisted.

##  3 4d VOA in the 5d Chern-Simons theory

To construct a set of physical operators on the domain wall, we need to
collect all fields, which live on the domain wall. In §3.1, we will describe
the first ingredient, the holomorphically twisted 4d $\mathcal{N}=1$ chiral
multiplet coming from the D6-D6’ strings. We can recast the 4d holomorphic
field theory in a first-order formalism, called a twisted formalism in [43],
and find that the 4d Lagrangian resembles that of the 2d $\beta\gamma$ system.
The authors of [13] derived the 4d $\beta\gamma$ system. In §3.1, §3.2, we
will translate and explain this mathematical notion in the more familiar
language to physicists using holomorphic descent, which is an analog of the
topological descent defined in Witten’s classical paper [1]. This is in the
line of [44, 42, 43], where the authors defined the topological holomorphic
descent and the product between a local operator and a descent operator. In
particular, see the beautiful exposition in the paper [44] by Beem, Ben-Zvi,
Bullimore, Dimofte, and Neitzke. The material given in these subsections can
be readily generalized to a higher even-dimensional supersymmetric field
theory. In §3.3, we identify the second ingredient: the boundary condition of
the 5d Chern-Simons gauge field on the domain wall. Combining these
ingredients, we construct the physical operators living on the domain wall in
§3.4. The 4d $\beta\gamma$ system on the domain wall, however, is anomalous.
The anomaly cancelation condition is a shift of the non-commutativity
parameter ${\epsilon}_{2}$ on one side of the domain wall, compared to that on
the other side, as we show in §3.5.

###  3.1 Holomorphic twist of 4d $\mathcal{N}=1$ theory

Let us study the D6-D6’ strings that are localized on the D6, D6’ intersection
$\mathbb{C}_{z}\times\mathbb{C}_{w}$. The D6-D6’ strings give the 4d
$\mathcal{N}=1$ chiral multiplet $\Phi=(\phi,\psi_{\alpha})$ and the D6’-D6
strings give the complex conjugate
$\bar{\Phi}=(\bar{\phi},\bar{\psi}_{\dot{\alpha}})$, where ${\alpha}$ and
$\dot{\alpha}$ are spinor indices of $SU(2)_{l}\times
SU(2)_{r}=SO(4)_{Lorentz}$. Since branes moving in the twisted supergravity
background inherit the twist of the background, we get the holomorphically
twisted 4d $\mathcal{N}=1$ chiral multiplet theory. We will define and study
the holomorphic twist of 4d $\mathcal{N}=1$ theory in this subsection.

4d $\mathcal{N}=1$ supersymmetry algebra is generated by four supercharges
$Q_{\alpha}$ and $\bar{Q}_{\dot{\alpha}}$.

$\displaystyle\left[Q_{\alpha},Q_{\beta}\right]=[Q_{\dot{\alpha}},Q_{\dot{\beta}}]=0,$
(18)
$\displaystyle[Q_{\alpha},Q_{\dot{\beta}}]=\sigma^{\mu}_{{\alpha}\dot{\beta}}P_{\mu},$

where $\mu$ is the 4d spacetime index, $\sigma^{0}=-\text{Id}$, and
$\sigma^{1,2,3}$ are Pauli matrices. The graded commutator777We unify the
convention for the commutator to distinguish it with the secondary product
$\\{~{},~{}\\}$, which we will soon introduce. $[~{},~{}]$ is defined by
$[a,b]=ab-(-1)^{F(a)F(b)}ba$, where $F(a)$ is a fermion number of $a$.

Under the holomorphic twist, the Lorentz symmetry algebra is redefined in such
a way that the supersymmetry generators are reorganized into a scalar $Q$ and
a 1-form ${\bf{Q}}$ under the new Lorentz symmetry algebra. They satisfy the
following relations.

$\displaystyle Q^{2}$ $\displaystyle=0,$ (19) $\displaystyle[Q,{\bf{Q}}]$
$\displaystyle=iP_{\bar{z}}d\bar{z}+iP_{\bar{w}}d\bar{w}.$

We may solve (19) utilizing (LABEL:oldsusy) and find the scalar supercharge

$\displaystyle Q=Q_{-},$ (20)

which defines the Q-cohomology and the 1-form supercharge ${\bf{Q}}$, which
anti-commutes with $Q$ to produce anti-holomorphic translation generators
$P_{\bar{z}}$, $P_{\bar{w}}$.

$\displaystyle{\bf{Q}}=\bar{Q}_{\dot{+}}d\bar{z}-\bar{Q}_{\dot{-}}d\bar{w}.$
(21)

Given a local operator ${\mathcal{O}}$, one may act ${\bf{Q}}$ n-times to
obtain an n-form operator ${\mathcal{O}}^{(n)}$

$\displaystyle{\bf{Q}}^{n}{\mathcal{O}}={\mathcal{O}}^{(n)}.$ (22)

There are two kinds of operators in the Q-cohomology. The first kind is a
local operator ${\mathcal{O}}(z)$. Their product

$\displaystyle{\mathcal{O}}_{1,2}(z_{1},z_{2})={\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})$
(23)

is again in the Q-cohomology. By Hartog’s theorem, in $d\geq 2$-complex
dimension, every function on $U\subset\mathbb{C}^{d}$ that is holomorphic away
from singularities can be uniquely extended to a holomorphic function on the
entire $U$. Therefore, there can not be a singular OPE between two local
operators. This is in contrast with a complex one dimension holomorphic
theory, which is the usual vertex operator algebra, where one obtains
interesting mode algebras of various holomorphic currents due to the singular
OPE’s between the currents.

Examples of the first-type operator in 4d $\mathcal{N}=1$ chiral multiplet
theory are

$\displaystyle\bar{\phi},\quad\psi_{+},$ (24)

including all of their holomorphic derivatives with respect to $\partial_{z}$
and $\partial_{w}$. See Appendix A for the derivation. Even though $\psi_{+}$
now transforms as $(2,0)$ form under the new Lorentz symmetry of the twisted
theory, we will keep the notation and not explicitly show the differential
form in the later discussion for a concise presentation. We will also do so
for the other components of the spinor $\psi_{-}$ and
$\bar{\psi}_{\dot{\alpha}}$, which are a scalar and $(0,1)$-form under the new
Lorentz symmetry.

The second type of operator in the Q-cohomology is more interesting. We can
construct it by applying the 1-form supercharge to the first-type operator in
the Q-cohomology. However, ${\mathcal{O}}^{(1)}$ is not Q-closed, but we can
check

$\displaystyle Q{\mathcal{O}}^{(1)}=\bar{\partial}{\mathcal{O}}.$ (25)

If we wedge ${\mathcal{O}}$ with the holomorphic top form $\Omega$ of
$\mathbb{C}^{2}$, we get

$\displaystyle Q(\Omega{\mathcal{O}}^{(1)})=d(\Omega{\mathcal{O}}).$ (26)

By integrating both sides along $S^{3}$, which does not have a boundary, we
get a Q-closed operator at $x$

$\displaystyle
Q\left[\int_{S^{3}_{x}}\Omega\wedge{\mathcal{O}}^{(1)}\right]=0,$ (27)

where $S^{3}_{x}$ is a 3-sphere centered at $x$. It may look nonlocal, as it
involves the integral over $S^{3}$, but it is a local operator at $x$ after
the integral. We can avoid a potential intersection with other second-type
operators centered at different points by shrinking $S^{3}$’s. For two second-
type operators sharing the same center, there may be a nontrivial algebraic
structure888The discussion in this subsection can be easily generalized to a
holomorphically twisted supersymmetric field theory on $\mathbb{C}^{2D}$ with
$D\geq 1$. For instance, one can replace (27) by  $\displaystyle
Q\left[\int_{S^{2D-1}_{x}}\Omega\wedge{\mathcal{O}}^{(D-1)}\right]=0,$ (28)
where $\Omega$ is the holomorphic top form of $\mathbb{C}^{2D}$.. We will
describe more about the algebra in the next subsection.

Examples of the second-type operator in 4d $\mathcal{N}=1$ chiral multiplet
theory are

$\displaystyle\int_{S^{3}}dzdw~{}\bar{\phi}^{(1)}$
$\displaystyle=\int_{S^{3}}dzdw~{}(\bar{\psi}_{\dot{+}}d\bar{z}-\bar{\psi}_{\dot{-}}d\bar{w}),$
(29) $\displaystyle\int_{S^{3}}dzdw~{}\psi_{+}^{(1)}$
$\displaystyle=\int_{S^{3}}dzdw~{}(\partial_{w}\phi d\bar{z}-\partial_{z}\phi
d\bar{w}).$

We have collected the supersymmetry transformations of the components of the
4d $\mathcal{N}=1$ chiral multiplet under $Q$ and ${\bf{Q}}$ in Appendix A.

Furthermore, we can define a secondary product between the first-type operator
and the second-type operator by wrapping the first-type operator with a
3-sphere that is used to define the second-type operator.

$\displaystyle\\{{\mathcal{O}}_{1},{\mathcal{O}}_{2}\\}(z_{0},w_{0})=\int_{S^{3}_{z_{0},w_{0}}(z,w)}~{}{\mathcal{O}}_{1}(z_{0},w_{0})~{}{\mathcal{O}}^{(1)}_{2}(z,w)dzdw,$
(30)

where $S^{3}_{z_{0},w_{0}}(z,w)$ is a 3-sphere centered at $(z_{0},w_{0})$ and
parametrized by two complex coordinates $z$, $w$.

Figure 2: It illustrates a product between the first and second-type
operators. We place ${\mathcal{O}}_{1}$ at the origin and wrap it with
${\mathcal{O}}_{2}^{(1)}$, which is supported on the 3-sphere $S^{3}$,
parametrized by $z,w$.

Again, $\\{{\mathcal{O}}_{1},{\mathcal{O}}_{2}\\}(z_{0},w_{0})$ is a local
operator in the Q-cohomology, supported at $(z_{0},w_{0})$. To compute the
secondary product, we need to make a contraction between ${\mathcal{O}}_{1}$
and ${\mathcal{O}}_{2}^{(1)}$. Hence, the result of the secondary product is
in general a first-type operator. It is the remaining operator, which is not
contracted. More details of the secondary product along with the study of the
holomorphic twist of a general 4d $\mathcal{N}=1$ gauge theory will be
discussed in [50].

It is helpful to write down the Lagrangian for the 4d $\mathcal{N}=1$ chiral
multiplet to extract the propagators used in the contraction.

$\displaystyle\int_{\mathbb{C}_{z}\times\mathbb{C}_{w}}\partial\phi\bar{\partial}\bar{\phi}+\psi_{+}\partial_{\bar{z}}\bar{\psi}_{\dot{-}}+\psi_{+}\partial_{\bar{w}}\bar{\psi}_{\dot{+}}+\ldots.$
(31)

From the kinetic terms, we read off the nonvanishing propagators that involve
the first-type operators in the Q-cohomology.

$\displaystyle\langle\bar{\phi}\partial\phi\rangle\sim\langle\psi_{+}\bar{\psi}\rangle\sim\frac{\omega_{BM}}{dzdw},$
(32)

where $\omega_{BM}$ is Bochner-Martinelli kernel

$\displaystyle\omega_{BM}=dzdw\frac{\bar{z}d\bar{w}-\bar{w}d\bar{z}}{|z|^{2}+|w|^{2}}.$
(33)

Bochner-Martinelli kernel is a four-dimensional analog of the $S^{1}$ residue
integral measure $\frac{dz}{z}$, used in 2d VOA mode algebra computation. It
has the following nice property that will play an important role in the next
subsection.

$\displaystyle\int_{S^{3}_{z_{0},w_{0}}}\omega_{BM}f(z,w)=f(z_{0},w_{0}).$
(34)

###  3.2 Higher vertex operator algebra

There is no way to extract a nontrivial algebraic structure out of the first-
type operators in the Q-cohomology since they are holomorphic functions, whose
OPE is again a holomorphic function without a singularity, due to Hartog’s
theorem. However, once we include the second-type operator, we start to see a
meromorphic structure that enables us to define a higher-dimensional analog of
the two-dimensional vertex operator algebra.

Let us insert an operator ${\mathcal{O}}_{1}$ at the origin of the space time
$\mathbb{C}_{z}\times\mathbb{C}_{w}$. We would like to study an OPE between
${\mathcal{O}}_{1}$ and another operator ${\mathcal{O}}_{2}$, which is either
first-type(a holomorphic function) or second-type($(0,1)-$form). Since the
origin is already occupied by ${\mathcal{O}}_{1}$, the support of
${\mathcal{O}}_{2}$ is in $\mathbb{C}^{2}\backslash\\{0\\}$. We may compute
the cohomology of a punctured $\mathbb{C}^{2}$ [11] to study the analytic
behavior of ${\mathcal{O}}_{2}$.

$\displaystyle\begin{cases}H^{0}\left(\mathbb{C}^{2}\backslash\\{0\\},\mathcal{O}_{\mathbb{C}^{2}\backslash\\{0\\}}\right)=\mathbb{C}[z,w]\\\
H^{1}\left(\mathbb{C}^{2}\backslash\\{0\\},\mathcal{O}_{\mathbb{C}^{2}\backslash\\{0\\}}\right)=\mathbb{C}[z^{-1},w^{-1}]\frac{1}{zw}\\\
H^{n}\left(\mathbb{C}^{2}\backslash\\{0\\},\mathcal{O}_{\mathbb{C}^{2}\backslash\\{0\\}}\right)=0&\text{for
}n>1,\end{cases}$ (35)

where $\mathcal{O}_{\mathbb{C}^{2}\backslash\\{0\\}}$ is the sheaf of
algebraic functions on ${\mathbb{C}^{2}\backslash\\{0\\}}$. The first-type
operators in the Q-cohomology fit in the first category of (35) and the
second-type operators in the Q-cohomology are in the second category. The
lesson from (35) is that by extending our attention to the derived setting, we
can define a meromorphic product between two operators, since Hartog’s theorem
only applies to the underived setting.

As emphasized in [11], the meromorphicity of 2d VOA is ensured by the similar
cohomology computation in complex one-dimension:

$\displaystyle
H^{0}\left(\mathbb{C}\backslash\\{0\\},\mathcal{O}_{\mathbb{C}\backslash\\{0\\}}\right)=\mathbb{C}[z,z^{-1}].$
(36)

Hence, we do not need supersymmetry to obtain the meromorphicity in two-
dimension. Before getting back to the 4d discussion, let us recall the
familiar example of the OPE and residue pairing in a simple 2d VOA; we will
find the analogous properties in the 4d soon. Consider 2d $\beta\gamma$
system, which has the following OPE:

$\displaystyle\beta(z)\cdot\gamma(0)\sim\frac{1}{z}+\ldots,$ (37)

where $\ldots$ are regular terms. Then, we may compute the following integral

$\displaystyle\int_{S^{1}_{0}}dz(\beta(z)\cdot\gamma(0))=\int_{S^{1}_{0}}\frac{dz}{z}=1,$
(38)

by using the familiar residue formula.

Similarly, we form an “OPE” between the first-type and second-type operators

$\displaystyle{\mathcal{O}}_{1}(z_{0},w_{0})\cdot{\mathcal{O}}_{2}^{(1)}(z,w)$
(39)

Then, we can compute the $S^{3}$ residue integral, using the nontrivial
propagators defined in (32).

$\displaystyle\int_{S^{3}_{z_{0},w_{0}}}dzdw~{}{\mathcal{O}}_{1}(z_{0},w_{0})\cdot{\mathcal{O}}_{2}^{(1)}(z,w).$
(40)

This is precisely the secondary product
$\\{{\mathcal{O}}_{1},{\mathcal{O}}_{2}\\}$, which we defined in (30).
Specializing to the 4d $\mathcal{N}=1$ theory of a chiral multiplet, we can
compute a simple secondary product:

$\displaystyle\\{\bar{\phi},\psi_{+}\\}$
$\displaystyle=\int_{S^{3}_{0}}dz\wedge
dw\wedge\bar{\phi}\cdot\psi_{+}^{(1)}=\int_{S^{3}_{0}}\omega_{BM}=1.$ (41)

We will soon analyze a product involving two second-type operators as well.

We can make the analogy more concrete by rewriting the Lagrangian in the
descent field utilizing the BV Lagrangian [13]. The physical Lagrangian (31)
admits a compact representation in terms of the following twisted superfields

$\displaystyle\gamma$
$\displaystyle=(\bar{\phi},\bar{\phi}^{(1)},\bar{\phi}^{(2)}),$ (42)
$\displaystyle\beta$ $\displaystyle=(\psi_{+},\psi_{+}^{(1)},\psi_{+}^{(2)}).$

By multiplying $\beta$, $\gamma$ so that the form degree of the integrand sums
up to $(2,2)$, we can rewrite (31) as a part of the bigger BV Lagrangian that
follows

$\displaystyle\int_{\mathbb{C}_{z}\times\mathbb{C}_{w}}dzdw\beta\bar{\partial}\gamma.$
(43)

This free field Lagrangian resembles 2d VOA $\beta\gamma$ system. As our main
intention to present this expression is to show the resemblance of the higher
VOA to the usual 2d VOA, we will not try to give details on the BV formalism.
The reader who is interested in this may refer to [43, 13].

Let us briefly summarize what we have so far. After passing to the
Q-cohomology, the holomorphic twist of 4d $\mathcal{N}=1$ chiral multiplet
theory is reorganized into (43). The algebra of operators in the holomorphic
sector of 4d $\mathcal{N}=1$ theory has a meromorphic structure, encoded in
the descent operators.

Let us now define a higher current that generates a higher VOA, which is an
analog of the affine current operator in the 2d VOA. Rather than giving an
abstract definition, we will explain using the 4d $\mathcal{N}=1$ chiral
multiplet example, as it will be our main interest. We define a precurrent $J$

$\displaystyle J=\bar{\phi}\psi_{+}$ (44)

and consider its first descendant $J^{(1)}$:

$\displaystyle
J^{(1)}=(-\bar{\psi}_{\dot{+}}d\bar{w}+\bar{\psi}_{\dot{-}}d\bar{z})\psi_{+}+\bar{\phi}(\partial_{w}\phi
d\bar{z}-\partial_{z}\phi d\bar{w}).$ (45)

This is not in Q-cohomology by the general discussion above, but we can obtain
a nontrivial Q-cohomology class by combining it with the holomorphic top form
on $\mathbb{C}^{2}$ and integrating the wedged product over $S^{3}$:

$\displaystyle\int_{S^{3}}dz\wedge dw\wedge J^{(1)}.$ (46)

We call $dz\wedge dw\wedge J^{(1)}$ as a higher current. Notice that $J^{(1)}$
captures some components of the flavor current operator in the 4d
$\mathcal{N}=1$ theory of a free chiral multiplet. In this sense, the
expression (46) is a charge $Q$ of a conserved global symmetry current $j$,
which is defined as

$\displaystyle Q=\int_{S^{2n-1}}\star j$ (47)

in the $2n-$dimensional spacetime, where $\star$ is the Hodge star of the
spacetime geometry. The higher current can be compared with $\star j$ in (46).
Indeed, it satisfies the conservation law up to a Q-exact term999See also the
relevant discussion in Section 3.5 of [42]. We thank Junya Yagi for useful
discussion on this point., which we can treat as zero

$\displaystyle d(dzdwJ^{(1)})=Q(dzdwJ^{(1)})\equiv 0.$ (48)

In principle, one can think about the multiplication operation by a monomial
$z^{m}w^{n}$ in the integrand of (46) and obtain a double graded associative
algebra of modes101010One should distinguish a higher VOA from a toroidal Lie
algebra [45], since the former is related to $(\mathbb{C}^{2})^{\times}\sim
S^{3}$ rather than $(\mathbb{C}^{\times})^{2}\sim T^{2}$ by construction (46).
Indeed, the original reference of the higher VOA [11] called it the sphere
algebra., which is a generalization of the single-graded algebra of modes of
the 2d VOA. Therefore, we can think (46) as a zero mode$(m=n=0)$ of the higher
current. This observation will be useful when we construct the algebra of
operators on the domain wall by combining all ingredients.

We have studied the holomorphically twisted 4d $\mathcal{N}=1$ chiral
multiplet, which comes from the D6-D6’ string. Since the domain wall separates
the 5d Chern-Simons theory, we also need to take care of the boundary
condition of the 5d gauge field on the domain wall to construct the algebra of
operators on the domain wall. It will be the second ingredient.

###  3.3 The boundary condition of the 5d gauge field

Before considering the boundary condition of the 5d gauge field $A_{5d}$, we
first notice that we can interpret the boundary condition of the 5d gauge
field as the background gauge field for the flavor symmetry $U(K)$ of the 4d
$\beta\gamma$ system. The induced non-dynamical coupling modifies the free
kinetic term of the 4d $\beta\gamma$ system in the following way,

$\displaystyle\int_{\mathbb{C}_{z}\times\mathbb{C}_{w}}dzdw\wedge\beta\left(\bar{\partial}+A_{5d}|_{\partial}\right)\gamma.$
(49)

Here, the anti-holomorphic derivative is replaced by the gauge covariant
derivative with respect to the 5d gauge field $A_{5d}$ at the boundary $t=0$,
$A_{5d}|_{\partial}$. The form of this minimal interaction indicates that the
5d gauge field $A_{5d}=A_{t}dt+A_{\bar{z}}d\bar{z}+A_{\bar{w}}d\bar{w}$ loses
its t-component $A_{t}$ when it approaches to the boundary.

Next, we will follow the standard approach used to study the boundary
condition of the 3d Chern-Simons theory. We will turn off the non-
commutativity parameter temporarily as it does not affect the following
analysis. We need to make sure the locality of the 5d Chern-Simons theory near
the boundary $t=0$. To show that, let us vary the action that leads to

$\displaystyle{\delta}S_{5d~{}CS}=\int_{\mathbb{R}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}dz\wedge
dw\wedge\text{Tr}\left({\delta}A\wedge dA+d({\delta}A\wedge A)\right).$ (50)

Using Stoke’s theorem, we may rewrite the second term as

$\displaystyle\int_{\mathbb{C}_{z}\times\mathbb{C}_{w}}dz\wedge
dw\wedge{\delta}A\wedge A.$ (51)

From this, we see two types of allowed boundary conditions

$\displaystyle A_{\bar{z}}=0\quad\text{or}\quad A_{\bar{w}}=0,$ (52)

which makes the boundary variation (51) zero. This resembles the familiar
holomorphic Dirichlet boundary condition of the 3d Chern-Simons theory. We
will choose to work with the second one $A_{\bar{w}}=0$.

By the equation of motion of the 5d Chern-Simons theory,

$\displaystyle(d_{t}+\bar{\partial}_{\mathbb{C}_{z}\times\mathbb{C}_{w}})A_{5d}=0,$
(53)

and the boundary condition that we have just fixed, we conclude the boundary
condition of the 5d gauge field at the domain wall is a holomorphic function
in $z$ and $w$, valued in Lie algebra $\mathfrak{gl}_{K}$:

$\displaystyle
A_{5d}|_{\partial}=A_{\bar{z}}(z,m)\in\mathbb{C}[z,w]\otimes\mathfrak{gl}_{K}.$
(54)

###  3.4 Physical observables on the domain wall

So far, we have collected the ingredients that live at the domain wall in the
5d Chern-Simons theory. Those are the holomorphic twist of 4d $\mathcal{N}=1$
chiral multiplet and the boundary condition of the 5d Chern-Simons theory. We
will combine those degrees of freedom and construct the gauge invariant
physical operators living on the domain wall. The relevant gauge symmetry is
$U(N)$, where $N$ is the number of D6’ branes. On the other hand, we treat
$U(K)$ as a flavor symmetry111111Since both D6 and D6’ branes are non-compact,
one may think both $U(N)$ and $U(K)$ groups to be non-dynamical. However, D6
is related to the bulk dynamics and D6’ is related to the boundary dynamics.
Since we are now studying the boundary dynamics on D6’, we may treat D6 heavy
and only freeze the associated gauge field. [39, 15] also discussed such an
assignment of the flavor and gauge symmetry in an analogous intersecting D4-D6
brane system., where $K$ is the number of D6 branes.

We have already seen a candidate for the gauge invariant algebra of operators
in §3.2

$\displaystyle{\mathcal{C}}[1]=\int_{S^{3}_{0}(z,w)}dz\wedge dw\wedge
J^{(1)}.$ (55)

We will shortly clarify the notation for the LHS of (55). Restoring all
indices of the precurrent J, we have

$\displaystyle J^{i}_{j}=\bar{\phi}^{i}_{a}\psi_{+}{}^{a}_{j},$ (56)

where $a$ is the $U(N)$ gauge symmetry and $i,j$ are the $U(K)$ flavor
symmetry index. As we have remarked in §3.2, we can multiply a holomorphic
function to the integrand of (55), the result is again well-defined. Now,
recall from §3.3 that the boundary condition of the 5d gauge field is
precisely a holomorphic function in $z$ and $w$, valued in the Lie algebra
$\mathfrak{gl}_{K}$. Therefore, we can combine (55) and

$\displaystyle A_{5d}|_{\partial}\in\mathbb{C}[z,w]\otimes\mathfrak{gl}_{K}$
(57)

to form a generating set of the algebra of operators on the domain wall.

Consequently, we propose that the generators of the algebra of operators on
the domain wall are

$\displaystyle{\mathcal{C}}[z^{m}w^{n}T^{A}]=\int_{S^{3}_{0}(z,w)}dzdw~{}z^{m}w^{n}T^{A}~{}J^{(1)}.$
(58)

$m,n$ are non-negative integers and $T^{A}\in\mathfrak{gl}_{K}$, where
$A\in\\{1,\ldots,K^{2}\\}$. A few remarks are in order.

First, if there is a 4d gauge field localized at the domain wall, we need to
introduce $b$, $c$ ghosts that would modify the precurrent $J$ and the higher
current $J^{(1)}$, as well. However, the gauge field is a 7d field that lives
in the 7d worldvolume of the $N$ D6’ branes. Unless we compactify the three
transverse directions normal to the domain wall, this can not be treated as
the 4d gauge field. Hence, we do not have to introduce the ghosts for the 7d
gauge field. Moreover, we do not introduce ghosts for constant gauge
transformations. Therefore, all we need to do is to take a trace in
$\mathfrak{gl}_{N}$, which we have already done in (56).121212We are thankful
to Kevin Costello, who patiently explained the relevant part of his paper
[15]. Second, notice that (58) is not a secondary product, defined in (30),
between ${\mathcal{C}}[1]$ and $A_{5d}\rvert_{\partial}$. Even though the
secondary product involves a second-type operator, the result of the product
is a first-type operator, which is not expected to generate a nontrivial
algebra. We will see why such an operator does not form a nontrivial algebra
in §4.

This generator is a four-dimensional version of the two-dimensional observable
considered in Proposition 15.3.1 of [15], which discussed the twisted
holography of $N$ M5 branes,

$\displaystyle{\mathcal{C}}_{2d}\left[A_{5d}\rvert_{\mathbb{C}_{z}}=z^{m}\partial_{z}^{n}T^{A}\right]=\int_{S^{1}_{0}(z)}dz~{}z^{m}\partial_{z}^{n}T^{A}~{}J.$
(59)

In this case, the stack of M5 branes forms a surface defect supported on
$\mathbb{C}_{z}$ in the 5d CS theory. $A_{5d}\rvert_{\mathbb{C}_{z}}$ is a
$\bar{z}$ component of the 5d gauge field on $\mathbb{C}_{z}$. $J$ is a
fermion bilinear $\psi\bar{\psi}$ and it is a local operator, which does not
involve any descent. The pair of fermions is a quantization of D6-D4 strings.
The D4 branes, which are avatars of the M5 branes in type IIA frame, intersect
with the D6 branes along $\mathbb{C}_{z}$. The author of [15] proved the
isomorphism between the algebra of operators on $N$ D4 branes, generated by
(59), and the $W_{K+\infty}$ algebra in the large $N$ limit. Note that we may
also write $w$ in (58) as $\partial_{z}$, due to the non-commutative
background on $\mathbb{C}_{z}\times\mathbb{C}_{w}$, which induces a non-
trivial commutator between $z$ and $w$

$\displaystyle\left[z,w\right]={\epsilon}_{2}\quad\text{or}\quad\left[z,\partial_{z}\right]={\epsilon}_{2}.$
(60)

###  3.5 5d/4d system and anomaly cancellation

Before we close the section, we would like to point out a subtle modification
of the 5d Chern-Simons theory on one side of the domain wall compared to
another side of the wall, due to the anomaly coming from the domain wall.
After the modification, the 5d/4d system is anomaly-free. Although we have
treated the $U(K)$ gauge field as a background field when we studied the
boundary dynamics, we still need to cancel the $U(K)$ gauge anomaly for the
bulk 5d $U(K)$ gauge theory to make sense.

The Lagrangian of the 5d-4d coupled system is

$\displaystyle\int_{\\{0\\}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}\beta\bar{\partial}_{A}\gamma+\frac{1}{\epsilon_{1}}\int_{\mathbb{R}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}dzdw\mathrm{Tr}\left(AdA+\frac{2}{3}A^{3}+2\epsilon_{2}A\\{A,A\\}\right)+\ldots.$
(61)

Recall that in the presence of the gauge field $A$, the partition function of
the $\beta\gamma$ system is a local section of a determinant line bundle
$\mathcal{L}_{\mathrm{det}}$ on the moduli space of connections of the base
manifold (here is $\mathbb{C}^{2}_{zw}$). In order to quantize $A$, we need to
have a well-defined partition function in the first place and the local
anomaly is the curvature of the determinant line bundle $F_{\mathrm{det}}$,
which is determined by the following formula [46, Theorem 10.35]

$\displaystyle\int_{D}F_{\mathrm{det}}=n\int_{\mathbb{C}_{z}\times\mathbb{C}_{w}\times
D}\mathrm{Tr}(F_{A}^{3}),$ (62)

where $D$ is an arbitrary 2 dimensional region in the moduli of connections,
$n$ is some constant independent of $D$, and $F_{A}=dA+A\wedge A$ is the
curvature form of $A$. To trivialize $\mathcal{L}_{\mathrm{det}}$, we start
from the partition function at $A=0$, find a path $L$ connecting $0$ and $A$,
and define the partition function at $A$ to be the parallel transport of
$Z_{A=0}$ along $L$. If we perturb $L$ to $L^{\prime}$ connecting $0$ and $A$,
then the difference between parallel transports along $L$ and $L^{\prime}$ is
$\int_{D}F_{\mathrm{det}}$, where $D$ is a disk bounded by $L$ and
$L^{\prime}$. Using Stokes formula, we can write it as

$\displaystyle
n\int_{L\times\mathbb{C}_{z}\times\mathbb{C}_{w}}\mathrm{CS}_{5d}(A)-n\int_{L^{\prime}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}\mathrm{CS}_{5d}(A),$
(63)

where $\mathrm{CS}_{5d}(A)$ is the ordinary 5d Chern-Simons form

$\displaystyle\mathrm{CS}_{5d}(A)=\mathrm{Tr}\left(A\wedge dA\wedge
dA+\frac{3}{2}A^{3}\wedge dA+\frac{3}{5}A^{5}\right).$ (64)

Since $A$ only has anti-holomorphic components along
$\mathbb{C}_{z}\times\mathbb{C}_{w}$, we can rewrite $A\wedge dA\wedge dA$ as
$A\wedge\partial A\wedge\partial A$, and all other parts of
$\mathrm{CS}_{5d}(A)$ vanish, and we see that the action functional

$\displaystyle\int_{\\{0\\}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}\beta\bar{\partial}_{A}\gamma-n\int_{\mathbb{R}_{t\geq
0}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}\mathrm{Tr}\left(A\wedge\partial
A\wedge\partial A\right)$ (65)

is free of gauge anomaly, where we put the boundary condition that $A\to 0$
when $t\to\infty$.

On the other hand, separating out the 5d holomorphic Chern-Simons action into
two domains, we get

$\displaystyle\int_{\\{0\\}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}\beta\bar{\partial}\gamma$
$\displaystyle+\frac{1}{\epsilon_{1}}\int_{\mathbb{R}^{-}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}dzdw\mathrm{Tr}\left(AdA+\frac{2}{3}A^{3}+2\epsilon_{2}A\\{A,A\\}\right)$
$\displaystyle+\frac{1}{{\epsilon}_{1}}\int_{\mathbb{R}^{+}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}dzdw\mathrm{Tr}\left(AdA+\frac{2}{3}A^{3}+2\epsilon^{\prime}_{2}A\\{A,A\\}\right),$
(66)

which can be rewritten as

$\displaystyle\frac{1}{\epsilon_{1}}\int_{\mathbb{R}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}dzdw\mathrm{Tr}\bigg{(}AdA+\frac{2}{3}A^{3}$
$\displaystyle+2\epsilon_{2}A\\{A,A\\}\bigg{)}+\int_{\\{0\\}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}\beta\bar{\partial}_{A}\gamma$
$\displaystyle-\frac{{\epsilon}^{\prime}_{2}-{\epsilon}_{2}}{{\epsilon}_{1}}\int_{\mathbb{R}^{+}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}}\mathrm{Tr}\left(A\wedge\partial
A\wedge\partial A\right).$ (67)

Therefore, we can cancel the anomaly of the domain wall by imposing the non-
commutativity parameter ${\epsilon}_{2}$ on the one side of the wall jumps to
a different non-commutativity parameter ${\epsilon}^{\prime}_{2}$ on the other
side of the wall:

$\displaystyle{\epsilon}_{2}^{\prime}={\epsilon}_{2}+n{\epsilon}_{1}.$ (68)

Note that $n\sim N$ up to a proportionality constant since the source of the
anomaly, which we have just canceled, is the fermions in the $N$ 4d
$\mathcal{N}=1$ chiral multiplets.

##  4 Algebra of operators on the domain wall

In this section, we will compute the commutation relation of two generic
elements of the algebra of physical operators on the domain wall. In §4.1, we
prove that the algebra is a deformation of the universal enveloping algebra of
Diff${}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K}$. For a concise
presentation, we simply summarize the idea here. The reader, who is interested
in the detail of the calculation, may refer to Appendix B. The computation is
only valid in a careful large $N$ limit and this is implicitly assumed in
§4.1. We explain how we took the large N limit in some detail in §4.2.
Finally, we conjecture the universal enveloping algebra that we discovered is
the classical part of its unique deformation
$U_{{\epsilon}_{1}}(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$,
which coincides with the tree-level part of the bulk algebra in §4.3,

###  4.1 The commutation relation

Now, we can compute the commutator of the physical observables. We will prove
that the mode algebra is the universal enveloping algebra of
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$. Consider

$\displaystyle\big{[}{\mathcal{C}}(z^{m}w^{n}T^{A}),$
$\displaystyle{\mathcal{C}}(z^{r}w^{s}T^{B})\big{]}=$ (69)
$\displaystyle\bigg{[}\int_{S^{3}_{0}(z_{1},w_{1})}dz_{1}dw_{1}J_{1}^{(1)}\left(z_{1}^{m}w_{1}^{n}T^{A}\right),\int_{S^{3}_{0}(z_{2},w_{2})}dz_{2}dw_{2}J_{2}^{(1)}\left(z_{2}^{r}w_{2}^{s}T^{B}\right)\bigg{]}.$

We can compute the commutator131313Note that the product used in the above
commutator is not a secondary product, since the one we are discussing is the
product between two descent operators. by contracting pairs of free fields
that give nonzero values. The nonzero propagators that appear in the
computation are

$\displaystyle\langle\partial\phi\bar{\phi}\rangle,\quad\langle\psi_{+}\bar{\psi}_{\dot{+}}\rangle,\quad\langle\psi_{+}\bar{\psi}_{\dot{-}}\rangle.$
(70)

There are two sorts of terms in the RHS: terms from a single contraction and
terms from double contractions. As shown in detail in the first part of
Appendix B, the single contracted terms reorganize into

$\displaystyle{\mathcal{C}}\left([z^{m}w^{n}T_{A},z^{r}w^{s}T_{B}]\right),$
(71)

where we used the $S^{3}$ residue integral in the computation

$\displaystyle\int_{S^{3}_{z_{0},w_{0}}(z_{1},w_{1})}\omega_{BM}(z_{1}-z_{0},w_{1}-w_{0})f(z_{1},w_{1})=f(z_{0},w_{0}).$
(72)

A few remarks follow. One may wonder why there is no gauge coupling $g$
dependence141414We are thankful to Christopher Beem and Kevin Costello for
discussion on this point. in (71), since there can be gauge boson loop
corrections to the propagators (70) that we use in the computation. The reason
that we do not see such corrections is that the Yang-Mills terms that provide
the gauge coupling are Q-exact151515Here, the relevant $Q$ is the supercharge
that defines the topological holomorphic sector of the 7d SYM on the
worldvolume of the $N$ D6’ branes. in our set-up. Therefore, all
$g-$dependence is Q-exact.

To compute the double contraction, it is useful to rewrite the integral
contour, appealing to the usual 2d CFT $S^{1}$ contour rearrangement used in
the computation of mode algebra in the affine Kac-Moody VOA. We perform the
following change of $S^{3}$ contours

$\displaystyle\left[S^{3}_{z_{0}}(z_{1})\right][S^{3}_{z_{0}}(z_{2})]-[S^{3}_{z_{0}}(z_{2})][S^{3}_{z_{0}}(z_{3})]=[S^{3}_{z_{0}}(z_{2})][S^{3}_{z_{2}}(z_{1}^{\prime})].$
(73)

Figure 3: The $S^{3}$ contour rearrangement. $S^{3}(z_{i})$ is a 3-sphere
embedded in $\mathbb{C}_{z}\times\mathbb{C}_{w}$ and parametrized by $z_{i}$
and $w_{i}$.

Using this contour and performing the double contractions in (69), we end up
with

$\displaystyle\int_{S^{3}_{0}(z_{2},w_{2})}\int_{S^{3}_{z_{2},w_{2}}(z_{1},w_{1})}\omega_{BM}(z_{1}-z_{2},w_{1}-w_{2})(\bar{w}_{12}d\bar{z}_{2}-\bar{z}_{12}d\bar{w}_{2})z_{1}^{m}w_{1}^{n}z_{2}^{r}w_{2}^{s}~{}\text{tr
}T^{A}T^{B}$ (74) $\displaystyle={}$
$\displaystyle\int_{S^{3}_{0}(z_{2})}(0)z_{2}^{m+r}w_{2}^{n+s}~{}\text{tr
}T^{A}T^{B}$ $\displaystyle={}$ $\displaystyle 0.$

We wrote down the details of the double contractions in the second part of
Appendix B. Therefore, we get

$\displaystyle\left[{\mathcal{C}}(z^{m}w^{n}T^{A}),{\mathcal{C}}(z^{r}w^{s}T^{B})\right]={\mathcal{C}}([z^{m}w^{n}T^{A},z^{r}w^{s}T^{B}]),$
(75)

where the commutator in the argument of ${\mathcal{C}}(\bullet)$ in the RHS is
that of $\mathfrak{gl}_{K}\otimes\text{Diff}_{{\epsilon}_{2}}\mathbb{C}$. From
this, we see that the domain wall algebra of operators satisfies the
commutation relations of the algebra
$U(\mathfrak{gl}_{K}\otimes\text{Diff}_{{\epsilon}_{2}}\mathbb{C})$.

We make a remark that
$U(\mathfrak{gl}_{K}\otimes\text{Diff}_{{\epsilon}_{2}}\mathbb{C})$ is an
${\epsilon}_{2}-$deformation of the usual double current algebra
$U(\mathfrak{gl}_{K}\otimes\mathbb{C}[z,w])$ and
$\mathfrak{gl}_{K}\otimes\mathbb{C}[z,w]$ is the 0-th cohomology of the
differential graded Lie algebra $A^{\bullet}_{2}\otimes\mathfrak{gl}_{K}$.
Here, $A^{\bullet}_{2}$ is the differential graded Lie algebra
$Rj_{*}(\mathcal{O}_{\mathbb{C}^{2}\setminus\\{0\\}})$, where
$j:\mathbb{C}^{2}\setminus\\{0\\}\hookrightarrow\mathbb{C}^{2}$ is the natural
open embedding and $Rj_{*}$ is the derived pushforward(see Definition 2.3 of
[11]). $A^{\bullet}_{2}\otimes\mathfrak{gl}_{K}$ admits a natural central
extension as $L_{\infty}$ algebra:

$\displaystyle\mathbb{C}\to\widetilde{\mathfrak{gl}}_{K}^{\bullet}\to
A^{\bullet}_{2}\otimes\mathfrak{gl}_{K}.$ (76)

This central extension is given by a 3-cocycle $\theta$ [12] defined as

$\displaystyle\theta:(A^{\bullet}_{2}\otimes\mathfrak{gl}_{K})^{\otimes 3}$
$\displaystyle\to\mathbb{C}$ $\displaystyle a_{0}\otimes a_{1}\otimes a_{2}$
$\displaystyle\mapsto\int_{S^{3}}\mathrm{Tr}(a_{0}\wedge\partial
a_{1}\wedge\partial a_{2}),$ (77)

where $S^{3}$ is a 3-sphere surrounding the origin $\\{0\\}$. Note that
different from 2d, this central extension does not appear in the basic
commutation relation.

The $L_{\infty}$ algebra $\widetilde{\mathfrak{gl}}_{K}^{\bullet}$ is
considered as a 4d generalization of the 2d affine Kac-Moody algebra. Note
that if we deform the usual commutative product structure on $\mathbb{C}^{2}$
to the Moyal product $\star_{{\epsilon}_{2}}$, then
$A^{\bullet}_{2}\otimes\mathfrak{gl}_{K}$ deforms to
$A^{\bullet}_{2,{\epsilon}_{2}}\otimes\mathfrak{gl}_{K}$, and the 3-cocycle
$\theta$ deforms to

$\displaystyle\theta_{{\epsilon}_{2}}(a_{0}\otimes a_{1}\otimes
a_{2})=\int_{S^{3}}\mathrm{Tr}(a_{0}\wedge_{{\epsilon}_{2}}\partial
a_{1}\wedge_{{\epsilon}_{2}}\partial a_{2}).$ (78)

Thus, in this way, we get a deformed 4d Kac-Moody algebra
$\widetilde{\mathfrak{gl}}_{K,{\epsilon}_{2}}^{\bullet}$, whose zeroth
cohomology is
$\mathbb{C}\oplus(\mathfrak{gl}_{K}\otimes\text{Diff}_{{\epsilon}_{2}}\mathbb{C})$.

Before closing this subsection, we would like to point out that Johansen
derived the 2d $W-$algebra, using second-type operators in 4d $\mathcal{N}=1$
holomorphic field theory on
$\mathcal{M}^{H}_{4}=\Sigma_{z}\times\mathbb{C}^{\times}_{w}$ in [9]. Here
$\Sigma_{z}$ is a Riemann surface parametrized by $z$ and
$\mathbb{C}^{\times}_{w}$ is a punctured complex plane parametrized by $w$.
The second-type operator that Johansen used is “2-dimensional”, different from
our “3-dimensional” second-type operator, as he integrated the first descent
operator over the Riemann surface $\Sigma_{z}$ by wedging it with a
holomorphic top form $dz$ of $\Sigma_{z}$:

$\displaystyle\int_{\Sigma_{z}\times\\{w\\}}dz\wedge{\bf{Q}}{\mathcal{O}}.$
(79)

Since this operator is a point on $\mathbb{C}^{\times}_{w}$ and “wraps”
$\Sigma_{z}$, when two such operators approach in $\mathbb{C}^{\times}_{w}$,
we can expect the meromorphic structure in the OPE. By effectively161616See
also [47, 48, 49] for another example, where the meromorphic structure
naturally arises in 4d $\mathcal{N}=2$ superconformal field theory without
relying on the second-type operators. reducing the spacetime dimension to 2d
via the second-type operator, Johansen avoided Hartog’s theorem in 4d.

We can see a hint of the 2d ${\mathcal{W}}_{K+\infty}$ algebra from a bare
form of our universal enveloping algebra. Taking
$\mathcal{M}^{H}_{4}=\Sigma\times\mathbb{C}^{\times}=\mathbb{C}\times\mathbb{C}^{\times}$,
and inserting it into our universal enveloping algebra
$U(\mathfrak{gl}_{K}\otimes{\mathcal{O}}_{{\epsilon}_{2}}(\mathcal{M}_{4}^{T}))$,
we obtain
$U(\mathfrak{gl}_{K}\otimes\mathbb{C}_{{\epsilon}_{2}}[z,w,w^{-1}])$. Setting
${\epsilon}_{2}=1$, we almost get the universal enveloping algebra, which is
one of the definition of the ${\mathcal{W}}_{K+\infty}$ algebra [15]. However,
importantly, we do not reproduce the central extension of
${\mathcal{W}}_{K+\infty}$ in this way, since our commutation relation lacks
the central extension due to the vanishing double contraction. It would be
interesting to clarify the relation between Johansen’s algebra and our
algebra.

###  4.2 The large N limit

Trying to avoid potential confusion, we have not discussed the large $N$ limit
yet. Let us describe how to take the large $N$ limit and explain the implicit
assumption that we make in the previous subsection. We will mostly follow
[15], where the author explained how to take the large $N$ limit of the
algebra of operators living on $N$ M5 branes. The difference is that we do not
need supergroups to proceed.

Let us denote by $Obs_{{\epsilon}_{2}}^{4d}(N)$ the algebra of gauge invariant
operators on the domain wall coming from a stack of $N$ D6’ branes. For
$N^{\prime}\geq N$, there is a natural block diagonal embedding
$\mathfrak{gl}_{N}\hookrightarrow\mathfrak{gl}_{N^{\prime}}$, which gives rise
to a map between algebras

$\displaystyle\rho^{N^{\prime}}_{N}:Obs_{{\epsilon}_{2}}^{4d}(N^{\prime})\to
Obs_{{\epsilon}_{2}}^{4d}(N).$ (80)

It maps the generator ${\mathcal{C}}(z^{r}w^{s}T^{A})$ to its restriction to
$\mathfrak{gl}_{N}$ fields. Note that $\rho^{N^{\prime}}_{N}$ is a
homomorphism of algebras since the computation of commutation relations in the
last section holds for all $N$. $\rho^{N^{\prime}}_{N}$ is also surjective as
generators are mapped to generators. Therefore, we see that the map defined in
the last section
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})\to
Obs_{{\epsilon}_{2}}^{4d}(N)$ is compatible with the transition map
$\rho^{N^{\prime}}_{N}$. The only relations are trace relations and by sending
$N\to\infty$ all trace relations are gone. Hence, the map
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})\to
Obs_{{\epsilon}_{2}}^{4d}(N)$ becomes an isomorphism in the large $N$ limit.

###  4.3 Conjecture

Let us discuss the implication of the result that we obtained. We proved that
the algebra of operators living on the domain wall is the universal enveloping
algebra of Diff${}_{{\epsilon}_{2}}{\mathbb{C}}\times\mathfrak{gl}_{K}$. It is
important to notice that this algebra has no ${\epsilon}_{1}$ parameter.
Classically, since both the 4d VOA and the boundary condition of the 5d gauge
field that were used to construct the set of generators of the enveloping
algebra do not depend on ${\epsilon}_{1}$, the resulting algebra constructed
out of these ingredients should not depend on ${\epsilon}_{1}$. Still, there
can be potential quantum corrections that depend on ${\epsilon}_{1}$ at the
level of both generators and relations, similar to the M5 brane example [15].
The quantum correction is reflected by changing the original structure
constants of the algebra
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})[\\![{\epsilon}_{1}]\\!]$
to a new one, i.e. this is a flat deformation of
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$ over the
base ring $\mathbb{C}[\\![{\epsilon}_{1}]\\!]$. Let us denote this deformed
algebra as
$U_{{\epsilon}_{1}}(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$.
Then, what is this algebra?

It is in general a hard question to characterize a deformation of the
universal enveloping algebra of a certain Lie algebra, but this particular
case of interest was explicitly worked out in the $N$ M2 brane twisted
holography [16]. One important result of [16] is the uniqueness of the
deformation of
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$171717More
precisely, the deformation space of
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K+R|R})$ for
all $R$ in a compatible way is isomorphic to $\mathbb{C}[\kappa]$, where
$\kappa$ is the central element $1\otimes\mathrm{Id}_{K+R|R}$.. Now, recall
from §2.1 that $Obs^{5d~{}CS}_{{\epsilon}_{1},{\epsilon}_{2}}$ the algebra of
operators in the 5d Chern-Simons theory is Koszul dual of a deformation of
$U(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$. This
leads us to conjecture that the quantum deformation of our classical result is
Koszul dual to $Obs^{5d~{}CS}_{{\epsilon}_{1},{\epsilon}_{2}}$ up to a
coordinate change
${\epsilon}_{1}\rightarrow{\epsilon}_{1}+f_{2}(\kappa){\epsilon}_{1}^{2}+f_{3}(\kappa){\epsilon}_{1}^{3}+\cdots$,
where $f_{i}(\kappa)\in\mathbb{C}[\kappa]$ are polynomials of $\kappa$,
establishing the conjectural twisted holography for the domain wall example.

Let us give more details about the unique deformation of the enveloping
algebra. Even though the deformation of the algebra is uniquely fixed, we can
not immediately deduce the algebraic relations of the deformed algebra. This
algebra actually appeared in another example of twisted M-theory. In [16], the
author considered $N$ M2 branes in the twisted M-theory background. Similar to
our case, the twisted M-theory background localizes to the 5d Chern-Simons
theory, whose algebra of operators is again
$Obs^{5d~{}CS}_{{\epsilon}_{1},{\epsilon}_{2}}$. The worldvolume theory of the
$N$ M2 branes is the 3d $\mathcal{N}=4$ $G=U(N)$ gauge theory with $K$
fundamental hypermultiplets and an adjoint hypermultiplet with Rozansky-Witten
twist and $\Omega$-deformation applied to two directions out of the 3d
worldvolume. The author computed $\mathcal{A}_{{\epsilon}_{1},{\epsilon}_{2}}$
the algebra of operators on the $N$ M2 branes and drew an isomorphism between
$\mathcal{A}_{{\epsilon}_{1},{\epsilon}_{2}}$ and the Koszul dual algebra of
$Obs^{5d~{}CS}_{{\epsilon}_{1},{\epsilon}_{2}}$. Hence, we can equally
understand
$U_{{\epsilon}_{1}}(\text{Diff}_{{\epsilon}_{2}}\otimes\mathfrak{gl}_{K})$ as
$\mathcal{A}_{{\epsilon}_{1},{\epsilon}_{2}}$. The generators and relations of
this algebra can be found in [23, 22].

The conjectural isomorphism between the quantum deformation of the algebra of
operators on the domain wall
$U_{{\epsilon}_{1}}(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$
and the Koszul dual of $Obs^{5d~{}CS}_{{\epsilon}_{1},{\epsilon}_{2}}$ leads
to the statement of the twisted holography of the domain wall in twisted
M-theory. It is rather surprising that the domain wall algebra is related to
the M2 brane algebra $A_{{\epsilon}_{1},{\epsilon}_{2}}$, since there is no
obvious structural similarity between the two quantum field theories on the M2
brane and the KK monopole intersection. It would be interesting to figure out
a more clear physical reason.

##  5 Conclusion and open questions

In this paper, we study the last available BPS defect in the twisted M-theory,
a domain wall-like defect, building on the already established line and
surface-like defects in the literature. We identified the algebra of physical
operators on the domain wall and establish the isomorphism from the boundary
algebra to the bulk operator algebra, which is the algebra of operators of the
5d Chern-Simons theory. The higher vertex operator algebra, which is the
algebraic structure under the holomorphically twisted 4d $\mathcal{N}=1$
theory, plays a key role in the derivation.

We engineered a domain wall-like defect in the 5d Chern-Simons theory by
introducing a new KK monopole in the twisted M-theory. The new KK monopole
along with the already existing Taub-NUT geometry forms a new $G_{2}$-holonomy
7-manifold, which is topologically twisted. In the type IIA frame, the
singular $G_{2}$-holonomy manifold is mapped to an intersecting D6-D6’ brane
configuration. The D6-D6’ strings are localized at the 4d intersection, where
the holomorphic twist is applied. This intrinsic degree of freedom living on
the wall is the $4d$ $\mathcal{N}=1$ bi-fundamental chiral multiplet. After
passing to the Q-cohomology, specified by the holomorphic twist, one gets a
higher version of the vertex operator algebra, which has a concise
presentation as the 4d $\beta\gamma$ system. The algebra of operators in the
4d $\beta\gamma$ system is generated by the integrated higher current.
Combining it with the boundary condition of the 5d gauge field on the domain
wall, we form a set of generators of the physical operators living on the
domain wall. We prove that this algebra is the universal enveloping algebra of
Diff${}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K}$ and conjecture an
isomorphism with the Koszul dual of the bulk algebra using the uniqueness of
deformation of U(Diff${}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K}$).

We can summarize the operator algebras on M2, M5, and the new KK monopole in
the twisted M-theory on
$\mathbb{R}_{t}\times\mathbb{C}_{z}\times\mathbb{C}_{w}\times\mathbb{C}_{{\epsilon}_{1}}\times\text{TN}_{K}$
in the following table.

BPS defect | Field theory I | Twist | Field theory II | Algebra
---|---|---|---|---
M2 | 3d $\mathcal{N}=4$ ADHM | 3T | 1d TQM | $U_{{\epsilon}_{1}}(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$
M5 | 6d $\mathcal{N}=(2,0)$ | 4T, 2H | 2d fermions | $U_{c=N}({\mathcal{O}}_{{\epsilon}_{2}}(\mathbb{C}\times\mathbb{C}^{\times})\otimes\mathfrak{gl}_{K})$
KKM | 4d $\mathcal{N}=1$ chiral | 3T, 4H | 4d $\beta\gamma$ system | $U_{{\epsilon}_{1}}(\text{Diff}_{{\epsilon}_{2}}\mathbb{C}\otimes\mathfrak{gl}_{K})$
Table 1: Summary of BPS defects, associated field theory, twists, protected
subsectors of the field theory, and algebras. ADHM means the $G=U(N)$ gauge
theory with 1 adjoint hypermultiplet and $K$ fundamental hypermultiplets. T
and H stand for topological and holomorphic twists. Field theories I and II
are the original theories of the relevant branes and their protected
subsectors. TQM is an abbreviation of topological quantum mechanics.

There is a notable difference between the last example and the first two
examples. Although the first two examples have an algebra of operators
constructed solely from the first-type operators, in other words, local
operators in the Q-cohomology, the algebra of operators of the last example
involves the second-type operators, in other words, ${\bf{Q}}-$descendant
operators. This difference stems from the fact that we can only find a
nontrivial algebraic structure in the derived setting in the complex dimension
$d\geq 2$, which enables us to avoid Hartog’s theorem about holomorphic
functions.

On the other hand, there is a similarity shared by all three examples: the
effective codimension of the elements of algebras is one. Let us explain more
about what we meant by the effective codimension. First, in 1d TQM we easily
see that the codimension of a local operator is 1. Second, in 2d VOA, when one
computes the mode algebra, one extracts a mode from a current operator by
using $S^{1}$ residue integral. The codimension of the associated $S^{1}$ in
2d spacetime is 1. Lastly, for higher VOA, when we derive the mode algebra on
the domain wall, we used $S^{3}$ contour to extract elements of the algebra.
The non-commutative structure comes from the codimension-1 property of $S^{3}$
in 4d spacetime.

Figure 4: From left to right, 1d TQM, 2d VOA, and 4d VOA operators, depicted
by the circles. The thick straight lines are the transverse directions of the
operators, showing the codimension-1 nature of the algebra of operators in
each case.

Let us now list some open questions that we hope to answer in the future.

  * •

What would be the canonical Koszul-dual pair interaction between the 5d gauge
theory and 4d holomorphic field theory at the level of modes? Can we derive a
deformed version of the commutation relation (75) by imposing the BRST
invariance of a certain set of Feynman diagrams, like [16]? It is not clear to
us how to define a local interaction between a ghost and a second-type
operator. Once it is identified, the computation will be as straightforward as
that of [16, 23].

  * •

Can we identify a supersymmetric or at least stable configuration of the
combined system of domain wall, line, and surface defects? [29, 30] studied
the network of M2 and M5 branes and identified the relevant fusion algebra
using free field realization of the M2 and M5 brane algebras and Feynman
diagram computation in 5d Chern-Simons theory. Incorporating the domain wall
defect in this line of analysis seems to be a very natural direction to
pursue.

  * •

In [51], Li and Yamazaki gave a complete answer for algebras acting on BPS
partition functions that count D6-D2-D0 bound states on toric Calabi-Yau
3-folds. We wonder if there is a connection between the algebra associated to
our 5d/4d/5d system and a potential BPS partition function that counts M2
branes wrapping associative 3-cycles in the $G_{2}$ manifold.

## Acknowledgements

We thank Christopher Beem, Kevin Costello, and Junya Yagi for illuminating
discussions. We especially thank Kevin Costello for his insightful comments on
our draft.

#### Funding information

Research of JO was supported by ERC Grants 682608 and 864828. Research at the
Perimeter Institute is supported by the Government of Canada through Industry
Canada and by the Province of Ontario through the Ministry of Economic
Development $\&$ Innovation.

##  Appendix A 4d $\mathcal{N}=1$ SUSY transformation of a chiral multiplet

In this appendix, we summarize the supersymmetry transformation of the 4d
$\mathcal{N}=1$ chiral multiplet $\Phi=(\phi,\psi_{\alpha},F)$ and its
conjugate $\bar{\Phi}=(\bar{\phi},\bar{\psi}_{\dot{\alpha}},\bar{F})$. In
particular, we only record the transformations, generated by the scalar $Q$,
which we used to define the Q-cohomology, and the 1-form supercharge
${\bf{Q}}$, which we used to define the second-type operators in the
Q-cohomology.

$\displaystyle Q\phi=\psi_{-},$ (81) $\displaystyle Q\psi_{+}=-F,$
$\displaystyle Q\psi_{-}=0,$ $\displaystyle QF=0,$ $\displaystyle
Q\bar{\phi}=0,$ $\displaystyle
Q\bar{\psi}_{\dot{+}}=-i\partial_{\bar{z}}\bar{\phi},$ $\displaystyle
Q\bar{\psi}_{\dot{-}}=i\partial_{\bar{w}}\bar{\phi},$ $\displaystyle
Q\bar{F}=i\partial_{\bar{w}}\bar{\psi}_{\dot{+}}+i\partial_{\bar{z}}\bar{\psi}_{\dot{-}}.$

Up to on-shell condition, there are three Q-closed operators:

$\displaystyle\psi_{+},\quad\psi_{-},\quad\bar{\phi}$ (82)

Since $\psi_{-}$ is Q-exact by the first equation, the operators in the
Q-cohomology are

$\displaystyle\bar{\phi},\quad\psi_{+}$ (83)

and their holomorphic derivatives with respect to $\partial_{z}$,
$\partial_{w}$.

Next, we record the action of 1-form supercharge on the on-shell chiral
multiplet fields.

$\displaystyle{\bf{Q}}\phi=0,$ (84)
$\displaystyle{\bf{Q}}\psi_{+}=d\bar{z}\partial_{w}\phi-d\bar{w}\partial_{z}\phi,$
$\displaystyle{\bf{Q}}\psi_{-}=d\bar{z}\partial_{\bar{z}}\phi+d\bar{w}\partial_{\bar{w}}\phi,$
$\displaystyle{\bf{Q}}\bar{\phi}=id\bar{z}\bar{\psi}_{\dot{+}}-id\bar{w}\bar{\psi}_{\dot{-}},$
$\displaystyle{\bf{Q}}\bar{\psi}_{\dot{+}}=-id\bar{w}\bar{F},$
$\displaystyle{\bf{Q}}\bar{\psi}_{\dot{-}}=-id\bar{z}\bar{F}.$

Since we only use the action of 1-form supercharge on (83), only second and
fourth lines are meaningful in the main text of this paper.

##  Appendix B Mode algebra computation

In this appendix, we will provide the details on the commutator computation
summarized in §4.1.

$\displaystyle\big{[}{\mathcal{C}}(z^{m}w^{n}T^{A}),{\mathcal{C}}(z^{r}w^{s}T^{B})\big{]}={\mathcal{C}}\big{(}\big{[}z^{m}w^{n}T^{A},z^{r}w^{s}T^{B}\big{]}\big{)}+0,$
(85)

where

$\displaystyle{\mathcal{C}}(z^{m}w^{n}T^{A})=\int_{S^{3}_{0}(z_{1},w_{1})}dz_{1}dw_{1}J^{(1)}_{1}(z_{1}^{m}w_{1}^{n}T^{A}),$
(86)
$\displaystyle{\mathcal{C}}(z^{r}w^{s}T^{B})=\int_{S^{3}_{0}(z_{2},w_{2})}dz_{2}dw_{2}J^{(1)}_{2}(z_{2}^{r}w_{2}^{s}T^{B}).$

We will first focus on expanding $J^{(1)}_{1}$ and $J^{(1)}_{2}$ and identify
the nonzero propagators that we will contract later.

$\displaystyle
J^{(1)}_{1}J^{(1)}_{2}=\big{[}(\bar{\psi}_{\dot{-}}d\bar{z}_{1}-\bar{\psi}_{\dot{+}}d\bar{w}_{1})$
$\displaystyle\psi_{+}+\bar{\phi}(\partial_{w_{1}}\phi
d\bar{z}_{1}-\partial_{z_{1}}\phi d\bar{w}_{1})\big{]}_{1}$ (87)
$\displaystyle\cdot\left[(\bar{\psi}_{\dot{-}}d\bar{z}_{2}-\bar{\psi}_{\dot{+}}d\bar{w}_{2})\psi_{+}+\bar{\phi}(\partial_{w_{2}}\phi
d\bar{z}_{2}-\partial_{z_{2}}\phi d\bar{w}_{2})\right]_{2}$

Out of 16 terms in the expansion, 8 terms can host nonzero contractions. Those
are

$\displaystyle
d\bar{z}_{1}d\bar{z}_{2}\left[(\bar{\psi}_{\dot{-}}\psi_{+})_{1}(\bar{\psi}_{\dot{-}}\psi_{+})_{2}+(\bar{\phi}\partial_{w_{1}}\phi)_{1}(\bar{\phi}\partial_{w_{2}}\phi)_{2}\right]$
(88) $\displaystyle+{}$ $\displaystyle
d\bar{w}_{1}d\bar{w}_{2}\left[(-\bar{\psi}_{\dot{+}}\psi_{+})_{1}(-\bar{\psi}_{\dot{+}}\psi_{+})_{2}+(-\bar{\phi}\partial_{z_{1}}\phi)_{1}(-\bar{\phi}\partial_{z_{2}}\phi)_{2}\right]$
$\displaystyle+{}$ $\displaystyle
d\bar{z}_{1}d\bar{w}_{2}\left[(\bar{\psi}_{\dot{-}}\psi_{+})_{1}(-\bar{\psi}_{\dot{+}}\psi_{+})_{2}+(\bar{\phi}\partial_{w_{1}}\phi)_{1}(-\bar{\phi}\partial_{z_{2}}\phi)_{2}\right]$
$\displaystyle+{}$ $\displaystyle
d\bar{w}_{1}d\bar{z}_{2}\left[(-\bar{\psi}_{\dot{+}}\psi_{+})_{1}(\bar{\psi}_{\dot{-}}\psi_{+})_{2}+(-\bar{\phi}\partial_{z_{1}}\phi)_{1}(\bar{\phi}\partial_{w_{2}}\phi)_{2}\right]$

Now, we will use

$\displaystyle\langle\partial_{z_{i}}\phi_{i}(z_{i},w_{i})\bar{\phi}_{j}(z_{j},w_{j})\rangle$
$\displaystyle=\frac{\bar{z}_{ij}}{d_{ij}^{2}},\quad\langle\partial_{w_{i}}\phi_{i}(z_{i},w_{i})\bar{\phi}_{j}(z_{j},w_{j})\rangle=\frac{\bar{w}_{ij}}{d_{ij}^{2}},$
(89)
$\displaystyle\langle\bar{\psi}_{\dot{+},i}(z_{i},w_{i})\psi_{+,j}(z_{j},w_{j})\rangle$
$\displaystyle=\frac{\bar{z}_{ij}}{d_{ij}^{2}},\quad\langle\bar{\psi}_{\dot{-},i}(z_{i},w_{i})\psi_{+,j}(z_{j},w_{j})\rangle=\frac{\bar{w}_{ij}}{d_{ij}^{2}},$

where

$\displaystyle d_{ij}^{2}=|z_{ij}|^{2}+|w_{ij}|^{2},$ (90)

in the following.

Single contraction

For each term in (LABEL:JJexpansion) that looks

$\displaystyle(AB)_{1}(AB)_{2},$ (91)

we may contract either $\langle A_{1}B_{2}\rangle$ or $\langle
B_{1}A_{2}\rangle$. Summing up all possible single contractions, we get

$\displaystyle\frac{1}{d_{12}^{2}}\bigg{(}$ $\displaystyle
d\bar{z}_{1}d\bar{z}_{2}\bar{w}_{12}(\psi_{+,1}\bar{\psi}_{\dot{-},2}+\bar{\psi}_{\dot{-},1}\psi_{+,2}+\partial_{w_{1}}\phi_{1}\bar{\phi}_{2}+\bar{\phi}_{1}\partial_{w_{2}}\phi_{2})$
(92)
$\displaystyle+d\bar{w}_{1}d\bar{w}_{2}\bar{z}_{12}(\psi_{+,1}\bar{\psi}_{\dot{+},2}+\bar{\psi}_{\dot{+},1}\psi_{+,2}+\partial_{z_{1}}\phi_{1}\bar{\phi}_{2}+\bar{\phi}_{1}\partial_{z_{2}}\phi_{2})$
$\displaystyle-d\bar{z}_{1}d\bar{w}_{2}(\bar{w}_{12}\psi_{+,1}\bar{\psi}_{\dot{+},2}+\bar{z}_{12}\bar{\psi}_{\dot{-},1}\psi_{+,2}+\bar{z}_{12}\partial_{w_{1}}\phi_{1}\bar{\phi}_{2}+\bar{w}_{12}\bar{\phi}_{1}\partial_{z_{2}}\phi_{2})$
$\displaystyle-d\bar{w}_{1}d\bar{z}_{2}(\bar{z}_{12}\psi_{+,1}\bar{\psi}_{\dot{-},2}+\bar{w}_{12}\bar{\psi}_{\dot{+},1}\psi_{+,2}+\bar{w}_{12}\partial_{z_{1}}\phi_{1}\bar{\phi}_{2}+\bar{z}_{12}\bar{\phi}_{1}\partial_{w_{2}}\phi_{2})\bigg{)}$

Next, rearrange the above so that we can find the following $S^{3}$ integral
kernel

$\displaystyle\omega_{BM}(z,w)=\frac{\bar{w}d\bar{z}-\bar{z}d\bar{w}}{(|z|^{2}+|w|^{2})^{2}}$
(93)

manifestly.

$\displaystyle
d\bar{z}_{1}\frac{\bar{w}_{12}d\bar{z}_{2}-\bar{z}_{12}d\bar{w}_{2}}{d_{12}^{2}}\bar{\psi}_{\dot{-},1}\psi_{+,2}+d\bar{w}_{1}\frac{\bar{z}_{12}d\bar{w}_{2}-\bar{w}_{12}d\bar{z}_{2}}{d_{12}^{2}}\bar{\psi}_{\dot{+},1}\psi_{+,2}$
(94) $\displaystyle+{}$ $\displaystyle
d\bar{z}_{1}\frac{\bar{w}_{12}d\bar{z}_{2}-\bar{z}_{12}d\bar{w}_{2}}{d_{12}^{2}}\partial_{w_{1}}\phi_{1}\bar{\phi}_{2}+d\bar{w}_{1}\frac{\bar{z}_{12}d\bar{w}_{2}-\bar{w}_{12}d\bar{z}_{2}}{d_{12}^{2}}\partial_{z_{1}}\phi_{1}\bar{\phi}_{2}$
$\displaystyle+{}$ $\displaystyle
d\bar{z}_{2}\frac{\bar{w}_{12}d\bar{z}_{1}-\bar{z}_{12}d\bar{w}_{1}}{d_{12}^{2}}\psi_{+,1}\bar{\psi}_{\dot{-},2}+d\bar{w}_{2}\frac{\bar{z}_{12}d\bar{w}_{1}-\bar{w}_{12}d\bar{z}_{1}}{d_{12}^{2}}\psi_{+,1}\bar{\psi}_{\dot{+},2}$
$\displaystyle+{}$ $\displaystyle
d\bar{z}_{2}\frac{\bar{w}_{12}d\bar{z}_{1}-\bar{z}_{12}d\bar{w}_{1}}{d_{12}^{2}}\bar{\phi}_{1}\partial_{w_{2}}\phi_{2}+d\bar{w}_{2}\frac{\bar{z}_{12}d\bar{w}_{1}-\bar{w}_{12}d\bar{z}_{1}}{d_{12}^{2}}\bar{\phi}_{1}\partial_{z_{2}}\phi_{2}$

Then, make the following change of variables

$\displaystyle(z_{2},w_{2})\rightarrow(\tilde{z}_{2},\tilde{w}_{2})=(z_{2}-z_{1},w_{2}-w_{1})$
(95)

in the first two lines of (LABEL:temp0) to get
$\omega_{BM}(\tilde{z}_{2},\tilde{w}_{2})$ and make the following change of
variables

$\displaystyle(z_{1},w_{1})\rightarrow(\tilde{z}_{1},\tilde{w}_{1})=(z_{1}-z_{2},w_{1}-w_{2})$
(96)

in the last two lines of (LABEL:temp0) to get
$\omega_{BM}(\tilde{z}_{1},\tilde{w}_{1})$. We have expanded $J^{(1)}_{1}\cdot
J^{(1)}_{2}$. Let us plug this back in

$\displaystyle{\mathcal{C}}(z^{m}w^{n}T^{A})\cdot{\mathcal{C}}(z^{r}w^{s}T^{B}).$
(97)

We will omit the elements of $\mathbb{C}[z,w]\otimes\mathfrak{gl}_{K}$ and
restore those at the end.

Then, we can rearrange (97) into

$\displaystyle\int_{S^{3}_{0}}\omega_{BM}(\tilde{z}_{2},\tilde{w}_{2})\int_{S^{3}_{0}}dz_{1}dw_{1}\left((\bar{\psi}_{\dot{-},1}d\bar{z}_{1}-\bar{\psi}_{\dot{+},1}d\bar{w}_{1})\psi_{+,2}+\bar{\phi}_{2}(\partial_{w_{1}}\phi_{1}d\bar{z}_{1}-\partial_{z_{1}}\phi_{1}d\bar{w}_{1})\right)$
(98) $\displaystyle+$
$\displaystyle\int_{S^{3}_{0}}\omega_{BM}(\tilde{z}_{1},\tilde{w}_{1})\int_{S^{3}_{0}}dz_{2}dw_{2}\left((\bar{\psi}_{\dot{-},2}d\bar{z}_{2}-\bar{\psi}_{\dot{+},2}d\bar{w}_{2})\psi_{+,1}+\bar{\phi}_{1}(\partial_{w_{2}}\phi_{2}d\bar{z}_{2}-\partial_{z_{2}}\phi_{2}d\bar{w}_{2})\right),$

where $S^{3}_{0}$ is a 3-sphere centered at the origin of
$\mathbb{C}_{z}\times\mathbb{C}_{w}$. Note that $\psi_{+,2}$ and
$\bar{\phi}_{2}$ in the first line of (LABEL:temp1) are functions of
$\tilde{z}_{2}+z_{1}$ and $\tilde{w}_{2}+w_{1}$ due to (95). Similarly,
$\psi_{+,1}$ and $\bar{\phi}_{1}$ are functions of $\tilde{z}_{1}+z_{2}$ and
$\tilde{w}_{1}+w_{2}$ due to (96). Therefore, so far, the integrands are bi-
local functions on two sets of variables. Evaluating the first $S^{3}$ residue
integrals in each line of (LABEL:temp1), we get

$\displaystyle\int_{S^{3}_{0}}dz_{1}dw_{1}\left((\bar{\psi}_{\dot{-},1}d\bar{z}_{1}-\bar{\psi}_{\dot{+},1}d\bar{w}_{1})\psi_{+,1}+\bar{\phi}_{1}(\partial_{w_{1}}\phi_{1}d\bar{z}_{1}-\partial_{z_{1}}\phi_{1}d\bar{w}_{1})\right)$
(99) $\displaystyle+$
$\displaystyle\int_{S^{3}_{0}}dz_{2}dw_{2}\left((\bar{\psi}_{\dot{-},2}d\bar{z}_{2}-\bar{\psi}_{\dot{+},2}d\bar{w}_{2})\psi_{+,2}+\bar{\phi}_{2}(\partial_{w_{2}}\phi_{2}d\bar{z}_{2}-\partial_{z_{2}}\phi_{2}d\bar{w}_{2})\right).$

Due to the residue integrals, all fields with subscript $i$ now depend on a
single set of variables $(z_{i},w_{i})$, so there is no bilocal behavior
anymore. Therefore, we observe that each integrand corresponds to
$J^{(1)}_{i}$ and rewrite (LABEL:temp11) as

$\displaystyle\int_{S^{3}}dz_{1}dw_{1}J_{1}^{(1)}+\int_{S^{3}}dz_{2}dw_{2}J_{2}^{(1)}$
(100)

Since the subscripts are now dummy variables, we can unify (100) as one term.
After restoring the omitted elements of
$\mathbb{C}[z,w]\otimes\mathfrak{gl}_{K}$, we get

$\displaystyle\int_{S^{3}}dzdwJ^{(1)}\left(z^{m}w^{n}T^{A}\right)\left(z^{r}w^{s}T^{B}\right).$
(101)

This is the result of the single contractions of a product of two algebra
generators (97). To get the commutator, we permute two elements of
$\mathbb{C}[z,w]\otimes\mathfrak{gl}_{K}$ in (101) and subtract the resulting
term from (101). We get

$\displaystyle\int_{S^{3}}dzdwJ^{(1)}\left[z^{m}w^{n}T^{A},z^{r}w^{s}T^{B}\right]=:{\mathcal{C}}\big{(}\big{[}z^{m}w^{n}T^{A},z^{r}w^{s}T^{B}\big{]}\big{)}.$
(102)

It reproduces the first term in the RHS of (85).

Double contractions

Let us start from the single-contracted expression (LABEL:temp0) and perform
another contractions.

$\displaystyle\frac{1}{d_{12}^{4}}\bigg{(}$ $\displaystyle
d\bar{z}_{1}(\bar{w}_{12}d\bar{z}_{2}-\bar{z}_{12}d\bar{w}_{2})\bar{w}_{12}+d\bar{w}_{1}(\bar{z}_{12}d\bar{w}_{12}-\bar{w}_{12}d\bar{z}_{12})\bar{z}_{12}$
(103) $\displaystyle+$ $\displaystyle
d\bar{z}_{1}(\bar{w}_{2}d\bar{z}_{12}-\bar{z}_{12}d\bar{w}_{2})\bar{w}_{12}+d\bar{w}_{1}(\bar{z}_{12}d\bar{w}_{2}-\bar{w}_{12}d\bar{z}_{2})\bar{z}_{12}$
$\displaystyle+$ $\displaystyle
d\bar{z}_{2}(\bar{w}_{12}d\bar{z}_{1}-\bar{z}_{12}d\bar{w}_{1})\bar{w}_{12}+d\bar{w}_{2}(\bar{z}_{12}d\bar{w}_{1}-\bar{w}_{12}d\bar{z}_{1})\bar{z}_{12}$
$\displaystyle+$ $\displaystyle
d\bar{z}_{2}(\bar{w}_{12}d\bar{z}_{1}-\bar{z}_{12}d\bar{w}_{1})(-\bar{w}_{12})+d\bar{w}_{12}(\bar{z}_{12}d\bar{w}_{1}-\bar{w}_{12}d\bar{z}_{1})(-\bar{z}_{12})\bigg{)},$

where in the third and fourth line we used

$\displaystyle\langle\psi_{+,1}\bar{\psi}_{\dot{-},2}\rangle=-\langle\bar{\psi}_{\dot{-},2}\psi_{+,1}\rangle=\langle\bar{\psi}_{\dot{-},1}\psi_{+,2}\rangle,\quad\langle\psi_{+,1}\bar{\psi}_{\dot{+},2}\rangle=-\langle\bar{\psi}_{\dot{+},2}\psi_{+,1}\rangle=\langle\bar{\psi}_{\dot{+},1}\psi_{+,2}\rangle,$
(104)
$\displaystyle\langle\bar{\phi}_{1}\partial_{w_{2}}\phi_{2}\rangle=\langle\partial_{w_{2}}\phi_{2}\bar{\phi}_{1}\rangle=-\langle\partial_{w_{1}}\phi_{1}\bar{\phi}_{2}\rangle,\quad\langle\bar{\phi}_{1}\partial_{z_{2}}\phi_{2}\rangle=\langle\partial_{z_{2}}\phi_{2}\bar{\phi}_{1}\rangle=-\langle\partial_{z_{1}}\phi_{1}\bar{\phi}_{2}\rangle.$

We can simplify (103) into

$\displaystyle
2(\bar{w}_{12}d\bar{z}_{2}-\bar{z}_{12}d\bar{w}_{2})\frac{\bar{w}_{12}d\bar{z}_{1}-\bar{z}_{12}d\bar{w}_{1}}{d_{12}^{4}}.$
(105)

By integrating (105) along the modified contour depicted in Figure 3, we can
finish the computation for the double contracted term of (85).

$\displaystyle
2\int_{S^{3}_{0}(z_{2},w_{2})}\int_{S^{3}_{z_{2},w_{2}}(z_{1},w_{1})}\frac{\bar{w}_{12}d\bar{z}_{1}-\bar{z}_{12}d\bar{w}_{1}}{d_{12}^{4}}(\bar{w}_{12}d\bar{z}_{2}-\bar{z}_{12}d\bar{w}_{2})(z_{1}^{m}w_{1}^{n}z_{2}^{r}w_{2}^{s}\text{
tr }T^{A}T^{B})$ (106) $\displaystyle={}$ $\displaystyle
2\int_{S^{3}_{0}(z_{2},w_{2})}\int_{S^{3}_{z_{2},w_{2}}(z_{1},w_{1})}\omega_{BM}(z_{1}-z_{2},w_{1}-w_{2})(\bar{w}_{12}d\bar{z}_{2}-\bar{z}_{12}d\bar{w}_{2})(z_{1}^{m}w_{1}^{n}z_{2}^{r}w_{2}^{s}\text{
tr }T^{A}T^{B})$ $\displaystyle={}$ $\displaystyle
2\int_{S^{3}_{0}(z_{1},w_{1})}(0)(z_{2}^{m}w_{2}^{n}z_{2}^{r}w_{2}^{s}\text{
tr }T^{A}T^{B})$ $\displaystyle={}$ $\displaystyle 0,$

where $S^{3}_{z_{i},w_{j}}(z_{j},w_{j})$ is a 3-sphere centered at
$(z_{i},w_{i})$ and parametrized by $(z_{j},w_{j})$. In the first equality, we
used the fact that $d^{4}_{12}=d^{2}_{12}$ on
$S^{3}_{z_{1},w_{1}}(z_{2},w_{2})$, assuming the radius of the 3-sphere is 1.
The result is independent on the radius, as it should. We conclude that there
is no central element in the commutation relation.

Therefore, combining with (102), we recover (85)

$\displaystyle\big{[}{\mathcal{C}}(z^{m}w^{n}T^{A}),{\mathcal{C}}(z^{r}w^{s}T^{B})\big{]}={\mathcal{C}}\big{(}\big{[}z^{m}w^{n}T^{A},z^{r}w^{s}T^{B}\big{]}\big{)}.$
(107)

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